Next: Introduction [Contents][Index]
UNU.RAN (Universal NonUniform RAndom Number generator) is a collection of algorithms for generating nonuniform pseudorandom variates as a library of C functions designed and implemented by the ARVAG (Automatic Random VAriate Generation) project group in Vienna, and released under the GNU Public License (GPL). It is especially designed for such situations where
Of course it is also well suited for standard distributions. However due to its more sophisticated programming interface it might not be as easy to use if you only look for a generator for the standard normal distribution. (Although UNU.RAN provides generators that are superior in many aspects to those found in quite a number of other libraries.)
UNU.RAN implements several methods for generating random numbers. The choice depends primary on the information about the distribution can be provided and – if the user is familar with the different methods – on the preferences of the user.
The design goals of UNU.RAN are to provide reliable, portable and robust (as far as this is possible) functions with a consisent and easy to use interface. It is suitable for all situation where experiments with different distributions including nonstandard distributions. For example it is no problem to replace the normal distribution by an empirical distribution in a model.
Since originally designed as a library for so called blackbox or universal algorithms its interface is different from other libraries. (Nevertheless it also contains special generators for standard distributions.) It does not provide subroutines for random variate generation for particular distributions. Instead it uses an objectoriented interface. Distributions and generators are treated as independent objects. This approach allows one not only to have different methods for generating nonuniform random variates. It is also possible to choose the method which is optimal for a given situation (e.g. speed, quality of random numbers, using for variance reduction techniques, etc.). It also allows to sample from nonstandard distribution or even from distributions that arise in a model and can only be computed in a complicated subroutine.
Sampling from a particular distribution requires the following steps:
There are four types of objects that can be manipulated independently:
Of course a library of standard distributions is included (and these can be further modified to get, e.g., truncated distributions). Moreover the library provides subroutines to build almost arbitrary distributions.
NULL
pointer is returned in the initialization step).
F
– Fdistributionbeta
– Beta distributioncauchy
– Cauchy distributionchi
– Chi distributionchisquare
– Chisquare distributionexponential
– Exponential distributionextremeI
– Extreme value type I (Gumbeltype) distributionextremeII
– Extreme value type II (Frechettype) distributiongamma
– Gamma distributiongig
– Generalized Inverse Gaussian distributiongig2
– Generalized Inverse Gaussian distributionhyperbolic
– Hyperbolic distributionig
– Inverse Gaussian distributionlaplace
– Laplace distributionlogistic
– Logistic distributionlognormal
– LogNormal distributionlomax
– Lomax distribution (Pareto distribution of second kind)normal
– Normal distributionpareto
– Pareto distribution (of first kind)powerexponential
– Powerexponential (Subbotin) distributionrayleigh
– Rayleigh distributionslash
– Slash distributionstudent
– Student’s t distributiontriangular
– Triangular distributionuniform
– Uniform distributionweibull
– Weibull distributionNext: Examples, Previous: UNU.RAN – Universal NonUniform RANdom number generators, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
Next: Installation, Up: Introduction [Contents][Index]
We designed this document in a way such that one can use UNU.RAN with reading as little as necessary. Read Installation for the instructions to install the library. Concepts of UNU.RAN, discribes the basics of UNU.RAN. It also has a short guideline for choosing an appropriate method. In Examples examples are given that can be copied and modified. They also can be found in the directory examples in the source tree.
Further information are given in consecutive chapters. Handling distribution objects, describes how to create and manipulate distribution objects. standard distributions, describes predefined distribution objects that are ready to use. Methods for generating nonuniform random variates describes the various methods in detail. For each of possible distribution classes (continuous, discrete, empirical, multivariate) there exists a short overview section that can be used to choose an appropriate method followed by sections that describe each of the particular methods in detail. These are merely for users with some knowledge about the methods who want to change methodspecific parameters and can be ignored by others.
Abbreviations and explanation of some basic terms can be found in Glossary.
Next: Using the library, Previous: Usage of this document, Up: Introduction [Contents][Index]
UNU.RAN was developed on an Intel architecture under Linux with the GNU C compiler but should compile and run on any computing environment. It requires an ANSI compliant C compiler.
Below find the installation instructions for unices.
UNU.RAN can be used with any uniform random number generator but (at the moment) some features work best with Pierre L’Ecuyer’s RngStreams library (see http://statmath.wu.ac.at/software/RngStreams/ for a description and downloading. For details on using uniform random number in UNU.RAN see Using uniform random number generators.
Install the required libraries first.
tar zxvf unuran1.9.0.tar.gz cd unuran1.9.0
sh ./configure prefix=<prefix>
where <prefix>
is the root of the installation tree.
When omitted /usr/local is used.
Use ./configure help
to get a list of other options.
In particular the following flags are important:
withurngrngstream
URNG: use Pierre L’Ecuyer’s RNGSTREAM library
[default=no
]
withurngprng
URNG: use Otmar Lendl’s PRNG library
[default=no
]
withurnggsl
URNG: use random number generators from GNU Scientific Library
[default=no
]
withurngdefault
URNG: global default URNG (builtinrngstream)
[default=builtin
]
We strongly recommend to use RngStreams library:
sh ./configure withurngrngstream withurngdefault=rngstream
Important: You must install the respective libraries
RngStreams, PRNG and GSL before
./configure
is executed.
enableshared
build shared libraries [default=no
]
unur_gen_info
for
information about generator objects. This is intented for
using in interactive computing environments.
This feature can be enabled / disabled by means of the
configure flag
enableinfo
INFO: provide function with information about
generator objects [default=yes
]
enabledeprecated
enable support for deprecated UNU.RAN routines
[default=no
]
enablecheckstruct
Debug: check validity of pointers to structures
[default=no
]
enablelogging
Debug: print informations about generator into logfile [default=no]
make make install
Obviously $(prefix)/include
and $(prefix)/lib
must be in the search path of your compiler. You can use environment
variables to add these directories to the search path. If you
are using the bash type (or add to your profile):
export LIBRARY_PATH="<prefix>/lib" export C_INCLURE_PATH="<prefix>/include"
If you want to make a shared library, then making such a library can be enabled using
sh ./configure enableshared
If you want to link against the shared library make sure that
it can be found when executing the binary that links to the
library. If it is not installed in the usual path, then the
easiest way is to set the LD_LIBRARY_PATH
environment
variable. See any operating system documentation about shared
libraries for more information, such as the ld(1) and
ld.so(8) manual pages.
make check
However, some of these tests requires the usage of the PRNG or RngStreams library and are only executed if these are installed enabled by the corresponding configure flag.
An extended set of tests is run by
make fullcheck
However some of these might fail occasionally due to roundoff errors or the mysteries of floating point arithmetic, since we have used some extreme settings to test the library.
UNU.RAN now relies on some aspects of IEEE
754 compliant floating point arithmetic. In particular,
1./0.
and 0./0.
must result in infinity
and NaN
(not a number), respectively, and must not
cause a floating point exception.
For allmost all modern compting architecture this is implemented
in hardware. For others there should be a special compiler flag
to get this feature (e.g., MIEEE
on DEC alpha or
mp
for the Intel C complier).
With UNU.RAN version 1.0.x some of the macro definitions in
file src/unuran_config.h are moved into file
config.h and are set/controlled by the
./configure
script.
Writting logging information into the logfile must now be enabled when running the configure script:
sh ./configure enablelogging
With UNU.RAN version 0.8.0 the interface for changing underlying distributions and running a reinitialization routine has been simplified. The old routines can be compiled into the library using the following configure flag:
sh ./configure enabledeprecated
Notice: Using these deprecated routines is not supported any more and this strong discouraged.
Wrapper functions for external sources of uniform random numbers are now enabled by configure flags and not by macros defined in file src/unuran_config.h.
The file src/unuran_config.h is not installed any more. It is now only included when the library is compiled. It should be removed from the global include path of the compiler.
Next: Concepts of UNU.RAN, Previous: Installation, Up: Introduction [Contents][Index]
The library is written in ANSI C and is intended to conform to the ANSI C standard. It should be portable to any system with a working ANSI C compiler.
The library does not rely on any nonANSI extensions in the interface it exports to the user. Programs you write using UNU.RAN can be ANSI compliant. Extensions which can be used in a way compatible with pure ANSI C are supported, however, via conditional compilation. This allows the library to take advantage of compiler extensions on those platforms which support them.
To avoid namespace conflicts all exported function names and
variables have the prefix unur_
, while exported macros have
the prefix UNUR_
.
If you want to use the library you must include the UNU.RAN header file
#include <unuran.h>
If you also need the test routines then also add
#include <unuran_tests.h>
If wrapper functions for external sources of uniform random number generators are used, the corresponding header files must also be included, e.g.,
#include <unuran_urng_rngstream.h>
If these header files are not installed on the standard search path
of your compiler you will also need to provide its location to the
preprocessor as a command line flag. The default location of the
unuran.h is /usr/local/include. A typical compilation
command for a source file app.c with the GNU C compiler
gcc
is,
gcc I/usr/local/include c app.c
This results in an object file app.o. The default include
path for gcc
searches /usr/local/include
automatically so the I
option can be omitted when UNU.RAN is
installed in its default location.
The library is installed as a single file, libunuran.a. A shared version of the library is also installed on systems that support shared libraries. The default location of these files is /usr/local/lib. To link against the library you need to specify the main library. The following example shows how to link an application with the library (and the the RNGSTREAMS library if you decide to use this source of uniform pseudorandom numbers),
gcc app.o lunuran lrngstreams lm
To run a program linked with the shared version of the library it
may be necessary to define the shell variable
LD_LIBRARY_PATH
to include the directory where the library
is installed. For example,
LD_LIBRARY_PATH=/usr/local/lib:$LD_LIBRARY_PATH
To compile a statically linked version of the program instead, use the
static
flag in gcc
,
gcc static app.o lunuran lrngstreams lm
The library header files automatically define functions to have
extern "C"
linkage when included in C++ programs.
Next: Contact the authors, Previous: Using the library, Up: Introduction [Contents][Index]
UNU.RAN is a C library for generating nonuniformly distributed random variates. Its emphasis is on the generation of nonstandard distribution and on streams of random variates of special purposes. It is designed to provide a consistent tool to sample from distributions with various properties. Since there is no universal method that fits for all situations, various methods for sampling are implemented.
UNU.RAN solves this complex task by means of an object oriented programming interface. Three basic objects are used:
UNUR_DISTR
UNUR_GEN
UNUR_PAR
Notice that the parameter objects only hold pointers to arrays but do not have their own copy of such an array. Especially, if a dynamically allocated array is used it must not be freed until the generator object has been created!
The idea behind these structures is that creatin distributions, choosing a generation method and draing samples are orthogonal (ie. independent) functions of the library. The parameter object is only introduced due to the necessity to deal with various parameters and switches for each of these generation methods which are required to adjust the algorithms to unusual distributions with extreme properties but have default values that are suitable for most applications. These parameters and the data for distributions are set by various functions.
Once a generator object has been created sampling (from the univariate continuous distribution) can be done by the following command:
double x = unur_sample_cont(generator);
Analogous commands exist for discrete and multivariate distributions. For detailed examples that can be copied and modified see Examples.
All information about a distribution are stored in objects
(structures) of type UNUR_DISTR
.
UNU.RAN has five different types of distribution objects:
cont
Continuous univariate distributions.
cvec
Continuous multivariate distributions.
discr
Discrete univariate distributions.
cemp
Continuous empirical univariate distribution, ie. given by a sample.
cvemp
Continuous empirical multivariate distribution, ie. given by a sample.
matr
Matrix distributions.
Distribution objects can be created from scratch by the following call
distr = unur_distr_<type>_new();
where <type>
is one of the five possible types from the
above table.
Notice that these commands only create an empty object which
still must be filled by means of calls for each type of
distribution object
(see Handling distribution objects).
The naming scheme of these functions is designed to indicate the
corresponding type of the distribution object and the task to be
performed. It is demonstated on the following example.
unur_distr_cont_set_pdf(distr, mypdf);
This command stores a PDF named mypdf
in the distribution
object distr
which must have the type cont
.
Of course UNU.RAN provides an easier way to use standard distributions.
Instead of using unur_distr_<type>_new
calls and fuctions
unur_distr_<type>_set_<…>
for setting data,
objects for standard distribution can be created by a single call.
Eg. to get an object for the normal distribution with mean 2 and
standard deviation 5 use
double parameter[2] = {2.0 ,5.0}; UNUR_DISTR *distr = unur_distr_normal(parameter, 2);
For a list of standard distributions see Standard distributions.
The information that a distribution object must contain depends heavily on the chosen generation method choosen.
Brackets indicate optional information while a tilde indicates that only an approximation must be provided. See Glossary, for unfamiliar terms.
Methods for continuous univariate distributions

Methods for continuous empirical univariate distributions EMPK: Requires an observed sample. 
Methods for continuous multivariate distributions NORTA: Requires rank correlation matrix and marginal distributions. 
Methods for continuous empirical multivariate distributions VEMPK: Requires an observed sample. 
Methods for discrete univariate distributions

Methods for matrix distributions MCORR: Distribution object for random correlation matrix. 
Markov Chain Methods for continuous multivariate distributions GIBBS: Tconcave logPDF and derivatives of logPDF. 
Because of tremendous variety of possible problems, UNU.RAN provides many methods. All information for creating a generator object has to be collected in a parameter object first. For example, if the task is to sample from a continuous distribution the method AROU might be a good choice. Then the call
UNUR_PAR *par = unur_arou_new(distribution);
creates an parameter object par
with a pointer to the
distribution object and default values for all necessary parameters
for method AROU.
Other methods can be used by replacing arou
with the name
of the desired methods (in lower case letters):
UNUR_PAR *par = unur_<method>_new(distribution);
This sets the default values for all necessary parameters for the
chosen method. These are suitable for almost all
applications. Nevertheless, it is possible to control the behavior
of the method using corresponding set
calls for each method.
This might be necessary to adjust the algorithm for an unusual
distribution with extreme properties, or just for fine tuning the
perforence of the algorithm.
The following example demonstrates how to change the maximum
number of iterations for method NINV to the value 50:
unur_ninv_set_max_iteration(par, 50);
All available methods are described in details in Methods for generating nonuniform random variates.
Now it is possible to create a generator object:
UNUR_GEN *generator = unur_init(par);
if (generator == NULL
) exit(EXIT_FAILURE);
Important: You must always check whether
unur_init
has
been executed successfully. Otherwise the NULL
pointer is returned
which causes a segmentation fault when used for sampling.
Important:
The call of
unur_init
destroys the parameter object!
Moreover, it is recommended to call
unur_init
immediately after
the parameter object par
has created and modified.
An existing generator object is a rather static construct.
Nevertheless, some of the parameters can still be modified by
chg
calls, e.g.
unur_ninv_chg_max_iteration(gen, 30);
Notice that it is important when parameters are changed because different functions must be used:
The function name includes the term set
and the first
argument must be of type UNUR_PAR
when the parameters are
changed before the generator object is created.
The function name includes the term chg
and the first
argument must be of type UNUR_GEN
when the parameters are
changed for an existing generator object.
For details see Methods for generating nonuniform random variates.
You can now use your generator object in any place of your program
to sample from your distribution. You only have to take care about
the type of variates it computes: double
, int
or a
vector (array of double
s).
Notice that at this point it does not matter whether you are
sampling from a gamma distribution, a truncated normal distribution
or even an empirical distribution.
It is possible for a generator object to change the parameters and
the domain of the underlying distribution. This must be done by
extracting this object by means of a
unur_get_distr
call and
changing the distribution using the correspondig set calls,
see Handling distribution objects.
The generator object must then be reinitialized by means
of the
unur_reinit
call.
Important: Currently not all methods allow reinitialization, see the description of the particular method (keyword Reinit).
When you do not need your generator object any more, you should destroy it:
unur_free(generator);
Each generator object can have its own uniform random number generator or share one with others. When created a parameter object the pointer for the uniform random number generator is set to the default generator. However, it can be changed at any time to any other generator:
unur_set_urng(par, urng);
or
unur_chg_urng(generator, urng);
respectively. See Using uniform random number generators, for details.
Previous: Concepts of UNU.RAN, Up: Introduction [Contents][Index]
If you have any problems with UNU.RAN, suggestions how to improve the library, or find a bug, please contact us via email unuran@statmath.wu.ac.at.
For news please visit out homepage at http://statmath.wu.ac.at/unuran/.
Next: String Interface, Previous: Introduction, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
The examples in this chapter should compile cleanly and can be found in the directory examples of the source tree of UNU.RAN. Assuming that UNU.RAN as well as the PRNG libraries have been installed properly (see Installation) each of these can be compiled (using the GCC in this example) with
gcc Wall O2 o example example.c lunuran lprng lm
Remark: lprng
must be omitted when the PRNG library
is not installed. Then however some of the examples might not work.
The library uses three objects:
UNUR_DISTR
, UNUR_PAR
and UNUR_GEN
.
It is not important to understand the details of these objects but
it is important not to changed the order of their creation.
The distribution object can be destroyed after the generator
object has been made. (The parameter object is freed automatically
by the
unur_init
call.) It is also important to check the result
of the
unur_init
call. If it has failed the NULL
pointer is
returned and causes a segmentation fault when used for sampling.
We give all examples with the UNU.RAN standard API and the more convenient string API.
Next: As short as possible (String API), Up: Examples [Contents][Index]
Select a distribution and let UNU.RAN do all necessary steps.
/*  */ /* File: example0.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Use a predefined standard distribution: */ /* Gaussian with mean zero and standard deviation 1. */ /* Since this is the standard form of the distribution, */ /* we need not give these parameters. */ distr = unur_distr_normal(NULL, 0); /* Use method AUTO: */ /* Let UNURAN select a suitable method for you. */ par = unur_auto_new(distr); /* Now you can change some of the default settings for the */ /* parameters of the chosen method. We don't do it here. */ /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Select a method, Previous: As short as possible, Up: Examples [Contents][Index]
Select a distribution and let UNU.RAN do all necessary steps.
/*  */ /* File: example0_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ /* Use a predefined standard distribution: */ /* Standard Gaussian distribution. */ /* Use method AUTO: */ /* Let UNURAN select a suitable method for you. */ gen = unur_str2gen("normal()"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Select a method (String API), Previous: As short as possible (String API), Up: Examples [Contents][Index]
Select method AROU and use it with default parameters.
/*  */ /* File: example1.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Use a predefined standard distribution: */ /* Gaussian with mean zero and standard deviation 1. */ /* Since this is the standard form of the distribution, */ /* we need not give these parameters. */ distr = unur_distr_normal(NULL, 0); /* Choose a method: AROU. */ /* For other (suitable) methods replace "arou" with the */ /* respective name (in lower case letters). */ par = unur_arou_new(distr); /* Now you can change some of the default settings for the */ /* parameters of the chosen method. We don't do it here. */ /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Arbitrary distributions, Previous: Select a method, Up: Examples [Contents][Index]
Select method AROU and use it with default parameters.
/*  */ /* File: example1_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ /* Use a predefined standard distribution: */ /* Standard Gaussian distribution. */ /* Choose a method: AROU. */ /* For other (suitable) methods replace "arou" with the */ /* respective name. */ gen = unur_str2gen("normal() & method=arou"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Arbitrary distributions (String API), Previous: Select a method (String API), Up: Examples [Contents][Index]
If you want to sample from a nonstandard distribution, UNU.RAN might be exactly what you need. Depending on the information is available, a method must be choosen for sampling, see Concepts of UNU.RAN for an overview and Methods for generating nonuniform random variates for details.
/*  */ /* File: example2.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* In this example we build a distribution object from scratch */ /* and sample from this distribution. */ /* */ /* We use method TDR (Transformed Density Rejection) which */ /* required a PDF and the derivative of the PDF. */ /*  */ /* Define the PDF and dPDF of our distribution. */ /* */ /* Our distribution has the PDF */ /* */ /* / 1  x*x if x <= 1 */ /* f(x) = < */ /* \ 0 otherwise */ /* */ /* The PDF of our distribution: */ double mypdf( double x, const UNUR_DISTR *distr ) /* The second argument (`distr') can be used for parameters */ /* for the PDF. (We do not use parameters in our example.) */ { if (fabs(x) >= 1.) return 0.; else return (1.x*x); } /* end of mypdf() */ /* The derivative of the PDF of our distribution: */ double mydpdf( double x, const UNUR_DISTR *distr ) { if (fabs(x) >= 1.) return 0.; else return (2.*x); } /* end of mydpdf() */ /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Create a new distribution object from scratch. */ /* It is a continuous distribution, and we need a PDF and the */ /* derivative of the PDF. Moreover we set the domain. */ /* Get empty distribution object for a continuous distribution */ distr = unur_distr_cont_new(); /* Assign the PDF and dPDF (defined above). */ unur_distr_cont_set_pdf( distr, mypdf ); unur_distr_cont_set_dpdf( distr, mydpdf ); /* Set the domain of the distribution (optional for TDR). */ unur_distr_cont_set_domain( distr, 1., 1. ); /* Choose a method: TDR. */ par = unur_tdr_new(distr); /* Now you can change some of the default settings for the */ /* parameters of the chosen method. We don't do it here. */ /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Change parameters of the method, Previous: Arbitrary distributions, Up: Examples [Contents][Index]
If you want to sample from a nonstandard distribution, UNU.RAN might be exactly what you need. Depending on the information is available, a method must be choosen for sampling, see Concepts of UNU.RAN for an overview and Methods for generating nonuniform random variates for details.
/*  */ /* File: example2_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* In this example we use a generic distribution object */ /* and sample from this distribution. */ /* */ /* The PDF of our distribution is given by */ /* */ /* / 1  x*x if x <= 1 */ /* f(x) = < */ /* \ 0 otherwise */ /* */ /* We use method TDR (Transformed Density Rejection) which */ /* required a PDF and the derivative of the PDF. */ /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ /* Use a generic continuous distribution. */ /* Choose a method: TDR. */ gen = unur_str2gen( "distr = cont; pdf=\"1x*x\"; domain=(1,1) & method=tdr"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Change parameters of the method (String API), Previous: Arbitrary distributions (String API), Up: Examples [Contents][Index]
Each method for generating random numbers allows several parameters to be modified. If you do not want to use default values, it is possible to change them. The following example illustrates how to change parameters. For details see Methods for generating nonuniform random variates.
/*  */ /* File: example3.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ double fparams[2]; /* array for parameters for distribution */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Use a predefined standard distribution: */ /* Gaussian with mean 2. and standard deviation 0.5. */ fparams[0] = 2.; fparams[1] = 0.5; distr = unur_distr_normal( fparams, 2 ); /* Choose a method: TDR. */ par = unur_tdr_new(distr); /* Change some of the default parameters. */ /* We want to use T(x)=log(x) for the transformation. */ unur_tdr_set_c( par, 0. ); /* We want to have the variant with immediate acceptance. */ unur_tdr_set_variant_ia( par ); /* We want to use 10 construction points for the setup */ unur_tdr_set_cpoints ( par, 10, NULL ); /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* It is possible with method TDR to truncate the distribution */ /* for an existing generator object ... */ unur_tdr_chg_truncated( gen, 1., 0. ); /* ... and sample again. */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Change uniform random generator, Previous: Change parameters of the method, Up: Examples [Contents][Index]
Each method for generating random numbers allows several parameters to be modified. If you do not want to use default values, it is possible to change them. The following example illustrates how to change parameters. For details see Methods for generating nonuniform random variates.
/*  */ /* File: example3_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ /* Use a predefined standard distribution: */ /* Gaussian with mean 2. and standard deviation 0.5. */ /* Choose a method: TDR with parameters */ /* c = 0: use T(x)=log(x) for the transformation; */ /* variant "immediate acceptance"; */ /* number of construction points = 10. */ gen = unur_str2gen( "normal(2,0.5) & method=tdr; c=0.; variant_ia; cpoints=10"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* It is possible with method TDR to truncate the distribution */ /* for an existing generator object ... */ unur_tdr_chg_truncated( gen, 1., 0. ); /* ... and sample again. */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Change parameters of underlying distribution, Previous: Change parameters of the method (String API), Up: Examples [Contents][Index]
All generator object use the same default uniform random number generator by default. This can be changed to any generator of your choice such that each generator object has its own random number generator or can share it with some other objects. It is also possible to change the default generator at any time. See Using uniform random number generators, for details.
The following example shows how the uniform random number generator can be set or changed for a generator object. It requires the RNGSTREAMS library to be installed and used. Otherwise the example must be modified accordingly.
/*  */ /* File: example_rngstreams.c */ /*  */ #ifdef UNURAN_SUPPORTS_RNGSTREAM /*  */ /* This example makes use of the RNGSTREAM library for */ /* for generating uniform random numbers. */ /* (see http://statmath.wu.ac.at/software/RngStreams/) */ /* To compile this example you must have set */ /* ./configure withurngrngstream */ /* (Of course the executable has to be linked against the */ /* RNGSTREAM library.) */ /*  */ /* Include UNURAN header files. */ #include <unuran.h> #include <unuran_urng_rngstreams.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ double fparams[2]; /* array for parameters for distribution */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Declare objects for uniform random number generators. */ UNUR_URNG *urng1, *urng2; /* uniform generator objects */ /* The RNGSTREAMS library sets a package seed. */ unsigned long seed[] = {111u, 222u, 333u, 444u, 555u, 666u}; RngStream_SetPackageSeed(seed); /* RngStreams only: */ /* Make a object for uniform random number generator. */ /* For details see */ /* http://statmath.wu.ac.at/software/RngStreams/ */ urng1 = unur_urng_rngstream_new("urng1"); if (urng1 == NULL) exit (EXIT_FAILURE); /* Use a predefined standard distribution: */ /* Beta with parameters 2 and 3. */ fparams[0] = 2.; fparams[1] = 3.; distr = unur_distr_beta( fparams, 2 ); /* Choose a method: TDR. */ par = unur_tdr_new(distr); /* Set uniform generator in parameter object */ unur_set_urng( par, urng1 ); /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* Now we want to switch to a different (independent) stream */ /* of uniform random numbers. */ urng2 = unur_urng_rngstream_new("urng2"); if (urng2 == NULL) exit (EXIT_FAILURE); unur_chg_urng( gen, urng2 ); /* ... and sample again. */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); /* We also should destroy the uniform random number generators.*/ unur_urng_free(urng1); unur_urng_free(urng2); exit (EXIT_SUCCESS); } /* end of main() */ /*  */ #else #include <stdio.h> #include <stdlib.h> int main(void) { printf("You must enable the RNGSTREAM library to run this example!\n\n"); exit (77); /* exit code for automake check routines */ } #endif /*  */
Next: Sample pairs of antithetic random variates, Previous: Change uniform random generator, Up: Examples [Contents][Index]
One a generator object has been created it allows to draw samples from the distribution with the given parameters. However, some methods allow to change the parameters of the underlying distribution and reinitialize the generator object again. Thus when the parameters of the distribution vary for each draw we save overhead for destroying the old object and creating a new one.
The following example shows how the parameters of a GIG distribution can be changed when method CSTD is used.
/*  */ /* File: example_reinit.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* In this example we show how the parameters of the underlying */ /* distribution can be changed for an existing generator object. */ /* We use the GIG distribution with method CSTD. */ /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Parameters of distribution. */ double dparam[3] = {0.5, 1., 5.}; /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Create initial GIG distribution object */ distr = unur_distr_gig(dparam, 3); /* Choose a method: CSTD. */ par = unur_cstd_new(distr); /* Create the generator object. */ gen = unur_init(par); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* It is possible for method CSTD to change the parameters of */ /* underlying distribution. However, we have to extract to */ /* pointer to the distribution. Be carefull with this pointer! */ distr = unur_get_distr(gen); /* Change the parameter(s). */ dparam[2] = 0.001; unur_distr_cont_set_pdfparams(distr,dparam,3); /* Do not forget to reinitialize the generator object. */ /* Check the return code. */ /* (and try to find a better error handling) */ if (unur_reinit(gen) != UNUR_SUCCESS) { fprintf(stderr, "ERROR: cannot reinitialize generator object\n"); exit (EXIT_FAILURE); } /* Draw a new sample. */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* Changing parameters can be repeated. */ dparam[2] = 1000; unur_distr_cont_set_pdfparams(distr,dparam,3); if (unur_reinit(gen) != UNUR_SUCCESS) { fprintf(stderr, "ERROR: cannot reinitialize generator object\n"); exit (EXIT_FAILURE); } for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: Sample pairs of antithetic random variates (String API), Previous: Change parameters of underlying distribution, Up: Examples [Contents][Index]
Using Method TDR it is easy to sample pairs of antithetic random variates.
/*  */ /* File: example_anti.c */ /*  */ #ifdef UNURAN_SUPPORTS_PRNG /*  */ /* This example makes use of the PRNG library for generating */ /* uniform random numbers. */ /* (see http://statmath.wu.ac.at/prng/) */ /* To compile this example you must have set */ /* ./configure withurngprng */ /* (Of course the executable has to be linked against the */ /* PRNG library.) */ /*  */ /* Example how to sample from two streams of antithetic random */ /* variates from Gaussian N(2,5) and Gamma(4) distribution, resp.*/ /*  */ /* Include UNURAN header files. */ #include <unuran.h> #include <unuran_urng_prng.h> /*  */ int main(void) { int i; /* loop variable */ double xn, xg; /* will hold the random number */ double fparams[2]; /* array for parameters for distribution */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen_normal, *gen_gamma; /* generator objects */ /* Declare objects for uniform random number generators. */ UNUR_URNG *urng1, *urng2; /* uniform generator objects */ /* PRNG only: */ /* Make a object for uniform random number generator. */ /* For details see http://statmath.wu.ac.at/prng/. */ /* The first generator: Gaussian N(2,5) */ /* uniform generator: We use the Mersenne Twister. */ urng1 = unur_urng_prng_new("mt19937(1237)"); if (urng1 == NULL) exit (EXIT_FAILURE); /* UNURAN generator object for N(2,5) */ fparams[0] = 2.; fparams[1] = 5.; distr = unur_distr_normal( fparams, 2 ); /* Choose method TDR with variant PS. */ par = unur_tdr_new( distr ); unur_tdr_set_variant_ps( par ); /* Set uniform generator in parameter object. */ unur_set_urng( par, urng1 ); /* Set auxilliary uniform random number generator. */ /* We use the default generator. */ unur_use_urng_aux_default( par ); /* Alternatively you can create and use your own auxilliary */ /* uniform random number generator: */ /* UNUR_URNG *urng_aux; */ /* urng_aux = unur_urng_prng_new("tt800"); */ /* if (urng_aux == NULL) exit (EXIT_FAILURE); */ /* unur_set_urng_aux( par, urng_aux ); */ /* Create the generator object. */ gen_normal = unur_init(par); if (gen_normal == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Destroy distribution object (gen_normal has its own copy). */ unur_distr_free(distr); /* The second generator: Gamma(4) with antithetic variates. */ /* uniform generator: We use the Mersenne Twister. */ urng2 = unur_urng_prng_new("anti(mt19937(1237))"); if (urng2 == NULL) exit (EXIT_FAILURE); /* UNURAN generator object for gamma(4) */ fparams[0] = 4.; distr = unur_distr_gamma( fparams, 1 ); /* Choose method TDR with variant PS. */ par = unur_tdr_new( distr ); unur_tdr_set_variant_ps( par ); /* Set uniform generator in parameter object. */ unur_set_urng( par, urng2 ); /* Set auxilliary uniform random number generator. */ /* We use the default generator. */ unur_use_urng_aux_default( par ); /* Alternatively you can create and use your own auxilliary */ /* uniform random number generator (see above). */ /* Notice that both generator objects gen_normal and */ /* gen_gamma can share the same auxilliary URNG. */ /* Create the generator object. */ gen_gamma = unur_init(par); if (gen_gamma == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Destroy distribution object (gen_normal has its own copy). */ unur_distr_free(distr); /* Now we can sample pairs of negatively correlated random */ /* variates. E.g.: */ for (i=0; i<10; i++) { xn = unur_sample_cont(gen_normal); xg = unur_sample_cont(gen_gamma); printf("%g, %g\n",xn,xg); } /* When you do not need the generator objects any more, you */ /* can destroy it. */ unur_free(gen_normal); unur_free(gen_gamma); /* We also should destroy the uniform random number generators.*/ unur_urng_free(urng1); unur_urng_free(urng2); exit (EXIT_SUCCESS); } /* end of main() */ /*  */ #else #include <stdio.h> #include <stdlib.h> int main(void) { printf("You must enable the PRNG library to run this example!\n\n"); exit (77); /* exit code for automake check routines */ } #endif /*  */
Next: More examples, Previous: Sample pairs of antithetic random variates, Up: Examples [Contents][Index]
Using Method TDR it is easy to sample pairs of antithetic random variates.
/*  */ /* File: example_anti_str.c */ /*  */ /* String API. */ /*  */ #ifdef UNURAN_SUPPORTS_PRNG /*  */ /* This example makes use of the PRNG library for generating */ /* uniform random numbers. */ /* (see http://statmath.wu.ac.at/prng/) */ /* To compile this example you must have set */ /* ./configure withurngprng */ /* (Of course the executable has to be linked against the */ /* PRNG library.) */ /*  */ /* Example how to sample from two streams of antithetic random */ /* variates from Gaussian N(2,5) and Gamma(4) distribution, resp.*/ /*  */ /* Include UNURAN header files. */ #include <unuran.h> #include <unuran_urng_prng.h> /*  */ int main(void) { int i; /* loop variable */ double xn, xg; /* will hold the random number */ /* Declare UNURAN generator objects. */ UNUR_GEN *gen_normal, *gen_gamma; /* PRNG only: */ /* Make a object for uniform random number generator. */ /* For details see http://statmath.wu.ac.at/prng/. */ /* Create the first generator: Gaussian N(2,5) */ gen_normal = unur_str2gen("normal(2,5) & method=tdr; variant_ps & urng=mt19937(1237)"); if (gen_normal == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Set auxilliary uniform random number generator. */ /* We use the default generator. */ unur_chgto_urng_aux_default(gen_normal); /* The second generator: Gamma(4) with antithetic variates. */ gen_gamma = unur_str2gen("gamma(4) & method=tdr; variant_ps & urng=anti(mt19937(1237))"); if (gen_gamma == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } unur_chgto_urng_aux_default(gen_gamma); /* Now we can sample pairs of negatively correlated random */ /* variates. E.g.: */ for (i=0; i<10; i++) { xn = unur_sample_cont(gen_normal); xg = unur_sample_cont(gen_gamma); printf("%g, %g\n",xn,xg); } /* When you do not need the generator objects any more, you */ /* can destroy it. */ /* But first we have to destroy the uniform random number */ /* generators. */ unur_urng_free(unur_get_urng(gen_normal)); unur_urng_free(unur_get_urng(gen_gamma)); unur_free(gen_normal); unur_free(gen_gamma); exit (EXIT_SUCCESS); } /* end of main() */ /*  */ #else #include <stdio.h> #include <stdlib.h> int main(void) { printf("You must enable the PRNG library to run this example!\n\n"); exit (77); /* exit code for automake check routines */ } #endif /*  */
Previous: Sample pairs of antithetic random variates (String API), Up: Examples [Contents][Index]
See Methods for continuous univariate distributions.
See Methods for continuous empirical univariate distributions.
See Methods for continuous empirical multivariate distributions.
See Methods for discrete univariate distributions.
Next: Handling distribution objects, Previous: Examples, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
The string interface (string API) provided by the
unur_str2gen
call is the easiest way to use UNU.RAN. This
function takes a character string as its argument. The string is
parsed and the information obtained is used to create a generator
object. It returns NULL
if this fails, either due to a syntax
error, or due to invalid data. In both cases unur_error
is
set to the corresponding error codes
(see Error reporting).
Additionally there exists the call
unur_str2distr
that only
produces a distribution object.
Notice that the string interface does not implement all features of the UNU.RAN library. For trickier tasks it might be necessary to use the UNU.RAN calls.
In Examples, all examples are given using both the UNU.RAN standard API and this convenient string API. The corresponding programm codes are equivalent.
Get a generator object for the distribution, method and uniform random number generator as described in the given string. See Syntax of String Interface, for details.
Get a distribution object for the distribution described in string. See Syntax of String Interface, and Distribution String, for details. However, only the block for the distribution object is allowed.
Make a generator object for the distribution, method and uniform
random number generator. The distribution can be given either as
string distrstr or as a distribution object distr.
The method must be given as a string methodstr.
For the syntax of these strings see
Syntax of String Interface.
However, the method
keyword is optional for these calls
and can be omitted. If methodstr is the empty (blank) string
or NULL
method AUTO is used.
The uniform random number generator is optional. If
urng is NULL
then the default uniform random number generator
is used.
Next: Distribution String, Up: String Interface [Contents][Index]
The given string holds information about the requested distribution and (optional) about the sampling method and the uniform random number generator invoked. The interpretation of the string is not casesensitive, all white spaces are ignored.
The string consists of up to three blocks, separated by ampersands
&
.
Each block consists of <key>=<value>
pairs, separated by
semicolons ;
.
The first key in each block is used to indicate each block. We have three different blocks with the following (first) keys:
distr
definition of the distribution (see Distribution String).
method
description of the transformation method (see Method String).
urng
uniform random number generation (see Uniform RNG String).
The distr
block must be the very first block and is
obligatory. All the other blocks are optional and can be arranged
in arbitrary order.
For details see the following description of each block.
In the following example
distr = normal(3.,0.75); domain = (0,inf) & method = tdr; c = 0
we have a distribution block for the truncated normal distribution with mean 3 and standard deviation 0.75 on domain (0,infinity); and block for choosing method TDR with parameter c set to 0.
The <key>=<value>
pairs that follow the first (initial) pair
in each block are used to set parameters.
The name of the parameter is given by the <key>
string. It is
deduced from the UNU.RAN set calls by taking the part after
…_set_
.
The <value>
string holds the parameters to be
set, separated by commata ,
.
There are three types of parameters:
"…"
or '…'
i.e. any sequence of characters enclosed by double quotes
"…"
or single quotes '…'
(there is no distinction between double quotes "
and
single quotes '
).
(…,…)
i.e. list of numbers, separated by commata ,
,
enclosed in parenthesis (...)
.
a sequence of characters that is not enclosed by double quotes
"…"
, single quotes '…'
,
or parenthesis (...)
.
It is interpreted as float or integer depending on the type of
the corresponding parameter.
The <value>
string (including the character =
) can be
omitted when no argument is required.
At the moment not all set
calls are supported.
The syntax for the <value>
can be directly derived from the
corresponding set
calls. To simplify the syntax additional
shortcuts are possible. The following table lists the parameters for
the set
calls that are supported by the string interface; the
entry in parenthesis gives the type of the argument as
<value>
string:
int (number):
The number is interpreted as an integer.
true
and on
are transformed to 1
,
false
and off
are transformed to 0
.
A missing argument is interpreted as 1
.
int, int (number, number or list):
The two numbers or the first two entries in the list are
interpreted as a integers.
inf
and inf
are transformed to INT_MAX
and
INT_MIN
respectively, i.e. the largest and smallest
integers that can be represented by the computer.
unsigned (number):
The number is interpreted as an unsigned hexadecimal integer.
double (number):
The number is interpreted as a floating point number.
inf
is transformed to UNUR_INFINITY
.
double, double (number, number or list):
The two numbers or the first two entries in the list are
interpreted as a floating point numbers.
inf
is transformed to UNUR_INFINITY
. However using
inf
in the list might not work for all versions of C. Then it
is recommended to use two single numbers instead of a list.
int, double* ([number,] list or number):
NULL
pointer is used instead an array as argument.
double*, int (list [,number]):
The list is interpreted as a double array.
The (second) number as its length.
If the length is omitted, it is replaced by the actual size of the
array. (Only in the distribution
block!)
char* (string):
The character string is passed as is to the corresponding set call.
Notice that missing entries in a list of numbers are interpreted as
0
. E.g, a the list (1,,3)
is read as (1,0,3)
, the
list (1,2,)
as (1,2,0)
.
The the list of key
strings in
Keys for Distribution String, and
Keys for Method String, for further details.
Next: Function String, Previous: Syntax of String Interface, Up: String Interface [Contents][Index]
The distr
block must be the very first block and is
obligatory. For that reason the keyword distr
is optional and
can be omitted (together with the =
character).
Moreover it is ignored while parsing the string. However, to
avoid some possible confusion it has to start with the
letter d
(if it is given at all).
The value of the distr
key is used to get the distribution
object, either via a unur_distr_<value>
call for a standard
distribution via a unur_distr_<value>_new
call to get an
object of a generic distribution.
However not all generic distributions are supported yet.
The parameters for the standard distribution are given
as a list. There must not be any character (other than white space)
between the name of the standard distribution and the opening
parenthesis (
of this list. E.g., to get a beta distribution,
use
distr = beta(2,4)
To get an object for a discrete distribution with probability vector (0.5,0.2,0.3), use
distr = discr; pv = (0.5,0.2,0.3)
It is also possible to set a PDF, PMF, or CDF using a string.
E.g., to create a continuous distribution with PDF proportional to
exp(sqrt(2+(x1)^2) + (x1))
and domain (0,inf) use
distr = cont; pdf = "exp(sqrt(2+(x1)^2) + (x1))"
Notice: If this string is used in an
unur_str2distr
or
unur_str2gen
call the double quotes "
must be protected by
\"
. Alternatively, single quotes may be used instead
distr = cont; pdf = 'exp(sqrt(2+(x1)^2) + (x1))'
For the details of function strings see Function String.
Up: Distribution String [Contents][Index]
List of standard distributions see Standard distributions
[distr =] beta(…)
⇒ see beta
– Beta distribution
[distr =] binomial(…)
⇒ see binomial
– Binomial distribution
[distr =] cauchy(…)
⇒ see cauchy
– Cauchy distribution
[distr =] chi(…)
⇒ see chi
– Chi distribution
[distr =] chisquare(…)
⇒ see chisquare
– Chisquare distribution
[distr =] exponential(…)
⇒ see exponential
– Exponential distribution
[distr =] extremeI(…)
⇒ see extremeI
– Extreme value type I (Gumbeltype) distribution
[distr =] extremeII(…)
⇒ see extremeII
– Extreme value type II (Frechettype) distribution
[distr =] F(…)
⇒ see F
– Fdistribution
[distr =] gamma(…)
⇒ see gamma
– Gamma distribution
[distr =] geometric(…)
⇒ see geometric
– Geometric distribution
[distr =] gig(…)
⇒ see gig
– Generalized Inverse Gaussian distribution
[distr =] gig2(…)
⇒ see gig2
– Generalized Inverse Gaussian distribution
[distr =] hyperbolic(…)
⇒ see hyperbolic
– Hyperbolic distribution
[distr =] hypergeometric(…)
⇒ see hypergeometric
– Hypergeometric distribution
[distr =] ig(…)
⇒ see ig
– Inverse Gaussian distribution
[distr =] laplace(…)
⇒ see laplace
– Laplace distribution
[distr =] logarithmic(…)
⇒ see logarithmic
– Logarithmic distribution
[distr =] logistic(…)
⇒ see logistic
– Logistic distribution
[distr =] lognormal(…)
⇒ see lognormal
– LogNormal distribution
[distr =] lomax(…)
⇒ see lomax
– Lomax distribution (Pareto distribution of second kind)
[distr =] negativebinomial(…)
⇒ see negativebinomial
– Negative Binomial distribution
[distr =] normal(…)
⇒ see normal
– Normal distribution
[distr =] pareto(…)
⇒ see pareto
– Pareto distribution (of first kind)
[distr =] poisson(…)
⇒ see poisson
– Poisson distribution
[distr =] powerexponential(…)
⇒ see powerexponential
– Powerexponential (Subbotin) distribution
[distr =] rayleigh(…)
⇒ see rayleigh
– Rayleigh distribution
[distr =] slash(…)
⇒ see slash
– Slash distribution
[distr =] student(…)
⇒ see student
– Student’s t distribution
[distr =] triangular(…)
⇒ see triangular
– Triangular distribution
[distr =] uniform(…)
⇒ see uniform
– Uniform distribution
[distr =] weibull(…)
⇒ see weibull
– Weibull distribution
List of generic distributions see Handling Distribution Objects
[distr =] cemp
⇒ see Continuous empirical univariate distributions
[distr =] cont
⇒ see Continuous univariate distributions
[distr =] discr
⇒ see Discrete univariate distributions
Notice:
Order statistics for continuous distributions (see Continuous univariate order statistics) are
supported by using the key orderstatistics
for distributions
of type CONT
.
List of keys that are available via the String API. For description see the corresponding UNU.RAN set calls.
name = "<string>"
⇒ see unur_distr_set_name
cemp
(Distribution Type) (see Continuous empirical univariate distributions)
data = (<list>) [, <int>]
⇒ see unur_distr_cemp_set_data
hist_bins = (<list>) [, <int>]
hist_domain = <double>, <double>  (<list>)
hist_prob = (<list>) [, <int>]
cont
(Distribution Type) (see Continuous univariate distributions)
cdf = "<string>"
center = <double>
domain = <double>, <double>  (<list>)
hr = "<string>"
logcdf = "<string>"
logpdf = "<string>"
mode = <double>
⇒ see unur_distr_cont_set_mode
pdf = "<string>"
pdfarea = <double>
pdfparams = (<list>) [, <int>]
orderstatistics = <int>, <int>  (<list>)
Make order statistics for given distribution. The first parameter
gives the sample size, the second parameter its rank.
(see see unur_distr_corder_new
)
discr
(Distribution Type) (see Discrete univariate distributions)
cdf = "<string>"
domain = <int>, <int>  (<list>)
mode [= <int>]
pmf = "<string>"
pmfparams = (<list>) [, <int>]
pmfsum = <double>
pv = (<list>) [, <int>]
⇒ see unur_distr_discr_set_pv
Next: Method String, Previous: Distribution String, Up: String Interface [Contents][Index]
In unuran it is also possible to define functions (e.g. CDF or PDF) as
strings. As you can see in Example 2 (Arbitrary distributions (String API)) it is very
easy to define the PDF of a distribution object by means of a string.
The possibilities using this string interface are more restricted than
using a pointer to a routine coded in C (Arbitrary distributions).
But the differences in evaluation time is small.
When a distribution object is defined using this string interface then
of course the same conditions on the given density or CDF must be
satisfied for a chosen method as for the standard API.
This string interface can be used for both within the UNU.RAN string
API using the
unur_str2gen
call, and for calls that define the
density or CDF for a particular distribution object as done with
(e.g.) the call
unur_distr_cont_set_pdfstr
.
Here is an example for the latter case:
unur_distr_cont_set_pdfstr(distr,"1x*x");
The syntax for the function string is case insensitive, white spaces are ingnored. The expressions are similar to most programming languages and mathematical programs (see also the examples below). It is especially influenced by C. The usual preceedence rules are used (from highest to lowest preceedence: functions, power, multiplication, addition, relation operators). Use parentheses in case of doubt or when these preceedences should be changed.
Relation operators can be used as indicator functions, i.e. the term
(x>1)
is evaluted as 1
if this relation is satisfied,
and as 0
otherwise.
The first unknown symbol (letter or word) is interpreted as the
variable of the function. It is recommended to use x
.
Only one variable can be used.
Important: The symbol e
is used twice, for Euler’s
constant (= 2.7182…) and as exponent.
The multiplication operator *
must not be omitted, i.e.
2 x
is interpreted as the string 2x
(which will result
in a syntax error).
Numbers Numbers are composed using digits and, optionally, a sign,
a decimal point, and an exponent indicated by

Constants

Special symbols

Relation operators (Indicator functions)

Arithmetic operators

Functions

Variable

1.231+7.9876*x1.234e3*x^2+3.335e5*x^3 sin(2*pi*x)+x^2 exp(((x3)/2.1)^2)
It is also possible to define functions using different terms
on separate domains. However, instead of constructs using
if … then … else …
indicator functions are available.
For example to define the density of triangular distribution with domain (1,1) and mode 0 use
(x>1)*(x<0)*(1+x) + (x>=0)*(x<1)*(1x)
Next: Uniform RNG String, Previous: Function String, Up: String Interface [Contents][Index]
The key method
is obligatory, it must be the first key and its
value is the name of a method suitable for the choosen standard
distribution. E.g., if method AROU is chosen, use
method = arou
Of course the all following keys dependend on the method choosen at
first. All corresponding set
calls of UNU.RAN are available
and the key is the string after the unur_<methodname>_set_
part of the command. E.g., UNU.RAN provides the command
unur_arou_set_max_sqhratio
to set a parameter of method AROU.
To call this function via the stringinterface, the
key max_sqhratio
can be used:
max_sqhratio = 0.9
Additionally the keyword debug
can be used to set debugging
flags (see Debugging, for details).
If this block is omitted, a suitable default method is used. Notice however that the default method may change in future versions of UNU.RAN.
Up: Method String [Contents][Index]
List of methods and keys that are available via the String API. For description see the corresponding UNU.RAN set calls.
method = arou
⇒ unur_arou_new
(see AROU – Automatic RatioOfUniforms method)
cpoints = <int> [, (<list>)]  (<list>)
⇒ see unur_arou_set_cpoints
darsfactor = <double>
⇒ see unur_arou_set_darsfactor
guidefactor = <double>
max_segments [= <int>]
max_sqhratio = <double>
pedantic [= <int>]
⇒ see unur_arou_set_pedantic
usecenter [= <int>]
⇒ see unur_arou_set_usecenter
usedars [= <int>]
⇒ see unur_arou_set_usedars
verify [= <int>]
⇒ see unur_arou_set_verify
method = ars
⇒ unur_ars_new
(see ARS – Adaptive Rejection Sampling)
cpoints = <int> [, (<list>)]  (<list>)
⇒ see unur_ars_set_cpoints
max_intervals [= <int>]
max_iter [= <int>]
⇒ see unur_ars_set_max_iter
pedantic [= <int>]
⇒ see unur_ars_set_pedantic
reinit_ncpoints [= <int>]
reinit_percentiles = <int> [, (<list>)]  (<list>)
verify [= <int>]
⇒ see unur_ars_set_verify
method = auto
⇒ unur_auto_new
(see AUTO – Select method automatically)
logss [= <int>]
⇒ see unur_auto_set_logss
method = cstd
⇒ unur_cstd_new
(see CSTD – Continuous STandarD distributions)
variant = <unsigned>
⇒ see unur_cstd_set_variant
method = dari
⇒ unur_dari_new
(see DARI – Discrete Automatic Rejection Inversion)
cpfactor = <double>
⇒ see unur_dari_set_cpfactor
squeeze [= <int>]
⇒ see unur_dari_set_squeeze
tablesize [= <int>]
⇒ see unur_dari_set_tablesize
verify [= <int>]
⇒ see unur_dari_set_verify
method = dau
⇒ unur_dau_new
(see DAU – (Discrete) AliasUrn method)
urnfactor = <double>
⇒ see unur_dau_set_urnfactor
method = dgt
⇒ unur_dgt_new
(see DGT – (Discrete) Guide Table method (indexed search))
guidefactor = <double>
⇒ see unur_dgt_set_guidefactor
variant = <unsigned>
⇒ see unur_dgt_set_variant
method = dsrou
⇒ unur_dsrou_new
(see DSROU – Discrete Simple RatioOfUniforms method)
cdfatmode = <double>
⇒ see unur_dsrou_set_cdfatmode
verify [= <int>]
⇒ see unur_dsrou_set_verify
method = dstd
⇒ unur_dstd_new
(see DSTD – Discrete STandarD distributions)
variant = <unsigned>
⇒ see unur_dstd_set_variant
method = empk
⇒ unur_empk_new
(see EMPK – EMPirical distribution with Kernel smoothing)
beta = <double>
⇒ see unur_empk_set_beta
kernel = <unsigned>
⇒ see unur_empk_set_kernel
positive [= <int>]
⇒ see unur_empk_set_positive
smoothing = <double>
⇒ see unur_empk_set_smoothing
varcor [= <int>]
⇒ see unur_empk_set_varcor
method = gibbs
⇒ unur_gibbs_new
(see GIBBS – Markov Chain  GIBBS sampler)
burnin [= <int>]
⇒ see unur_gibbs_set_burnin
c = <double>
⇒ see unur_gibbs_set_c
thinning [= <int>]
⇒ see unur_gibbs_set_thinning
variant_coordinate
variant_random_direction
method = hinv
⇒ unur_hinv_new
(see HINV – Hermite interpolation based INVersion of CDF)
boundary = <double>, <double>  (<list>)
⇒ see unur_hinv_set_boundary
cpoints = (<list>), <int>
⇒ see unur_hinv_set_cpoints
guidefactor = <double>
max_intervals [= <int>]
order [= <int>]
⇒ see unur_hinv_set_order
u_resolution = <double>
method = hitro
⇒ unur_hitro_new
(see HITRO – Markov Chain  HITandrun sampler with RatioOfuniforms)
adaptive_multiplier = <double>
burnin [= <int>]
⇒ see unur_hitro_set_burnin
r = <double>
⇒ see unur_hitro_set_r
thinning [= <int>]
⇒ see unur_hitro_set_thinning
use_adaptiveline [= <int>]
use_adaptiverectangle [= <int>]
use_boundingrectangle [= <int>]
v = <double>
⇒ see unur_hitro_set_v
variant_coordinate
variant_random_direction
method = hrb
⇒ unur_hrb_new
(see HRB – Hazard Rate Bounded)
upperbound = <double>
⇒ see unur_hrb_set_upperbound
verify [= <int>]
⇒ see unur_hrb_set_verify
method = hrd
⇒ unur_hrd_new
(see HRD – Hazard Rate Decreasing)
verify [= <int>]
⇒ see unur_hrd_set_verify
method = hri
⇒ unur_hri_new
(see HRI – Hazard Rate Increasing)
p0 = <double>
⇒ see unur_hri_set_p0
verify [= <int>]
⇒ see unur_hri_set_verify
method = itdr
⇒ unur_itdr_new
(see ITDR – Inverse Transformed Density Rejection)
cp = <double>
⇒ see unur_itdr_set_cp
ct = <double>
⇒ see unur_itdr_set_ct
verify [= <int>]
⇒ see unur_itdr_set_verify
xi = <double>
⇒ see unur_itdr_set_xi
method = mvtdr
⇒ unur_mvtdr_new
(see MVTDR – MultiVariate Transformed Density Rejection)
boundsplitting = <double>
maxcones [= <int>]
⇒ see unur_mvtdr_set_maxcones
stepsmin [= <int>]
⇒ see unur_mvtdr_set_stepsmin
verify [= <int>]
⇒ see unur_mvtdr_set_verify
method = ninv
⇒ unur_ninv_new
(see NINV – Numerical INVersion)
max_iter [= <int>]
⇒ see unur_ninv_set_max_iter
start = <double>, <double>  (<list>)
⇒ see unur_ninv_set_start
table [= <int>]
⇒ see unur_ninv_set_table
u_resolution = <double>
usebisect
⇒ see unur_ninv_set_usebisect
usenewton
⇒ see unur_ninv_set_usenewton
useregula
⇒ see unur_ninv_set_useregula
x_resolution = <double>
method = nrou
⇒ unur_nrou_new
(see NROU – Naive RatioOfUniforms method)
center = <double>
⇒ see unur_nrou_set_center
r = <double>
⇒ see unur_nrou_set_r
u = <double>, <double>  (<list>)
⇒ see unur_nrou_set_u
v = <double>
⇒ see unur_nrou_set_v
verify [= <int>]
⇒ see unur_nrou_set_verify
method = pinv
⇒ unur_pinv_new
(see PINV – Polynomial interpolation based INVersion of CDF)
boundary = <double>, <double>  (<list>)
⇒ see unur_pinv_set_boundary
keepcdf [= <int>]
⇒ see unur_pinv_set_keepcdf
max_intervals [= <int>]
order [= <int>]
⇒ see unur_pinv_set_order
searchboundary = <int>, <int>  (<list>)
smoothness [= <int>]
⇒ see unur_pinv_set_smoothness
u_resolution = <double>
use_upoints [= <int>]
usecdf
⇒ see unur_pinv_set_usecdf
usepdf
⇒ see unur_pinv_set_usepdf
method = srou
⇒ unur_srou_new
(see SROU – Simple RatioOfUniforms method)
cdfatmode = <double>
⇒ see unur_srou_set_cdfatmode
pdfatmode = <double>
⇒ see unur_srou_set_pdfatmode
r = <double>
⇒ see unur_srou_set_r
usemirror [= <int>]
⇒ see unur_srou_set_usemirror
usesqueeze [= <int>]
⇒ see unur_srou_set_usesqueeze
verify [= <int>]
⇒ see unur_srou_set_verify
method = ssr
⇒ unur_ssr_new
(see SSR – Simple Setup Rejection)
cdfatmode = <double>
⇒ see unur_ssr_set_cdfatmode
pdfatmode = <double>
⇒ see unur_ssr_set_pdfatmode
usesqueeze [= <int>]
⇒ see unur_ssr_set_usesqueeze
verify [= <int>]
⇒ see unur_ssr_set_verify
method = tabl
⇒ unur_tabl_new
(see TABL – a TABLe method with piecewise constant hats)
areafraction = <double>
boundary = <double>, <double>  (<list>)
⇒ see unur_tabl_set_boundary
cpoints = <int> [, (<list>)]  (<list>)
⇒ see unur_tabl_set_cpoints
darsfactor = <double>
⇒ see unur_tabl_set_darsfactor
guidefactor = <double>
max_intervals [= <int>]
max_sqhratio = <double>
nstp [= <int>]
⇒ see unur_tabl_set_nstp
pedantic [= <int>]
⇒ see unur_tabl_set_pedantic
slopes = (<list>), <int>
⇒ see unur_tabl_set_slopes
usedars [= <int>]
⇒ see unur_tabl_set_usedars
useear [= <int>]
⇒ see unur_tabl_set_useear
variant_ia [= <int>]
⇒ see unur_tabl_set_variant_ia
variant_splitmode = <unsigned>
verify [= <int>]
⇒ see unur_tabl_set_verify
method = tdr
⇒ unur_tdr_new
(see TDR – Transformed Density Rejection)
c = <double>
⇒ see unur_tdr_set_c
cpoints = <int> [, (<list>)]  (<list>)
⇒ see unur_tdr_set_cpoints
darsfactor = <double>
⇒ see unur_tdr_set_darsfactor
guidefactor = <double>
⇒ see unur_tdr_set_guidefactor
max_intervals [= <int>]
max_sqhratio = <double>
pedantic [= <int>]
⇒ see unur_tdr_set_pedantic
reinit_ncpoints [= <int>]
reinit_percentiles = <int> [, (<list>)]  (<list>)
usecenter [= <int>]
⇒ see unur_tdr_set_usecenter
usedars [= <int>]
⇒ see unur_tdr_set_usedars
usemode [= <int>]
⇒ see unur_tdr_set_usemode
variant_gw
⇒ see unur_tdr_set_variant_gw
variant_ia
⇒ see unur_tdr_set_variant_ia
variant_ps
⇒ see unur_tdr_set_variant_ps
verify [= <int>]
⇒ see unur_tdr_set_verify
method = utdr
⇒ unur_utdr_new
(see UTDR – Universal Transformed Density Rejection)
cpfactor = <double>
⇒ see unur_utdr_set_cpfactor
deltafactor = <double>
pdfatmode = <double>
⇒ see unur_utdr_set_pdfatmode
verify [= <int>]
⇒ see unur_utdr_set_verify
method = vempk
⇒ unur_vempk_new
(see VEMPK – (Vector) EMPirical distribution with Kernel smoothing)
smoothing = <double>
⇒ see unur_vempk_set_smoothing
varcor [= <int>]
⇒ see unur_vempk_set_varcor
method = vnrou
⇒ unur_vnrou_new
(see VNROU – Multivariate Naive RatioOfUniforms method)
r = <double>
⇒ see unur_vnrou_set_r
v = <double>
⇒ see unur_vnrou_set_v
verify [= <int>]
⇒ see unur_vnrou_set_verify
Previous: Method String, Up: String Interface [Contents][Index]
The value of the urng
key is passed to the PRNG interface (see
PRNG manual
for details).
However it only works when using the PRNG library is enabled,
see Installation for details. There are no other keys.
IMPORTANT: UNU.RAN creates a new uniform random number generator for
the generator object. The pointer to this uniform generator
has to be read and saved via a
unur_get_urng
call in order to
clear the memory before the UNU.RAN generator object is
destroyed.
If this block is omitted the UNU.RAN default generator is used (which must not be destroyed).
Next: Methods for generating nonuniform random variates, Previous: String Interface, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
Objects of type UNUR_DISTR
are used for handling
distributions. All data about a distribution are stored in this
object. UNU.RAN provides functions that return instances of such
objects for standard distributions
(see Standard distributions).
It is then possible to change these distribution objects by
various set calls. Moreover, it is possible to build a
distribution object entirely from scratch. For this purpose
there exists unur_distr_<type>_new
calls that
return an empty object of this type for each object type
(eg. univariate contiuous) which can be filled with the
appropriate set calls.
UNU.RAN distinguishes between several types of distributions, each of which has its own sets of possible parameters (for details see the corresponding sections):
Notice that there are essential data about a distribution, eg. the PDF, a list of (shape, scale, location) parameters for the distribution, and the domain of (the possibly truncated) distribution. And there exist parameters that are/can be derived from these, eg. the mode of the distribution or the area below the given PDF (which need not be normalized for many methods). UNU.RAN keeps track of parameters which are known. Thus if one of the essential parameters is changed all derived parameters are marked as unknown and must be set again if these are required for the chosen generation method. Additionally to set calls there are calls for updating derived parameters for objects provided by the UNU.RAN library of standard distributions (one for each parameter to avoid computational overhead since not all parameters are required for all generator methods).
All parameters of distribution objects can be read by corresponding get calls.
Every generator object has its own copy of a distribution object
which is accessible by a
unur_get_distr
call. Thus the
parameter for this distribution can be read. However,
never extract the distribution object out of a
generator object and run one of the set calls on it to modify
the distribution. (How should the poor generator object know
what has happend?) Instead there exist calls for each of the
generator methods that change particular parameters of the
internal copy of the distribution object.
UNU.RAN collects all data required for a particular generation method in a distribution object. There are two ways to get an instance of a distributions object:
unur_distr_<type>_new
call,
where <type>
is the type of the distribution as
listed in the below subsections.
unur_distr_<name>_new
call
to get prebuild distribution from the UNU.RAN library of standard
distributions.
Here <name>
is the name of the
standard distribution in Standard distributions.
In either cases the corresponding
unur_distr_<type>_set_<param>
calls to set the
necessary parameters <param>
(case 1), or
change the values of the standard distribution in case 2 (if
this makes sense for you). In the latter case <type>
is the type to which the standard distribution belongs to.
These set
calls return UNUR_SUCCESS
when the
correspondig parameter has been set successfully. Otherwise an
error code is returned.
The parameters of a distribution are divided into essential and derived parameters.
Notice, that there are some restrictions in setting parameters
to avoid possible confusions.
Changing essential parameters marks derived parameters as
unknown
. Some of the parameters cannot be changed any
more when already set; some parameters block each others.
In such a case a new instance of a distribution object has to be
build.
Additionally unur_distr_<type>_upd_<param>
calls can
be used for updating derived parameters for objects provided by
the UNU.RAN library of standard distributions.
All parameters of a distribution object get be read by means of
unur_distr_<type>_get_<param>
calls.
Every distribution object be identified by its name
which
is a string of arbitrary characters provided by the user. For
standard distribution it is automatically set to
<name>
in the corresponding new
call. It can
be changed to any other string.
The calls in this section can be applied to all distribution objects.
free
an instance of a generator object.
type
of a generator object.
dimension
of a generator object.
name
(identifier string) of a generator object.
Destroy the distribution object.
Set and get name of distribution. The name can be an arbitrary character string. It can be used to identify generator objects for the user. It is used by UNU.RAN when printing information of the distribution object into a log files.
Get number of components of a random vector (its dimension) the distribution.
For univariate distributions it returns dimension 1
.
For matrix distributions it returns the number of components
(i.e., number of rows times number of columns).
When the respective numbers of rows and columns are needed use
unur_distr_matr_get_dim
instead.
Get type of distribution. Possible types are
UNUR_DISTR_CONT
univariate continuous distribution
UNUR_DISTR_CEMP
empirical continuous univariate distribution (i.e. a sample)
UNUR_DISTR_CVEC
continuous mulitvariate distribution
UNUR_DISTR_CVEMP
empirical continuous multivariate distribution (i.e. a vector sample)
UNUR_DISTR_DISCR
discrete univariate distribution
UNUR_DISTR_MATR
matrix distribution
Alternatively the unur_distr_is_<TYPE>
calls can be used.
TRUE
if distribution is a continuous univariate distribution.
TRUE
if distribution is a continuous multivariate distribution.
TRUE
if distribution is an empirical continuous univariate distribution,
i.e. a sample.
TRUE
if distribution is an empirical continuous multivariate
distribution.
TRUE
if distribution is a discrete univariate distribution.
TRUE
if distribution is a matrix distribution.
Store a pointer to an external object. This might be usefull if the PDF, PMF, CDF or other functions used to implement a particular distribution a parameter set that cannot be stored as doubles (e.g. pointers to some structure that holds information of the distribution).
Important: When UNU.RAN copies this distribution object into the generator object, then the address extobj that this pointer contains is simply copied. Thus the generator holds an address of a nonprivate object! Once the generator object has been created any change in the external object might effect the generator object.
Warning: External objects must be used with care. Once the generator object has been created or the distribution object has been copied you must not destroy this external object.
Get the pointer to the external object.
Important: Changing this object must be done with with extreme care.
Next: Continuous univariate order statistics, Previous: Functions for all kinds of distribution objects, Up: Handling distribution objects [Contents][Index]
The calls in this section can be applied to continuous univariate distributions.
new
instance of a continuous univariate
distribution.
cdf
),
probability density function (PDF, pdf
) and the
derivative of the density function (dpdf
).
The following is important:
pdf
need not be normalized, i.e.,
any integrable nonnegative function can be used.
dpdf
must the derivate of the function provided
as pdf
.
cdf
must be a distribution function, i.e. it
must be monotonically increasing with range [0,1].
cdf
and pdf
are used together for a
pariticular generation method, then pdf
must be the
derivate of the cdf
, i.e., it must be normalized.
logpdf
) and the derivative of the logarithm of the
density function (dlogpdf
).
Some methods use the logarithm of the density if available.
pdfparams
) and the
area below the graph (pdfarea
) of the given density.
mode
(or pole) of the distribution.
center
of the distribution.
It is used by some generation methods to adjust the parameters
of the generation algorithms to gain better performance. It can
be seens as the location of the “central part” of the
distribution.
hr
) of the distribution instead of its pdf
.
cdf
, pdf
, dpdf
,
and hr
can be provided as str
ings instead of
function pointers.
domain
of the distribution. Notice that
the library also can handle truncated distributions, i.e.,
distributions that are derived from (standard) distributions by
simply restricting its domain to a subset. However, there is a
subtle difference between changing the domain of a distribution
object by a
unur_distr_cont_set_domain
call and changing the
(truncated) domain for an existing generator object. The domain
of the distribution object is used to create the generator
object with hats, squeezes, tables, etc. Whereas truncating the
domain of an existing generator object need not necessarily
require a recomputation of these data. Thus by a
unur_<method>_chg_truncated
call (if available) the
sampling region is restricted to the subset of the domain of the
given distribution object. However, generation methods that
require a recreation of the generator object when the domain is
changed have a unur_<method>_chg_domain
call instead.
For these calls there are of course no restrictions on the given
domain (i.e., it is possible to increase the domain of the
distribution) (see Methods for generating nonuniform random variates, for details).
Create a new (empty) object for univariate continuous distribution.
Set respective pointer to the probability density function (PDF),
the derivative of the probability density function (dPDF), the
cumulative distribution function (CDF), and the inverse CDF of the
distribution.
Each of these function pointers must be of type
double funct(double x, const UNUR_DISTR *distr)
.
Due to the fact that some of the methods do not require a normalized PDF the following is important:
unur_distr_cont_set_pdfarea
call.
For a truncated distribution this must be of course the integral of
the PDF in the given truncated domain.
For distributions from the UNU.RAN library of standard
distributions this is done automatically by the
unur_distr_cont_upd_pdfarea
call.
It is important to note that all these functions must return a
result for all values of x. Eg., if the domain of a given
PDF is the interval [1,1], then the given function must return
0.0
for all points outside this interval.
In case of an overflow the PDF should return
UNUR_INFINITY
.
It is not possible to change such a function. Once the PDF or
CDF is set it cannot be overwritten. This also holds when the
logPDF is given or when the PDF
is given by the
unur_distr_cont_set_pdfstr
or
unur_distr_cont_set_logpdfstr
call.
A new distribution object has to be used instead.
Get the respective pointer to the PDF, the derivative of the
PDF, the CDF, and the inverse CDF of the distribution. The
pointer is of type double funct(double x, const UNUR_DISTR *distr)
.
If the corresponding function is not available for the distribution,
the NULL
pointer is returned.
Evaluate the PDF, derivative of the PDF, the CDF, and the inverse
CDF at x and u,respectively.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the distribution,
UNUR_INFINITY
is returned and unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
IMPORTANT: In the case of a truncated standard distribution these calls always return the respective values of the untruncated distribution!
Analogous calls for the logarithm of the density distribution functions.
This function provides an alternative way to set a PDF and its
derivative of the distribution.
pdfstr is a character string that contains the formula
for the PDF, see Function String, for details.
The derivative of the given PDF is computed automatically.
See also the remarks for the
unur_distr_cont_set_pdf
call.
It is not possible to call this funtion twice or to call this
function after a
unur_distr_cont_set_pdf
call.
This function provides an alternative way to set a CDF; analogously
to the
unur_distr_cont_set_pdfstr
call.
The PDF and its derivative of the given CDF are computed automatically.
Get pointer to respective string for PDF, derivate of PDF, and CDF of distribution that is given as string (instead of a function pointer). This call allocates memory to produce this string. It should be freed when it is not used any more.
Sets array of parameters for distribution. There is an upper limit
for the number of parameters n_params
. It is given by the
macro UNUR_DISTR_MAXPARAMS
in unuran_config.h. (It is set to
5 by default but can be changed to any appropriate nonnegative number.)
If n_params is negative or exceeds this limit no parameters
are copied into the distribution object and unur_errno
is set to
UNUR_ERR_DISTR_NPARAMS
.
For standard distributions from the UNU.RAN library the parameters
are checked. Moreover, the domain is updated automatically unless it
has been changed before by a
unur_distr_cont_set_domain
call.
If the given parameters are invalid for the standard distribution,
then no parameters are set and an error code is returned.
Notice, that the given parameter list for such a distribution is
handled in the same way as in the corresponding new
calls, i.e. optional parameters for the PDF that are not present in
the given list are (re)set to their default values.
Important: If the parameters of a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
are changed, then neither its mode nor the normalization
constant are updated. Please use the respective calls
unur_distr_cont_upd_mode
and
unur_distr_cont_upd_pdfarea
.
Moreover, if the domain has been changed by a
unur_distr_cont_set_domain
it is not automatically updated, either.
Updating the normalization constant is in particular very important,
when the CDF of the distribution is used.
Get number of parameters of the PDF and set pointer params to
array of parameters. If no parameters are stored in the object, an
error code is returned and params
is set to NULL
.
Important: Do not change the entries in params!
This function provides an interface for additional vector parameters for a continuous distribution.
It sets the parameter with number par.
par indicates directly which of the parameters is set and
must be a number between 0
and UNUR_DISTR_MAXPARAMS
1
(the upper limit of possible parameters defined in
unuran_config.h; it is set to 5 but can be changed to any
appropriate nonnegative number.)
The entries of a this parameter are given by the array param_vec of size n_param_vec.
If param_vec is NULL
then the corresponding entry is cleared.
If an error occurs no parameters are copied into the parameter
object unur_errno
is set to UNUR_ERR_DISTR_DATA
.
Get parameter of the PDF with number par.
The pointer to the parameter array is stored in param_vecs, its
size is returned by the function.
If the requested parameter is not set, then an error code is returned
and params
is set to NULL
.
Important: Do not change the entries in param_vecs!
Analogous calls for the logarithm of the density and distribution functions.
Set the left and right borders of the domain of the
distribution. This can also be used to truncate an existing
distribution. For setting the boundary to
+/ infinity
use +/ UNUR_INFINITY
.
If right is not strictly greater than left no domain
is set and unur_errno
is set to UNUR_ERR_DISTR_SET
.
Important: For some technical reasons it is assumed that the density
is unimodal and thus monotone on either side of the mode! This is used in
the case when the given mode is outside of the original domain. Then the
mode is set to the corresponding boundary of the new domain.
If this result is not the desired it must be changed by using a
unur_distr_cont_set_mode
call (or a
unur_distr_cont_upd_mode
call). The same holds for the center of the distribution.
Get the left and right borders of the domain of the
distribution. If the domain is not set +/ UNUR_INFINITY
is
assumed and returned. No error is reported in this case.
Get the left and right borders of the (truncated) domain of the
distribution. For nontruncated distribution this call is
equivalent to the
unur_distr_cont_get_domain
call.
This call is only useful in connection with a
unur_get_distr
call
to get the boundaries of the sampling region of a generator object.
Set pointer to the hazard rate (HR) of the distribution.
The hazard rate (or failure rate) is a mathematical way of describing aging. If the lifetime X is a random variable with density f(x) and CDF F(x) the hazard rate h(x) is defined as h(x) = f(x) / (1F(x)). In other words, h(x) represents the (conditional) rate of failure of a unit that has survived up to time x with probability 1F(x). The key distribution is the exponential distribution as it has constant hazard rate of value 1. Hazard rates tending to infinity describe distributions with subexponential tails whereas distributions with hazard rates tending to zero have heavier tails than the exponential distribution.
It is important to note that all these functions must return a
result for all floats x. In case of an overflow the PDF should
return UNUR_INFINITY
.
Important: Do not simply use f(x) / (1F(x)), since this is numerically very unstable and results in numerical noise if F(x) is (very) close to 1. Moreover, if the density f(x) is known a generation method that uses the density is more appropriate.
It is not possible to change such a function. Once the HR is set it
cannot be overwritten. This also holds when the HR is given by the
unur_distr_cont_set_hrstr
call. A new distribution object has to
be used instead.
Get the pointer to the hazard rate of the distribution. The
pointer is of type
double funct(double x, const UNUR_DISTR *distr)
.
If the corresponding function is not available for the distribution,
the NULL
pointer is returned.
Evaluate the hazard rate at x.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the distribution,
UNUR_INFINITY
is returned and unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
This function provides an alternative way to set a hazard rate and its
derivative of the distribution.
hrstr is a character string that contains the formula
for the HR, see Function String, for details.
See also the remarks for the
unur_distr_cont_set_hr
call.
It is not possible to call this funtion twice or to call this
function after a
unur_distr_cont_set_hr
call.
Get pointer to string for HR of distribution that is given via the string interface. This call allocates memory to produce this string. It should be freed when it is not used any more.
The following paramters must be set whenever one of the essential parameters has been set or changed (and the parameter is required for the chosen method).
Set mode of distribution. The mode must be contained in
the domain of distribution. Otherwise the mode is not set and
unur_errno
is set to UNUR_ERR_DISTR_SET
.
For distributions with unbounded density, this call is used to set
the pole of the PDF. Notice that the PDF should then return
UNUR_INFINITY
at the pole.
Notice that the mode is adjusted when the domain is set, see the
remark for the
unur_distr_cont_set_domain
call.
Recompute the mode of the distribution. This call works
properly for distribution objects from the UNU.RAN library of
standard distributions when the corresponding function is
available. Otherwise a (slow) numerical mode finder based on
Brent’s algorithm is used. If it failes unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
Get mode of distribution. If the mode is not marked as known,
unur_distr_cont_upd_mode
is called to compute the mode. If this
is not successful UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
(There is no difference between the case where no routine for
computing the mode is available and the case where no mode exists
for the distribution at all.)
Set center of the distribution. The center is used by some methods to shift the distribution in order to decrease numerical roundoff error. If not given explicitly a default is used.
Important: This call does not check whether the center is contained in the given domain.
Default: The mode, if set by a
unur_distr_cont_set_mode
or
unur_distr_cont_upd_mode
call; otherwise 0
.
Get center of the distribution. It always returns some point
as there always exists a default for the center, see
unur_distr_cont_set_center
.
Set the area below the PDF. If area
is nonpositive, no
area is set and unur_errno
is set to
UNUR_ERR_DISTR_SET
.
For a distribution object created by the
UNU.RAN library of standard distributions you always should use
the
unur_distr_cont_upd_pdfarea
.
Otherwise there might be
ambiguous sideeffects.
Recompute the area below the PDF of the distribution.
It only works for distribution objects from the
UNU.RAN library of standard distributions when the
corresponding function is available. Otherwise unur_errno
is
set to UNUR_ERR_DISTR_DATA
.
This call also sets the normalization constant such that the given PDF is the derivative of a given CDF, i.e. the area is 1. However, for truncated distributions the area is smaller than 1.
The call does not work for distributions from the UNU.RAN library of standard distributions with truncated domain when the CDF is not available.
Get the area below the PDF of the distribution. If this area is
not known,
unur_distr_cont_upd_pdfarea
is called to compute
it. If this is not successful UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
Next: Continuous empirical univariate distributions, Previous: Continuous univariate distributions, Up: Handling distribution objects [Contents][Index]
These are special cases of a continuous univariate distributions and thus they have most of these parameters (with the exception that functions cannot be changed). Additionally,
rank
of the order
statistics.
Create an object for order statistics of sample size
n and rank k.
distribution must be a pointer to a univariate continuous
distribution.
The resulting generator object is of the same type as of a
unur_distr_cont_new
call.
(However, it cannot be used to make an order statistics out of an
order statistics.)
To have a PDF for the order statistics, the given distribution object must contain a CDF and a PDF. Moreover, it is assumed that the given PDF is the derivative of the given CDF. Otherwise the area below the PDF of the order statistics is not computed correctly.
Important: There is no warning when the computed area below the PDF of the order statistics is wrong.
Get pointer to distribution object for underlying distribution.
Change sample size n and rank k of order statistics.
In case of invalid data, no parameters are changed.
The area below the PDF can be set to that of the underlying
distribution by a
unur_distr_corder_upd_pdfarea
call.
Get sample size n and rank k of order statistics. In case of error an error code is returned.
Additionally most of the set and get calls for continuous
univariate distributions work. The most important exceptions are
that the PDF and CDF cannot be changed and
unur_distr_cont_upd_mode
uses in any way a (slow) numerical
method that might fail.
Get the respective pointer to the PDF, the derivative of the
PDF and the CDF of the distribution, respectively. The pointer is of type
double funct(double x, UNUR_DISTR *distr)
.
If the corresponding function is not available for the distribution,
the NULL
pointer is returned.
See also
unur_distr_cont_get_pdf
.
(Macro)
Evaluate the PDF, derivative of the PDF. and the CDF,
respectively, at x.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the distribution,
UNUR_INFINITY
is returned and unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
See also
unur_distr_cont_eval_pdf
.
(Macro)
IMPORTANT: In the case of a truncated standard distribution these calls always return the respective values of the untruncated distribution!
Set array of parameters for underlying distribution.
See
unur_distr_cont_set_pdfparams
for details.
(Macro)
Get number of parameters of the PDF of the underlying distribution
and set pointer params to array of parameters.
See
unur_distr_cont_get_pdfparams
for details.
(Macro)
Set the left and right borders of the domain of the
distribution.
See
unur_distr_cont_set_domain
for details.
(Macro)
Get the left and right borders of the domain of the
distribution.
See
unur_distr_cont_get_domain
for details.
(Macro)
Get the left and right borders of the (truncated) domain of the
distribution.
See
unur_distr_cont_get_truncated
for details.
(Macro)
The following paramters must be set whenever one of the essential parameters has been set or changed (and the parameter is required for the chosen method).
Set mode of distribution.
See also
unur_distr_corder_set_mode
.
(Macro)
Recompute the mode of the distribution numerically. Notice that
this routine is slow and might not work properly in every case.
See also
unur_distr_cont_upd_mode
for further details.
(Macro)
Get mode of distribution.
See
unur_distr_cont_get_mode
for details.
(Macro)
Set the area below the PDF.
See
unur_distr_cont_set_pdfarea
for details.
(Macro)
Recompute the area below the PDF of the distribution.
It only works for order statistics for distribution objects from
the UNU.RAN library of standard distributions when the
corresponding function is available.
unur_distr_cont_upd_pdfarea
assumes that the PDF of the underlying
distribution is normalized, i.e. it is the derivative of its CDF.
Otherwise the computed area is wrong and there is no warning
about this failure.
See
unur_distr_cont_upd_pdfarea
for further details.
(Macro)
Get the area below the PDF of the distribution.
See
unur_distr_cont_get_pdfarea
for details.
(Macro)
Next: Continuous multivariate distributions, Previous: Continuous univariate order statistics, Up: Handling distribution objects [Contents][Index]
Empirical univariate distributions are derived from observed data. There are two ways to create such a generator object:
unur_distr_cemp_set_data
call.
unur_distr_cemp_set_hist
call.
How these data are used to sample from the empirical distribution depends from the chosen generation method.
Create a new (empty) object for empirical univariate continuous distribution.
Set observed sample for empirical distribution.
Read data from file filename.
It reads the first number from each line.
Numbers are parsed by means of the C standard routine strtod
.
Lines that do not start with +
, 
, .
, or a
digit are ignored. (Beware of lines starting with a blank!)
In case of an error (file cannot be opened, invalid string for double in line) no data are copied into the distribution object and an error code is returned.
Get number of samples and set pointer sample to array of
observations. If no sample has been given, an error code
is returned and sample
is set to NULL
.
Important: Do not change the entries in sample!
Set a histogram with bins of equal width. prob is an array of length n_prob that contains the probabilities for the bins (in ascending order). xmin and xmax give the lower and upper bound of the histogram, respectively. The bins are assumed to have equal width.
Remark: This is shortcut for calling
unur_distr_cemp_set_hist_prob
and
unur_distr_cemp_set_hist_domain
.
Notice: All sampling methods either use raw data or histogram.
It is possible to set both types of data; however, it is not
checked whether the given histogran corresponds to possibly given
raw data.
Set probabilities of a histogram with n_prob bins.
Hence prob must be an array of length n_prob that
contains the probabilities for the bins in ascending order.
It is important also to set the location of the bins either
with a
unur_distr_cemp_set_hist_domain
for bins of equal
width or
unur_distr_cemp_set_hist_bins
when the bins have
different width.
Notice: All sampling methods either use raw data or histogram. It is possible to set both types of data; however, it is not checked whether the given histogram corresponds to possibly given raw data.
Set a domain of a histogram with bins of equal width. xmin and xmax give the lower and upper bound of the histogram, respectively.
Set location of bins of a histogram with n_bins bins.
Hence bins must be an array of length n_bins.
The domain of the distribution is automatically set by
this call and overrides any calls to
unur_distr_cemp_set_hist_domain
.
Important:
The probabilities of the bins of the distribution must be
already be set by a
unur_distr_cemp_set_hist_prob
(or a
unur_distr_cemp_set_hist
call) and the value of
n_bins must equal n_prob+1
from the
corresponding value of the respective call.
Next: Continuous univariate full conditional distribution, Previous: Continuous empirical univariate distributions, Up: Handling distribution objects [Contents][Index]
The following calls handle multivariate distributions. However, the requirements of particular generation methods is not as unique as for univariate distributions. Moreover, random vector generation methods are still under development. The below functions are a first attempt to handle this situation.
Notice that some of the parameters – when given carelessly – might contradict to others. For example: Some methods require the marginal distribution and some methods need a standardized form of the marginal distributions, where the actual mean and variance is stored in the mean vector and the covariance matrix, respectively.
We also have to mention that some methods might abuse some of the parameters. Please read the discription of the chosen sampling method carfully.
The following kind of calls exists:
new
instance of a continuous multivariate
distribution;
pdf
) and the
gradient of the density function (dpdf
).
The following is important:
pdf
need not be normalized, i.e.,
any integrable nonnegative function can be used.
dpdf
must the derivate of the function provided
as pdf
.
logpdf
) and the gradient of the logarithm of the
density function (dlogpdf
).
Some methods use the logarithm of the density if available.
pdfparams
) and the
volume below the graph (pdfvol
) of the given density.
mode
and mean
of the distribution.
center
of the distribution.
It is used by some generation methods to adjust the parameters
of the generation algorithms to gain better performance. It can
be seens as the location of the “central part” of the
distribution.
covar
iance matrix of the distribution and
its cholesky
and inv
verse matrices.
rankcorr
elation matrix of the distribution.
marginal
distributions.
Create a new (empty) object for multivariate continuous
distribution. dim is the number of components of the random
vector (i.e. its dimension). It is also possible to use dimension 1.
Notice, however, that this is treated as a distribution of random
vectors with only one component and not as a distribution of
real numbers. For the latter
unur_distr_cont_new
should be used
to create an object for a univariate distribution.
Set respective pointer to the PDF of the distribution.
This function must be of type
double funct(const double *x, UNUR_DISTR *distr)
,
where x must be a pointer to a double array of appropriate
size (i.e. of the same size as given to the
unur_distr_cvec_new
call).
It is not necessary that the given PDF is normalized, i.e. the
integral need not be 1.
Nevertheless the volume below the PDF can be provided by a
unur_distr_cvec_set_pdfvol
call.
It is not possible to change the PDF. Once the PDF is set it cannot be overwritten. This also holds when the logPDF is given. A new distribution object has to be used instead.
Set pointer to the gradient of the PDF. The type of this function must be
int funct(double *result, const double *x, UNUR_DISTR *distr)
,
where result and x must be pointers to double arrays of
appropriate size (i.e. of the same size as given to the
unur_distr_cvec_new
call).
The gradient of the PDF is stored in the array result.
The function should return an error code in case of an error and must
return UNUR_SUCCESS
otherwise.
The given function must be the gradient of the function
given by a
unur_distr_cvec_set_pdf
call.
It is not possible to change the gradient of the PDF. Once the dPDF is set it cannot be overwritten. This also holds when the gradient of the logPDF is given. A new distribution object has to be used instead.
Set pointer to partial derivatives of the PDF. The type of this function must be
double funct(const double *x, int coord, UNUR_DISTR *distr)
,
where x must be a pointer to a double array of appropriate
size (i.e. of the same size as given to the
unur_distr_cvec_new
call). coord is the coordinate for which the partial dervative should be
computed.
Notice that coord must be an integer from {0,…,dim1}.
It is not possible to change the partial derivative of the PDF. Once the pdPDF is set it cannot be overwritten. This also holds when the partial derivative of the logPDF is given. A new distribution object has to be used instead.
Get the pointer to the PDF of the distribution. The
pointer is of type
double funct(const double *x, UNUR_DISTR *distr)
.
If the corresponding function is not available for the
distribution, the NULL
pointer is returned.
Get the pointer to the gradient of the PDF of the
distribution. The pointer is of type
int double funct(double *result, const double *x, UNUR_DISTR *distr)
.
If the corresponding function is not available for the
distribution, the NULL
pointer is returned.
Evaluate the PDF of the distribution at x.
x must be a pointer to a double array of appropriate size
(i.e. of the same size as given to the
unur_distr_cvec_new
call)
that contains the vector for which the function has to be evaluated.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the
distribution, UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_DATA
.
Evaluate the gradient of the PDF of the distribution at
x.
The result is stored in the double array result.
Both result and x must be pointer to double arrays of
appropriate size (i.e. of the same size as given to the
unur_distr_cvec_new
call).
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the
distribution, an error code is returned and unur_errno
is set to UNUR_ERR_DISTR_DATA
(result is left unmodified).
Evaluate the partial derivative of the PDF of the distribution
at x for the coordinate coord.
x must be a pointer to a double array of appropriate size
(i.e. of the same size as given to the
unur_distr_cvec_new
call)
that contains the vector for which the function has to be evaluated.
Notice that coord must be an integer from {0,…,dim1}.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the
distribution, UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_DATA
.
Analogous calls for the logarithm of the density function.
Set mean vector for multivariate distribution.
mean must be a pointer to an array of size dim
, where
dim
is the dimension returned by
unur_distr_get_dim
.
A NULL
pointer for mean is interpreted as the zero
vector (0,…,0).
Important: If the parameters of a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
are changed, then neither its mode nor the normalization
constant are updated. Please use the respective calls
unur_distr_cvec_upd_mode
and
unur_distr_cvec_upd_pdfvol
.
Get the mean vector of the distribution. The function returns a
pointer to an array of size dim
.
If the mean vector is not marked as known the NULL
pointer is
returned and unur_errno
is set to
UNUR_ERR_DISTR_GET
.
Important: Do not modify the array that holds the mean vector!
Set covariance matrix for multivariate distribution.
covar must be a pointer to an array of size
dim
x dim
, where dim
is the dimension returned
by
unur_distr_get_dim
.
The rows of the matrix have to be stored
consecutively in this array.
covar must be a variancecovariance matrix of the
distribution, i.e. it must be symmetric and positive definit and
its diagonal entries (i.e. the variance of the components of the
random vector) must be strictly positive.
The Cholesky factor is computed (and stored) to verify the positive
definiteness condition.
Notice that the inverse of the given covariance matrix is
automatically computed when it is requested by some routine.
Notice that the computation of this inverse matrix is unstable in
case of high correlations and/or high dimensions. Thus it might
fail and methods that require this inverse cannot be used.
As an alternative the inverse of the covariance matrix can be
directly set by a
unur_distr_cvec_set_covar_inv
call.
A NULL
pointer for covar is interpreted as the
identity matrix.
Important: This entry is abused in some methods which do not require the covariance matrix. It is then used to perform some transformation to obtain better performance.
Important: In case of an error (e.g. because covar is not a valid covariance matrix) an error code is returned. Moreover, the covariance matrix is not set and is marked as unknown. A previously set covariance matrix is then no longer available.
Important: If the parameters of a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
are changed, then neither its mode nor the normalization
constant are updated. Please use the respective calls
unur_distr_cvec_upd_mode
and
unur_distr_cvec_upd_pdfvol
.
Remark: UNU.RAN does not check whether the an eventually
set covariance matrix and a rankcorrelation matrix do not
contradict each other.
Set inverse of the covariance matrix for multivariate distribution.
covar_inv must be a pointer to an array of size
dim
x dim
, where dim
is the dimension returned
by
unur_distr_get_dim
.
The rows of the matrix have to be stored
consecutively in this array.
covar_inv must be symmetric and positive definit. Only the symmetry of the matrix is checked.
A NULL
pointer for covar_inv is interpreted as the identity matrix.
Important: In case of an error (because covar_inv is not symetric) an error code is returned. Moreover, the inverse of the covariance matrix is not set and is marked as unknown. A previously set inverse matrix is then no longer available.
Remark: UNU.RAN does not check whether the given matrix is positive definit.
Remark: UNU.RAN does not check whether the matrix covar_inv is the inverse of the eventually set covariance matrix.
Get covariance matrix of distribution, its Cholesky factor,
and its inverse, respectively. The function returns a
pointer to an array of size dim
x dim
.
The rows of the matrix are stored consecutively in this array.
If the requested matrix is not marked as known the NULL
pointer is returned and unur_errno
is set to
UNUR_ERR_DISTR_GET
.
Important: Do not modify the array that holds the covariance matrix!
Remark: The inverse of the covariance matrix is computed if it is not already stored.
Set rankcorrelation matrix (Spearman’s correlation) for
multivariate distribution.
rankcorr must be a pointer to an array of size
dim
x dim
, where dim
is the dimension returned
by
unur_distr_get_dim
.
The rows of the matrix have to be stored
consecutively in this array.
rankcorr must be a rankcorrelation matrix of the
distribution, i.e. it must be symmetric and positive definite
and its diagonal entries must be equal to 1
.
The Cholesky factor is computed (and stored) to verify the positive definiteness condition.
A NULL
pointer for rankcorr is interpreted as the identity matrix.
Important: In case of an error (e.g. because rankcorr is not a valid rankcorrelation matrix) an error code is returned. Moreover, the rankcorrelation matrix is not set and is marked as unknown. A previously set rankcorrelation matrix is then no longer available.
Remark: UNU.RAN does not check whether the an eventually set covariance matrix and a rankcorrelation matrix do not contradict each other.
Get rankcorrelation matrix and its cholesky factor, respectively,
of distribution. The function
returns a pointer to an array of size dim
x dim
.
The rows of the matrix are stored consecutively in this array.
If the requested matrix is not marked as known the NULL
pointer is returned and unur_errno
is set to
UNUR_ERR_DISTR_GET
.
Important: Do not modify the array that holds the rankcorrelation matrix!
Sets marginal distributions of the given distribution to the same marginal distribution object. The marginal distribution must be an instance of a continuous univariate distribution object. Notice that the marginal distribution is copied into the distribution object.
Analogously to the above
unur_distr_cvec_set_marginals
call.
However, now an array marginals of the pointers to each of
the marginal distributions must be given. It must be an
array of size dim
, where dim
is the dimension
returned by
unur_distr_get_dim
.
Notice: Local copies for each of the entries are stored in
the distribution object. If some of these entries are
identical (i.e. contain the same pointer), then for each of these a
new copy is made.
Similar to the above
unur_distr_cvec_set_marginal_array
call.
However, now the pointers to the particular marginal distributions
can be given as parameter and does not require an array of
pointers. Additionally the given distribution objects are
immediately destroyed. Thus calls like
unur_distr_normal
can be
used as arguments.
(With
unur_distr_cvec_set_marginal_array
the result of such call
has to be stored in a pointer since it has to be freed afterwarts
to avoid memory leaks!)
The number of pointers to in the list of function arguments
must be equal to the dimension of the distribution,
i.e. the dimension returned by
unur_distr_get_dim
.
If one of the given pointer to marginal distributions is the NULL
pointer then the marginal distributions of distribution are
not set (or previous settings are not changed) and an error code is
returned.
Important: All distribution objects given in the argument list are destroyed!
Get pointer to the nth marginal distribution
object from the given multivariate distribution.
If this does not exist, NULL
is returned.
The marginal distributions are enumerated from 1
to dim
, where dim
is the dimension
returned by
unur_distr_get_dim
.
Sets array of parameters for distribution. There is an upper limit
for the number of parameters n_params
. It is given by the
macro UNUR_DISTR_MAXPARAMS
in unuran_config.h. (It is set to
5 by default but can be changed to any appropriate nonnegative number.)
If n_params is negative or exceeds this limit no parameters
are copied into the distribution object and unur_errno
is set to
UNUR_ERR_DISTR_NPARAMS
.
For standard distributions from the UNU.RAN library the parameters
are checked. Moreover, the domain is updated automatically.
If the given parameters are invalid for the standard distribution,
then no parameters are set and an error code is returned.
Notice that the given parameter list for such a distribution is
handled in the same way as in the corresponding new
calls, i.e. optional parameters for the PDF that are not present in
the given list are (re)set to their default values.
Important: If the parameters of a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
are changed, then neither its mode nor the normalization
constant are updated. Please use the respective calls
unur_distr_cvec_upd_mode
and
unur_distr_cvec_upd_pdfvol
.
Get number of parameters of the PDF and set pointer params to
array of parameters. If no parameters are stored in the object, an
error code is returned and params
is set to NULL
.
Important: Do not change the entries in params!
This function provides an interface for additional vector parameters for a multivariate distribution besides mean vector and covariance matrix which have their own calls.
It sets the parameter with number par.
par indicates directly which of the parameters is set and
must be a number between 0
and UNUR_DISTR_MAXPARAMS
1
(the upper limit of possible parameters defined in
unuran_config.h; it is set to 5 but can be changed to any
appropriate nonnegative number.)
The entries of a this parameter are given by the array param_vec of size n_params. Notice that using this interface an An (n x m)matrix has to be stored in an array of length n_params = n times m; where the rows of the matrix are stored consecutively in this array.
Due to great variety of possible parameters for a multivariate distribution there is no simpler interface.
If param_vec is NULL
then the corresponding entry is cleared.
Important: If the parameters of a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
are changed, then neither its mode nor the normalization
constant are updated. Please use the respective calls
unur_distr_cvec_upd_mode
and
unur_distr_cvec_upd_pdfvol
.
If an error occurs no parameters are copied into the parameter
object unur_errno
is set to UNUR_ERR_DISTR_DATA
.
Get parameter of the PDF with number par.
The pointer to the parameter array is stored in param_vecs, its
size is returned by the function.
If the requested parameter is not set, then an error code is returned
and params
is set to NULL
.
Important: Do not change the entries in param_vecs!
Set rectangular domain for distribution with lowerleft
and upperright vertices. Both must be pointer to an
array of the size returned by
unur_distr_get_dim
.
A NULL
pointer is interpreted as the zero vector (0,…,0).
For setting a coordinate of the boundary to
+/ infinity
use +/ UNUR_INFINITY
.
The lowerleft vertex must be strictly smaller than
upperright in each component. Otherwise no domain
is set and unur_errno
is set to UNUR_ERR_DISTR_SET
.
By default the domain of a distribution is unbounded. Thus one can use this call to truncate an existing distribution.
Important: Changing the domain of distribution
marks derived parameters like the mode or the center as unknown and
must be set after changing the domain. This is important for
the already set (or default) value for the center does not
fall into the given domain.
Notice that calls of the PDF and derived functions return 0.
when the parameter is not contained in the domain.
Check whether x falls into the domain of distribution.
The following paramters must be set whenever one of the essential parameters has been set or changed (and the parameter is required for the chosen method).
Set mode of the distribution. mode must be a pointer to an
array of the size returned by
unur_distr_get_dim
.
A NULL
pointer for mode is interpreted as the zero
vector (0,…,0).
Recompute the mode of the distribution. This call works
properly for distribution objects from the UNU.RAN library of
standard distributions when the corresponding function is
available. If it failes unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
Get mode of the distribution. The function returns a pointer to
an array of the size returned by
unur_distr_get_dim
.
If the mode is not marked as known the NULL
pointer is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
(There is no difference between the case where no routine for
computing the mode is available and the case where no mode exists
for the distribution at all.)
Important: Do not modify the array that holds the mode!
Set center of the distribution. center must be a pointer to an
array of the size returned by
unur_distr_get_dim
.
A NULL
pointer for center is interpreted as the zero
vector (0,…,0).
The center is used by some methods to shift the distribution in order to decrease numerical roundoff error. If not given explicitly a default is used. Moreover, it is used as starting point for several numerical search algorithm (e.g. for the mode). Then center must be a pointer where the call to the PDF returns a nonzero value. In particular center must contained in the domain of the distribution.
Default: The mode, if given by a
unur_distr_cvec_set_mode
call;
else the mean, if given by a
unur_distr_cvec_set_mean
call;
otherwise the null vector (0,…,0).
Get center of the distribution. The function returns a pointer to
an array of the size returned by
unur_distr_get_dim
.
It always returns some point as there always exists a default for
the center, see
unur_distr_cvec_set_center
.
Important: Do not modify the array that holds the center!
Set the volume below the PDF. If vol is nonpositive, no
volume is set and unur_errno
is set to
UNUR_ERR_DISTR_SET
.
Recompute the volume below the PDF of the distribution.
It only works for distribution objects from the
UNU.RAN library of standard distributions when the
corresponding function is available. Otherwise unur_errno
is
set to UNUR_ERR_DISTR_DATA
.
This call also sets the normalization constant such that the given PDF is the derivative of a given CDF, i.e. the volume is 1.
Get the volume below the PDF of the distribution. If this volume is
not known,
unur_distr_cont_upd_pdfarea
is called to compute
it. If this is not successful UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
Next: Continuous empirical multivariate distributions, Previous: Continuous multivariate distributions, Up: Handling distribution objects [Contents][Index]
Full conditional distribution for a given continuous multivariate distributiion. The condition is a position vector and either a variable that is variated or a vector that indicates the direction on which the random vector can variate.
There is a subtle difference between using direction vector and using the kth variable. When a direction vector is given the PDF of the conditional distribution is defined by f(t) = PDF(pos + t * dir). When a variable is selected the full conditional distribution with all other variables fixed is used.
This is a special case of a continuous univariate distribution and thus they have most of these parameters (with the exception that functions cannot be changed). Additionally,
This distibution type is primarily used for evaluation the conditional distribution and its derivative (as required for, e.g., the Gibbs sampler). The density is not normalized (i.e. does not integrate to one). Mode and area are not available and it does not make sense to use any call to set or change parameters except the ones given below.
Create an object for full conditional distribution for the given distribution. The condition is given by a position vector pos and either the kth variable that is variated or the vector dir that contains the direction on which the random vector can variate.
distribution must be a pointer to a multivariate continuous
distribution.
pos must be a pointer to an array of size dim
, where
dim
is the dimension of the underlying distribution object.
dir must be a pointer to an array if size dim
or NULL
.
k must be in the range 0, …, dim1
.
If the kth variable is used, dir must be set to NULL
.
Notice: There is a subtle difference between using direction
vector dir and using the kth variable.
When dir is given, the current position pos is mapped into
0 of the conditional distribution and the derivative is taken from
the function PDF(pos+t*dir) w.r.t. t.
On the other hand, when the coordinate k is used (i.e., when
dir is set to NULL
), the full conditional distribution of the
distribution is considered (as used for the Gibbs sampler).
In particular, the current point is just projected into the
onedimensional subspace without mapping it into the point 0.
Notice: If a coordinate k is used, then the kth partial derivative is used if it as available. Otherwise the gradient is computed and the kth component is returned.
The resulting generator object is of the same type as of a
unur_distr_cont_new
call.
Set/change condition for conditional distribution. Change values of fixed variables to pos and use direction dir or kth variable of conditional distribution.
pos must be a pointer to an array of size dim
, where
dim
is the dimension of the underlying distribution object.
dir must be a pointer to an array if size dim
or NULL
.
k must be in the range 0, …, dim1
.
If the kth variable is used, dir must be set to NULL
.
Notice: There is a subtle difference between using direction
vector dir and using the kth variable.
When dir is given, the current position pos is mapped into
0 of the conditional distribution and the derivative is taken from
the function PDF(pos+t*dir) w.r.t. t.
On the other hand, when the coordinate k is used (i.e., when
dir is set to NULL
), the full conditional distribution of the
distribution is considered (as used for the Gibbs sampler).
In particular, the current point is just projected into the
onedimensional subspace without mapping it into the point 0.
Get condition for conditional distribution.
The values for the fixed variables are stored in pos, which
must be a pointer to an array of size dim
.
The condition is stored in dir and k, respectively.
Important: Do not change the entries in pos and dir!
Get pointer to distribution object for underlying distribution.
Next: MATRix distributions, Previous: Continuous univariate full conditional distribution, Up: Handling distribution objects [Contents][Index]
Empirical multivariate distributions are just lists of vectors (with the same dimension). Thus there are only calls to insert these data. How these data are used to sample from the empirical distribution depends from the chosen generation method.
Create a new (empty) object for an empirical multivariate
continuous distribution. dim is the number of components of
the random vector (i.e. its dimension). It must be at least 2;
otherwise
unur_distr_cemp_new
should be used to create an object
for an empirical univariate distribution.
Set observed sample for empirical distribution.
sample is an array of doubles of size
dim
x n_sample, where
dim
is the dimension of the distribution returned by
unur_distr_get_dim
.
The data points must be stored consecutively in sample, i.e.,
data points (x1, y1), (x2, y2), … are given as an array
{x1, y1, x2, y2, …}.
Read data from file filename.
It reads the first dim
numbers from each line, where
dim
is the dimension of the distribution returned by
unur_distr_get_dim
.
Numbers are parsed by means of the C standard routine strtod
.
Lines that do not start with +
, 
, .
, or a
digit are ignored. (Beware of lines starting with a blank!)
In case of an error (file cannot be opened, too few entries in a line, invalid string for double in line) no data are copied into the distribution object and an error code is returned.
Get number of samples and set pointer sample to array of
observations. If no sample has been given, an error code
is returned and sample is set to NULL
.
If successful sample points to an array of length
dim
x n_sample
, where
dim
is the dimension of the distribution returned by
unur_distr_get_dim
and n_sample
the return value of the
function.
Important: Do not modify the array sample.
Next: Discrete univariate distributions, Previous: Continuous empirical multivariate distributions, Up: Handling distribution objects [Contents][Index]
Distributions for random matrices. Notice that UNU.RAN uses
arrays of double
s to handle matrices. The rows of
the matrix are stored consecutively.
Create a new (empty) object for a matrix distribution. n_rows
and n_cols are the respective numbers of rows and columns of
the random matrix (i.e. its dimensions). It is also possible to
have only one number or rows and/or columns.
Notice, however, that this is treated as a distribution of random
matrices with only one row or column or component and not as a
distribution of vectors or real numbers. For the latter
unur_distr_cont_new
or
unur_distr_cvec_new
should be
used to create an object for a univariate distribution and a
multivariate (vector) distribution, respectively.
Get number of rows and columns of random matrix (its dimension).
It returns the total number of components. If successfull
UNUR_SUCCESS
is returned.
Previous: MATRix distributions, Up: Handling distribution objects [Contents][Index]
The calls in this section can be applied to discrete univariate distributions.
new
instance of a discrete univariate
distribution.
cdf
) and
probability mass function (PMF, pmf
).
The following is important:
pmf
need not be normalized, i.e.,
any summable nonnegative function on the set of intergers can be
used.
cdf
must be a distribution function, i.e. it
must be monotonically increasing with range [0,1].
cdf
and pdf
are used together for a
pariticular generation method, then pmf
must be
normalized, i.e. it must sum to 1.
cdf
and pdf
can be
provided as str
ings instead of function pointers.
pv
), i.e. an array of double
s.
It can be automatically computed if the pmf
is
given but pv
is not.
pmfparams
) and the
total sum (pmfsum
) of the given PMF or PV.
mode
of the distribution.
domain
of the distribution.
Create a new (empty) object for a univariate discrete distribution.
There are two interfaces for discrete univariate distributions: Either provide a (finite) probability vector (PV). Or provide a probability mass function (PMF). For the latter case there are also a couple of derived parameters that are not required when a PV is given.
It is not possible to set both a PMF and a PV directly. However, the
PV can be computed from the PMF (or the CDF if no PMF is available)
by means of a
unur_distr_discr_make_pv
call.
If both the PV and the PMF are given in the distribution object it
depends on the generation method which of these is used.
Set finite probability vector (PV) for the distribution. It is not necessary that the entries in the given PV sum to 1. n_pv must be positive. However, there is no testing whether all entries in pv are nonnegative.
If no domain has been set, then the left boundary is set to
0
, by default. If n_pv is too large, e.g. because
left boundary + n_pv exceeds the range of integers,
then the call fails.
Notice that it is not possible to set both a PV and a PMF or CDF.
If the PMF or CDF is set first one cannot set the PV.
If the PMF or CDF is set first after a PV is set, the latter is
removed (and recomputed using
unur_distr_discr_make_pv
when required).
Compute a PV when a PMF or CDF is given. However, when the domain is not given or is too large and the sum over the PMF is given then the (right) tail of the distribution is chopped off such that the probability for the tail region is less than 1.e8. If the sum over the PMF is not given a PV of maximal length is computed.
The maximal size of the created PV is bounded by the macro
UNUR_MAX_AUTO_PV
that is defined in unuran_config.h.
If successful, the length of the generated PV is returned.
If the sum over the PMF on the chopped tail is not neglible small
(i.e. greater than 1.e8 or unknown) than the
negative of the length of the PV is returned and
unur_errno
is set to UNUR_ERR_DISTR_SET
.
Notice that the left boundary of the PV is set to 0
by
default when a discrete distribution object is created from
scratch.
If computing a PV fails for some reasons, an error code is returned and
unur_errno
is set to UNUR_ERR_DISTR_SET
.
Get length of PV of the distribution and set pointer
pv to array of probabilities. If no PV is given,
an error code is returned and pv is set to NULL
.
(It does not call
unur_distr_discr_make_pv
!)
Set respective pointer to the PMF and the CDF of the distribution.
These functions must be of type
double funct(int k, const UNUR_DISTR *distr)
.
It is important to note that all these functions must return a
result for all integers k. E.g., if the domain of a given
PMF is the interval {1,2,3,…,100}, than the given function
must return 0.0
for all points outside this interval.
The default domain for the PMF or CDF is [0
, INT_MAX
].
The domain can be changed using a
unur_distr_discr_set_domain
call.
It is not possible to change such a function. Once the PMF or CDF is set it cannot be overwritten. A new distribution object has to be used instead.
Notice that it is not possible to set both a PV and a PMF or CDF.
If the PMF or CDF is set first one cannot set the PV.
If the PMF or CDF is set first after a PV is set, the latter is
removed (and recomputed using
unur_distr_discr_make_pv
when required).
Set inverse CDF of the distribution.
invcdf must be a pointer must be of type
int funct(double x, const UNUR_DISTR *distr)
,
i.e., it should return a double
.
Evaluate the PV, PMF, and the CDF, respectively, at k.
Notice that distribution must not be the NULL
pointer.
If no PV is set for the distribution, then
unur_distr_discr_eval_pv
behaves like
unur_distr_discr_eval_pmf
.
If the corresponding function is not available for the distribution,
UNUR_INFINITY
is returned and unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
IMPORTANT: In the case of a truncated standard distribution these calls always return the respective values of the untruncated distribution!
Evaluate the inverse CDF at u.
Notice that distribution must not be the NULL
pointer.
If the corresponding function is not available for the distribution,
INT_MAX
is returned and unur_errno
is set to
UNUR_ERR_DISTR_DATA
.
IMPORTANT: In the case of a truncated standard distribution these calls always return the respective values of the untruncated distribution!
This function provides an alternative way to set a PMF of the
distribution.
pmfstr is a character string that contains the formula
for the PMF, see Function String, for details.
See also the remarks for the
unur_distr_discr_set_pmf
call.
It is not possible to call this funtion twice or to call this
function after a
unur_distr_discr_set_pmf
call.
This function provides an alternative way to set a CDF; analogously
to the
unur_distr_discr_set_pmfstr
call.
Get pointer to respective string for PMF and CDF of distribution that is given via the string interface. This call allocates memory to produce this string. It should be freed when it is not used any more.
Set array of parameters for distribution. There is an upper limit
for the number of parameters n_params. It is given by the
macro UNUR_DISTR_MAXPARAMS
in unuran_config.h. (It is set to
5 but can be changed to any appropriate nonnegative number.)
If n_params is negative or exceeds this limit no parameters
are copied into the distribution object and unur_errno
is set to UNUR_ERR_DISTR_NPARAMS
.
For standard distributions from the UNU.RAN library the parameters
are checked. Moreover, the domain is updated automatically unless it
has been changed before by a
unur_distr_discr_set_domain
call.
If the given parameters are invalid for the standard distribution,
then no parameters are set and an error code is returned.
Notice that the given parameter list for such a distribution is
handled in the same way as in the corresponding new
calls, i.e. optional parameters for the PDF that are not present in
the given list are (re)set to their default values.
Important: Integer parameter must be given as double
s.
Get number of parameters of the PMF and set pointer
params to array of parameters. If no parameters are stored
in the object, an error code is returned and params
is set to
NULL
.
Set the left and right borders of the domain of the
distribution. This can also be used to truncate an existing
distribution. For setting the boundary to
+/ infinity
use
INT_MIN
and INT_MAX
, respectively.
If right is not strictly greater than left no domain
is set and unur_errno
is set to UNUR_ERR_DISTR_SET
.
It is allowed to use this call to increase the domain.
If the PV of the discrete distribution is used,
than the right boudary is ignored (and internally set to
left + size of PV  1).
Notice that INT_MIN
and INT_MAX
are interpreted as
(minus/plus) infinity.
Default: [0
, INT_MAX
].
Get the left and right borders of the domain of the
distribution. If the domain is not set explicitly
the interval [INT_MIN
, INT_MAX
] is assumed and returned.
When a PV is given then the domain is set automatically to
[0
,size of PV  1].
The following paramters must be set whenever one of the essential parameters has been set or changed (and the parameter is required for the chosen method).
Set mode of distribution.
Recompute the mode of the distribution. This call works properly
for distribution objects from the
UNU.RAN library of standard distributions
when the corresponding function is available.
Otherwise a (slow) numerical mode finder is used. It only works properly
for unimodal probability mass functions. If it failes
unur_errno
is set to UNUR_ERR_DISTR_DATA
.
Get mode of distribution. If the mode is not marked as known,
unur_distr_discr_upd_mode
is called to compute the mode. If this
is not successful INT_MAX
is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
(There is no difference between the case where no routine for
computing the mode is available and the case where no mode exists
for the distribution at all.)
Set the sum over the PMF. If sum
is nonpositive, no
sum is set and unur_errno
is set to
UNUR_ERR_DISTR_SET
.
For a distribution object created by the
UNU.RAN library of standard distributions you always should use
the
unur_distr_discr_upd_pmfsum
.
Otherwise there might be
ambiguous sideeffects.
Recompute the sum over the PMF of the distribution.
In most cases the normalization constant is recomputed and thus the
sum is 1. This call works for distribution objects from the UNU.RAN
library of standard distributions when the corresponding function
is available. When a PV, a PMF with finite domain, or a CDF is
given, a simple generic function which uses a naive summation loop
is used. If this computation is not possible, an error code is
returned and unur_errno
is set to UNUR_ERR_DISTR_DATA
.
The call does not work for distributions from the UNU.RAN library of standard distributions with truncated domain when the CDF is not available.
Get the sum over the PMF of the distribution. If this sum is
not known,
unur_distr_discr_upd_pmfsum
is called to compute
it. If this is not successful UNUR_INFINITY
is returned and
unur_errno
is set to UNUR_ERR_DISTR_GET
.
Next: Using uniform random number generators, Previous: Handling distribution objects, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
Sampling from a particular distribution with UNU.RAN requires the following steps:
unur_init
.
Important: Initialization of the generator object might fail.
unur_init
returns a NULL
pointer then, which must not be
used for sampling.
unur_get_distr
call and
changing the distribution using the correspondig set calls,
see Handling distribution objects.
The generator object must then be reinitialized by means of
the
unur_reinit
call.
Important: Currently not all methods allow reinitialization, see the description of the particular method (keyword Reinit).
Important: Reinitialization of the generator object might fail.
Thus one must check the return code of the
unur_reinit
call.
Important: When reinitialization fails then sampling routines
always return UNUR_INFINITY
(for continuous distributions) or 0
(for
discrete distributions), respectively.
However, it is still possible to change the underlying distribution
and try to reinitialize again.
Next: AUTO – Select method automatically, Up: Methods for generating nonuniform random variates [Contents][Index]
Routines for all generator objects.
Initialize a generator object. All necessary information must be stored in the parameter object.
Important: If an error has occurred a NULL
pointer is
return. This must not be used for the sampling routines (this causes a
segmentation fault).
Always check whether the call was successful or not!
Important: This call destroys the parameter object automatically. Thus it is not necessary/allowed to free it.
Update an existing generator object after the underlying
distribution has been modified (using
unur_get_distr
together
with corresponding set calls.
It must be executed before sampling using this generator
object is continued as otherwise it produces an invalid sample or
might even cause a segmentation fault.
Important: Currently not all methods allow reinitialization, see the description of the particular method (keyword Reinit).
Important: Reinitialization of the generator object might fail. Thus one must check the return code:
UNUR_SUCCESS (0x0u)
success (no error)
UNUR_ERR_NO_REINIT
reinit routine not implemented.
some error has occured while trying to reinitialize the generator object.
Important: When reinitialization fails then sampling routines
always return UNUR_INFINITY
(for continuous distributions)
or 0
(for discrete distributions), respectively.
However, it is still possible to change the underlying distribution
and try to reinitialize again.
Important: When one tries to run
unur_reinit
but reinitialization
is not implemented, then the generator object cannot be used any more
and must be destroyed and a new one has to be built from scratch.
Sample from generator object. The three routines depend on the type of the generator object (discrete or continuous univariate distribution, multivariate distribution, or random matrix).
Notice: UNU.RAN uses arrays of double
s to handle
matrices. There the rows of the matrix are stored consecutively.
Notice: The routines
unur_sample_vec
and
unur_sample_matr
return UNUR_SUCCESS
if generation was successful and
some error code otherwise.
Important: These routines do not check whether
generator is an invalid NULL
pointer.
Compute the U quantile of a continuous distribution using a generator object that implements an (approximate) inversion methods.
The following methods are currently available:
double
.
Important: This routine does not check whether
generator is an invalid NULL
pointer.
In case of an error UNUR_INFINITY or INT_MAX (depending on the type of generator) is returned.
Destroy (free) the given generator object.
Get a string with informations about the given generator.
These informations allow some fine tuning of the generation method.
If help is TRUE
, some hints on setting parameters are given.
This function is intented for using in interactive environments (like R).
If an error occurs, then NULL
is returned.
Get the number of dimension of a (multivariate) distribution.
For a univariate distribution 1
is return.
Get identifier string for generator.
Get identifier for generating method. These identifiers are declared in src/methods/unur_metthods.h.
Return TRUE
if the generator object implements an inversion method,
and FALSE
otherwise.
Get pointer to distribution object from generator object. This function can be used to change the parameters of the distribution and reinitialize the generator object. Notice that currently not all generating methods have a reinitialize routine. This function should be used with extreme care. Changing the distribution is changed and using the generator object without reinitializing might cause wrong samples or segmentation faults. Moreover, if the corresponding generator object is freed, the pointer must not be used.
Important: The returned distribution object must not
be freed. If the distribution object is changed then one must
run
unur_reinit
!
Set flag whether the generator object should make a private copy of the given distribution object or just stores the pointer to this distribution object. Values for use_privatecopy:
TRUE
make a private copy (default)
FALSE
do not make a private copy and store pointer to given (external) distribution object.
By default, generator objects keep their own private copy of the given distribution object. Thus the generator object can be handled independently from other UNU.RAN objects (with uniform random number generators as the only exception). When the generator object is initialized the given distribution object is cloned and stored.
However, in some rare situations it can be useful when only the
pointer to the given distribution object is stored without making a
private copy. A possible example is when only one random variate has
to be drawn from the distribution.
This behavior can be achieved when use_localcopy is set to
FALSE
.
Warning! Using a pointer to the external distribution object instead of a private copy must be done with extreme care! When the distrubtion object is changed or freed then the generator object does not work any more, might case a segmentation fault, or (even worse) produces garbage. On the other hand, when the generator object is initialized or used to draw a random sampling the distribution object may be changed.
Notice:
The prototypes of all unur_<method>_new
calls use a
const
qualifier for the distribution argument.
However, if use_privatecopy is set to FALSE
this qualifier is
discarded and the distribution might be changed.
Important!
If use_localcopy is set to FALSE
and the corresponding
distribution object is changed then one must run
unur_reinit
on the generator object.
(Notice that currently not all generation methods support
reinitialization.)
Default: use_privatecopy is TRUE
.
Next: Methods for continuous univariate distributions, Previous: Routines for all generator objects, Up: Methods for generating nonuniform random variates [Contents][Index]
AUTO selects a an appropriate method for the given distribution object automatically. There are no parameters for this method, yet. But it is planned to give some parameter to describe the task for which the random variate generator is used for and thus make the choice of the generating method more appropriate. Notice that the required sampling routine for the generator object depends on the type of the given distribution object.
The chosen method also depends on the sample size for which the
generator object will be used. If only a few random variates
the order of magnitude of the sample size should be set via a
unur_auto_set_logss
call.
IMPORTANT: This is an experimental version and the method chosen may change in future releases of UNU.RAN.
For an example see Example: As short as possible.
Create a generator object for the given distribution object.
Get default parameters for generator.
Set the order of magnitude for the size of the sample that will be generated by the generator, i.e., the the common logarithm of the sample size.
Default is 10.
Notice: This feature will be used in future releases of UNU.RAN only.
Next: Methods for continuous empirical univariate distributions, Previous: AUTO – Select method automatically, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for continuous univariate distributions

/*  */ /* File: example_cont.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from a continuous univariate */ /* distribution. */ /* */ /* We build a distribution object from scratch and sample. */ /*  */ /* Define the PDF and dPDF of our distribution. */ /* */ /* Our distribution has the PDF */ /* */ /* / 1  x*x if x <= 1 */ /* f(x) = < */ /* \ 0 otherwise */ /* */ /* The PDF of our distribution: */ double mypdf( double x, const UNUR_DISTR *distr ) /* The second argument (`distr') can be used for parameters */ /* for the PDF. (We do not use parameters in our example.) */ { if (fabs(x) >= 1.) return 0.; else return (1.x*x); } /* end of mypdf() */ /* The derivative of the PDF of our distribution: */ double mydpdf( double x, const UNUR_DISTR *distr ) { if (fabs(x) >= 1.) return 0.; else return (2.*x); } /* end of mydpdf() */ /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Create a new distribution object from scratch. */ /* Get empty distribution object for a continuous distribution */ distr = unur_distr_cont_new(); /* Fill the distribution object  the provided information */ /* must fulfill the requirements of the method choosen below. */ unur_distr_cont_set_pdf(distr, mypdf); /* PDF */ unur_distr_cont_set_dpdf(distr, mydpdf); /* its derivative */ unur_distr_cont_set_mode(distr, 0.); /* mode */ unur_distr_cont_set_domain(distr, 1., 1.); /* domain */ /* Choose a method: TDR. */ par = unur_tdr_new(distr); /* Set some parameters of the method TDR. */ unur_tdr_set_variant_gw(par); unur_tdr_set_max_sqhratio(par, 0.90); unur_tdr_set_c(par, 0.5); unur_tdr_set_max_intervals(par, 100); unur_tdr_set_cpoints(par, 10, NULL); /* Create the generator object. */ gen = unur_init(par); /* Notice that this call has also destroyed the parameter */ /* object `par' as a side effect. */ /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
/*  */ /* File: example_cont_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from a continuous univariate */ /* distribution. */ /* We use a generic distribution object and sample. */ /* */ /* The PDF of our distribution is given by */ /* */ /* / 1  x*x if x <= 1 */ /* f(x) = < */ /* \ 0 otherwise */ /* */ /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ /* Use a generic continuous distribution. */ /* Choose a method: TDR. */ gen = unur_str2gen( "distr = cont; pdf=\"1x*x\"; domain=(1,1); mode=0. & \ method=tdr; variant_gw; max_sqhratio=0.90; c=0.5; \ max_intervals=100; cpoints=10"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: ARS – Adaptive Rejection Sampling, Up: Methods for continuous univariate distributions [Contents][Index]
Tconcave PDF, dPDF
mode
Setup: slow, Sampling: fast
not implemented
AROU is a variant of the ratioofuniforms method that uses the fact that the transformed region is convex for many distributions. It works for all Tconcave distributions with T(x) = 1/sqrt(x).
It is possible to use this method for correlation induction by
setting an auxiliary uniform random number generator via the
unur_set_urng_aux
call. (Notice that this must be done after a
possible
unur_set_urng
call.)
When an auxiliary generator is used then the number of used
uniform random numbers that is used up for one generated random
variate is constant and equal to 1.
There exists a test mode that verifies whether the conditions for
the method are satisfied or not while sampling. It can be
switched on by calling
unur_arou_set_verify
and
unur_arou_chg_verify
respectively.
Notice however that sampling is (much) slower then.
For densities with modes not close to 0 it is suggested to set
either the mode or the center of the distribution by the
unur_distr_cont_set_mode
or
unur_distr_cont_set_center
call.
The latter is the approximate location of the mode or the mean
of the distribution. This location provides some information
about the main part of the PDF and is used to avoid numerical
problems.
Get default parameters for generator.
If usedars is set to TRUE
, “derandomized adaptive rejection
sampling” (DARS) is used in setup.
Segments where the area between hat and squeeze is too
large compared to the average area between hat and squeeze
over all intervals are split.
This procedure is repeated until the ratio between area below squeeze
and area below hat exceeds the bound given by
unur_arou_set_max_sqhratio
call or the maximum number of segments is
reached. Moreover, it also aborts when no more segments can be
found for splitting.
Segments are split such that the angle of the segments are halved (corresponds to arcmean rule of method TDR (see TDR – Transformed Density Rejection)).
Default is TRUE
.
Set factor for “derandomized adaptive rejection sampling”.
This factor is used to determine the segments that are “too
large”, that is, all segments where the area between squeeze and
hat is larger than factor times the average area over all
intervals between squeeze and hat.
Notice that all segments are split when factor is set to
0.
, and that there is no splitting at all when factor
is set to UNUR_INFINITY
.
Default is 0.99
. There is no need to change this parameter.
Set upper bound for the
ratio (area inside squeeze) / (area inside envelope).
It must be a number between 0 and 1.
When the ratio exceeds the given number no further construction
points are inserted via adaptive rejection sampling.
Use 0
if no construction points should be added after the
setup.
Use 1
if adding new construction points should not be
stopped until the maximum number of construction points is reached.
Default is 0.99
.
Get the current ratio (area inside squeeze) / (area inside envelope)
for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the hat for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the squeeze for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Set maximum number of segements. No construction points are added after the setup when the number of segments succeeds max_segs.
Default is 100
.
Set construction points for enveloping polygon.
If stp is NULL
, then a heuristical rule of thumb is used to
get n_stp construction points.
This is the default behavior when this routine is not called.
The (default) number of construction points is 30
, then.
Use the center as construction point.
Default is TRUE
.
Set factor for relative size of the guide table for indexed search
(see also method DGT DGT – (Discrete) Guide Table method (indexed search)). It must be greater than or equal
to 0
.
When set to 0
, then sequential search is used.
Default is 2
.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Sometimes it might happen that
unur_init
has been executed
successfully. But when additional construction points are added by
adaptive rejection sampling, the algorithm detects that the
PDF is not Tconcave.
With pedantic being TRUE
, the
sampling routine is then exchanged by a routine that simply returns
UNUR_INFINITY
. Otherwise the new point is not added to the
list of construction points. At least the hat function remains
Tconcave.
Setting pedantic to FALSE
allows sampling from a
distribution which is “almost” Tconcave and small errors are
tolerated. However it might happen that the hat function cannot be
improved significantly. When the hat function that has been
constructed by the
unur_init
call is extremely large then it
might happen that the generation times are extremely high
(even hours are possible in extremely rare cases).
Default is FALSE
.
Next: CEXT – wrapper for Continuous EXTernal generators, Previous: AROU – Automatic RatioOfUniforms method, Up: Methods for continuous univariate distributions [Contents][Index]
concave logPDF, derivative of logPDF
mode
Setup: fast, Sampling: slow
supported
ARS is an acceptance/rejection method that uses the concavity
of the logdensity function to construct hat function and
squeezes automatically.
It is very similar to method TDR (see TDR – Transformed Density Rejection) with variant GW,
parameter c = 0
, and DARS switched off.
Moreover, method ARS requires the logPDF and its derivative
dlogPDF to run. On the other hand, it is designed to draw only a
(very) small samples and it is much more robust against
densities with very large or small areas below the PDF as
it occurs, for example, in conditional distributions of
(high dimensional) multivariate distributions.
Additionally, it can be reinitialized when the underlying
distribution has been modified.
Thus it is well suited for Gibbs sampling.
Notice, that method ARS is a restricted version of TDR. If the full functionally of Transformed Density Rejection is needed use method TDR – Transformed Density Rejection.
Method ARS is designed for distributions with logconcave densities. To use this method you need a distribution object with the logarithm of the PDF and its derivative given.
The number of construction points as well as a set of such
points can be provided using
unur_ars_set_cpoints
.
Notice that addition construction points are added by means of
adaptive rejection sampling until the maximal number of
intervals given by
unur_ars_set_max_intervals
is reached.
A generated distribution object can be reinitialized using the
unur_reinit
call. When
unur_reinit
is called construction
points for the new generator are necessary. There are two options:
Either the same construction points as for the initial generator
(given by a
unur_ars_set_cpoints
call) are used (this is the
default), or percentiles of the old hat function can be used.
This can be set or changed using
unur_ars_set_reinit_percentiles
and
unur_ars_chg_reinit_percentiles
.
This feature is usefull when the underlying distribution object
is only moderately changed. (An example is Gibbs sampling with
small correlations.)
There exists a test mode that verifies whether the conditions for
the method are satisfied or not. It can be switched on by calling
unur_ars_set_verify
and
unur_ars_chg_verify
respectively.
Notice however that sampling is (much) slower then.
Method ARS aborts after a given number of iterations and return
UNUR_INFINITY to prevent (almost) infinite loops. This might
happen when the starting hat is much too large and it is not
possible to insert new construction points due to severe
numerical errors or (more likely) the given PDF is not
logconcave. This maximum number of iterations can be set by
means of a
unur_ars_set_max_iter
call.
Get default parameters for generator.
Set maximum number of intervals. No construction points are added after the setup when the number of intervals suceeds max_ivs. It is increased automatically to twice the number of construction points if this is larger.
Default is 200
.
Set construction points for the hat function. If cpoints is
NULL
then a heuristic rule of thumb is used to get n_cpoints
construction points. This is the default behavior.
n_cpoints should be at least 2
, otherwise defaults are used.
The default number of construction points is 2.
By default, when the generator object is reinitialized, it
used the same construction points as for the initialization
procedure.
Often the underlying distribution object has been changed only
moderately. For example, the full conditional distribution of a
multivariate distribution.
In this case it might be more appropriate to use
percentilesm of the hat function for the last (unchanged)
distribution. percentiles must then be a pointer to an
ordered array of numbers between 0.01
and 0.99
.
If percentiles is NULL
, then a heuristic rule of thumb is
used to get n_percentiles values for these percentiles.
Notice that n_percentiles must be at least 2
,
otherwise defaults are used.
(Then the first and third quartiles are used by default.)
When reinit fails with the given construction points or the percentiles
of the old hat function, another trial is undertaken with ncpoints
construction points. ncpoints must be at least 10
.
Default: 30
The rejection loop stops after max_iter iterations and return UNUR_INFINITY.
Default: 10000
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Sometimes it might happen that
unur_init
has been executed
successfully. But when additional construction points are added by
adaptive rejection sampling, the algorithm detects that the
PDF is not logconcave.
With pedantic being TRUE
, the
sampling routine is exchanged by a routine that simply returns
UNUR_INFINITY
. Otherwise the new point is not added to the
list of construction points. At least the hat function remains
logconcave.
Setting pedantic to FALSE
allows sampling from a
distribution which is “almost” logconcave and small errors are
tolerated. However it might happen that the hat function cannot be
improved significantly. When the hat functions that has been
constructed by the
unur_init
call is extremely large then it
might happen that the generation times are extremely high
(even hours are possible in extremely rare cases).
Default is FALSE
.
Get the logarithm of area below the hat for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Evaluate the inverse of the CDF of the hat distribution at u.
If u is out of the domain [0,1] then unur_errno
is set
to UNUR_ERR_DOMAIN
and the respective bound of
the domain of the distribution are returned (which is
UNUR_INFINITY
or UNUR_INFINITY
in the case of
unbounded domains).
Next: CSTD – Continuous STandarD distributions, Previous: ARS – Adaptive Rejection Sampling, Up: Methods for continuous univariate distributions [Contents][Index]
routine for sampling continuous random variates
depends on external generator
supported
Method CEXT is a wrapper for external generators for continuous univariate distributions. It allows the usage of external random variate generators within the UNU.RAN framework.
The following steps are required to use some external generator within the UNU.RAN framework (some of these are optional):
unur_cext_new
call.
The argument distribution is optional and can be replaced
by NULL
. However, it is required if you want to pass
parameters of the generated distribution to the external
generator or for running some validation tests provided by
UNU.RAN.
int (*init)(UNUR_GEN *gen)
and plug it into the generator
object using the
unur_cext_set_init
call. Notice that the
init routine must return UNUR_SUCCESS
when it has
been executed successfully and UNUR_FAILURE
otherwise.
It is possible to get the size of and the pointer to the array
of parameters of the underlying distribution object by the
respective calls
unur_cext_get_ndistrparams
and
unur_cext_get_distrparams
.
Parameters for the external generator that are computed in the
init routine can be stored in a single array or structure
which is available by the
unur_cext_get_params
call.
Using an init routine is optional and can be omitted.
double (*sample)(UNUR_GEN *gen)
and plug it into the
generator object using the
unur_cext_set_sample
call.
Uniform random numbers are provided by the
unur_sample_urng
call. Do not use your own implementation of a uniform random
number generator directly. If you want to use your own random
number generator we recommend to use the UNU.RAN interface (see
see Using uniform random number generators).
The array or structure that contains parameters for the external
generator that are computed in the init routine are
available using the
unur_cext_get_params
call.
Using a sample routine is of course obligatory.
It is possible to change the parameters and the domain of the
chosen distribution and run
unur_reinit
to reinitialize the
generator object. The init routine is then called again.
Here is a short example that demonstrates the application of this method by means of the exponential distribution:
/*  */ /* File: example_cext.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* This example shows how an external generator for the */ /* exponential distribution with one scale parameter can be */ /* used within the UNURAN framework. */ /* */ /* Notice, that this example does not provide the simplest */ /* solution. */ /*  */ /* Initialization routine. */ /* */ /* Here we simply read the scale parameter of the exponential */ /* distribution and store it in an array for parameters of */ /* the external generator. */ /* [ Of course we could do this in the sampling routine as */ /* and avoid the necessity of this initialization routine. ] */ int exponential_init (UNUR_GEN *gen) { /* Get pointer to parameters of exponential distribution */ double *params = unur_cext_get_distrparams(gen); /* The scale parameter is the first entry (see manual) */ double lambda = (params) ? params[0] : 1.; /* Get array to store this parameter for external generator */ double *genpar = unur_cext_get_params(gen, sizeof(double)); genpar[0] = lambda; /* Executed successfully */ return UNUR_SUCCESS; } /*  */ /* Sampling routine. */ /* */ /* Contains the code for the external generator. */ double exponential_sample (UNUR_GEN *gen) { /* Get scale parameter */ double *genpar = unur_cext_get_params(gen,0); double lambda = genpar[0]; /* Sample a uniformly distributed random number */ double U = unur_sample_urng(gen); /* Transform into exponentially distributed random variate */ return ( log(1.  U) * lambda ); } /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Use predefined exponential distribution with scale param. 2 */ double fpar[1] = { 2. }; distr = unur_distr_exponential(fpar, 1); /* Use method CEXT */ par = unur_cext_new(distr); /* Set initialization and sampling routines. */ unur_cext_set_init(par, exponential_init); unur_cext_set_sample(par, exponential_sample); /* Create the generator object. */ gen = unur_init(par); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Get default parameters for new generator.
Set initialization routine for external generator. Inside the
Important: The routine init must return
UNUR_SUCCESS
when the generator was initialized successfully
and UNUR_FAILURE
otherwise.
Parameters that are computed in the init routine can be
stored in an array or structure that is avaiable by means of the
unur_cext_get_params
call. Parameters of the underlying
distribution object can be obtained by the
unur_cext_get_distrparams
call.
Set sampling routine for external generator.
Important:
Use unur_sample_urng(gen)
to get a uniform random number.
The pointer to the array or structure that contains the parameters
that are precomputed in the init routine are available by
unur_cext_get_params(gen,0)
.
Additionally one can use the
unur_cext_get_distrparams
call.
Get pointer to memory block for storing parameters of external
generator. A memory block of size size is automatically (re)
allocated if necessary and the pointer to this block is stored in
the generator object. If one only needs the pointer to this
memory block set size to 0
.
Notice, that size is the size of the memory block and not the length of an array.
Important: This rountine should only be used in the initialization and sampling routine of the external generator.
Get size of and pointer to array of parameters of underlying distribution in generator object.
Important: These rountines should only be used in the initialization and sampling routine of the external generator.
Next: HINV – Hermite interpolation based INVersion of CDF, Previous: CEXT – wrapper for Continuous EXTernal generators, Up: Methods for continuous univariate distributions [Contents][Index]
standard distribution from UNU.RAN library (see Standard distributions) or continuous distribution with inverse CDF.
Setup: fast, Sampling: depends on distribution and generator
supported
CSTD is a wrapper for special generators for continuous
univariate standard distributions. It only works for
distributions in the UNU.RAN library of standard distributions
(see Standard distributions)
or for continuous distributions where the inverse CDF is given.
If a distribution object is provided that is build from scratch,
it must provide the inverse CDF. Then CSTD implements the
inversion method. Otherwise, the NULL
pointer is returned.
For some distributions more than one special generator is possible.
Create a distribution object for a standard distribution
from the UNU.RAN library
(see Standard distributions),
or create a continuous distribution object and set the function
for the inverse CDF using
unur_distr_cont_set_invcdf
.
For some distributions more than one special generator
(variants) is possible. These can be choosen by a
unur_cstd_set_variant
call. For possible variants
see Standard distributions.
However the following are common to all distributions:
UNUR_STDGEN_DEFAULT
the default generator.
UNUR_STDGEN_FAST
the fastest available special generator.
UNUR_STDGEN_INVERSION
the inversion method (if available).
Notice that the variant UNUR_STDGEN_FAST
for a special
generator may be slower than one of the universal algorithms!
Additional variants may exist for particular distributions.
Sampling from truncated distributions (which can be constructed by
changing the default domain of a distribution by means of
unur_distr_cont_set_domain
or
unur_cstd_chg_truncated
calls)
is possible but requires the inversion method. Moreover the CDF
of the distribution must be implemented.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for new generator. It requires a distribution object for a continuous univariant distribution from the UNU.RAN library of standard distributions (see Standard distributions).
Using a truncated distribution is allowed only if the inversion method
is available and selected by the
unur_cstd_set_variant
call immediately
after creating the parameter object.
Use a
unur_distr_cont_set_domain
call to get a truncated distribution.
To change the domain of a (truncated) distribution of a generator use the
unur_cstd_chg_truncated
call.
Set variant (special generator) for sampling from a given distribution. For possible variants see Standard distributions.
Common variants are UNUR_STDGEN_DEFAULT
for the default generator,
UNUR_STDGEN_FAST
for (one of the) fastest implemented
special generators, and UNUR_STDGEN_INVERSION
for the
inversion method (if available).
If the selected variant number is not implemented, then an error code is
returned and the variant is not changed.
Change left and right border of the domain of the (truncated) distribution. This is only possible if the inversion method is used. Otherwise this call has no effect and an error code is returned.
Notice that the given truncated domain must be a subset of the domain of the given distribution. The generator always uses the intersection of the domain of the distribution and the truncated domain given by this call.
It is not required to run
unur_reinit
after this call has been used.
Important: If the CDF is (almost) the same for left and
right and (almost) equal to 0
or 1
, then the truncated
domain is not chanced and the call returns an error code.
Notice: If the parameters of the distribution has been changed it is recommended to set the truncated domain again, since the former call might change the domain of the distribution but not update the values for the boundaries of the truncated distribution.
Next: HRB – Hazard Rate Bounded, Previous: CSTD – Continuous STandarD distributions, Up: Methods for continuous univariate distributions [Contents][Index]
CDF
PDF, dPDF
Setup: (very) slow, Sampling: (very) fast
supported
HINV is a variant of numerical inversion, where the inverse CDF is approximated using Hermite interpolation, i.e., the interval [0,1] is split into several intervals and in each interval the inverse CDF is approximated by polynomials constructed by means of values of the CDF and PDF at interval boundaries. This makes it possible to improve the accuracy by splitting a particular interval without recomputations in unaffected intervals. Three types of splines are implemented: linear, cubic, and quintic interpolation. For linear interpolation only the CDF is required. Cubic interpolation also requires PDF and quintic interpolation PDF and its derivative.
These splines have to be computed in a setup step. However, it only works for distributions with bounded domain; for distributions with unbounded domain the tails are chopped off such that the probability for the tail regions is small compared to the given uresolution.
The method is not exact, as it only produces random variates of the approximated distribution. Nevertheless, the maximal numerical error in "udirection" (i.e. UCDF(X), for X = "approximate inverse CDF"(U) UCDF(X)) can be set to the required resolution (within machine precision). Notice that very small values of the uresolution are possible but may increase the cost for the setup step.
As the possible maximal error is only estimated in the setup it may be necessary to set some special design points for computing the Hermite interpolation to guarantee that the maximal uerror can not be bigger than desired. Such points are points where the density is not differentiable or has a local extremum. Notice that there is no necessity to do so. However, if you do not provide these points to the algorithm there might be a small chance that the approximation error is larger than the given uresolution, or that the required number of intervals is larger than necessary.
HINV works for continuous univariate distribution objects with
given CDF and (optional) PDF. It uses Hermite interpolation of
order 1, 3 [default] or 5. The order can be set by means of
unur_hinv_set_order
.
For distributions with unbounded domains the tails are chopped
off such that the probability for the tail regions is small
compared to the given uresulution. For finding these cut points
the algorithm starts with the region [1.e20,1.e20]
. For
the exceptional case where this might be too small (or one knows
this region and wants to avoid this search heuristics) it can be
directly set via a
unur_hinv_set_boundary
call.
It is possible to use this method for generating from truncated
distributions. It even can be changed for an existing generator
object by an
unur_hinv_chg_truncated
call.
This method is not exact, as it only produces random variates of
the approximated distribution. Nevertheless, the numerical error
in "udirection" (i.e. UCDF(X), for
X = "approximate inverse CDF"(U) UCDF(X)) can be controlled
by means of
unur_hinv_set_u_resolution
.
The possible maximal error is only estimated in the setup. Thus
it might be necessary to set some special design points for
computing the Hermite interpolation to guarantee that the
maximal uerror can not be bigger than desired. Such points
(e.g. extremal points of the PDF, points with infinite
derivative) can be set using using the
unur_hinv_set_cpoints
call.
If the mode for a unimodal distribution is set in the distribution
object this mode is automatically used as designpoint if the
unur_hinv_set_cpoints
call is not used.
As already mentioned the maximal error of this approximation is
only estimated. If this error is crucial for an application we
recommend to compute this error using
unur_hinv_estimate_error
which runs a small Monte Carlo simulation.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
The values given by the last
unur_hinv_chg_truncated
call will be
then changed to the values of the domain of the underlying distribution
object. Moreover, starting construction points (nodes) that are given by
a
unur_hinv_set_cpoints
call are ignored when
unur_reinit
is
called.
It is important to note that for a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
the normalization constant has to be updated using the
unur_distr_cont_upd_pdfarea
call whenever its parameters have been
changed by means of a
unur_distr_cont_set_pdfparams
call.
Get default parameters for generator.
Set order of Hermite interpolation. Valid orders are
1
, 3
, and 5
.
Notice that order greater than 1
requires the density
of the distribution, and order greater than 3
even
requires the derivative of the density. Using order 1
results for most distributions in a huge number of intervals
and is therefore not recommended. If the maximal error in
udirection is very small (say smaller than 1.e10
),
order 5
is recommended as it leads to considerably
fewer design points, as long there are no poles or heavy tails.
Remark: When the target distribution has poles or (very) heavy
tails order 5
(i.e., quintic interpolation) is
numerically less stable and more sensitive to roundoff errors than
order 3
(i.e., cubic interpolation).
Default is 3
if the density is given and 1
otherwise.
Set maximal error in udirection. However, the given uerror must not
be smaller than machine epsilon (DBL_EPSILON
) and should not be
too close to this value. As the resolution of most uniform random
number sources is 2^(32) = 2.3e10
, a value of 1.e10
leads to an inversion algorithm that could be called exact. For most
simulations slightly bigger values for the maximal error are enough
as well.
Remark: The uerror might become larger than u_resolution due
to rescaling of floating point numbers when the domain of the
distribution is truncated by a
unur_hinv_chg_truncated
call.
Default is 1.e10
.
Set starting construction points (nodes) for Hermite interpolation.
As the possible maximal error is only estimated in the setup it may be necessary to set some special design points for computing the Hermite interpolation to guarantee that the maximal uerror can not be bigger than desired. We suggest to include as special design points all local extrema of the density, all points where the density is not differentiable, and isolated points inside of the domain with density 0. If there is an interval with density constant equal to 0 inside of the given domain of the density, both endpoints of this interval should be included as special design points. Notice that there is no necessity to do so. However, if these points are not provided to the algorithm the approximation error might be larger than the given uresolution, or the required number of intervals could be larger than necessary.
Important: Notice that the given points must be in increasing order and they must be disjoint.
Important: The boundary point of the computational region must not be given in this list! Points outside the boundary of the computational region are ignored.
Default is for unimodal densities  if known  the mode of the density, if it is not equal to the border of the domain.
Set the left and right boundary of the computational interval.
Of course +/ UNUR_INFINITY
is not allowed.
If the CDF at left and right is not close to the
respective values 0.
and 1.
then this interval is
increased by a (rather slow) search algorithm.
Important: This call does not change the domain of the given distribution itself. But it restricts the domain for the resulting random variates.
Default is 1.e20
.
Set factor for relative size of the guide table for indexed search
(see also method DGT DGT – (Discrete) Guide Table method (indexed search)). It must be greater than or equal
to 0
.
When set to 0
, then sequential search is used.
Default is 1
.
Set maximum number of intervals. No generator object is created if the necessary number of intervals for the Hermite interpolation exceeds max_ivs. It is used to prevent the algorithm to eat up all memory for very badly shaped CDFs.
Default is 1000000
(1.e6).
Get number of nodes (design points) used for Hermite interpolation in the generator object. The number of intervals is the number of nodes minus 1. It returns an error code in case of an error.
Evaluate Hermite interpolation of inverse CDF at u.
If u is out of the domain [0,1] then unur_errno
is set
to UNUR_ERR_DOMAIN
and the respective bound of
the domain of the distribution are returned (which is
UNUR_INFINITY
or UNUR_INFINITY
in the case of
unbounded domains).
Notice: When the domain has been truncated by a
unur_hinv_chg_truncated
call then the inverse CDF of the
truncated distribution is returned.
Changes the borders of the domain of the (truncated) distribution.
Notice that the given truncated domain must be a subset of the domain of the given distribution. The generator always uses the intersection of the domain of the distribution and the truncated domain given by this call. The tables of splines are not recomputed. Thus it might happen that the relative error for the generated variates from the truncated distribution is greater than the bound for the nontruncated distribution. This call also fails when the CDF values of the boundary points are too close, i.e. when only a few different floating point numbers would be computed due to roundoff errors with floating point arithmetic.
Remark: The uerror might become larger than the u_resolution
given by a
unur_hinv_set_u_resolution
call due to rescaling of
floating point numbers when the domain of the distribution is
truncated.
When failed an error code is returned.
Important: Always check the return code since the domain is not changed in case of an error.
Estimate maximal uerror and mean absolute error (MAE) for generator by means of a (quasi) MonteCarlo simulation with sample size samplesize. The results are stored in max_error and MAE, respectively.
It returns UNUR_SUCCESS
if successful.
Next: HRD – Hazard Rate Decreasing, Previous: HINV – Hermite interpolation based INVersion of CDF, Up: Methods for continuous univariate distributions [Contents][Index]
bounded hazard rate
upper bound for hazard rate
Setup: fast, Sampling: slow
supported
Generates random variate with given hazard rate which must be bounded from above. It uses the thinning method with a constant dominating hazard function.
HRB requires a hazard function for a continuous distribution
together with an upper bound. The latter has to be set using the
unur_hrb_set_upperbound
call. If no such upper bound is given
it is assumed that the upper bound can be achieved by evaluating
the hazard rate at the left hand boundary of the domain of the
distribution. Notice, however, that for decreasing hazard rate
the method HRD (see Hazard Rate Decreasing) is much
faster and thus the prefered method.
It is important to note that the domain of the distribution can
be set via a
unur_distr_cont_set_domain
call.
However, the left border must not be negative. Otherwise it is
set to 0
. This is also the default if no domain is
given at all. For computational reasons the right border is
always set to UNUR_INFINITY
independently of the given
domain. Thus for domains bounded from right the function for
computing the hazard rate should return UNUR_INFINITY
right of this domain.
For distributions with increasing hazard rate method HRI (see Hazard Rate Increasing) is required.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that the upper bound given by the
unur_hrb_set_upperbound
call
cannot be changed and must be valid for the changed distribution.
Get default parameters for generator.
Set upper bound for hazard rate. If this call is not used it is assumed that the the maximum of the hazard rate is achieved at the left hand boundary of the domain of the distribution.
Turn verifying of algorithm while sampling on/off.
If the hazard rate is not bounded by the given bound, then
unur_errno
is set to UNUR_ERR_GEN_CONDITION
.
Default is FALSE
.
Next: HRI – Hazard Rate Increasing, Previous: HRB – Hazard Rate Bounded, Up: Methods for continuous univariate distributions [Contents][Index]
decreasing (nonincreasing) hazard rate
Setup: fast, Sampling: slow
supported
Generates random variate with given nonincreasing hazard rate. It is necessary that the distribution object contains this hazard rate. Decreasing hazard rate implies that the corresponding PDF of the distribution has heavier tails than the exponential distribution (which has constant hazard rate).
HRD requires a hazard function for a continuous distribution with nonincreasing hazard rate. There are no parameters for this method.
It is important to note that the domain of the distribution can
be set via a
unur_distr_cont_set_domain
call. However, only
the left hand boundary is used. For computational reasons the
right hand boundary is always reset to UNUR_INFINITY
.
If no domain is given by the user then the left hand boundary is
set to 0
.
For distributions which do not have decreasing hazard rates but are bounded from above use method HRB (see Hazard Rate Bounded). For distributions with increasing hazard rate method HRI (see Hazard Rate Increasing) is required.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for generator.
Turn verifying of algorithm while sampling on/off.
If the hazard rate is not bounded by the given bound, then
unur_errno
is set to UNUR_ERR_GEN_CONDITION
.
Default is FALSE
.
Next: ITDR – Inverse Transformed Density Rejection, Previous: HRD – Hazard Rate Decreasing, Up: Methods for continuous univariate distributions [Contents][Index]
increasing (nondecreasing) hazard rate
Setup: fast, Sampling: slow
supported
Generates random variate with given nonincreasing hazard rate. It is necessary that the distribution object contains this hazard rate. Increasing hazard rate implies that the corresponding PDF of the distribution has heavier tails than the exponential distribution (which has constant hazard rate).
The method uses a decomposition of the hazard rate into a main part which is constant for all x beyond some point p0 and a remaining part. From both of these parts points are sampled using the thinning method and the minimum of both is returned. Sampling from the first part is easier as we have a constant dominating hazard rate. Thus p0 should be large. On the other hand, if p0 is large than the thinning algorithm needs many iteration. Thus the performance of the the algorithm deponds on the choice of p0. We found that values close to the expectation of the generated distribution result in good performance.
HRI requires a hazard function for a continuous distribution
with nondecreasing hazard rate.
The parameter p0 should be set to a value close to the
expectation of the required distribution using
unur_hri_set_p0
.
If performance is crucial one may try other
values as well.
It is important to note that the domain of the distribution can
be set via a
unur_distr_cont_set_domain
call. However, only
the left hand boundary is used. For computational reasons the
right hand boundary is always reset to UNUR_INFINITY
.
If no domain is given by the user then the left hand boundary is
set to 0
.
For distributions with decreasing hazard rate method HRD (see Hazard Rate Decreasing) is required. For distributions which do not have increasing or decreasing hazard rates but are bounded from above use method HRB (see Hazard Rate Bounded).
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that the upper bound given by the
unur_hrb_set_upperbound
call
cannot be changed and must be valid for the changed distribution.
Notice that the parameter p0 which has been set by a
unur_hri_set_p0
call cannot be changed and must be valid for the changed distribution.
Get default parameters for generator.
Set design point for algorithm. It is used to split the domain of the
distribution. Values for p0 close to the expectation of the
distribution results in a relatively good performance of the algorithm.
It is important that the hazard rate at this point must be greater
than 0
and less than UNUR_INFINITY
.
Default: left boundary of domain + 1.
Turn verifying of algorithm while sampling on/off.
If the hazard rate is not bounded by the given bound, then
unur_errno
is set to UNUR_ERR_GEN_CONDITION
.
Default is FALSE
.
Next: NINV – Numerical INVersion, Previous: HRI – Hazard Rate Increasing, Up: Methods for continuous univariate distributions [Contents][Index]
monotone PDF, dPDF, pole
splitting point between pole and tail region, cvalues
Setup: moderate, Sampling: moderate
supported
ITDR is an acceptance/rejection method that works for monotone densities. It is especially designed for PDFs with a single pole. It uses different hat functions for the pole region and for the tail region. For the tail region Transformed Density Rejection with a single construction point is used. For the pole region a variant called Inverse Transformed Density Rejection is used. The optimal splitting point between the two regions and the respective maximum local concavity and inverse local concavity (see Glossary) that guarantee valid hat functions for each regions are estimated. This splitting point is set to the intersection point of local concavity and inverse local concavity. However, it is assumed that both, the local concavity and the inverse local concavity do not have a local minimum in the interior of the domain (which is the case for all standard distributions with a single pole). In other cases (or when the builtin search routines do not compute nonoptimal values) one can provide the splitting point, and the cvalues.
Method ITDR requires a distribution object with given PDF
and its derivative and the location of the pole (or mode).
The PDF must be monotone and may contain a pole.
It must be set via the
unur_distr_cont_set_pdf
and
unur_distr_cont_set_dpdf
calls. The PDF should return
UNUR_INFINITY for the pole. Alternatively, one can also
set the logarithm of the PDF and its derivative via the
unur_distr_cont_set_logpdf
and
unur_distr_cont_set_dlogpdf
calls. This is in especially useful since then the setup and
search routines are numerically more stable. Moreover, for many
distributions computing the logarithm of the PDF is less
expensive then computing the PDF directly.
The pole of the distribution is given by a
unur_distr_cont_set_mode
call. Notice that distributions with
“heavy” poles may have numerical problems caused by the
resultion of the floating point numbers used by computers.
While the minimal distance between two different floating point
numbers is about 1.e320
near 0.
it increases
to 1.e16
near 1.
Thus any random variate
generator implemented on a digital computer in fact draws samples
from a discrete distribution that approximates the desired
continuous distribution. For distributions with “heavy” poles
not at 0 this approximation may be too crude and thus every
goodnessoffit test will fail.
Besides this theoretic problem that cannot be resolved we
have to take into consideration that roundoff errors occur more
frequently when we have PDFs with poles far away from
0.
Method ITDR tries to handles this situation as good as
possible by moving the pole into 0.
Thus do not use a wrapper for your PDF that hides this shift
since the information about the resolution of the floating point
numbers near the pole gets lost.
Method ITDR uses different hats for the pole region and for the
tail region. The splitting point between these two regions, the
optimal cvalue and design points for constructing the hats
using Transformed Density Rejection are computed automatically.
(The results of these computations can be read using the
respective calls
unur_itdr_get_xi
unur_itdr_get_cp
, and
unur_itdr_get_ct
for the intersection point between local
concavity and inverse local concavity, the cvalue for the
pole and the tail region.)
However, one can also analyze the local concavity and inverse
local concavity set the corresponding values using
unur_itdr_set_xi
unur_itdr_set_cp
, and
unur_itdr_set_ct
calls.
Notice, that cvalues greater than 1/2 can be set to
0.5
. Although this results in smaller acceptance
probabities sampling from the hat distribution is much faster
than for other values of c. Depending on the expenses of
evaluating the PDF the resulting algorithm is usually faster.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
However, the values given by
unur_itdr_set_xi
unur_itdr_set_cp
,
or
unur_itdr_set_ct
calls are then ignored when
unur_reinit
is
called.
Get default parameters for generator.
Sets points where local concavity and inverse local concavity are (almost) equal. It is used to estimate the respective cvalues for pole region and hat regions and to determine the splitting point bx between pole and tail region. If no such point is provided it will be computed automatically.
Default: not set.
Sets parameter cp for transformation T for inverse
density in pole region.
It must be at most 0 and greater than 1.
A value of 0.5
is treated separately and usually results in
faster marginal generation time (at the expense of smaller
acceptance probabilities.
If no cpvalue is given it is estimated automatically.
Default: not set.
Sets parameter ct for transformation T for
density in tail region.
It must be at most 0. For densities with unbounded domain
it must be greater than 1.
A value of 0.5
is treated separately and usually results in
faster marginal generation time (at the expense of smaller
acceptance probabilities.
If no ctvalue is given it is estimated automatically.
Default: not set.
Get intersection point xi, and cvalues cp and ct,
respectively.
(In case of an error UNUR_INFINITY
is returned.)
Get area below hat.
(In case of an error UNUR_INFINITY
is returned.)
Turn verifying of algorithm while sampling on/off.
If the condition
PDF(x) <= hat(x)
is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However, notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change the verifying of algorithm while sampling on/off.
Next: NROU – Naive RatioOfUniforms method, Previous: ITDR – Inverse Transformed Density Rejection, Up: Methods for continuous univariate distributions [Contents][Index]
CDF
Setup: optional, Sampling: (very) slow
supported
NINV implementations of some methods for numerical inversion: Newton’s method, regula falsi (combined with interval bisectioning), and bisection method. Regula falsi and bisection method require only the CDF while Newton’s method also requires the PDF. To speed up marginal generation times a table with suitable starting points can be created during the setup. The performance of the algorithm can adjusted by the desired accuracy of the method. It is possible to use this method for generating from truncated distributions. The truncated domain can be changed for an existing generator object.
Method NINV generates random variates by numerical inversion and requires a continuous univariate distribution objects with given CDF. Three variants are available:
Newton’s method additionally requires the PDF of the distribution and cannot be used otherwise (NINV automatically switches to regula falsi then). Default algorithm is regula falsi. It is slightly slower but numerically much more stable than Newton’s algorithm. Interval bisectioning is the slowest method and should only be considered as a last resort when the other methods fails.
It is possible to draw samples from truncated distributions.
The truncated domain can even be changed for an existing generator
object by an
unur_ninv_chg_truncated
call.
Marginal generation times can be sped up by means of a table
with suitable starting points which can be created during the
setup. Using such a table can be switched on by means of a
unur_ninv_set_table
call where the table size is given as a
parameter. The table is still useful when the (truncated) domain
is changed often, since it is computed for the
domain of the given distribution. (It is not possible to enlarge
this domain.) If it is necessary to recalculate the table during
sampling, the command
unur_ninv_chg_table
can be used.
As a rule of thumb using such a table is appropriate when the
number of generated points exceeds the table size by a factor of
100.
The default number of iterations of NINV should be enough for all
reasonable cases. Nevertheless, it is possible to adjust the maximal
number of iterations with the commands
unur_ninv_set_max_iter
and
unur_ninv_chg_max_iter
.
In particular this might be necessary when the PDF has a pole or
the distribution has extremely heavy tails.
It is also possible to set/change the accuracy of the method
(which also heavily influencies the generation time).
We use two measures for the approximation error which can be
used independently: xerror and uerror
(see The Inversion Method for more details).
It is possible to set the maximal tolerated error using
with
unur_ninv_set_x_resolution
and
with
unur_ninv_set_u_resolution
resp., and change it with the
respective calls
unur_ninv_chg_x_resolution
and
unur_ninv_chg_x_resolution
.
The algorithm tries to satisfy both accuracy goals (and
raises an error flag it this fails).
One of these accuracy checks can be disabled by setting the
accuracy goal to a negative value.
NINV tries to use proper starting values for both the regula
falsi and bisection method, and for Newton’s method. Of course
the user might have more knowledge about the properties of the
target distribution and is able to share his wisdom with NINV
using the respective commands
unur_ninv_set_start
and
unur_ninv_chg_start
.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
The values given by the last
unur_ninv_chg_truncated
call will be
then changed to the values of the domain of the underlying distribution
object. It is important to note that for a distribution from the
UNU.RAN library of standard distributions
(see Standard distributions)
the normalization constant has to be updated using the
unur_distr_cont_upd_pdfarea
call whenever its parameters have been
changed by means of a
unur_distr_cont_set_pdfparams
call.
It might happen that NINV aborts
unur_sample_cont
without
computing the correct value (because the maximal number
iterations has been exceeded). Then the last approximate value
for x is returned (with might be fairly false) and
unur_error
is set to UNUR_ERR_GEN_SAMPLING
.
Get default parameters for generator.
Switch to regula falsi combined with interval bisectioning. (This the default.)
Switch to Newton’s method.
Notice that it is numerically less stable than regula falsi.
It it is not possible to invert the CDF for a particular uniform random
number U when calling
unur_sample_cont
unur_error
is set
to UNUR_ERR_GEN_SAMPLING
.
Thus it is recommended to check unur_error
before
using the result of the sampling routine.
Switch to bisection method. This is a slow algorithm and should only be used as a last resort.
Set and change number of maximal iterations.
Default is 100
.
Set and change the maximal tolerated relative xerror. If x_resolution is negative then checking of the xerror is disabled.
Default is 1.e8
.
Set and change the maximal tolerated (abolute) uerror. If u_resolution is negative then checking of the uerror is disabled.
Default is 1
(disabled).
Set starting points. If not set, suitable values are chosen automatically.
Newton:  left:  starting point 
Regula falsi:  left, right:  boundary of starting interval 
If the starting points are not set then the follwing points are used by default:
Newton:  left:  CDF(left) = 0.5 
Regula falsi:  left:  CDF(left) = 0.1 
right:  CDF(right) = 0.9 
If left == right, then UNU.RAN always uses the default starting points!
Change the starting points for numerical inversion.
If left==right, then UNU.RAN uses the default starting points
(see
unur_ninv_set_start
).
Generates a table with no_of_points points containing suitable starting values for the iteration. The value of no_of_points must be at least 10 (otherwise it will be set to 10 automatically).
The table points are chosen such that the CDF at these points form an equidistance sequence in the interval (0,1).
If a table is used, then the starting points given by
unur_ninv_set_start
are ignored.
No table is used by default.
Recomputes a table as described in
unur_ninv_set_table
.
Changes the borders of the domain of the (truncated) distribution.
Notice that the given truncated domain must be a subset of the domain of the given distribution. The generator always uses the intersection of the domain of the distribution and the truncated domain given by this call. Moreover the starting point(s) will not be changed.
Important: If the CDF is (almost) the same for left and
right and (almost) equal to 0
or 1
, then the truncated
domain is not chanced and the call returns an error code.
Notice: If the parameters of the distribution has been changed by a
unur_distr_cont_set_pdfparams
call it is recommended to set the
truncated domain again, since the former call might change the
domain of the distribution but not update the values for the
boundaries of the truncated distribution.
Get approximate approximate value of inverse CDF at u.
If u is out of the domain [0,1] then unur_errno
is set
to UNUR_ERR_DOMAIN
and the respective bound of
the domain of the distribution are returned (which is
UNUR_INFINITY
or UNUR_INFINITY
in the case of
unbounded domains).
Notice: This function always evaluates the inverse CDF of
the given distribution. A call to
unur_ninv_chg_truncated
call
has no effect.
Next: PINV – Polynomial interpolation based INVersion of CDF, Previous: NINV – Numerical INVersion, Up: Methods for continuous univariate distributions [Contents][Index]
mode, center, bounding rectangle for acceptance region
Setup: slow or fast, Sampling: moderate
supported
NROU is an implementation of the (generalized) ratioofuniforms
method which uses (minimal) bounding rectangles, see
The RatioofUniforms Method. It uses a positive control parameter
r for adjusting the algorithm to the given distribution to
improve performance and/or to make this method applicable.
Larger values of r increase the class of distributions
for which the method works at the expense of a higher rejection
constant. For computational reasons r=1 should be used if
possible (this is the default).
Moreover, this implementation uses the center
mu
of
the distribution (see
unur_distr_cont_get_center
for
details of its default values).
For the special case with r=1 the coordinates of the minimal bounding rectangles are given by
v^{+} = sup_x sqrt(PDF(x)),
u^{} = inf_x (x mu) sqrt(PDF(x)),
u^{+} = sup_x (x mu) sqrt(PDF(x)),
where mu is the center of the distribution. For other values of r we have
v^{+} = sup_x (PDF(x))^{1/(r+1)},
u^{} = inf_x (x mu) (PDF(x))^{r/(r+1)},
u^{+} = sup_x (x mu) (PDF(x))^{r/(r+1)}.
These bounds can be given directly. Otherwise they are computed automatically by means of a (slow) numerical routine. Of course this routine can fail, especially when this rectangle is not bounded.
It is important to note that the algorithm works with PDF(x mu) instead of PDF(x). This is important as otherwise the acceptance region can become a very long and skinny ellipsoid along a diagonal of the (huge) bounding rectangle.
For using the NROU method UNU.RAN needs the PDF of the
distribution. Additionally, the parameter r can be set via
a
unur_vnrou_set_r
call. Notice that the acceptance
probability decreases when r is increased. On the other
hand is is more unlikely that the bounding rectangle does not
exist if r is small.
A bounding rectangle can be given by the
unur_vnrou_set_u
and
unur_vnrou_set_v
calls.
Important: The bounding rectangle has to be
provided for the function
PDF(xcenter)!
Notice that center
is the center of the given
distribution, see
unur_distr_cont_set_center
.
If in doubt or if this value is not optimal, it can be changed
(overridden) by a
unur_nrou_set_center
call.
If the coordinates of the bounding rectangle are not provided by the user then the minimal bounding rectangle is computed automatically.
By means of
unur_vnrou_set_verify
and
unur_vnrou_chg_verify
one can run the sampling algorithm in a checking mode, i.e., in
every cycle of the rejection loop it is checked whether the used
rectangle indeed enclosed the acceptance region of the
distribution. When in doubt (e.g., when it is not clear whether
the numerical routine has worked correctly) this can be used to
run a small Monte Carlo study.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that derived parameters like the mode must also be (re) set
if the parameters or the domain has be changed.
Notice, however, that then the values that has been set by
unur_vnrou_set_u
and
unur_vnrou_set_v
calls are removed and
the coordinates of the bounding box are computed numerically.
Get default parameters for generator.
Sets left and right boundary of bounding rectangle. If no values are given, the boundary of the minimal bounding rectangle is computed numerically.
Notice: Computing the minimal bounding rectangle may fail under some circumstances. Moreover, for multimodal distributions the bounds might be too small as only local extrema are computed. Nevertheless, for T_c concave distributions with c=1/2 it should work.
Important: The bounding rectangle that has to be provided is for the function PDF(xcenter)!
Default: not set.
Set upper boundary for bounding rectangle. If this value is not given then sqrt(PDF(mode)) is used instead.
Notice: When the mode is not given for the distribution object, then it will be computed numerically.
Default: not set.
Sets the parameter r of the generalized ratioofuniforms method.
Notice: This parameter must satisfy r>0.
Default: 1
.
Set the center mu of the PDF. If not set the center of the given distribution object is used.
Default: see
unur_distr_cont_set_center
.
Turn verifying of algorithm while sampling on/off.
If the condition
PDF(x) <= hat(x)
is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However, notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change the verifying of algorithm while sampling on/off.
Next: SROU – Simple RatioOfUniforms method, Previous: NROU – Naive RatioOfUniforms method, Up: Methods for continuous univariate distributions [Contents][Index]
domain, center, CDF, derivative of PDF
Setup: (very) slow, Sampling: (very) fast
not implemented
PINV is a variant of numerical inversion, where the inverse CDF is approximated using Newton’s interpolating formula. The interval [0,1] is split into several subintervals. In each of these the inverse CDF is constructed at nodes (CDF(x),x) for some points x in this subinterval. If the PDF is given, then the CDF is computed numerically from the given PDF using adaptive GaussLobatto integration with 5 points. Subintervals are split until the requested accuracy goal is reached.
The method is not exact, as it only produces random variates of the approximated distribution. Nevertheless, the maximal tolerated approximation error can be set to be the resolution (but of course is bounded by the machine precision). We use the uerror UCDF(X) to measure the error for X = "approximate inverse CDF"(U). Notice that very small values of the uresolution are possible but increase the cost for the setup step. We call the maximal tolerated uerror the uresolution of the algorithm in the sequel.
Both the order of the interpolating polynomial and the uresolution can be selected.
The interpolating polynomials have to be computed in a setup step. However, it only works for distributions with bounded domain; for distributions with unbounded domain the tails are cut off such that the probability for the tail regions is small compared to the given uresolution.
The construction of the interpolation polynomial only works when the PDF is unimodal or when the PDF does not vanish between two modes.
There are some restrictions for the given distribution:
1.e12
.
Regions with very small PDF values or heavy tails might lead to an abortion of the setup or (even worse) the approximation error might become larger than requested, since the (computation of the) interpolating polynomial becomes numerically unstable.
Remark: We also have implemented experimental variants. However, we observed that these variants are more sensitive to roundoff errors, especially in the right hand tail and we do not recommend their usage unless there are severe reasons.
We have used a smoothness parameter to control this feature. However, besides numerical problems we observed that this variant requires more intervals and thus larger setup times and higher memory consumptions.
PINV works for continuous univariate distribution objects with
given PDF. The corresponding distribution object should contain a
typical point of the distribution, i.e., a point where the PDF
is not too small, e.g., (a point near) the mode.
However, it is important that the center is not the
pole of the distribution (or a point too close to the pole).
It can be set using a
unur_distr_cont_set_center
or
a
unur_distr_cont_set_mode
call. If neither is set, or if the
given center cannot be used, then a simple search routine tries
to find an appropriate point for the center.
It is recommended that the domain of the distribution with
bounded domain is specified using a
unur_distr_cont_set_domain
call. Otherwise, the boundary is searched numerically which
might be rather expensive, especially when this boundary point
is 0
.
When sampling from truncated distributions with extreme
truncation points, it is recommended to provide the logdensity
using
unur_distr_cont_set_logpdf
and the mode.
Then the PDF is rescaled such that the PDF at the mode is 1.
Thus the algorithm is numerically more stable.
The inverse CDF is interpolated using Newton polynomials.
The order of this polynomial can be set by means of a
unur_pinv_set_order
call.
The smoothness of the interpolant at interval boundaries can be
controlled using a
unur_pinv_set_smoothness
call.
Then Hermite interpolation instead of Newton interpolation is
used. (The former can be seen as a limiting case of Newton
interpolation with double (or triple) points.)
However, using higher smoothness is not recommended
unless differentiability at the interval boundaries is
important.
For distributions with unbounded domains the tails are cut
off such that the probability for the tail regions is small
compared to the given uresolution. For finding these cut points
the algorithm starts with the region [1.e100,1.e100]
. For
the exceptional case where this does not work these starting
points can be changed via a
unur_pinv_set_boundary
call.
This method is not exact, as it only produces random variates of
the approximated distribution. Nevertheless, the numerical error
in "udirection" (i.e., UCDF(X), for
X = "approximate inverse CDF"(U) UCDF(X)) can be controlled
by means of
unur_pinv_set_u_resolution
.
However, the maximal error of this approximation is only
estimated. For very small uresolutions the actual approximation
error might be (slightly) larger than the requested uresolution.
(Of course the size of this value depends on the given PDF.)
If this error is crucial for an application we recommend to
compute this error using
unur_pinv_estimate_error
which runs a
small Monte Carlo simulation.
See also the documentation for function
unur_pinv_set_u_resolution
and the remark given there.
The number of required subintervals heavily depends on the order
of the interpolating polynomial and the requested uresolution:
it increases when order or uresolution are decreased.
It can be checked using a
unur_pinv_get_n_intervals
call.
The maximum number of such subintervals is fixed but can be
increased using a
unur_pinv_set_max_intervals
call.
If this maximum number is too small then the setup aborts with
a corresponding error message.
It is also possible to use the CDF of the distribution instead
of the PDF. Then the distribution object must contain a pointer
to the CDF. Moreover, this variant of the algorithm has to be
switched on using an
unur_pinv_set_usecdf
call.
Notice, however, that the setup for this variant is numerically
less stable than using integration of the PDF (the default
variant). Thus using the CDF is not recommended.
Get default parameters for generator.
Set order of interpolation. Valid orders are between 3
and
17
. Higher orders result in fewer intervals for the
approximations.
Default: 5
.
Set smoothness of interpolant. By construction the interpolant is piecewise polynomial and thus smooth on each of the intervals where these polynomials are constructed. At the interval boundaries, however, it usually not be differentiable. Method PINV also implements variants of Newton interpolation where the first (or second) derivative of the interpolating polynomial coincides with the respective derivative of the inverse CDF at the nodes. The the interpolant is (twice) differentiable even at the interval boundaries. These variants can be seen as limiting case of Newton interpolation with double (or triple) points as nodes and are known as Hermite interpolation.
Possible values for smoothness:
Value  Effect  Requirements 

0  continuous  requires PDF (or CDF) 
1  differentiable  requires PDF (optional: CDF), order of polynomial must be odd 
2  twice differentiable  requires PDF and its derivative (optional: CDF), order must be 5, 8, 11, 14 or 17 
If the order of the polynomial does not satisfy the given condition, then it is increased to the next larger possible value.
Remark: A higher smoothness parameter usually results in a higher number of intervals and thus a higher setup time and memory consumption. We also observed that higher smoothness parameters make the algorithm more sensible for roundoff error. Then the setup fails.
Remark: If the interpolating polynomial cannot be constructed for the requested smoothness on a particular interval, then the smoothness parameter is reduced for that interval.
Remark:
For order 3
and smoothness 1
(cubic Hermite
interpolation) monotonicity is guaranteed by checking a simple
monotonicity condition for the coefficients of the polynomials.
Remark:
Using smoothness larger than 0
is
not recommended unless differentiability at the interval
boundaries is important for ones application.
Default: 0
.
Set maximal tolerated uerror. Values of u_resolution must
at least 1.e15
and 1.e5
at most.
Notice that the resolution of most uniform random number sources is
2^{32}
= 2.3e10
. Thus a value of 1.e10
leads to an inversion algorithm that could be called exact. For most
simulations slightly bigger values for the maximal error are enough
as well.
Smaller values for u_resolution increase the number of
subinterval that are necessary for the approximation of the inverse
CDF. For very small values (less then 1.e12
) this number
might exceed the maximum number of such intervals. However, this
number can be increased using a
unur_pinv_set_max_intervals
call.
Remark:
We ran many experiments and found that the observed uerror was
always smaller than the given u_resolution whenever this
value was 1.e12
. For values smaller than 1e13
the
maximal observed uerror was slightly larger. One use 1.e15
if best approximation is required. However, then the actual uerror
can be as large as 1.e14
.
Warning! These figures are based on our experiments (with some tolerance added to be on the safe side). There is no guarantee for these error estimates for a particular distribution.
Default is 1.e10
.
If use_upoints is TRUE
, then the nodes of the interpolating
polynomial are constructed by means of Chebyshev points in uscale
not in xscale. This results is a better approximation but almost
doubles the number of PDF or CDF evaluations during the setup.
(This is an experimental feature.)
Default: FALSE
Use PDF (if available) to compute approximate inverse CDF.
This is the default.
Use CDF (if available) to compute approximate inverse CDF. This variant is intend for running experiments with method PINV.
Remark:
We ran many experiments and found that for small values of the
given u_resolution (less than 1.e12
) the setup fails
for distributions with heavy tails. We found that using the PDF
(instead of the CDF) is numerically more stable.
This is especially the case when the smoothness parameter is set
to 1
or 2
.
Using the CDF is not recommended.
Set left and right point for finding the cutoff points
for the "computational domain", i.e., the domain that covers the
essential part of the distribution.
The cutoff points are computed such that the tail probabilities
are smaller than given by
unur_pinv_set_u_resolution
.
It is usually safe to use a large interval.
However, +/ UNUR_INFINITY
is not allowed.
Important: This call does not change the domain of the given distribution itself. But it restricts the domain for the resulting random variates.
Default: intersection of [1.e100,+1.e100]
and the given
domain of the distribution.
If left or right is set to FALSE
then the respective
boundary as given by a
unur_pinv_set_boundary
call is used
without any further computations.
However, these boundary points might cause numerical problems
during the setup when PDF returns 0
“almost everywhere”.
If set to TRUE
(the default) then the computational interval is
shortened to a more sensible region by means of a search algorithm.
Switching off this search is useful, e.g., for the Gamma(2)
distribution where the left border 0
is fixed and finite.
Remark: The searching algorithm assumes that the support of the distribution is connected.
Remark:
Do not set this parameter to FALSE
except when searching for
cutoff points fails and one wants to try with precomputed values.
Default: TRUE
.
Set maximum number of intervals. max_ivs must be at least
100
and at most 1000000
.
Default is 10000
.
Get number of intervals used for interpolation in
the generator object.
It returns 0
in case of an error.
If the PDF is given, then the CDF is computed numerically
from the given PDF using adaptive GaussLobatto integration.
Thus a table of CDF points is stored to keep the number of
evaluations of the PDF minimal. Usually this table is discarded
when the setup is completed.
If keepcdf is TRUE
, then this table is kept and can be used
to compute the CDF of the underlying distribution by means of
function
unur_pinv_eval_approxcdf
.
This option is ignored when
unur_pinv_set_usecdf
is called.
Default: FALSE
Evaluate interpolation of inverse CDF at u.
If u is out of the domain (0,1) then unur_errno
is set
to UNUR_ERR_DOMAIN
and the respective bound of
the domain of the distribution are returned (which is
UNUR_INFINITY
or UNUR_INFINITY
in the case of
unbounded domains).
Evaluate (approximate) CDF at x. If the PDF of the
distribution is given, then adaptive GaussLobatto integration is
used to compute the CDF.
If the PDF is used to create the generator object, then the
table of integral values must not removed at the end of setup and thus
unur_pinv_set_keepcdf
must be called.
Estimate maximal uerror and mean absolute error (MAE) for generator by means of MonteCarlo simulation with sample size samplesize. The results are stored in max_error and MAE, respectively.
It returns UNUR_SUCCESS
if successful.
Next: SSR – Simple Setup Rejection, Previous: PINV – Polynomial interpolation based INVersion of CDF, Up: Methods for continuous univariate distributions [Contents][Index]
Tconcave PDF, mode, area
Setup: fast, Sampling: slow
supported
[LJa01] [LJa02] [HLD04: Sect.6.3.1; Sect.6.3.2; Sect.6.4.1; Alg.6.4; Alg.6.5; Alg.6.7]
SROU is based on the ratioofuniforms method (see The RatioofUniforms Method) that uses universal inequalities for constructing a (universal) bounding rectangle. It works for all Tconcave distributions, including logconcave and Tconcave distributions with T(x) = 1/sqrt(x).
Moreover an (optional) parameter r
can be given, to
adjust the generator to the given distribution. This parameter
is strongly related to the parameter c
for transformed
density rejection (see TDR – Transformed Density Rejection) via the formula
c = r/(r+1). The rejection constant increases with higher
values for r
. On the other hand, the given density must
be
T_c
concave for the corresponding c.
The default setting for r
is 1 which results in a very
simple code. (For other settings, sampling uniformly from the
acceptance region is more complicated.)
Optionally the CDF at the mode can be given to increase the
performance of the algorithm. Then the rejection constant is
reduced by 1/2 and (if r=1
) even a universal squeeze can
(but need not be) used.
A way to increase the performance of the algorithm when the
CDF at the mode is not provided is the usage of the mirror
principle (only if r=1
). However, using squeezes and using
the mirror principle is only recommended when the PDF is
expensive to compute.
The exact location of the mode and/or the area below the PDF can be replace by appropriate bounds. Then the algorithm still works but has larger rejection constants.
SROU works for any continuous univariate distribution object with
given
T_c
concave PDF with
c<1,
)
mode and area below PDF. Optional the CDF at the mode
can be given to increase the performance of the algorithm by
means of the
unur_srou_set_cdfatmode
call. Additionally
squeezes can be used and switched on via
unur_srou_set_usesqueeze
(only if r=1
).
A way to increase the performance of the algorithm when the
CDF at the mode is not provided is the usage of the mirror
principle which can be swithced on by means of a
unur_srou_set_usemirror
call (only if r=1
) .
However using squeezes and using
the mirror principle is only recommended when the PDF is
expensive to compute.
The parameter r
can be given, to adjust the generator to
the given distribution. This parameter is strongly related
parameter c
for transformed density rejection via the
formula c = r/(r+1).
The parameter r
can be any value larger than or equal to
1. Values less then 1 are automatically set to 1.
The rejection constant depends on the chosen parameter
r
but not on the particular distribution. It is 4 for
r
equal to 1 and higher for higher values of r
.
It is important to note that different algorithms for different
values of r
: If r
equal to 1 this is much faster
than the algorithm for r
greater than 1.
The default setting for r
is 1.
If the (exact) area below the PDF is not known, then an upper
bound can be used instead (which of course increases the rejection
constant). But then the squeeze flag must not be set and
unur_srou_set_cdfatmode
must not be used.
If the exact location of the mode is not known, then use the
approximate location and provide the (exact) value of the PDF at
the mode by means of the
unur_srou_set_pdfatmode
call. But then
unur_srou_set_cdfatmode
must not be used. Notice, that a (slow)
numerical mode finder will be used if no mode is given at all.
It is even possible to give an upper bound for the PDF only.
However, then the (upper bound for the) area below the PDF has to be
multiplied by the ratio between the upper bound and the lower bound of
the PDF at the mode. Again setting the squeeze flag and using
unur_srou_set_cdfatmode
is not allowed.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that derived parameters like the mode must also be (re) set
if the parameters or the domain has be changed.
Moreover, if the PDF at the mode has been provided by a
unur_srou_set_pdfatmode
call, additionally
unur_srou_chg_pdfatmode
must be used (otherwise this call is
not necessary since then this figure is computed directly from
the PDF).
There exists a test mode that verifies whether the conditions
for the method are satisfied or not while sampling. It can be
switched on by calling
unur_srou_set_verify
and
unur_srou_chg_verify
respectively. Notice however that
sampling is (a little bit) slower then.
Get default parameters for generator.
Set parameter r for transformation. Only values greater than or equal to 1 are allowed. The performance of the generator decreases when r is increased. On the other hand r must not be set to small, since the given density must be T_cconcave for c = r/(r+1).
Notice: If r is set to 1
a simpler and much
faster algorithm is used then for r greater than one.
For computational reasons values of r that are greater than
1
but less than 1.01
are always set to 1.01
.
Default is 1
.
Set CDF at mode.
When set, the performance of the algorithm is increased by factor 2.
However, when the parameters of the distribution are changed
unur_srou_chg_cdfatmode
has to be used to update this value.
Default: not set.
Set pdf at mode. When set, the PDF at the mode is never changed. This is to avoid additional computations, when the PDF does not change when parameters of the distributions vary. It is only useful when the PDF at the mode does not change with changing parameters of the distribution.
IMPORTANT:
This call has to be executed after a possible call of
unur_srou_set_r
.
Default: not set.
Set flag for using universal squeeze (default: off). Using squeezes is only useful when the evaluation of the PDF is (extremely) expensive. Using squeezes is automatically disabled when the CDF at the mode is not given (then no universal squeezes exist).
Squeezes can only be used if r=1
.
Default is FALSE
.
Set flag for using mirror principle (default: off). Using the mirror principle is only useful when the CDF at the mode is not known and the evaluation of the PDF is rather cheap compared to the marginal generation time of the underlying uniform random number generator. It is automatically disabled when the CDF at the mode is given. (Then there is no necessity to use the mirror principle. However disabling is only done during the initialization step but not at a reinitialization step.)
The mirror principle can only be used if r=1
.
Default is FALSE
.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change CDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Change PDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Next: TABL – a TABLe method with piecewise constant hats, Previous: SROU – Simple RatioOfUniforms method, Up: Methods for continuous univariate distributions [Contents][Index]
Tconcave PDF, mode, area
Setup: fast, Sampling: slow
supported
SSR is an acceptance/rejection method that uses universal inequalities for constructing (universal) hats and squeezes (see The Rejection Method). It works for all Tconcave distributions with T(x) = 1/sqrt(x).
It requires the PDF, the (exact) location of the mode and the area below the given PDF. The rejection constant is 4 for all Tconcave distributions with unbounded domain and is less than 4 when the domain is bounded. Optionally the CDF at the mode can be given to increase the performance of the algorithm. Then the rejection constant is at most 2 and a universal squeeze can (but need not be) used. However, using squeezes is not recommended unless the evaluation of the PDF is expensive.
The exact location of the mode and/or the area below the PDF can be replace by appropriate bounds. Then the algorithm still works but has larger rejection constants.
SSR works for any continuous univariate distribution object with
given Tconcave PDF (with
T(x) = 1/sqrt(x),)
mode and area below PDF. Optional the CDF at the mode
can be given to increase the performance of the algorithm by
means of the
unur_ssr_set_cdfatmode
call. Additionally
squeezes can be used and switched on via
unur_ssr_set_usesqueeze
.
If the (exact) area below the PDF is not known, then an upper
bound can be used instead (which of course increases the rejection
constant). But then the squeeze flag must not be set and
unur_ssr_set_cdfatmode
must not be used.
If the exact location of the mode is not known, then use the
approximate location and provide the (exact) value of the PDF at
the mode by means of the
unur_ssr_set_pdfatmode
call. But then
unur_ssr_set_cdfatmode
must not be used. Notice, that a (slow)
numerical mode finder will be used if no mode is given at all.
It is even possible to give an upper bound for the PDF only.
However, then the (upper bound for the) area below the PDF has to be
multiplied by the ratio between the upper bound and the lower bound of
the PDF at the mode. Again setting the squeeze flag and using
unur_ssr_set_cdfatmode
is not allowed.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that derived parameters like the mode must also be (re) set
if the parameters or the domain has be changed.
Moreover, if the PDF at the mode has been provided by a
unur_ssr_set_pdfatmode
call, additionally
unur_ssr_chg_pdfatmode
must be used (otherwise this call is
not necessary since then this figure is computed directly from
the PDF).
Important:
If any of mode, PDF or CDF at the mode, or the area below the mode
has been changed, then
unur_reinit
must be executed.
(Otherwise the generator produces garbage).
There exists a test mode that verifies whether the conditions for
the method are satisfied or not while sampling. It can be
switched on/off by calling
unur_ssr_set_verify
and
unur_ssr_chg_verify
respectively.
Notice, however, that sampling is (a little bit) slower then.
Get default parameters for generator.
Set CDF at mode.
When set, the performance of the algorithm is increased by factor 2.
However, when the parameters of the distribution are changed
unur_ssr_chg_cdfatmode
has to be used to update this value.
Default: not set.
Set pdf at mode. When set, the PDF at the mode is never changed. This is to avoid additional computations, when the PDF does not change when parameters of the distributions vary. It is only useful when the PDF at the mode does not change with changing parameters for the distribution.
Default: not set.
Set flag for using universal squeeze (default: off). Using squeezes is only useful when the evaluation of the PDF is (extremely) expensive. Using squeezes is automatically disabled when the CDF at the mode is not given (then no universal squeezes exist).
Default is FALSE
.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change CDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Change PDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Next: TDR – Transformed Density Rejection, Previous: SSR – Simple Setup Rejection, Up: Methods for continuous univariate distributions [Contents][Index]
PDF, all local extrema, cutoff values for the tails
approximate area
Setup: (very) slow, Sampling: fast
not implemented
TABL (called Ahrens method in [HLD04] ) is an acceptance/rejection method (see The Rejection Method) that uses a decomposition of the domain of the distribution into many short subintervals. Inside of these subintervals constant hat and squeeze functions are utilized. Thus it is easy to use the idea of immediate acceptance for points below the squeeze. This reduces the expected number of uniform random numbers per generated random variate to less than two. Using a large number of subintervals only little more than one random number is necessary on average. Thus this method becomes very fast.
Due to the constant hat function this method only works for
distributions with bounded domains. Thus for unbounded domains
the left and right tails have to be cut off. This is no problem
when the probability of falling into these tail regions is
beyond computational relevance (e.g. smaller than 1.e12
).
For easy construction of hat and squeeze functions it is necessary to know the regions of monotonicity (called slopes) or equivalently all local maxima and minima of the density. The main problem for this method in the setup is the choice of the subintervals. A simple and close to optimal approach is the "equal area rule" [HLD04: Cha.5.1] . There the subintervals are selected such that the area below the hat is the same for each subinterval which can be realized with a simple recursion. If more subintervals are necessary it is possible to split either randomly chosen intervals (adaptive rejection sampling, ARS) or those intervals, where the ratio between squeeze and hat is smallest. This version of the setup is called derandomized ARS (DARS). With the default settings TABL is first calculating approximately 30 subintervals with the equal area rule. Then DARS is used till the desired fit of the hat is reached.
A convenient measure to control the quality of the fit of hat
and squeeze is the ratio (area below squeeze)/(area below hat)
called sqhratio
which must be smaller or equal to one.
The expected number of iterations in the rejection algorithm
is known to be smaller than 1/sqhratio and the expected number
of evaluations of the density is bounded by 1/sqhratio  1
.
So values of the sqhratio close to one (e.g. 0.95
or
0.99
) lead to many subintervals. Thus a better fitting
hat is constructed and the sampling algorithm becomes fast; on
the other hand large tables are needed and the setup is very
slow. For moderate values of sqhratio (e.g. 0.9
or
0.8
) the sampling is slower but the required tables are
smaller and the setup is not so slow.
It follows from the above explanations that TABL is always requiring a slow setup and that it is not very well suited for heavytailed distributions.
For using the TABL method UNU.RAN needs a bounded interval to which the generated variates can be restricted and information about all local extrema of the distribution. For unimodal densities it is sufficient to provide the mode of the distribution. For the case of a builtin unimodal distribution with bounded domain all these information is present in the distribution object and thus no extra input is necessary (see example_TABL1 below).
For a builtin unimodal distribution with unbounded domain we
should specify the cutoff values for the tails. This can be
done with the
unur_tabl_set_boundary
call (see example_TABL2
below). For the case that we do not set these boundaries the
default values of +/ 1.e20
are used. We can see in
example_TABL1 that this still works fine for many standard
distributions.
For the case of a multimodal distribution we have to set the
regions of monotonicity (called slopes) explicitly using the
unur_tabl_set_slopes
command (see example_TABL3 below).
To controll the fit of the hat and the size of the tables and
thus the speed of the setup and the sampling it is most
convenient to use the
unur_tabl_set_max_sqhratio
call. The
default is 0.9
which is a sensible value for most
distributions and applications. If very large samples of a
distribution are required or the evaluation of a density is very
slow it may be useful to increase the sqhratio to
eg. 0.95
or even 0.99
. With the
unur_tabl_get_sqhratio
call we can check which sqhratio was
really reached. If that value is below the desired value it is
necessary to increase the maximal number of subintervals, which
defaults to 1000
, using the
unur_tabl_set_max_intervals
call.
The
unur_tabl_get_n_intervals
call can be used to find out the
number of subintervals the setup calculated.
It is also possible to set the number of intervals and their
respective boundaries by means of the
unur_tabl_set_cpoints
call.
It is also possible to use method TABL for correlation induction
(variance reduction) by setting of an auxiliary uniform random
number generator via the
unur_set_urng_aux
call. (Notice that
this must be done after a possible
unur_set_urng
call.)
However, this only works when immediate acceptance is switched
off by a
unur_tabl_set_variant_ia
call.
Get default parameters for generator.
Use immediate acceptance when use_ia is set to TRUE
.
This technique requires less uniform. If it is set to FALSE
,
“classical” acceptance/rejection from hat distribution
is used.
Notice: Auxiliary uniform random number generators for correlation induction (variance reduction) can only be used when “classical” acceptance/rejection is used.
Default: TRUE
.
Set construction points for the hat function. If stp is NULL
than a heuristic rule of thumb is used to get n_stp
construction points. This is the default behavior.
The default number of construction points is 30
.
Set number of construction points for the hat function. n_stp
must be greater than zero. After the setup there are about
n_stp construction points. However it might be larger when a
small fraction is given by the
unur_tabl_set_areafraction
call.
It also might be smaller for some variants.
Default is 30
.
If useear is set to TRUE
, the “equal area rule” is used,
the given slopes are partitioned in such a way that the area below
the hat function in each subinterval (“stripe”) has the same
area (except the last the last interval which can be smaller).
The area can be set by means of the
unur_tabl_set_areafraction
call.
Default is TRUE
.
Set parameter for the equal area rule. During the setup a piecewise constant hat is constructed, such that the area below each of these pieces (strips) is the same and equal to the (given) area below the PDF times fraction (which must be greater than zero).
Important: If the area below the PDF is not set in the distribution object, then 1 is assumed.
Default is 0.1
.
If usedars is set to TRUE
, “derandomized adaptive rejection
sampling” (DARS) is used in the setup.
Intervals, where the area between hat and squeeze is too
large compared to the average area between hat and squeeze
over all intervals, are split.
This procedure is repeated until the ratio between squeeze and hat
exceeds the bound given by
unur_tabl_set_max_sqhratio
call or the
maximum number of intervals is reached. Moreover, it also aborts
when no more intervals can be found for splitting.
For finding splitting points the arcmean rule (a mixture of arithmetic mean and harmonic mean) is used.
Default is TRUE
.
Set factor for “derandomized adaptive rejection sampling”.
This factor is used to determine the segments that are “too
large”, that is, all segments where the area between squeeze and
hat is larger than factor times the average area over all
intervals between squeeze and hat.
Notice that all segments are split when factor is set to
0.
, and that there is no splitting at all when factor
is set to UNUR_INFINITY
.
Default is 0.99
. There is no need to change this parameter.
There are three variants for adaptive rejection sampling. These differ in the way how an interval is split:
1
use the generated point to split the interval.
2
use the mean point of the interval.
3
use the arcmean point; suggested for distributions with heavy tails.
Default is splitmode 2
.
Set upper bound for the ratio (area below squeeze) / (area below hat). It must be a number between 0 and 1. When the ratio exceeds the given number no further construction points are inserted via DARS in the setup.
For the case of ARS (unur_tabl_set_usedars() must be set to FALSE
):
Use 0
if no construction points should be added after the setup.
Use 1
if added new construction points should not be stopped
until the maximum number of construction points is reached.
If max_ratio is close to one, many construction points are used.
Default is 0.9
.
Get the current ratio (area below squeeze) / (area below hat)
for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the hat for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the squeeze for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Set maximum number of intervals. No construction points are added in or after the setup when the number of intervals suceeds max_ivs.
Default is 1000
.
Get the current number of intervals. (In case of an error 0 is returned.)
Set slopes for the PDF.
A slope <a,b> is an interval [a,b] or [b,a] where the PDF is
monotone and PDF(a) >= PDF(b).
The list of slopes is given by an array slopes where each
consecutive tuple (i.e. (slopes[0], slopes[1])
,
(slopes[2], slopes[3])
, etc.) defines one slope.
Slopes must be sorted (i.e. both slopes[0]
and
slopes[1]
must not be greater than any entry of the slope
(slopes[2], slopes[3])
, etc.)
and must not be overlapping. Otherwise no slopes are set and
unur_errno is set to UNUR_ERR_PAR_SET
.
Notice: n_slopes is the number of slopes (and not the length of the array slopes).
Notice that setting slopes resets the given domain for the distribution. However, in case of a standard distribution the area below the PDF is not updated.
Set factor for relative size of the guide table for indexed search
(see also method DGT DGT – (Discrete) Guide Table method (indexed search)). It must be greater than or equal
to 0
.
When set to 0
, then sequential search is used.
Default is 1
.
Set the left and right boundary of the computation interval.
The piecewise hat is only constructed inside this interval. The
probability outside of this region must not be of
computational relevance.
Of course +/ UNUR_INFINITY
is not allowed.
Default is 1.e20,1.e20
.
Change the borders of the domain of the (truncated) distribution.
Notice that the given truncated domain must be a subset of the domain of the given distribution. The generator always uses the intersection of the domain of the distribution and the truncated domain given by this call. The hat function will not be changed.
Important: The ratio between the area below the hat and the area below the squeeze changes when the sampling region is restricted. In particalur it becomes (very) large when sampling from the (far) tail of the distribution. Then it is better to create a generator object for the tail of distribution only.
Important:
This call does not work for variant IA
(immediate
acceptance). In this case UNU.RAN switches automatically to
variant RH
(use “classical” acceptance/rejection from hat
distribution) and does revert to the variant originally set by the
user.
Important: It is not a good idea to use adaptave rejection sampling while sampling from a domain that is a strict subset of the domain that has been used to construct the hat. For that reason adaptive adding of construction points is automatically disabled by this call.
Important: If the CDF of the hat is (almost) the same
for left and right and (almost) equal to 0
or
1
, then the truncated domain is not changed and the call
returns an error code.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Sometimes it might happen that
unur_init
has been executed
successfully. But when additional construction points are added by
adaptive rejection sampling, the algorithm detects that the
PDF is not monotone in the given slopes.
With pedantic being TRUE
, the sampling routine is exchanged
by a routine that simply returns UNUR_INFINITY
indicating an
error.
Default is FALSE
.
Next: UTDR – Universal Transformed Density Rejection, Previous: TABL – a TABLe method with piecewise constant hats, Up: Methods for continuous univariate distributions [Contents][Index]
Tconcave PDF, dPDF
mode
Setup: slow, Sampling: fast
supported
TDR is an acceptance/rejection method that uses the concavity of a
transformed density to construct hat function and squeezes
automatically. Such PDFs are called Tconcave. Currently the
following transformations are implemented and can be selected by
setting their c
values by a
unur_tdr_set_c
call:
c = 0
T(x) = log(x)
c = 0.5
T(x) = 1/sqrt(x) (Default)
In future releases the transformations T(x) = (x)^c will be available for any c with 0 > c > 1. Notice that if a PDF is Tconcave for a c then it also Tconcave for every c’<c. However the performance decreases when c’ is smaller than c. For computational reasons we suggest the usage of c = 0.5 (this is the default). For c <= 1 the hat is not bounded any more if the domain of the PDF is unbounded. But in the case of a bounded domain using method TABL is preferred to a TDR with c < 1 (except in a few special cases).
We offer three variants of the algorithm.
GW
squeezes between construction points
PS
squeezes proportional to hat function (Default)
IA
same as variant PS but uses a compositon method with “immediate acceptance” in the region below the squeeze.
GW
has a slightly faster setup but higher marginal generation
times.
PS
is faster than GW
. IA
uses less uniform
random numbers and is therefore faster than PS
.
It is also possible to evaluate the inverse of the CDF of the hat distribution
directly using the
unur_tdr_eval_invcdfhat
call.
There are lots of parameters for these methods, see below.
It is possible to use this method for correlation induction by
setting an auxiliary uniform random number generator via the
unur_set_urng_aux
call. (Notice that this must be done after a
possible
unur_set_urng
call.)
When an auxiliary generator is used then the number of
uniform random numbers from the first URNG that are used for one
generated random variate is constant and given in the following table:
GW ... 2
PS ... 2
IA ... 1
There exists a test mode that verifies whether the conditions for
the method are satisfied or not. It can be switched on by calling
unur_tdr_set_verify
and
unur_tdr_chg_verify
respectively.
Notice however that sampling is (much) slower then.
For densities with modes not close to 0 it is suggested to set
either the mode or the center of the distribution by the
unur_distr_cont_set_mode
or
unur_distr_cont_set_center
call.
The latter is the approximate location of the mode or the mean
of the distribution. This location provides some information
about the main part of the PDF and is used to avoid numerical
problems.
It is possible to use this method for generating from truncated
distributions. It even can be changed for an existing generator
object by an
unur_tdr_chg_truncated
call.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Important: The ratio between the area below the hat and the area below the squeeze changes when the sampling region is restricted. Especially it becomes (very) small when sampling from the (far) tail of the distribution. Then it is better to create a new generator object for the tail of the distribution only.
Get default parameters for generator.
Set parameter c for transformation T.
Currently only values between 0 and 0.5 are allowed.
If c
is between 0 and 0.5 it is set to 0.5.
Default is 0.5
.
Use original version with squeezes between construction points as proposed by Gilks & Wild (1992).
Use squeezes proportional to the hat function. This is faster than the original version. This is the default.
Use squeezes proportional to the hat function together with a composition method that required less uniform random numbers.
If usedars is set to TRUE
, “derandomized adaptive rejection
sampling” (DARS) is used in setup.
Intervals where the area between hat and squeeze is too
large compared to the average area between hat and squeeze
over all intervals are split.
This procedure is repeated until the ratio between area below squeeze
and area below hat exceeds the bound given by
unur_tdr_set_max_sqhratio
call or the maximum number of intervals is
reached. Moreover, it also aborts when no more intervals can be
found for splitting.
For finding splitting points the following rules are used (in this order, i.e., is if the first rule cannot be applied, the next one is used):
Notice, however, that for unbounded intervals neither rule 1 nor rule 3 can be used.
As an additional feature, it is possible to choose amoung these
rules.
If usedars is set to 1
or TRUE
the expected point
(rule 1) is used (it switches to rule 2 for a particular
interval if rule 1 cannot be applied).
If it is set to 2
the arcmean rule is used.
If it is set to 3
the mean is used.
Notice that rule 3 can only be used if the domain of the
distribution is bounded. It is faster than the other two methods
but for heavytailed distribution and large domain the hat
converges extremely slowly.
The default depends on the given construction points.
If the user has provided such points via a
unur_tdr_set_cpoints
call, then usedars is set to FALSE
by default, i.e.,
there is no further splitting.
If the user has only given the number of construction points (or
only uses the default number), then usedars is set to TRUE
(i.e., use rule 1).
Set factor for “derandomized adaptive rejection sampling”.
This factor is used to determine the intervals that are “too
large”, that is, all intervals where the area between squeeze and
hat is larger than factor times the average area over all
intervals between squeeze and hat.
Notice that all intervals are split when factor is set to
0.
, and that there is no splitting at all when factor
is set to UNUR_INFINITY
.
Default is 0.99
. There is no need to change this parameter.
Set construction points for the hat function. If stp is NULL
than a heuristic rule of thumb is used to get n_stp
construction points. This is the default behavior.
The default number of construction points is 30.
By default, when the generator object is reinitialized, it
used the same construction points as for the initialization
procedure.
Often the underlying distribution object has been changed only
moderately. For example, the full conditional distribution of a
multivariate distribution.
In this case it might be more appropriate to use
percentilesm of the hat function for the last (unchanged)
distribution. percentiles must then be a pointer to an
ordered array of numbers between 0.01
and 0.99
.
If percentiles is NULL
, then a heuristic rule of thumb is
used to get n_percentiles values for these percentiles.
Notice that n_percentiles must be at least 2
,
otherwise defaults are used.
(Then the first and third quartiles are used by default.)
When reinit fails with the given construction points or the percentiles
of the old hat function, another trial is undertaken with ncpoints
construction points. ncpoints must be at least 10
.
Default: 50
Change the borders of the domain of the (truncated) distribution.
Notice that the given truncated domain must be a subset of the
domain of the given distribution. The generator always uses the
intersection of the domain of the distribution and the truncated
domain given by this call. The hat function will not be changed and
there is no need to run
unur_reinit
.
Important:
The ratio between the area below the hat and the area below the
squeeze changes when the sampling region is restricted. In particular
it becomes (very) large when sampling from the (far) tail of the
distribution. Then it is better to create a generator object for the
tail of distribution only.
Important:
This call does not work for variant IA
(immediate
acceptance). In this case UNU.RAN switches automatically to
variant PS
.
Important: It is not a good idea to use adaptave rejection sampling while sampling from a domain that is a strict subset of the domain that has been used to construct the hat. For that reason adaptive adding of construction points is automatically disabled by this call.
Important: If the CDF of the hat is (almost) the same
for left and right and (almost) equal to 0
or
1
, then the truncated domain is not changed and the call
returns an error code.
Set upper bound for the ratio (area below squeeze) / (area below hat). It must be a number between 0 and 1. When the ratio exceeds the given number no further construction points are inserted via adaptive rejection sampling. Use 0 if no construction points should be added after the setup. Use 1 if added new construction points should not be stopped until the maximum number of construction points is reached.
Default is 0.99
.
Get the current ratio (area below squeeze) / (area below hat)
for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the hat for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Get the area below the squeeze for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Set maximum number of intervals. No construction points are added after the setup when the number of intervals suceeds max_ivs. It is increased automatically to twice the number of construction points if this is larger.
Default is 100
.
Use the center as construction point. Default is TRUE
.
Use the (exact!) mode as construction point.
Notice that the behavior of the algorithm is different to simply
adding the mode in the list of construction points via a
unur_tdr_set_cpoints
call. In the latter case the mode is treated
just like any other point. However, when usemode
is TRUE
, the
tangent in the mode is always set to 0. Then the hat of the
transformed density can never cut the xaxis which must never
happen if c < 0, since otherwise the hat would not be bounded.
Default is TRUE
.
Set factor for relative size of the guide table for indexed search
(see also method DGT DGT – (Discrete) Guide Table method (indexed search)). It must be greater than or equal
to 0
.
When set to 0
, then sequential search is used.
Default is 2.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Sometimes it might happen that
unur_init
has been executed
successfully. But when additional construction points are added by
adaptive rejection sampling, the algorithm detects that the
PDF is not Tconcave.
With pedantic being TRUE
, the
sampling routine is exchanged by a routine that simply returns
UNUR_INFINITY
. Otherwise the new point is not added to the
list of construction points. At least the hat function remains
Tconcave.
Setting pedantic to FALSE
allows sampling from a
distribution which is “almost” Tconcave and small errors are
tolerated. However it might happen that the hat function cannot be
improved significantly. When the hat functions that has been
constructed by the
unur_init
call is extremely large then it
might happen that the generation times are extremely high
(even hours are possible in extremely rare cases).
Default is FALSE
.
Evaluate the inverse of the CDF of the hat distribution at u.
As a side effect the values of the hat, the density, and the squeeze
at the computed point x are stored in hx, fx, and
sqx, respectively. However, these computations are suppressed
if the corresponding variable is set to NULL
.
If u is out of the domain [0,1] then unur_errno
is set
to UNUR_ERR_DOMAIN
and the respective bound of
the domain of the distribution are returned (which is
UNUR_INFINITY
or UNUR_INFINITY
in the case of
unbounded domains).
Important:
This call does not work for variant IA
(immediate
acceptance). In this case the hat CDF is evaluated as if
variant PS
is used.
Notice: This function always evaluates the inverse CDF of
the hat distribution. A call to
unur_tdr_chg_truncated
call
has no effect.
Previous: TDR – Transformed Density Rejection, Up: Methods for continuous univariate distributions [Contents][Index]
Tconcave PDF, mode, approximate area
Setup: moderate, Sampling: Moderate
supported
UTDR is based on the transformed density rejection and uses three almost optimal points for constructing hat and squeezes. It works for all Tconcave distributions with T(x) = 1/sqrt(()x).
It requires the PDF and the (exact) location of the mode. Notice that if no mode is given at all, a (slow) numerical mode finder will be used. Moreover the approximate area below the given PDF is used. (If no area is given for the distribution the algorithm assumes that it is approximately 1.) The rejection constant is bounded from above by 4 for all Tconcave distributions.
UTDR works for any continuous univariate distribution object with given Tconcave PDF (with T(x) = 1/sqrt(x),) mode and approximate area below PDF.
When the PDF does not change at the mode for varying parameters, then
this value can be set with
unur_utdr_set_pdfatmode
to avoid some
computations. Since this value will not be updated any more when the
parameters of the distribution are changed,
the
unur_utdr_chg_pdfatmode
call is necessary to do this manually.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that derived parameters like the mode must also be (re) set
if the parameters or the domain has be changed.
Moreover, if the PDF at the mode has been provided by a
unur_utdr_set_pdfatmode
call, additionally
unur_utdr_chg_pdfatmode
must be used (otherwise this call is
not necessary since then this figure is computed directly from
the PDF).
There exists a test mode that verifies whether the conditions for
the method are satisfied or not. It can be switched on by calling
unur_utdr_set_verify
and
unur_utdr_chg_verify
respectively.
Notice however that sampling is slower then.
Get default parameters for generator.
Set pdf at mode. When set, the PDF at the mode is never changed. This is to avoid additional computations, when the PDF does not change when parameters of the distributions vary. It is only useful when the PDF at the mode does not change with changing parameters for the distribution.
Default: not set.
Set factor for position of left and right construction point. The cp_factor is used to find almost optimal construction points for the hat function. There is no need to change this factor in almost all situations.
Default is 0.664
.
Set factor for replacing tangents by secants. higher factors increase the rejection constant but reduces the risk of serious roundoff errors. There is no need to change this factor it almost all situations.
Default is 1.e5
.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change PDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Next: Methods for continuous multivariate distributions, Previous: Methods for continuous univariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for continuous empirical univariate distributions EMPK: Requires an observed sample. 
/*  */ /* File: example_emp.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from an empirial continuous univariate */ /* distribution. */ /*  */ int main(void) { int i; double x; /* data points */ double data[15] = { 0.1, 0.05, 0.5, 0.08, 0.13,\ 0.21,0.44, 0.43, 0.33, 0.3, \ 0.18, 0.2, 0.37, 0.29, 0.9 }; /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Create a distribution object and set empirical sample. */ distr = unur_distr_cemp_new(); unur_distr_cemp_set_data(distr, data, 15); /* Choose a method: EMPK. */ par = unur_empk_new(distr); /* Set smooting factor. */ unur_empk_set_smoothing(par, 0.8); /* Create the generator object. */ gen = unur_init(par); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
/*  */ /* File: example_emp_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from an empirial continuous univariate */ /* distribution. */ /*  */ int main(void) { int i; double x; /* Declare UNURAN generator object. */ UNUR_GEN *gen; /* generator object */ /* Create the generator object. */ gen = unur_str2gen("distr = cemp; \ data=(0.10, 0.05,0.50, 0.08, 0.13, \ 0.21,0.44,0.43,0.33,0.30, \ 0.18, 0.20,0.37,0.29,0.90) & \ method=empk; smoothing=0.8"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: EMPL – EMPirical distribution with Linear interpolation, Up: Methods for continuous empirical univariate distributions [Contents][Index]
observed sample
Setup: slow (as sample is sorted), Sampling: fast (depends on kernel)
not implemented
EMPK generates random variates from an empirical distribution that is given by an observed sample. The idea is that simply choosing a random point from the sample and to return it with some added noise results in a method that has very nice properties, as it can be seen as sampling from a kernel density estimate. If the underlying distribution is continuous, especially the fine structur of the resulting empirical distribution is much better than using only resampling without noise.
Clearly we have to decide about the density of the noise (called kernel) and about the standard deviation of the noise. The mathematical theory of kernel density estimation shows us that we are comparatively free in choosing the kernel. It also supplies us with a simple formula to compute the optimal standarddeviation of the noise, called bandwidth (or window width) of the kernel.
The variance of the estimated density is slightly larger than that of the observed sample. However, this can be easily corrected if required.
There is also a correction (mirroring technique) for distributions with nonnegative support.
A simple robust reference method is implemented to find a good standard deviation of the noise (i.e. the bandwidth of kernel density estimation). For some cases (e.g. densities with two or more sharp distinct peaks) there kernel density estimation can be adjusted by changing the smoothness factor and the so called beta factor.
EMPK uses empirical distributions. The main parameter is the
choice if of kernel density. The most important kernels can be
set by
unur_empk_set_kernel
.
Additionally generators for other
kernels can be used by using
unur_empk_set_kernelgen
instead.
Additionally variance correction and a correction for
nonnegative variates can be switched on.
The two other parameters (smoothing factor and beta factor) are
only useful for people knowing the theory of kernel density
estimation. It is not necessary to change them if
the true underlying distribution is somehow comparable with a
bellshaped curve, even skewed or with some not too sharp extra peaks.
In all these cases the simple robust reference method implemented to
find a good standard deviation of the noise (i.e. the bandwidth of
kernel density estimation) should give sensible results.
However, it might be necessary to overwrite this automatic method
to find the bandwidth eg. when resampling from data with
two or more sharp distinct peaks. Then the distribution has nearly
discrete components as well and our automatic method may
easily choose too large a bandwidth which results in an
empirical distribution which is oversmoothed (i.e. it has
lower peaks than the original distribution). Then it
is recommended to decrease the bandwidth using the
unur_empk_set_smoothing
call. A smoothing factor of 1
is the default. A smoothing factor of 0
leads to naive
resampling of the data. Thus an appropriate value between these
extremes should be choosen. We recommend to consult a reference
on kernel smoothing when doing so; but it is not a simple problem
to determine an optimal bandwidth for distributions with sharp peaks.
In general, for most applications it is perfectly ok to use the default values offered. Unless you have some knowledge on density estimation we do not recommend to change anything. There are two exceptions:
unur_empk_set_smoothing(par, 0.)
unur_empk_set_kernel(par, UNUR_DISTR_BOXCAR);
to change the used noise distribution from the default Gaussian distribution to the uniform distribution.
Get default parameters for generator.
Select one of the supported kernel distributions. Currently the following kernels are supported:
UNUR_DISTR_GAUSSIAN
Gaussian (normal) kernel
UNUR_DISTR_EPANECHNIKOV
Epanechnikov kernel
UNUR_DISTR_BOXCAR
Boxcar (uniform, rectangular) kernel
UNUR_DISTR_STUDENT
t3 kernel (Student’s distribution with 3 degrees of freedom)
UNUR_DISTR_LOGISTIC
logistic kernel
For other kernels (including kernels with Student’s distribution
with other than 3 degrees of freedom) use the
unur_empk_set_kernelgen
call.
It is not possible to call
unur_empk_set_kernel
twice.
Default is the Gaussian kernel.
Set generator for the kernel used for density estimation.
alpha is used to compute the optimal bandwidth from the point of view of minimizing the mean integrated square error (MISE). It depends on the kernel K and is given by
alpha(K) = Var(K)^(2/5){ \int K(t)^2 dt}^(1/5)
For standard kernels (see above) alpha is computed by the algorithm.
kernvar is the variance of the used kernel. It is only required for the variance corrected version of density estimation (which is used by default); otherwise it is ignored. If kernelvar is nonpositive, variance correction is disabled. For standard kernels (see above) kernvar is computed by the algorithm.
It is not possible to call
unur_empk_set_kernelgen
after a standard kernel
has been selected by a
unur_empk_set_kernel
call.
Notice that the uniform random number generator of the kernel
generator is overwritten during the
unur_init
call and at each
unur_chg_urng
call with the uniform generator used for the empirical
distribution.
Default is the Gaussian kernel.
beta is used to compute the optimal bandwidth from the point of view of minimizing the mean integrated square error (MISE). beta depends on the (unknown) distribution of the sampled data points. By default Gaussian distribution is assumed for the sample (beta = 1.3637439). There is no requirement to change beta.
Default: 1.3637439
Set and change the smoothing factor.
The smoothing factor controlles how “smooth” the resulting density
estimation will be. A smoothing factor equal to 0
results in naive
resampling. A very large smoothing factor (together with the
variance correction) results in a density which is approximately
equal to the kernel.
Default is 1 which results in a smoothing parameter minimising
the MISE (mean integrated squared error) if the data are not too
far away from normal. If a large smoothing factor is used, then
variance correction must be switched on.
Default: 1
Switch variance correction in generator on/off.
If varcor is TRUE
then the variance of the used
density estimation is the same as the sample variance. However this
increases the MISE of the estimation a little bit.
Default is FALSE
.
If positive is TRUE
then only nonnegative random variates are
generated. This is done by means of a mirroring technique.
Default is FALSE
.
Next: HIST – HISTogramm of empirical distribution, Previous: EMPK – EMPirical distribution with Kernel smoothing, Up: Methods for continuous empirical univariate distributions [Contents][Index]
observed sample
Setup: slow (as sample is sorted), Sampling: very fast (inversion)
not implemented
EMPL generates random variates from an empirical distribution that is given by an observed sample. This is done by linear interpolation of the empirical CDF. Although this method is suggested in the books of Law and Kelton (2000) and Bratly, Fox, and Schrage (1987) we do not recommend this method at all since it has many theoretical drawbacks: The variance of empirical distribution function does not coincide with the variance of the given sample. Moreover, when the sample increases the empirical density function does not converge to the density of the underlying random variate. Notice that the range of the generated point set is always given by the range of the given sample.
This method is provided in UNU.RAN for the sake of completeness. We always recommend to use method EMPK (see EMPirical distribution with Kernel smoothing).
If the data seem to be far away from having a bell shaped histogram, then we think that naive resampling is still better than linear interpolation.
EMPL creates and samples from an empiral distribution by linear interpolation of the empirical CDF. There are no parameters to set.
Important: We do not recommend to use this method! Use method EMPK (see EMPirical distribution with Kernel smoothing) instead.
Get default parameters for generator.
Previous: EMPL – EMPirical distribution with Linear interpolation, Up: Methods for continuous empirical univariate distributions [Contents][Index]
histogram
Setup: moderate, Sampling: fast
not implemented
Method HIST generates random variates from an empirical distribution that is given as histogram. Sampling is done using the inversion method.
If observed (raw) data are provided we recommend method EMPK (see EMPirical distribution with Kernel smoothing) instead of compting a histogram as this reduces information.
Method HIST uses empirical distributions that are given as a histgram. There are no optional parameters.
Get default parameters for generator.
Next: Markov chain samplers for continuous multivariate distributions, Previous: Methods for continuous empirical univariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for continuous multivariate distributions NORTA: Requires rank correlation matrix and marginal distributions. 
Next: MVTDR – MultiVariate Transformed Density Rejection, Up: Methods for continuous multivariate distributions [Contents][Index]
standard distribution from UNU.RAN library (see Standard distributions).
depends on distribution and generator
supported
MVSTD is a wrapper for special generators for multivariate
continuous standard distributions. It only works for
distributions in the UNU.RAN library of standard distributions
(see Standard distributions).
If a distribution object is provided that is build from scratch,
or if no special generator for the given standard distribution is
provided, the NULL
pointer is returned.
Create a distribution object for a standard distribution from the UNU.RAN library (see Standard distributions).
Sampling from truncated distributions (which can be constructed by
changing the default domain of a distribution by means of
unur_distr_cvec_set_domain_rect
call) is not possible.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for new generator. It requires a distribution object for a multivariate continuous distribution from the UNU.RAN library of standard distributions (see Standard distributions). Using a truncated distribution is not possible.
Next: NORTA – NORmal To Anything, Previous: MVSTD – MultiVariate continuous STandarD distributions, Up: Methods for continuous multivariate distributions [Contents][Index]
logconcave (log)PDF, gradient of (log)PDF
mode
Setup: slow, Sampling: depends on dimension
not implemented
MVTDR a multivariate version of the Transformed Density Rection (see TDR – Transformed Density Rejection) that works for logconcave densities. For this method the domain of the distribution is partitioned into cones with the mode (or the center) of the distribution as their (common) vertex. The hat function is then constructed as tangent planes of the transformed density in each of these cones. The respective construction points lie on the central lines in the cones through the vertex. The point is chosen such that the hat is minimal among all such points (see the given references for more details).
The cones are created by starting with the orthants of the reals space. These are then iteratively split when the volume below the hat in such cones is too large. Thus an increasing number of cones results in a better fitting hat function. Notice however, that the required number of cones increases exponentially with the number of dimension. Moreover, due to the construction the rejection does not converge to 1 and remains strictly larger than 1.
For distributions with bounded domains the cones are cut to pyramids that cover the domain.
Create a multivariate generator object that contains the PDF and its gradient. This object also should contain the mode of the distribution (or a point nearby should be provided as center of the distribution).
The method has three parameter to adjust the method for the given distribution:
stepsmin
Minimal number of iterations for splitting cones.
Notice that we start with 2^dim initial cones and that we arrive
at 2^(dim+stepsmin) cones after these splits. So this number
must be set with care. It can be set by a
unur_mvtdr_set_stepsmin
call.
boundsplitting
Cones where the volume below the hat is relatively large
(i.e. larger than the average volume over all cones times
boundsplitting
are further split.
This parameter can set via a
unur_mvtdr_set_boundsplitting
call.
maxcones
The maximum number of generated cones. When this number is
reached, the initialization routine is stopped. Notice that the
rejection constant can be still prohibitive large.
This parameter can set via a
unur_mvtdr_set_maxcones
call.
Setting of these parameter can be quite tricky. The default
settings lead to hat functions where the volume below the hat is
similar in each cone. However, there might be some problems with
distributions with higher correlations, since then too few cones
are created. Then it might be necessary to increase the values
for stepsmin
and maxcones
and to set
boundsplitting
to 0
.
The number of cones and the total volume below the hat can be
controlled using the respective calls
unur_mvtdr_get_ncones
and
unur_mvtdr_get_hatvol
.
Notice, that the rejection constant is
bounded from below by some figure (larger than 1) that depends
on the dimension.
Unfortunately, the algorithm cannot detect the quality of the constructed hat.
Get parameters for generator.
Set minimum number of triangulation step for each starting cone. stepsmin must be nonnegative.
Default: 5
.
Set bound for splitting cones. All cones are split which have a
volume below the hat that is greater than bound_splitting times
the average over all volumes. However, the number given by the
unur_mvtdr_set_maxcones
is not exceeded.
Notice that the later number is always reached
if bound_splitting is less than 1.
Default: 1.5
Set maximum number of cones.
Notice that this number is always increased to 2^{dim+stepsmin} where dim is the dimension of the distribution object and stepsmin the given mimimum number of triangulation steps.
Notice: For higher dimensions and/or higher correlations between the coordinates of the random vector the required number of cones can be very high. A too small maximum number of cones can lead to a very high rejection constant.
Default: 10000
.
Get the number of cones used for the hat function of the
generator.
(In case of an error 0
is returned.)
Get the volume below the hat for the generator.
(In case of an error UNUR_INFINITY
is returned.)
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Next: VNROU – Multivariate Naive RatioOfUniforms method, Previous: MVTDR – MultiVariate Transformed Density Rejection, Up: Methods for continuous multivariate distributions [Contents][Index]
rank correlation matrix, marginal distributions
Setup: slow, Sampling: depends on dimension
not implemented
NORTA (NORmal to anything) is a model to get random vectors with given marginal distributions and rank correlation.
Important: Notice that marginal distribution and (rank) correlation structure do not uniquely define a multivariate distribution. Thus there are many other (more or less sensible) models.
In the NORTA model multinormal random variates with the given (Spearman’s) rank correlations are generated. In a second step the (standard normal distributed) marginal variates are transformed by means of the CDF of the normal distribution to get uniform marginals. The resulting random vectors have uniform marginals and the desired rank correlation between its components. Such a random vector is called ’copula’.
By means of the inverse CDF the uniform marginals are then transformed into the target marginal distributions. This transformation does not change the rank correlation.
For the generation of the multinormal distribution the (Spearman’s) rank correlation matrix is transformed into the corresponding (Pearson) correlation matrix. Samples from the resulting multinormal distribution are generated by means of the Cholesky decomposition of the covariance matrix.
It can happen that the desired rank correlation matrix is not feasible, i.e., it cannot occur as rank correlation matrix of a multinormal distribution. The resulting "covariance" matrix is not positive definite. In this case an eigenvector correction method is used. Then all nonpositive eigenvalues are set to a small positive value and hence the rank correlation matrix of the generated random vectors is "close" to the desired matrix.
Create a multivariate generator object and set marginal
distributions using
unur_distr_cvec_set_marginals
unur_distr_cvec_set_marginal_array
, or
unur_distr_cvec_set_marginal_list
.
(Do not use the corresponding calls for the standard
marginal distributions).
When the domain of the multivariate distribution is set by of a
unur_distr_cvec_set_domain_rect
call then the domain of each
of the marginal distributions is truncated by the respective
coordinates of the given rectangle.
If copulae are required (i.e. multivariate distributions with
uniform marginals) such a generator object can be created by
means of
unur_distr_copula
.
There are no optional parameters for this method.
Get default parameters for generator.
Previous: NORTA – NORmal To Anything, Up: Methods for continuous multivariate distributions [Contents][Index]
mode, center, bounding rectangle for acceptance region
Setup: fast or slow, Sampling: slow
supported
VNROU is an implementation of the multivariate
ratioofuniforms method which uses a (minimal) bounding
hyperrectangle, see also The RatioofUniforms Method. It uses an
additional parameter r that can be used for adjusting the
algorithm to the given distribution to improve performance
and/or to make this method applicable. Larger values of
r increase the class of distributions for which the
method works at the expense of higher rejection
constants. Moreover, this implementation uses the center
mu
of the distribution (which is set to the mode or
mean by default, see
unur_distr_cvec_get_center
for details of
its default values).
The minimal bounding has then the coordinates
v^{+} = sup_x (f(x))^{1/r d+1},
u^{}_i = inf_x_i (x_i mu_i) (f(x))^{r/r d+1},
u^{+}_i = sup_x_i (x_i mu_i) (f(x))^{r/r d+1},
where x_i is the ith coordinate of point x; mu_i is the ith coordinate of the center mu. d denotes the dimension of the distribution. These bounds can either be given directly, or are computed automatically by means of an numerical routine by Hooke and Jeeves [HJa61] called direct search (see src/utils/hooke.c for further references and details). Of course this algorithm can fail, especially when this rectangle is not bounded.
It is important to note that the algorithm works with PDF(xcenter) instead of PDF(x), i.e. the bounding rectangle has to be provided for PDF(xcenter). This is important as otherwise the acceptance region can become a very long and skinny ellipsoid along a diagonal of the (huge) bounding rectangle.
VNROU is based on the rejection method (see The Rejection Method), and it is important to note that the acceptance probability decreases exponentially with dimension. Thus even for moderately many dimensions (e.g. 5) the number of repetitions to get one random vector can be prohibitively large and the algorithm seems to stay in an infinite loop.
For using the VNROU method UNU.RAN needs the PDF of the
distribution. Additionally, the parameter r can be set via
a
unur_vnrou_set_r
call. Notice that the acceptance
probability decreases when r is increased. On the other
hand is is more unlikely that the bounding rectangle does not
exist if r is small.
A bounding rectangle can be given by the
unur_vnrou_set_u
and
unur_vnrou_set_v
calls.
Important: The bounding rectangle has to be
provided for the function
PDF(xcenter)!
Notice that center
is the center of the given
distribution, see
unur_distr_cvec_set_center
.
If in doubt or if this value is not optimal, it can be changed
(overridden) by a
unur_distr_cvec_set_center
call.
If the coordinates of the bounding rectangle are not provided by the user then the minimal bounding rectangle is computed automatically.
By means of
unur_vnrou_set_verify
and
unur_vnrou_chg_verify
one can run the sampling algorithm in a checking mode, i.e., in
every cycle of the rejection loop it is checked whether the used
rectangle indeed enclosed the acceptance region of the
distribution. When in doubt (e.g., when it is not clear whether
the numerical routine has worked correctly) this can be used to
run a small Monte Carlo study.
Important:
The rejection constant (i.e. the expected number of iterations
for generationg one random vector) can be extremely high, in
particular when the dimension is 4 or higher.
Then the algorithm will perform almost infinite loops.
Thus it is recommended to read the volume below the hat function
by means of the
unur_vnrou_get_volumehat
call. The returned
number divided by the volume below the PDF (which is 1 in case
of a normalized PDF) gives the rejection constant.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that the coordinates of a bounding rectangle given by
unur_vnrou_set_u
and
unur_vnrou_set_v
calls are used also
when the generator is reused. These can be changed by means of
unur_vnrou_chg_u
and
unur_vnrou_chg_v
calls.
(If no such coordinates have been given, then they are computed
numerically during the reinitialization proceedure.)
Get default parameters for generator.
Sets left and right boundaries of bounding hyperrectangle. If no values are given, the boundary of the minimal bounding hyperrectangle is computed numerically.
Important: The boundaries are those of the density shifted by the center of the distribution, i.e., for the function PDF(xcenter)!
Notice: Computing the minimal bounding rectangle may fail under some circumstances. Moreover, for multimodal distributions the bounds might be too small as only local extrema are computed. Nevertheless, for logconcave distributions it should work.
Default: not set (i.e. computed automatically)
Change left and right boundaries of bounding hyperrectangle.
Set upper boundary for bounding hyperrectangle. If no values are given, the density at the mode is evaluated. If no mode is given for the distribution it is computed numerically (and might fail).
Default: not set (i.e. computed automatically)
Change upper boundary for bounding hyperrectangle.
Sets the parameter r of the generalized multivariate ratioofuniforms method.
Notice: This parameter must satisfy r>0.
Default: 1
.
Turn verifying of algorithm while sampling on/off.
If the condition PDF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change the verifying of algorithm while sampling on/off.
Get the volume of below the hat. For normalized densities, i.e. when the volume below PDF is 1, this value equals the rejection constant for the vnrou method.
In case of an error UNUR_INFINITY is returned.
Next: Methods for continuous empirical multivariate distributions, Previous: Methods for continuous multivariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Markov chain samplers generate sequences of random vectors which have the target distribution as stationary distribution. There generated vectors are (more or less) correlated and it might take a long time until the sequence has converged to the given target distribution.
Beware: MCMC sampling can be dangerous!
Markov Chain Methods for continuous multivariate distributions GIBBS: Tconcave logPDF and derivatives of logPDF. 
Next: HITRO – Markov Chain  HITandrun sampler with RatioOfuniforms, Up: Markov chain samplers for continuous multivariate distributions [Contents][Index]
Tconcave logPDF, derivatives of logPDF
Setup: fast, Sampling: moderate
not implemented
Method GIBBS implements a Gibbs sampler for a multivariate distribution with given joint density and its gradient. When running such a Markov chain all coordinates are updated cyclically using full conditional distributions. After each step the state of the chain is returned (i.e., a random point is returned whenever a single coordinate has been updated). It is also possible to return only points after all coordinates have been updated by "thinning" the chain. Moreover, to reduce autocorrelation this thinning factor can be any integer. Notice, however, that the sampling time for a chain of given length is increased by the same factor, too.
GIBBS also provides a variant of the Gibbs sampler where in each step a point from the full conditional distribution along some random direction is sampled. This direction is chosen uniformly from the sphere in each step. This method is also known as HitandRun algorithm for nonuniform distributions.
Our experiences shows that the original Gibbs sampler with sampling along coordinate axes is superior to random direction sampling as long as the correlations between the components of the random vector are not too high.
For both variants transformed density rejection (see methods
see TDR – Transformed Density Rejection and see ARS – Adaptive Rejection Sampling) is used to
sample from the full conditional distributions. In opposition to
the univariate case, it is important that the factor c
is
as large as possible. I.e., for a logconcave density c
must be set to 0.
, since otherwise numerical underflow
might stop the algorithm.
Important: GIBBS does not generate independent random points. The starting point of the Gibbs chain must be in a "typical" region of the target distribution. If such a point is not known or would be too expensive, then the first part of the chain should be discarded (burnin of the chain).
For using the GIBBS method UNU.RAN needs the logarithm of the PDF of the multivariate joint distribution and its gradient or partial derivatives.
It provides two variants:
The coordinates are updated cyclically.
It requires the partial derivatives of the (logarithm of the)
PDF of the target distribution,
see
unur_distr_cvec_set_pdlogpdf
.
Otherwise, the gradient of the logPDF
(see
unur_distr_cvec_set_dlogpdf
)
is used, which is more expensive.
This variant can be selected using
unur_gibbs_set_variant_coordinate
.
In each step is a direction is sampled uniformly from the sphere and the next point in the chain is sampled from the full conditional distribution along this direction.
It requires the gradient of the logPDF and thus each step is more expensive than each step for coordinate direction sampling.
This variant can be selected using
unur_gibbs_set_variant_random_direction
.
It is important that the c
parameter for the TDR method
is as large as possible. For logconcave distribution it must be
set to 0
, since otherwise numerical underflow can cause
the algorithm to stop.
The starting point of the Gibbs chain must be "typical" for the
target distribution. If such a point is not known or would be
too expensive, then the first part of the chain should be
discarded (burnin of the chain). When using the
unur_gibbs_set_burnin
call this is done during the setup
of the Gibbs sampler object.
In case of a fatal error in the generator for conditional distributions the methods generates points that contain UNUR_INFINITY.
Warning: The algorithm requires that all full
conditionals for the given distribution object are
Tconcave. However, this property is not checked.
If this property is not satisfied, then generation from the
conditional distributions becomes (very) slow and might fail or
(even worse) produces random vectors from an incorrect
distribution.
When using
unur_gibbs_set_burnin
then the setup already
might fail. Thus when in doubt whether GIBBS can be used for
the targent distribution it is a good idea to use a burnin for
checking.
Remark: It might happen (very rarely) that the chain becomes stuck due to numerical errors. (This is in particular the case when the given PDF does not fulfill the condition of this method.) When this happens during burnin then the setup is aborted (i.e. it fails). Otherwise the chain restarts again from its starting point.
Warning: Be carefull with debugging flags. If it
contains flag 0x01000000u
it produces a lot of output for
each step in the algorithm.
(This flag is switched of in the default debugging flags).
...........................................................................
Coordinate Direction Sampling: Sampling along the coordinate directions (cyclic).
This is the default.
Random Direction Sampling: Sampling along the random directions.
Set parameter c for transformation
T
of the
transformed density rejection method.
Currently only values between 0
and 0.5
are
allowed. If c
is between 0
and 0.5
it is set
to 0.5
.
For c =0
(for logconcave densities) method ARS
(see ARS – Adaptive Rejection Sampling) is used which is very robust against badly
normalized PDFs. For other values method TDR (see TDR – Transformed Density Rejection) is used.
The value for c should be as large as possible to avoid
fatal numerical underflows. Thus for logconcave distributions
c must be set to 0.
Default is 0
.
Sets the starting point of the Gibbs sampler. x0 must be
a "typical" point of the given distribution.
If such a "typical" point is not known and a starting point is
merely guessed, the first part of the Gibbs chain should be
discarded (burnin), e.g.\ by mean of the
unur_gibbs_set_burnin
call.
Default is the result of
unur_distr_cvec_get_center
for the
given distribution object.
Sets the thinning parameter. When thinning is set to k then every kth point from the iteration is returned by the sampling algorithm.
Notice: This parameter must satisfy thinning>=1.
Default: 1
.
If a "typical" point for the target distribution is not known but merely guessed, the first part of the Gibbs chain should be discarded (burnin). This can be done during the initialization of the generator object. The length of the burnin can is then burnin.
When method GIBBS is not applicable for the target distribution then the initialization already might fail during the burnin. Thus this reduces the risk of running a generator that returns UNUR_INFINITY cased by some fatal error during sampling.
The thinning factor set by a
unur_gibbs_set_thinning
call has
no effect on the length of the burnin, i.e., for the burnin
always a thinning factor 1
is used.
Notice: This parameter must satisfy thinning>=0.
Default: 0
.
Get and change the current state of the Gibbs chain.
Reset state of chain to starting point.
Notice: Currently this function does not reset the generators for conditional distributions. Thus it is not possible to get the same Gibbs chain even when the underlying uniform random number generator is reset.
Previous: GIBBS – Markov Chain  GIBBS sampler, Up: Markov chain samplers for continuous multivariate distributions [Contents][Index]
mode, center, bounding rectangle for acceptance region
Setup: fast, Sampling: fast
not implemented
HITRO is an implementation of a hitandrun sampler that runs on the acceptance region of the multivariate ratioofuniforms method, see The RatioofUniforms Method.
The RatioofUniforms transforms the region below the density into some region that we call "region of acceptance" in the following. The minimal bounding hyperrectangle of this region is given by
v^{+} = sup_x (f(x))^{1/r d+1},
u^{}_i = inf_x_i (x_i mu_i) (f(x))^{r/r d+1},
u^{+}_i = sup_x_i (x_i mu_i) (f(x))^{r/r d+1},
where d denotes the dimension of the distribution; x_i is the ith coordinate of point x; mu_i is the ith coordinate of the center mu of the distribution, i.e., a point in the "main region" of the distribution. Using the center is important, since otherwise the acceptance region can become a very long and skinny ellipsoid along a diagonal of the (huge) bounding rectangle.
For each step of the HitandRun algorithm we have to choose some direction. This direction together with the current point of the chain determines a straight line. Then a point is sampled uniformly on intersection of this line and the region of acceptance. This is done by rejection from a uniform distribution on a line segment that covers it. Depending of the chosen variant the endpoints of this covering line are computed either by means of a (not necessary minimal) bounding hyperrectangle, or just the "covering plate" of the bounding hyperrectangle.
The required bounds of the hyperrectable can be given directly by the user. Otherwise, these are computed automatically by means of a numerical routine by Hooke and Jeeves [HJa61] called direct search (see src/utils/hooke.c for further references and details). However, this expensive computation can be avoided by determine these bounds "on the fly" by the following adaptive algorithm: Start with some (small) hyperrectangle and enlarge it whenever the endpoints of the covering line segment are not contained in the acceptance region of the RatioofUnfiorms method. This approach works reliable as long as the region of acceptance is convex.
The performance of the uniform sampling from the line segment is much improved if the covering line is adjusted (shortened) whenever a point is rejected (adaptive sampling). This technique reduces the expected number of iterations enormously.
Method HITRO requires that the region of acceptance of the RatioofUniforms method is bounded. The shape of this region can be controlled by a parameter r. Higher values of r result in larger classes of distributions with bounded region of acceptance. (A distribution that has such a bounded region for some r also has a bounded region for every r’ greater than r.) On the other hand the acceptance probability decreases with increasing r. Moreover, roundoff errors are more likely and (for large values of r) might result in a chain with a stationary distribution different from the target distribution.
Method HITRO works optimal for distributions whose region of
acceptance is convex. This is in particular the case for all
logconcave distributions when we set r = 1
.
For bounded but nonconvex regions of acceptance convergence is
yet not guarenteed by mathematical theory.
Method HITRO requires the PDF of the target distribution (derivatives are not necessary).
The acceptance region of the RatioofUniforms transformation
must be bounded. Its shape is controlled by parameter r.
By default this parameter is set to 1
as this guarentees
a convex region of acceptance when the PDF of the given
distribution is logconcave. It should only be set to a
different (higher!) value using
unur_vnrou_set_r
if otherwise
x_i (f(x))^{r/r d+1}
were not
bounded for each coordinate.
There are two variants of the HITRO sampler:
The coordinates are updated cyclically.
This can be seen as a Gibbs sampler running on the acceptance
region of the RatioofUniforms method.
This variant can be selected using
unur_hitro_set_variant_coordinate
.
In each step is a direction is sampled uniformly from the sphere.
This variant can be selected using
unur_hitro_set_variant_random_direction
.
Notice that each iteration of the coordinate direction sampler is cheaper than an iteration of the random direction sampler.
Sampling uniformly from the line segment can be adjusted in several ways:
When adaptive line sampling is switched on, the covering line is shortened whenever a point is rejected. However, when the region of acceptance is not convex the line segment from which we have to sample might not be connected. We found that the algorithm still works but at the time being there is no formal proof that the generated Markov chain has the required stationary distribution.
Adaptive line sampling can switch on/off by means of the
unur_hitro_set_use_adaptiveline
call.
For computing the covering line we can use the bounding hyperrectangle or just its upper bound. The latter saves computing time during the setup and when computing the covering during at each iteration step at the expense of a longer covering line. When adaptive line sampling is used the total generation time for the entire chain is shorter when only the "covering plate" is used.
Notice: When coordinate sampling is used the entire bounding rectangle is used.
Using the entire bounding hyperrectangle can be switched on/off
by means of the
unur_hitro_set_use_boundingrectangle
call.
A bounding rectangle can be given by the
unur_vnrou_set_u
and
unur_vnrou_set_v
calls.
Otherwise, the minimal bounding rectangle is computed
automatically during the setup by means of a numerical
algorithm. However, this is (very) slow especially in higher
dimensions and it might happen that this algorithm (like
any other numerical algorithm) does not return a correct result.
Alternatively the bounding rectangle can be computed
adaptively. In the latter case
unur_vnrou_set_u
and
unur_vnrou_set_v
can be used to provide a starting rectangle
which must be sufficiently small.
Then both endpoints of the covering line segment are always
check whether they are outside the acceptance region of the
RatioofUniforms method. If they are not, then the line segment
and the ("bounding") rectangle are enlarged using a factor that
can be given using the
unur_hitro_set_adaptive_multiplier
call.
Notice, that running this method in the adaptive rectangle mode requires that the region of acceptance is convex when random directions are used, or the given PDF is unimodal when coordinate direction sampling is used. Moreover, it requires two additional calls to the PDF in each iteration step of the chain.
Using addaptive bounding rectangles can be switched on/off
by means of the
unur_hitro_set_use_adaptiverectangle
call.
The algorithm takes of a bounded rectangular domain given by a
unur_distr_cvec_set_domain_rect
call, i.e. the PDF is set to
zero for every x outside the given domain.
However, it is only the coordinate direction sampler where the
boundary values are directly used to get the endpoins of the
coverline line for the line sampling step.
Important: The bounding rectangle has to be
provided for the function
PDF(xcenter)!
Notice that center
is the center of the given
distribution, see
unur_distr_cvec_set_center
.
If in doubt or if this value is not optimal, it can be changed
(overridden) by a
unur_distr_cvec_set_center
call.
Get default parameters for generator.
Coordinate Direction Sampling: Sampling along the coordinate directions (cyclic).
Notice: For this variant the entire bounding rectangle is
always used independent of the
unur_hitro_set_use_boundingrectangle
call.
This is the default.
Random Direction Sampling: Sampling along the random directions.
When adaptive is set to TRUE
adaptive line sampling is
applied, otherwise simple rejection is used.
Notice: When adaptive line sampling is switched off, the entire bounding rectangle must be used since otherwise the sampling time can be arbitrarily slow.
Warning: When adaptive line sampling is switched off, sampling can be arbitrarily slow. In particular this happens when random direction sampling is used for distributions with rectangular domains. Then the algorithm can be trapped into a vertex (or even edge).
Default is TRUE
.
When rectangle is set to TRUE
the entire bounding rectangle is used
for computing the covering line. Otherwise, only an upper bound for the
acceptance region is used.
Notice: When coordinate sampling is used the entire bounding rectangle has is always used and this call has no effect.
Default: FALSE
for random direction samplig, TRUE
for coordinate
direction sampling.
When adaptive is set to FALSE
the bounding rectangle is
determined during the setup. Either, it is computed automatically by
a (slow) numerical method, or it must be provided by
unur_vnrou_set_u
and
unur_vnrou_set_v
calls.
If adaptive is set to TRUE
the bounding rectangle is computed
adaptively. In this case the
unur_vnrou_set_u
and
unur_vnrou_set_v
calls can be used to provide a starting
rectangle. This should be sufficiently small.
If not given then we assume
v_max = 1,
u_min=(0.001,0.001,...,0.001),
and
u_max=(0.001,0.001,...,0.001).
Adaptive enlargements of the bounding hyperrectangle can be
controlled set setting an enlargement factor given
by a
unur_hitro_set_adaptive_multiplier
call.
Using adaptive computation of the bounding rectangle reduces the setup time significantly (when it is not given by the user) at the expense of two additional PDF evaluations during each iteration step.
Important: Using adaptive bounding rectangles requires that the region of acceptance is convex when random directions are used, or a unimodal PDF when coordinate direction sampling is used.
Default: FALSE
for random direction samplig, TRUE
for coordinate
direction sampling.
Sets the parameter r of the generalized multivariate ratioofuniforms method.
Notice: This parameter must satisfy r>0.
Default: 1
.
Set upper boundary for bounding hyperrectangle. If not set not set the mode of the distribution is used.
If adaptive bounding rectangles the value is used for the
starting rectangle. If not given (and the mode of the distribution
is not known) then vmax=1e3
is used.
If deterministic bounding rectangles these values are the given values are used for the rectangle. If no value is given (and the mode of the distribution is not known), the upper bound of the minimal bounding hyperrectangle is computed numerically (slow).
Default: not set.
Sets left and right boundaries of bounding hyperrectangle.
If adaptive bounding rectangles these values are used for the
starting rectangle. If not given then
umin={b,b,…,b}
and
umax={b,b,…,b}
with b=1.e3
is used.
If deterministic bounding rectangles these values are the given values are used for the rectangle. If no values are given, the boundary of the minimal bounding hyperrectangle is computed numerically (slow).
Important: The boundaries are those of the density shifted by the center of the distribution, i.e., for the function PDF(xcenter)!
Notice: Computing the minimal bounding rectangle may fail under some circumstances. Moreover, for multimodal distributions the bounds might be too small as only local extrema are computed. Nevertheless, for logconcave distributions it should work.
Default: not set.
Adaptive enlargements of the bounding hyperrectangle can be controlled set setting the enlargement factor. This must be greater than 1. Values close to 1 result in small adaptive steps and thus reduce the risk of too large bounding rectangles. On the other hand many adaptive steps might be necessary.
Notice: For practical reasons this call does not accept
values for factor less than 1.0001
. If this value is
UNUR_INFINITY this results in infinite loops.
Default: 1.1
Sets the starting point of the HITRO sampler in the original
scale. x0 must be a "typical" point of the given distribution.
If such a "typical" point is not known and a starting point is
merely guessed, the first part of the HITRO chain should be
discarded (burnin), e.g.\ by mean of the
unur_hitro_set_burnin
call.
Important: The PDF of the distribution must not vanish at the given point x0.
Default is the result of
unur_distr_cvec_get_center
for the
given distribution object.
Sets the thinning parameter. When thinning is set to
k then every kth point from the iteration is returned by
the sampling algorithm.
If thinning has to be set such that each coordinate is updated
when using coordinate direction sampling, then thinning
should be dim+1
(or any multiple of it) where
dim
is the dimension of the distribution object.
Notice: This parameter must satisfy thinning>=1.
Default: 1
.
If a "typical" point for the target distribution is not known but merely guessed, the first part of the HITRO chain should be discarded (burnin). This can be done during the initialization of the generator object. The length of the burnin can is then burnin.
The thinning factor set by a
unur_hitro_set_thinning
call has
no effect on the length of the burnin, i.e., for the burnin
always a thinning factor 1
is used.
Notice: This parameter must satisfy thinning>=0.
Default: 0
.
Get and change the current state of the HITRO chain.
Notice: The state variable contains the point in the
dim+1
dimensional point in the (tansformed) region of
acceptance of the RatioofUniforms method. Its coordinate
are stored in the following order:
state[] = {v, u1, u2, …, udim}
.
If the state can only be changed if the given state is inside this region.
Reset state of chain to starting point.
Notice: Currently this function does not reset the generators for conditional distributions. Thus it is not possible to get the same HITRO chain even when the underlying uniform random number generator is reset.
Next: Methods for discrete univariate distributions, Previous: Markov chain samplers for continuous multivariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for continuous empirical multivariate distributions VEMPK: Requires an observed sample. 
/*  */ /* File: example_vemp.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from an empirial continuous */ /* multivariate distribution. */ /*  */ int main(void) { int i; /* 4 data points of dimension 2 */ double data[] = { 1. ,1., /* 1st data point */ 1.,1., /* 2nd data point */ 1.,1., /* 3rd data point */ 1.,1. }; /* 4th data point */ double result[2]; /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Create a distribution object with dimension 2. */ distr = unur_distr_cvemp_new( 2 ); /* Set empirical sample. */ unur_distr_cvemp_set_data(distr, data, 4); /* Choose a method: VEMPK. */ par = unur_vempk_new(distr); /* Use variance correction. */ unur_vempk_set_varcor( par, 1 ); /* Create the generator object. */ gen = unur_init(par); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { unur_sample_vec(gen, result); printf("(%f,%f)\n", result[0], result[1]); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
(not implemented)
observed sample
Setup: slow, Sampling: slow (depends on dimension)
not implemented
VEMPK generates random variates from a multivariate empirical distribution that is given by an observed sample. The idea is that simply choosing a random point from the sample and to return it with some added noise results in a method that has very nice properties, as it can be seen as sampling from a kernel density estimate. Clearly we have to decide about the density of the noise (called kernel) and about the covariance matrix of the noise. The mathematical theory of kernel density estimation shows us that we are comparatively free in choosing the kernel. It also supplies us with a simple formula to compute the optimal standarddeviation of the noise, called bandwidth (or window width) of the kernel.
Currently only a Gaussian kernel with the same covariance matrix as the given sample is implemented. However it is possible to choose between a variance corrected version or those with optimal MISE. Additionally a smoothing factor can be set to adjust the estimated density to nonbellshaped data densities.
VEMPK uses empirical distributions. The main parameter would be the choice if of kernel density. However, currently only Gaussian kernels are supported. The parameters for the density are computed by a simple but robust method. However, it is possible to control its behavior by changing the smoothing factor. Additionally, variance correction can be swithed on (at the price of suboptimal MISE).
Get default parameters for generator.
Set and change the smoothing factor. The smoothing factor controlles how “smooth” the resulting density estimation will be. A smoothing factor equal to 0 results in naive resampling. A very large smoothing factor (together with the variance correction) results in a density which is approximately equal to the kernel. Default is 1 which results in a smoothing parameter minimising the MISE (mean integrated squared error) if the data are not too far away from normal. If a large smoothing factor is used, then variance correction must be switched on.
Default: 1
Switch variance correction in generator on/off.
If varcor is TRUE
then the variance of the used
density estimation is the same as the sample variance. However this
increases the MISE of the estimation a little bit.
Default is FALSE
.
Next: Methods for random matrices, Previous: Methods for continuous empirical multivariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for discrete univariate distributions

/*  */ /* File: example_discr.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from a discrete univariate distribution.*/ /*  */ int main(void) { int i; double param = 0.3; double probvec[10] = {1.0, 2.0, 3.0, 4.0, 5.0,\ 6.0, 7.0, 8.0, 4.0, 3.0}; /* Declare the three UNURAN objects. */ UNUR_DISTR *distr1, *distr2; /* distribution objects */ UNUR_PAR *par1, *par2; /* parameter objects */ UNUR_GEN *gen1, *gen2; /* generator objects */ /* First distribution: defined by PMF. */ distr1 = unur_distr_geometric(¶m, 1); unur_distr_discr_set_mode(distr1, 0); /* Choose a method: DARI. */ par1 = unur_dari_new(distr1); gen1 = unur_init(par1); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen1 == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Second distribution: defined by (finite) PV. */ distr2 = unur_distr_discr_new(); unur_distr_discr_set_pv(distr2, probvec, 10); /* Choose a method: DGT. */ par2 = unur_dgt_new(distr2); gen2 = unur_init(par2); if (gen2 == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* print some random integers */ for (i=0; i<10; i++){ printf("number %d: %d\n", i*2, unur_sample_discr(gen1) ); printf("number %d: %d\n", i*2+1, unur_sample_discr(gen2) ); } /* Destroy all objects. */ unur_distr_free(distr1); unur_distr_free(distr2); unur_free(gen1); unur_free(gen2); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
/*  */ /* File: example_discr_str.c */ /*  */ /* String API. */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Example how to sample from a discrete univariate distribution.*/ /*  */ int main(void) { int i; /* loop variable */ /* Declare UNURAN generator objects. */ UNUR_GEN *gen1, *gen2; /* generator objects */ /* First distribution: defined by PMF. */ gen1 = unur_str2gen("geometric(0.3); mode=0 & method=dari"); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen1 == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Second distribution: defined by (finite) PV. */ gen2 = unur_str2gen( "distr=discr; pv=(1,2,3,4,5,6,7,8,4,3) & method=dgt"); if (gen2 == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* print some random integers */ for (i=0; i<10; i++){ printf("number %d: %d\n", i*2, unur_sample_discr(gen1) ); printf("number %d: %d\n", i*2+1, unur_sample_discr(gen2) ); } /* Destroy all objects. */ unur_free(gen1); unur_free(gen2); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Next: DAU – (Discrete) AliasUrn method, Up: Methods for discrete univariate distributions [Contents][Index]
Tconcave PMF, mode, approximate area
Setup: moderate, Sampling: fast
supported
DARI is based on rejection inversion, which can be seen as an adaptation of transformed density rejection to discrete distributions. The used transformation is 1/sqrt(x) .
DARI uses three almost optimal points for constructing the (continuous) hat. Rejection is then done in horizontal direction. Rejection inversion uses only one uniform random variate per trial.
DARI has moderate setup times (the PMF is evaluated nine times), and good marginal speed, especially if an auxiliary array is used to store values during generation.
DARI works for all T_1/2 concave distributions. It requires the PMF and the location of the mode. Moreover the approximate sum over the PMF is used. (If no sum is given for the distribution the algorithm assumes that it is approximately 1.) The rejection constant is bounded from above by 4 for all Tconcave distributions.
DARI works for discrete distribution object with given PMF.
The sum over probabilities should be approximately
one. Otherwise it must be set by a
unur_distr_discr_set_pmfsum
call to its (approximate) value.
The size of an auxiliary table can be set by
unur_dari_set_tablesize
.
The expected number of evaluations can be reduced by switching
the use of squeezes by means of
unur_dari_set_squeeze
.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Notice, that derived parameters like the mode must also be (re) set
if the parameters or the domain has be changed.
There exists a test mode that verifies whether the conditions for
the method are satisfied or not. It can be switched on by calling
unur_dari_set_verify
and
unur_dari_chg_verify
respectively.
Notice however that sampling is (much) slower then.
Get default parameters for generator.
Turn utilization of the squeeze of the algorithm on/off. This squeeze does not resamble the squeeze of the continuous TDR method. It was especially designed for rejection inversion.
The squeeze is not necessary if the size of the auxiliary table is big enough (for the given distribution). Using a squeeze is suggested to speed up the algorithm if the domain of the distribution is very big or if only small samples are produced.
Default: no squeeze.
Set the size for the auxiliary table, that stores constants
computed during generation.
If size is set to 0
no table is used.
The speedup can be impressive if the PMF is expensive to
evaluate and the “main part of the distribution” is concentrated
in an interval shorter than the size of the table.
Default is 100
.
Set factor for position of the left and right construction point, resp. The cp_factor is used to find almost optimal construction points for the hat function. The cp_factor must be positive and should not exceed 2. There is no need to change this factor in almost all situations.
Default is 0.664
.
Turn verifying of algorithm while sampling on/off.
If the condition is violated for some x then unur_errno
is set to UNUR_ERR_GEN_CONDITION
. However notice that this
might happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Next: DEXT – wrapper for Discrete EXTernal generators, Previous: DARI – Discrete Automatic Rejection Inversion, Up: Methods for discrete univariate distributions [Contents][Index]
probability vector (PV)
Setup: slow (linear with the vectorlength), Sampling: very fast
supported
DAU samples from distributions with arbitrary but finite probability vectors (PV) of length N. The algorithmus is based on an ingeneous method by A.J. Walker and requires a table of size (at least) N. It needs one random numbers and only one comparison for each generated random variate. The setup time for constructing the tables is O(N).
By default the probability vector is indexed starting at
0
. However this can be changed in the distribution object by
a
unur_distr_discr_set_domain
call.
The method also works when no probability vector but a PMF is
given. However then additionally a bounded (not too large) domain
must be given or the sum over the PMF (see
unur_distr_discr_make_pv
for details).
Create an object for a discrete distribution either by setting a
probability vector or a PMF. The performance can be slightly
influenced by setting the size of the used table which can be
changed by
unur_dau_set_urnfactor
.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for generator.
Set size of urn table relative to length of the probability vector. It must not be less than 1. Larger tables result in (slightly) faster generation times but require a more expensive setup. However sizes larger than 2 are not recommended.
Default is 1
.
Next: DGT – (Discrete) Guide Table method (indexed search), Previous: DAU – (Discrete) AliasUrn method, Up: Methods for discrete univariate distributions [Contents][Index]
routine for sampling discrete random variates
depends on external generator
supported
Method DEXT is a wrapper for external generators for discrete univariate distributions. It allows the usage of external random variate generators within the UNU.RAN framework.
The following steps are required to use some external generator within the UNU.RAN framework (some of these are optional):
unur_dext_new
call.
The argument distribution is optional and can be replaced
by NULL
. However, it is required if you want to pass
parameters of the generated distribution to the external
generator or for running some validation tests provided by
UNU.RAN.
int (*init)(UNUR_GEN *gen)
and plug it into the generator
object using the
unur_dext_set_init
call. Notice that the
init routine must return UNUR_SUCCESS
when it has
been executed successfully and UNUR_FAILURE
otherwise.
It is possible to get the size of and the pointer to the array
of parameters of the underlying distribution object by the
respective calls
unur_dext_get_ndistrparams
and
unur_dext_get_distrparams
.
Parameters for the external generator that are computed in the
init routine can be stored in a single array or structure
which is available by the
unur_dext_get_params
call.
Using an init routine is optional and can be omitted.
int (*sample)(UNUR_GEN *gen)
and plug it into the
generator object using the
unur_dext_set_sample
call.
Uniform random numbers are provided by the
unur_sample_urng
call. Do not use your own implementation of a uniform random
number generator directly. If you want to use your own random
number generator we recommend to use the UNU.RAN interface (see
see Using uniform random number generators).
The array or structure that contains parameters for the external
generator that are computed in the init routine are
available using the
unur_dext_get_params
call.
Using a sample routine is of course obligatory.
It is possible to change the parameters and the domain of the
chosen distribution and run
unur_reinit
to reinitialize the
generator object. The init routine is then called again.
Here is a short example that demonstrates the application of this method by means of the geometric distribution:
/*  */ /* File: example_dext.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* This example shows how an external generator for the */ /* geometric distribution can be used within the UNURAN */ /* framework. */ /* */ /* Notice, that this example does not provide the simplest */ /* solution. */ /*  */ /* Initialization routine. */ /* */ /* Here we simply read the parameter of the geometric */ /* distribution and store it in an array for parameters of */ /* the external generator. */ /* [ Of course we could do this in the sampling routine as */ /* and avoid the necessity of this initialization routine. ] */ int geometric_init (UNUR_GEN *gen) { /* Get pointer to parameters of geometric distribution */ double *params = unur_dext_get_distrparams(gen); /* The parameter is the first entry (see manual) */ double p = params[0]; /* Get array to store this parameter for external generator */ double *genpar = unur_dext_get_params(gen, sizeof(double)); genpar[0] = p; /* Executed successfully */ return UNUR_SUCCESS; } /*  */ /* Sampling routine. */ /* */ /* Contains the code for the external generator. */ int geometric_sample (UNUR_GEN *gen) { /* Get scale parameter */ double *genpar = unur_dext_get_params(gen,0); double p = genpar[0]; /* Sample a uniformly distributed random number */ double U = unur_sample_urng(gen); /* Transform into geometrically distributed random variate */ return ( (int) (log(U) / log(1.p)) ); } /*  */ int main(void) { int i; /* loop variable */ int K; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Use predefined geometric distribution with parameter 1/10 */ double fpar[1] = { 0.1 }; distr = unur_distr_geometric(fpar, 1); /* Use method DEXT */ par = unur_dext_new(distr); /* Set initialization and sampling routines. */ unur_dext_set_init(par, geometric_init); unur_dext_set_sample(par, geometric_sample); /* Create the generator object. */ gen = unur_init(par); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* It is possible to reuse the distribution object to create */ /* another generator object. If you do not need it any more, */ /* it should be destroyed to free memory. */ unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the standard Gaussian distribution. */ /* Eg.: */ for (i=0; i<10; i++) { K = unur_sample_discr(gen); printf("%d\n",K); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
Get default parameters for new generator.
Set initialization routine for external generator. Inside the
Important: The routine init must return
UNUR_SUCCESS
when the generator was initialized successfully
and UNUR_FAILURE
otherwise.
Parameters that are computed in the init routine can be
stored in an array or structure that is avaiable by means of the
unur_dext_get_params
call. Parameters of the underlying
distribution object can be obtained by the
unur_dext_get_distrparams
call.
Set sampling routine for external generator.
Important:
Use unur_sample_urng(gen)
to get a uniform random number.
The pointer to the array or structure that contains the parameters
that are precomputed in the init routine are available by
unur_dext_get_params(gen,0)
.
Additionally one can use the
unur_dext_get_distrparams
call.
Get pointer to memory block for storing parameters of external
generator. A memory block of size size is automatically (re)
allocated if necessary and the pointer to this block is stored in
the generator object. If one only needs the pointer to this
memory block set size to 0
.
Notice, that size is the size of the memory block and not the length of an array.
Important: This rountine should only be used in the initialization and sampling routine of the external generator.
Get size of and pointer to array of parameters of underlying distribution in generator object.
Important: These rountines should only be used in the initialization and sampling routine of the external generator.
Next: DSROU – Discrete Simple RatioOfUniforms method, Previous: DEXT – wrapper for Discrete EXTernal generators, Up: Methods for discrete univariate distributions [Contents][Index]
probability vector (PV)
Setup: slow (linear with the vectorlength), Sampling: very fast
supported
DGT samples from arbitrary but finite probability vectors. Random numbers are generated by the inversion method, i.e.,
Step (2) is the crucial step. Using sequential search requires
O(E(X)) comparisons, where E(X) is the expectation of
the distribution. Indexed search, however, uses a guide table to
jump to some I’ <= I near I to find X in constant
time. Indeed the expected number of comparisons is reduced to 2,
when the guide table has the same size as the probability vector
(this is the default). For larger guide tables this number
becomes smaller (but is always larger than 1), for smaller
tables it becomes larger. For the limit case of table size 1 the
algorithm simply does sequential search (but uses a more expensive
setup then method DSS (see DSS – (Discrete) Sequential Search method). On the other hand the
setup time for guide table is O(N), where N denotes the
length of the probability vector (for size 1 no preprocessing is
required). Moreover, for very large guide tables memory effects might
even reduce the speed of the algorithm. So we do not recommend to
use guide tables that are more than three times larger than the
given probability vector. If only a few random numbers have to be
generated, (much) smaller table sizes are better.
The size of the guide table relative to the length of the given
probability vector can be set by a
unur_dgt_set_guidefactor
call.
There exist two variants for the setup step which can be set by a
unur_dgt_set_variant
call: Variants 1 and 2.
Variant 2 is faster but more sensitive to roundoff errors when the
guide table is large. By default variant 2 is used for short
probability vectors (N<1000) and variant 1 otherwise.
By default the probability vector is indexed starting at
0
. However this can be changed in the distribution object by
a
unur_distr_discr_set_domain
call.
The method also works when no probability vector but a PMF is
given. However, then additionally a bounded (not too large) domain
must be given or the sum over the PMF. In the latter case the
domain of the distribution is trucated (see
unur_distr_discr_make_pv
for details).
Create an object for a discrete distribution either by setting a
probability vector or a PMF. The performance can be slightly
influenced by setting the size of the used table which can be
changed by
unur_dgt_set_guidefactor
.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for generator.
Set size of guide table relative to length of PV. Larger guide tables result in faster generation time but require a more expensive setup. Sizes larger than 3 are not recommended. If the relative size is set to 0, sequential search is used. However, this is not recommended, except in exceptional cases, since method DSS (see DSS – (Discrete) Sequential Search method) is has almost no setup and is thus faster (but requires the sum over the PV as input parameter).
Default is 1
.
Set variant for setup step. Possible values are 1
or
2
.
Variant 2
is faster but more sensitive to roundoff errors
when the guide table is large.
By default variant 2
is used for short probability
vectors (N<1000) and variant 1
otherwise.
Next: DSS – (Discrete) Sequential Search method, Previous: DGT – (Discrete) Guide Table method (indexed search), Up: Methods for discrete univariate distributions [Contents][Index]
Tconcave PMF, mode, sum over PMF
Setup: fast, Sampling: slow
supported
DSROU is based on the ratioofuniforms method (see The RatioofUniforms Method) but uses universal inequalities for constructing a (universal) bounding rectangle. It works for all Tconcave distributions with T(x) = 1/sqrt(x) .
The method requires the PMF, the (exact) location of the mode and the sum over the given PDF. The rejection constant is 4 for all Tconcave distributions. Optionally the CDF at the mode can be given to increase the performance of the algorithm. Then the rejection constant is reduced to 2.
The method works for Tconcave discrete distributions with given PMF. The sum over of the PMF or an upper bound of this sum must be known.
Optionally the CDF at the mode can be given to increase the
performance using
unur_dsrou_set_cdfatmode
.
However, this must not be called if the sum over the
PMF is replaced by an upper bound.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
If any of mode, CDF at mode, or the sum over the PMF has been
changed, then
unur_reinit
must be executed.
(Otherwise the generator produces garbage).
There exists a test mode that verifies whether the conditions
for the method are satisfied or not while sampling. It can be
switched on or off by calling
unur_dsrou_set_verify
and
unur_dsrou_chg_verify
respectively.
Notice however that sampling is (a little bit) slower then.
Get default parameters for generator.
Set CDF at mode.
When set, the performance of the algorithm is increased by factor 2.
However, when the parameters of the distribution are changed
unur_dsrou_chg_cdfatmode
has to be used to update this value.
Notice that the algorithm detects a mode at the left boundary of
the domain automatically and it is not necessary to use this call
for a monotonically decreasing PMF.
Default: not set.
Turn verifying of algorithm while sampling on/off.
If the condition squeeze(x) <= PMF(x) <= hat(x) is
violated for some x then unur_errno
is set to
UNUR_ERR_GEN_CONDITION
. However notice that this might
happen due to roundoff errors for a few values of
x (less than 1%).
Default is FALSE
.
Change CDF at mode of distribution.
unur_reinit
must be executed before sampling from the
generator again.
Next: DSTD – Discrete STandarD distributions, Previous: DSROU – Discrete Simple RatioOfUniforms method, Up: Methods for discrete univariate distributions [Contents][Index]
probability vector (PV) and sum over PV; or probability mass function(PMF), sum over PV and domain; or or cumulative distribution function (CDF)
Setup: fast, Sampling: very slow (linear in expectation)
supported
DSS samples from arbitrary discrete distributions. Random numbers are generated by the inversion method, i.e.,
Step (2) is the crucial step. Using sequential search requires O(E(X)) comparisons, where E(X) is the expectation of the distribution. Thus this method is only recommended when only a few random variates from the given distribution are required. Otherwise, table methods like DGT (see DGT – (Discrete) Guide Table method (indexed search)) or DAU (see DAU – (Discrete) AliasUrn method) are much faster. These methods also need not the sum over the PMF (or PV) as input. On the other hand, however, these methods always compute a table.
DSS runs with the PV, the PMF, or the CDF of the distribution. It uses actually uses the first one in this list (in this ordering) that could be found.
It works with a discrete distribution object with contains at least the PV, the PMF, or the CDF.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for generator.
Previous: DSS – (Discrete) Sequential Search method, Up: Methods for discrete univariate distributions [Contents][Index]
standard distribution from UNU.RAN library (see Standard distributions) or discrete distribution with inverse CDF.
Setup: fast, Sampling: depends on distribution and generator
supported
DSTD is a wrapper for special generators for discrete univariate
standard distributions. It only works for distributions in the
UNU.RAN library of standard distributions
(see Standard distributions)
or for discrete distributions where the inverse CDF is given.
If a distribution object is provided that is build from scratch,
it must provide the inverse CDF. Then CSTD implements the
inversion method. Otherwise, the NULL
pointer is returned.
For some distributions more than one special generator is possible.
Create a distribution object for a standard distribution
from the UNU.RAN library
(see Standard distributions),
or create a discrete distribution object and set the function
for the inverse CDF using
unur_distr_discr_set_invcdf
.
For some distributions more than one special generator
(variants) is possible. These can be choosen by a
unur_dstd_set_variant
call. For possible variants
See Standard distributions.
However the following are common to all distributions:
UNUR_STDGEN_DEFAULT
the default generator.
UNUR_STDGEN_FAST
the fastest available special generator.
UNUR_STDGEN_INVERSION
the inversion method (if available).
Notice that the variant UNUR_STDGEN_FAST
for a special
generator might be slower than one of the universal algorithms!
Additional variants may exist for particular distributions.
Sampling from truncated distributions (which can be constructed by
changing the default domain of a distribution by means of
unur_distr_discr_set_domain
or unur_dstd_chg_truncated calls)
is possible but requires the inversion method. Moreover the CDF
of the distribution must be implemented.
It is possible to change the parameters and the domain of the chosen
distribution and run
unur_reinit
to reinitialize the generator object.
Get default parameters for new generator. It requires a distribution object for a discrete univariant distribution from the UNU.RAN library of standard distributions (see Standard distributions).
Using a truncated distribution is allowed only if the inversion method
is available and selected by the
unur_dstd_set_variant
call immediately
after creating the parameter object.
Use a
unur_distr_discr_set_domain
call to get a truncated
distribution.
Set variant (special generator) for sampling from a given distribution. For possible variants see Standard distributions.
Common variants are UNUR_STDGEN_DEFAULT
for the default generator,
UNUR_STDGEN_FAST
for (one of the) fastest implemented
special generators, and UNUR_STDGEN_INVERSION
for the
inversion method (if available).
If the selected variant number is not implemented, then an error code is
returned and the variant is not changed.
Change left and right border of the domain of the (truncated) distribution. This is only possible if the inversion method is used. Otherwise this call has no effect and an error code is returned.
Notice that the given truncated domain must be a subset of the domain of the given distribution. The generator always uses the intersection of the domain of the distribution and the truncated domain given by this call.
It is not required to run
unur_reinit
after this call has been used.
Important: If the CDF is (almost) the same for left and
right and (almost) equal to 0
or 1
, then the truncated
domain is not chanced and the call returns an error code.
Notice: If the parameters of the distribution has been changed it is recommended to set the truncated domain again, since the former call might change the domain of the distribution but not update the values for the boundaries of the truncated distribution.
Next: Methods for uniform univariate distributions, Previous: Methods for discrete univariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
Methods for matrix distributions MCORR: Distribution object for random correlation matrix. 
Distribution object for random correlation matrix
Setup: fast, Sampling: depends on dimension
supported
MCORR generates a random correlation matrix (Pearson’s correlation). Two methods are used:
Notice that due to roundoff errors the generated matrices might not be positive definite in extremely rare cases (especially when the given eigenvalues are amost 0).
There are many other possibilites (distributions) of sampling the random rows from a sphere. The chosen methods are simple but does not result in a uniform distriubution of the random correlation matrices.
It only works with distribution objects of random correlation matrices (see Random Correlation Matrix).
Create a distibution object for random correlation matrices by a
unur_distr_correlation
call
(see Random Correlation Matrix).
When a correlation matrix with given eigenvalues should be
generated, these eigenvalues can be set by a
unur_mcorr_set_eigenvalues
call.
Otherwise, a faster algorithm is used that generates correlation matrices with random eigenstructure.
Notice that due to roundoff errors, there is a (small) chance that the resulting matrix is not positive definite for a Cholesky decomposition algorithm, especially when the dimension of the distribution is high.
It is possible to change the given eigenvalues using
unur_mcorr_chg_eigenvalues
and run
unur_reinit
to
reinitialize the generator object.
Get default parameters for generator.
Sets the (optional) eigenvalues of the correlation matrix. If set, then the Marsaglia and Olkin algorithm will be used to generate random correlation matrices with given eigenvalues.
Important: the given eigenvalues of the correlation matrix must be strictly positive and sum to the dimension of the matrix. If nonpositive eigenvalues are attempted, no eigenvalues are set and an error code is returned. In case, that their sum is different from the dimension, an implicit scaling to give the correct sum is performed.
Change the eigenvalues of the correlation matrix.
One must run
unur_reinit
to reinitialize the generator
object then.
Next: Meta Methods for univariate distributions, Previous: Methods for random matrices, Up: Methods for generating nonuniform random variates [Contents][Index]
UNIF is a simple wrapper that makes it possible to use a uniform random number generator as a UNU.RAN generator. There are no parameters for this method.
Create a generator object with NULL
as argument. The created generator
object returns raw random numbers from the underlying uniform
random number generator.
Get default parameters for generator.
UNIF does not need a distribution object. dummy is not used and
can (should) be set to NULL
. It is used to keep the API consistent.
Previous: Methods for uniform univariate distributions, Up: Methods for generating nonuniform random variates [Contents][Index]
/*  */ /* File: example_mixt.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Mixture of a Gaussian and a Cauchy distribution. */ /*  */ int main(void) { /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *comp[2]; /* array of generator objects (components)*/ UNUR_GEN *gen; /* generator object for mixture */ double prob[2]; /* array of probabilities */ int i; /* loop variable */ double x; /* will hold the random number */ /* Create generators for components */ distr = unur_distr_normal(NULL,0); /* Gaussian distribution */ par = unur_pinv_new(distr); /* choose method PINV */ comp[0] = unur_init(par); /* initialize */ unur_distr_free(distr); /* free distribution obj. */ distr = unur_distr_cauchy(NULL,0); /* Cauchy distribution */ par = unur_tdr_new(distr); /* choose method TDR */ comp[1] = unur_init(par); /* initialize */ unur_distr_free(distr); /* free distribution obj. */ /* Probabilities for components (need not sum to 1) */ prob[0] = 0.4; prob[1] = 0.3; /* Create mixture */ par = unur_mixt_new(2,prob,comp); /* Initialize generator object */ gen = unur_init(par); /* we do not need the components any more */ for (i=0; i<2; i++) unur_free(comp[i]); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
/*  */ /* File: example_mixt_inv.c */ /*  */ /* Include UNURAN header file. */ #include <unuran.h> /*  */ /* Mixture of truncated Gaussian on (INFINITY,0] and */ /* Exponential distribution on [0,INFINITY) */ /*  */ int main(void) { /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *comp[2]; /* array of generator objects (components)*/ UNUR_GEN *gen; /* generator object for mixture */ double prob[2]; /* array of probabilities */ int i; /* loop variable */ double x; /* will hold the random number */ /* Create generators for components */ distr = unur_distr_normal(NULL,0); /* Gaussian distribution */ unur_distr_cont_set_domain(distr,UNUR_INFINITY,0); par = unur_pinv_new(distr); /* choose method PINV */ comp[0] = unur_init(par); /* initialize */ unur_distr_free(distr); /* free distribution obj. */ distr = unur_distr_exponential(NULL,0); /* Exponential distr. */ par = unur_pinv_new(distr); /* choose method PINV */ comp[1] = unur_init(par); /* initialize */ unur_distr_free(distr); /* free distribution obj. */ /* Probabilities for components (need not sum to 1) */ prob[0] = 0.4; prob[1] = 0.3; /* Create mixture */ par = unur_mixt_new(2,prob,comp); /* We want to use inversion for the mixture as well. */ /* (Thus the above order of the components is important!) */ unur_mixt_set_useinversion(par,TRUE); /* Initialize generator object */ gen = unur_init(par); /* we do not need the components any more */ for (i=0; i<2; i++) unur_free(comp[i]); /* It is important to check if the creation of the generator */ /* object was successful. Otherwise `gen' is the NULL pointer */ /* and would cause a segmentation fault if used for sampling. */ if (gen == NULL) { fprintf(stderr, "ERROR: cannot create generator object\n"); exit (EXIT_FAILURE); } /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* When you do not need the generator object any more, you */ /* can destroy it. */ unur_free(gen); exit (EXIT_SUCCESS); } /* end of main() */ /*  */
MIXT allows to sample from a mixture of univariate distributions.
Let f_1,...,f_n be PDFs of various distributions called the components and (p_1,...,p_n) be a probability vector. Then f(x) = p_1 * f_1(x) + ...+ p_n * f_n(x) is the PDF of the so called mixture of these distributions.
Method MIXT takes generator objects for the components and a probability vector and creates a generator object for this mixture.
The sampling part works as follows:
When the (interior of the) domains of the the components are disjoint then it is possible to sample from the mixture by inversion, provided that the following conditions are met:
Create generator objects for the components of the mixture and
store the corresponding pointers in an array.
Store all probabilities an a double array of the same size.
Create the parameter object for the generator of the mixture
distribution by means of
unur_mixt_new
.
The components of the mixture can be any continuous or discrete
univariate distributions. This also includes generators for
empirical distributions and mixtures of distributions.
In particular, mixtures can also be defined recursively.
Remark:
The components of the mixture can be continuous or discrete
distributions. The resulting mixture, however, is always a
continuous distribution and thus
unur_sample_cont
must be used!
The inversion method can be switched on by means of
unur_mixt_set_useinversion
call.
However, the conditions for this method must then be met.
Otherwise, initialization of the mixture object fails.
Get default parameters for the generator for a mixture of the distributions given in the array comp (components) of length n. The probabilities are given by prob.
The generators in comp must be objects for (continuous or discrete) univariate distributions
If useinv is TRUE
, then the inversion method is used for
sampling from the mixture distribution.
However, the following conditions must be satisfied:
If one of these conditions is violated, then initialization of the mixture object fails.
Default is FALSE
.
Next: UNU.RAN Library of standard distributions, Previous: Methods for generating nonuniform random variates, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
UNU.RAN is designed to work with many sources of (pseudo) random numbers or low discrepancy numbers (so called quasirandom numbers) for almost all tasks in discrete event simulation, (quasi) Monte Carlo integration or any other stochastic methods. Hence UNU.RAN uses pointers to access uniform (pseudo) random number generators (URNG).
Each UNU.RAN (nonuniform random variate) generator object has a pointer to a URNG object. Thus each UNU.RAN generator object may have its own (independent) URNG or several generator objects can share the same URNG.
If no URNG is provided for a parameter or generator object a default generator is used which is the same for all generators. This URNG is defined in unuran_config.h at compile time and can be changed at runtime.
UNU.RAN uses a unified interface for all sources of random numbers.
Unfortunately, the API for random number generators, like the
GSL (GNU Scientific Library), Otmar Lendl’s prng
(Pseudo random number generators), or a single function
implemented by the user herself, are quite different.
Hence an object of type UNUR_URNG
is introduced to store
the URNG. Thus it is possible to handle different sources of
such URNGs with the unified API. It is inspired from similar to
Pierre L’Ecuyers RngStreams library:
The routine to create a URNG depends on the chosen random number generator (i.e. library). Nevertheless, there exist wrapper functions to simplify this task.
Currently the following sources of uniform random numbers are directly supported (i.e., there exist wrapper functions). Of course other random number generation libraries can be used.
FVOID
URNGs of type double uniform(void *state)
.
The argument state can be simply ignored in the
implementation of uniform
when a global state variable is
used.
UNU.RAN contains some buildin URNGs of this type in directory
src/uniform/.
PRNG
URNGs from Otmar Lendl’s prng
library. It provides a very
flexible way to sample form arbitrary URNGs by means of an object
oriented programing paradigma. Similarly to the UNU.RAN library
independent generator objects can be build and used.
This library has been developed by the pLab group at the university of Salzburg (Austria, EU) and implemented by Otmar Lendl. It is available from http://statmath.wu.ac.at/prng/ or from the pLab site at http://random.mat.sbg.ac.at/.
This interface must be compiled into UNU.RAN using the
configure flag withurngprng
.
RNGSTREAM
Pierre L’Ecuyer’s RngStream
library for multiple
independent streams of pseudorandom numbers.
A GNUstyle package is available from
http://statmath.wu.ac.at/software/RngStreams/.
This interface must be compiled into UNU.RAN using the
configure flag withurngrngstream
.
GSL
URNG from the GNU Scientific Library (GSL). It is available from http://www.gnu.org/software/gsl/.
This interface must be compiled into UNU.RAN using the
configure flag withurnggsl
.
Each UNU.RAN generator object has a pointer to a uniform
(pseudo) random number generator (URNG). It can be set via the
unur_set_urng
call. It is also possible to read this pointer
via
unur_get_urng
or change the URNG for an existing generator
object by means of
unur_chg_urng
.
It is important to note that these calls only copy the pointer
to the URNG object into the generator object.
If no URNG is provided for a parameter or generator object a default
URNG is used which is the same for all generators. This URNG is
defined in unuran_config.h at compile time. A pointer to
this default URNG can be obtained via
unur_get_default_urng
.
Nevertheless, it is also possible to change this default URNG by
another one at runtime by means of the
unur_set_default_urng
call. However, this only takes effect for new parameter objects.
Some generating methods provide the possibility of correlation
induction. For this feature a second auxiliary URNG is required.
It can be set and changed by
unur_set_urng_aux
and
unur_chg_urng_aux
calls, respectively. Since the auxiliary
URNG is by default the same as the main URNG, the
auxiliary URNG must be set after any
unur_set_urng
or
unur_chg_urng
call! Since in special cases mixing of two URNG
might cause problems, we supply a default auxiliary generator
that can be used by a
unur_use_urng_aux_default
call (after
the main URNG has been set). This default auxiliary generator
can be changed with analogous calls as the (main) default
uniform generator.
Uniform random number generators form different sources have
different programming interfaces. Thus UNU.RAN stores all
information about a particular uniform random number generator
in a structure of type UNUR_URNG
. Before a URNG can be
used with UNU.RAN an appropriate object has to be created ba a
unur_urng_new
call.
This call takes two arguments: the pointer to the sampling
routine of the generator and a pointer to a possible argument
that stores the state of the generator. The function must be of
type double (*sampleunif)(void *params)
, but functions
without any argument also work.
Additionally one can set pointers to functions for reseting or
jumping the streams generated by the URNG by the corresponding
set
calls.
UNU.RAN provides a unified API to all sources of random numbers. Notice, however, that not all functions work for all random number generators (as the respective library has not implemented the corresponding feature).
There are wrapper functions for some libraries of uniform random number generators to simplify the task of creating a UNU.RAN object for URNGs. These functions must be compiled into UNU.RAN using the corresponding configure flags (see description of the respective interface below).
Get the pointer to the default URNG. The default URNG is used by all
generators where no URNG was set explicitly by a
unur_set_urng
call.
Change the default URNG that is used for new parameter objects. It returns the pointer to the old default URNG that has been used.
Analogous calls for default auxiliary generator.
Use the URNG urng
for the new generator. This overrides the
default URNG. It also sets the auxiliary URNG to urng
.
Important: For multivariate distributions that use
marginal distributions this call does not work properly.
It is then better first to create the generator object (by
a
unur_init
call) and then change the URNG by means of
unur_chg_urng
.
Change the URNG for the given generator. It returns the pointer to
the old URNG that has been used by the generator.
It also changes the auxiliary URNG to urng
and thus it
overrides the last
unur_chg_urng_aux
call.
Get the pointer to the URNG that is used by the generator. This is usefull if two generators should share the same URNG.
Use the auxiliary URNG urng_aux
for the new generator.
(Default is the default URNG or the URNG from the last
unur_set_urng
call. Thus if the auxiliary generator should be
different to the main URNG,
unur_set_urng_aux
must be called after
unur_set_urng
.
The auxiliary URNG is used as second stream of uniform random
number for correlation induction.
It is not possible to set an auxiliary URNG for a method that does
not need one. In this case an error code is returned.
Use the default auxiliary URNG.
(It must be set after
unur_get_urng
.
)
It is not possible to set an auxiliary URNG for a method that does
not use one (i.e. the call returns an error code).
Switch to default auxiliary URNG.
(It must be set after
unur_get_urng
.
)
It is not possible to set an auxiliary URNG for a method that does
not use one (i.e. the call returns an error code).
Change the auxiliary URNG for the given generator. It returns
the pointer to the old auxiliary URNG that has been used by the
generator. It has to be called after each
unur_chg_urng
when the
auxiliary URNG should be different from the main URNG.
It is not possible to change the auxiliary URNG for a method that
does not use one (i.e. the call NULL
).
Get the pointer to the auxiliary URNG that is used by the generator. This is usefull if two generators should share the same URNG.
Notice: Some of the below function calls do not work for every source of random numbers since not every library has implemented these features.
Get a uniform random number from urng.
If the NULL
pointer is given, the default uniform generator is
used.
Get a uniform random number from the underlying uniform
random number generator of generator gen.
If the NULL
pointer is given, the default uniform generator is
used.
Set array X of length dim with uniform random numbers
sampled from generator urng. If urng is the NULL
pointer, the default uniform generator is used.
Important: If urng is based on a point set generator (this is the case for generators of low discrepance point sets as used in quasiMonte Carlo methods) it has a “natural dimension” s. In this case either only the first s entries of X are filled (if s < dim), or the first dim coordinates of the generated point are filled.
The called returns the actual number of entries filled. In case of
an error 0
is returned.
Reset urng object. The routine tries two ways to reset the generator (in this order):
unur_urng_set_reset
call.
unur_urng_seed
call (which
requires a seeding function given by a
unur_urng_set_seed
call).
If neither of the two methods work resetting of the generator is not possible and an error code is returned.
If the NULL
pointer is given, the default uniform generator is
reset.
Jump into defined state ("sync") of the generator. This is useful
when point generators are used where the coordinates are
sampled via
unur_urng_sample
.
Then this call can be used to
jump to the first coordinate of the next generated point.
Set seed for generator urng.
It returns an error code if this is not possible for the given
URNG. If the NULL
pointer is given, the default uniform generator is
seeded (if possible).
Notice: Seeding should be done only once for a particular
generator (except for resetting it to the initial state).
Expertise is required when multiple seeds are used to get independent
streams. Thus we recommend appropriate libraries for this task,
e.g. Pierre L’Ecuyer’s RngStreams package. For this library
only a package seed can be set and thus the
unur_urng_seed
call
will not have any effect to generators of this type. Use
unur_urng_reset
or
unur_urng_rngstream_new
instead, depending
whether one wants to reset the stream or get a new stream that is
independent from the previous ones.
Switch to antithetic random numbers in urng. It returns an error code if this is not possible for the given URNG.
If the NULL
pointer is given, the antithetic flag of the default
uniform generator is switched (if possible).
Jump to start of the next substream of urng. It returns an error code if this is not possible for the given URNG.
If the NULL
pointer is given, the default uniform generator is set
to the start of the next substream (if possible).
Jump to start of the current substream of urng. It returns an error code if this is not possible for the given URNG.
If the NULL
pointer is given, the default uniform generator is set
to the start of the current substream (if possible).
Analogous to
unur_urng_sync
unur_urng_seed
,
unur_urng_anti
unur_urng_reset
,
unur_urng_nextsub
and
unur_urng_resetsub
but act on the URNG object used by the generator object.
Warning: These calls should be used with care as it influences all generator objects that share the same URNG object!
Notice: These functions are provided to built a
UNUR_URNG object for a particular external random number
generator from scratch. For some libraries that contain random
number generators (like the GSL) there are special calls,
e.g.
unur_urng_gsl_new
to get such an object. Then there is no
need to change the UNUR_URNG object as it already contains all
available features.
If you have a particular library for random number generators you can either write wrapper function like those in src/uniform/urng_gsl.c or write an email to the authors of UNU.RAN to write it for you.
Get a new URNG object. sampleunif is a function to the uniform sampling routine, state a pointer to its arguments which usually contains the state variables of the generator.
Functions sampleunif with a different type for p or without an argument at all also work. A typecast might be necessary to avoid compiler warnings or error messages.
For functions sampleunif that does not have any argument
should use NULL
for state.
Important: sampleunif must not be the NULL
pointer.
There are appropriate calls that simplifies the task of creating URNG objects for some libraries with uniform random number generators, see below.
Destroy urng object. It returns an error code if this is not possible.
If the NULL
is given, this function does nothing.
Warning: This call must be used with care. The urng
object must not be used by any existing generator object!
It is designed to work in conjunction with the wrapper functions
to create URNG objects for generators of a particular library.
Thus an object created by an
unur_urng_prng_new
call can be
simply destroyed by an
unur_urng_free
call.
Set function to fill array X of length dim with random numbers generated by generator urng (if available).
Set function for jumping into a defined state (“sync”).
Set function to seed generator urng (if available).
Set function to switch the antithetic flag of generator urng (if available).
Set function for reseting the uniform random number generator urng (if available).
Set function that allows jumping to start of the next substream of urng (if available).
Set function that allows jumping to start of the current substream of urng (if available).
Set function for destroying urng (if available).
Next: Interface to GSL uniform random number generators, Up: Using uniform random number generators [Contents][Index]
Simple interface for URNGs of type double uniform(void *state)
.
UNU.RAN contains some buildin URNGs of this type:
unur_urng_MRG31k3p
Combined multiple recursive generator by Pierre L’Ecuyer and Renee Touzin.
unur_urng_fish
Linear congruential generator by Fishman and Moore.
unur_urng_mstd
Linear congruential generator "Minimal Standard" by Park and Miller.
Notice, however, that these generators are provided as a fallback for the case that no stateoftheart uniform random number generators (e.g. see Pierre L’Ecuyer’s Rngstream library) are used.
Create an URNG object using
unur_urng_fvoid_new
.
By this call a pointer to the sampling routine and (optional) a
pointer to a reset routine are copied into the URNG object.
Other functions, like seeding the URNG, switching to antithetic
random number, or jumping to next substream, can be added to the
URNG object by the respective calls, e.g. by
unur_urng_set_seed
.
The following routines are supported for URNG objects of this
type:
unur_urng_sample
unur_urng_sample_array
unur_urng_seed
[optional]
unur_urng_reset
[optional]
unur_urng_free
Make a URNG object for a generator that consists of a single function call urand.
If there is no reset function use NULL
for the second argument.
Next: Interface to GSL generators for quasirandom points, Previous: Simple interface for uniform random number generators, Up: Using uniform random number generators [Contents][Index]
Interface to the uniform random number generators from the GNU Scientific Library (GSL). Documentation and source code of this library is available from http://www.gnu.org/software/gsl/.
The interface to the GSL must be compiled into UNU.RAN using the
configure flag withurnggsl
.
Notice that the GSL has to be installed before running
./configure
.
When using this interface unuran_urng_gsl.h must be included in the corresponding C file, i.e., one must add the line
#include <unuran_urng_gsl.h>
Moreover, one must not forget to link the executable against libgsl.
The following routines are supported for URNG objects of type GSL:
/*  */ /* File: example_gsl.c */ /*  */ #ifdef UNURAN_SUPPORTS_GSL /*  */ /* This example makes use of the GSL library for generating */ /* uniform random numbers. */ /* (see http://www.gnu.org/software/gsl/) */ /* To compile this example you must have set */ /* ./configure withurnggsl */ /* (Of course the executable has to be linked against the */ /* GSL library.) */ /*  */ /* Include UNURAN header files. */ #include <unuran.h> #include <unuran_urng_gsl.h> /*  */ int main(void) { int i; /* loop variable */ double x; /* will hold the random number */ /* Declare the three UNURAN objects. */ UNUR_DISTR *distr; /* distribution object */ UNUR_PAR *par; /* parameter object */ UNUR_GEN *gen; /* generator object */ /* Declare objects for uniform random number generators. */ UNUR_URNG *urng; /* uniform generator objects */ /* GNU Scientific Library only: */ /* Make a object for uniform random number generator. */ urng = unur_urng_gsl_new(gsl_rng_mt19937); if (urng == NULL) exit (EXIT_FAILURE); /* Create a generator object using this URNG */ distr = unur_distr_normal( NULL, 0 ); par = unur_tdr_new(distr); unur_set_urng( par, urng ); gen = unur_init(par); if (gen == NULL) exit (EXIT_FAILURE); unur_distr_free(distr); /* Now you can use the generator object `gen' to sample from */ /* the distribution. Eg.: */ for (i=0; i<10; i++) { x = unur_sample_cont(gen); printf("%f\n",x); } /* Destroy objects */ unur_free(gen); unur_urng_free(urng); exit (EXIT_SUCCESS); } /* end of main() */ /*  */ #else #include <stdio.h> #include <stdlib.h> int main(void) { printf("You must enable the GSL to run this example!\n\n"); exit (77); /* exit code for automake check routines */ } #endif /*  */
Make object for URNGs from the GSL (GNU Scientific Library). urngtype is the type of the chosen generator as described in the GSL manual (see Section Random Number Generation). This library is available from http://www.gnu.org/software/gsl/.
Similar to
unur_urng_gsl_new
but it uses a pointer to a
generator object as returned by gsl_rng_alloc(rng_type)
;
see GSL manual for details.
Notice: There is a subtle but important difference between
these two calls. When a generator object is created by a
unur_urng_gsl_new
call, then resetting of the generator works.
When a generator object is created by a
unur_urng_gslptr_new
call, then resetting only works after a
unur_urng_seed(urng,myseed)
call.
Next: Interface to Otmar Lendl’s pseudorandom number generators, Previous: Interface to GSL uniform random number generators, Up: Using uniform random number generators [Contents][Index]
Interface to the generators for quasirandom points (also called low discrepancy point sets) from the GNU Scientific Library (GSL). Documentation and source code of this library is available from http://www.gnu.org/software/gsl/.
The interface to the GSL must be compiled into UNU.RAN using the
configure flag withurnggsl
.
Notice that the GSL has to be installed before running
./configure
.
When using this interface unuran_urng_gsl.h must be included in the corresponding C file, i.e., one must add the line
#include <unuran_urng_gsl.h>
Moreover, one must not forget to link the executable against libgsl.
The following routines are supported for URNG objects of this type:
unur_urng_sync
is used to jump to the first coordinate of
the next point generated by the generator.
Make object for quasirandom point generators for dimension dim from the GSL (GNU Scientific Library). qrngtype is the type of the chosen generator as described in the GSL manual (see section QuasiRandom Sequences). This library is available from http://www.gnu.org/software/gsl/.
Next: Interface to L’Ecuyer’s RNGSTREAM random number generators, Previous: Interface to GSL generators for quasirandom points, Up: Using uniform random number generators [Contents][Index]
URNGs from Otmar Lendl’s prng
library. It provides a very
flexible way to sample form arbitrary URNGs by means of an object
oriented programing paradigma. Similarly to the UNU.RAN library
independent generator objects can be build and used.
This library has been developed by the pLab group at the university of Salzburg (Austria, EU) and implemented by Otmar Lendl. It is available from http://statmath.wu.ac.at/prng/ or from the pLab site at http://random.mat.sbg.ac.at/.
The interface to the PRNG library must be compiled into UNU.RAN using the
configure flag withurngprng
.
Notice that the PRNG library has to be installed before running
./configure
.
When using this interface unuran_urng_prng.h must be included in the corresponding C file, i.e., one must add the line
#include <unuran_urng_prng.h>
Moreover, one must not forget to link the executable against libprng.
The following routines are supported for URNG objects of type PRNG:
unur_urng_sample
unur_urng_sample_array
unur_urng_seed
(availability depends on chosen PRNG generator!)
unur_urng_reset
unur_urng_free
Make object for URNGs from Otmar Lendl’s prng package. prngstr is a string that contains the necessary information to create a uniform random number generator. For the format of this string see the prng user manual.
The prng library provides a very flexible way to sample form arbitrary URNGs by means of an object oriented programing paradigma. Similarly to the UNU.RAN library independent generator objects can be build and used. The library has been developed and implemented by Otmar Lendl as member of the pLab group at the university of Salzburg (Austria, EU).
It is available via anonymous ftp from http://statmath.wu.ac.at/prng/ or from the pLab site at http://random.mat.sbg.ac.at/.
Similar to
unur_urng_prng_new
but it uses a pointer to a
generator object as returned by prng_new(prngstr)
;
see prng manual for details.
Next: Combine point set generator with random shifts, Previous: Interface to Otmar Lendl’s pseudorandom number generators, Up: Using uniform random number generators [Contents][Index]
URNGs from Pierre L’Ecuyer’s RngStream library for multiple independent streams of pseudorandom numbers. This library provides multiple independent streams of pseudorandom numbers which itselves can be splitted into many substreams. It is available from http://www.iro.umontreal.ca/~lecuyer/myftp/streams00/c/. A GNUstyle package is available from http://statmath.wu.ac.at/software/RngStreams/.
The interface to the RngStream library must be compiled into UNU.RAN using the
configure flag withurngrngstream
.
Notice that the RngStream library has to be installed before running
./configure
.
When using this interface unuran_urng_rngstream.h must be included in the corresponding C file, i.e., one must add the line
#include <unuran_urng_rngstream.h>
Moreover, one must not forget to link the executable against the
RngStream library (i.e., when using the GNUstyle package
in UNIX like environments one has to add lrngstreams
when linking an executable).
Notice that the rngstream library uses a package seed,
that means one should seed the uniform random number generator
only once in an application using the routine
RngStream_SetPackageSeed
:
unsigned long seed[] = {111u, 222u, 333u, 444u, 555u, 666u}; RngStream_SetPackageSeed(seed);
The following routines are supported for URNG objects of this type:
unur_urng_sample
unur_urng_sample_array
unur_urng_reset
unur_urng_nextsub
unur_urng_resetsub
unur_urng_anti
unur_urng_free
Make object for URNGs from Pierre L’Ecuyer’s RngStream library. urngstr is an arbitrary string to label a stream. It need not be unique.
Similar to
unur_urng_rngstream_new
but it uses a pointer to a
generator object as returned by RngStream_CreateStream()
.
Previous: Interface to L’Ecuyer’s RNGSTREAM random number generators, Up: Using uniform random number generators [Contents][Index]
Generators of type RANDOMSHIFT combine a point set generator with generators to apply random shifts as proposed in [CPa76] :
Create a URNG object for a point set generator and a URNG object
for a generator to create shift vectors at random.
The meta URNG object can then be created using
unur_urng_randomshift_new
.
Notice that only pointers to the two underlying URNG generator
objects are copied into the newly created meta generator. Thus
manipulating the meta URNG also changes the underlying URNGs
and vice versa.
The following routines are supported for URNG objects of type RANDOMSHIFT:
unur_urng_sample
unur_urng_sample_array
unur_urng_reset
unur_urng_sync
unur_urng_randomshift_nextshift
unur_urng_free
unur_urng_sync
is used to jump to the first coordinate of
the next point generated by the generator.
unur_urng_randomshift_nextshift
allows to replace the shift
vector by another randomly chosen shift vector.
Important:
unur_urng_sync
is only available if it is
if it is implemented for the underlying point set generator.
Important:
unur_urng_reset
is only available if it is
available for both underlying generators.
Make object for URNG with randomly shifted point sets. qrng is a generated that generates point sets of dimension dim. srng is a generated that generates random numbers or vectors.
Notice: Only pointers to the respective objects qrng and srng are copied into the created meta generator. Thus manipulating the meta URNG also changes the underlying URNGs and vice versa.
Get the next (randomly chosen) vector for shifting the points set, and the underlying point generator qrng is reset.
Next: Error handling and Debugging, Previous: Using uniform random number generators, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
Although it is not its primary target, many distributions are already implemented in UNU.RAN. This section presents these available distributions and their parameters.
The syntax to get a distribuion object for distributions
<dname>
is:
<dname>
(double* params, int n_params) ¶params is an array of doubles of size n_params holding the parameters.
E.g. to get an object for the gamma distribution (with shape parameter) use
unur_distr_gamma( params, 1 );
Distributions may have default parameters with need not be given explicitely. E.g. The gamma distribution has three parameters: the shape, scale and location parameter. Only the (first) shape parameter is required. The others can be omitted and are then set by default values.
/* alpha = 5; default: beta = 1, gamma = 0 */ double fpar[] = {5.}; unur_distr_gamma( fpar, 1 ); /* alpha = 5, beta = 3; default: gamma = 0 */ double fpar[] = {5., 3.}; unur_distr_gamma( fpar, 2 ); /* alpha = 5, beta = 3, gamma = 2 double fpar[] = {5., 3., 2.}; unur_distr_gamma( fpar, 3 );
Important: Naturally the computational accuracy
limits the possible parameters. There shouldn’t be problems
when the parameters of a distribution are in a “reasonable” range but
e.g. the normal distribution N(10^15,1) won’t yield the desired results.
(In this case it would be better generating N(0,1) and then
transform the results.)
Of course computational inaccuracy is not specific to UNU.RAN
and should always be kept in mind when working with computers.
Important: The routines of the standard library are included for nonuniform random variate generation and not to provide special functions for statistical computations.
The following keywords are used in the tables:
probability density function, with variable x.
probability mass function, with variable k.
normalization constant for given PDF and PMF, resp. They must be multiplied by constant to get the “real” PDF and PMF.
gives information whether the CDF is implemented in UNU.RAN.
domain PDF and PMF, resp.
list of parameters for distribution, where n_std is the number
of parameters for the standard form of the distribution and
n_total the total number for the (nonstandard form of the)
distribution. list is the list of parameters in the order as
they are stored in the array of parameters. Optional parameter that
can be omitted are enclosed in square brackets […]
.
A detailed list of these parameters gives then the range of valid parameters and defaults for optional parameters that are used when these are omitted.
gives reference for distribution (see Bibliography).
lists available special generators for the distribution.
The first number is the variant that to be set by
unur_cstd_set_variant
and
unur_dstd_set_variant
call, respectively.
If no variant is set the default variant DEF
is used.
In the table the respective abbreviations DEF
and INV
are used for UNUR_STDGEN_DEFAULT
and
UNUR_STDGEN_INVERSION
.
Also the references for these methods are given (see Bibliography).
Notice that these generators might be slower than universal methods.
If DEF
is ommited, the first entry is the default generator.
Next: UNU.RAN Library of continuous multivariate distributions, Up: UNU.RAN Library of standard distributions [Contents][Index]
F
– Fdistributionbeta
– Beta distributioncauchy
– Cauchy distributionchi
– Chi distributionchisquare
– Chisquare distributionexponential
– Exponential distributionextremeI
– Extreme value type I (Gumbeltype) distributionextremeII
– Extreme value type II (Frechettype) distributiongamma
– Gamma distributiongig
– Generalized Inverse Gaussian distributiongig2
– Generalized Inverse Gaussian distributionhyperbolic
– Hyperbolic distributionig
– Inverse Gaussian distributionlaplace
– Laplace distributionlogistic
– Logistic distributionlognormal
– LogNormal distributionlomax
– Lomax distribution (Pareto distribution of second kind)normal
– Normal distributionpareto
– Pareto distribution (of first kind)powerexponential
– Powerexponential (Subbotin) distributionrayleigh
– Rayleigh distributionslash
– Slash distributionstudent
– Student’s t distributiontriangular
– Triangular distributionuniform
– Uniform distributionweibull
– Weibull distribution
Next: beta
– Beta distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
F
– Fdistribution(x^(nu_1/21)) / (1+nu_1/nu_2 x)^((nu_1+nu_2)/2)
(nu_1/nu_2)^(nu_1/2) / B(nu_1/2,nu_2/2)
0 < x < infinity
No.  name  default  
[0]  nu_1  > 0  (scale)  
[1]  nu_2  > 0  (scale) 
Next: cauchy
– Cauchy distribution, Previous: F
– Fdistribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
beta
– Beta distribution(xa)^(p1) * (bx)^(q1)
1 / (B(p,q) * (ba)^(p+q1))
a < x < b
No.  name  default  
[0]  p  > 0  (scale)  
[1]  q  > 0  (scale)  
[2]  a  0  (location, scale)  
[3]  b  > a  1  (location, scale) 
Next: chi
– Chi distribution, Previous: beta
– Beta distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
cauchy
– Cauchy distribution1/(1 + ((xtheta)/lambda)^2)
1/(pi * lambda)
infinity < x < infinity
No.  name  default  
[0]  theta  0  (location)  
[1]  lambda  > 0  1  (scale) 
INV
Inversion method
Next: chisquare
– Chisquare distribution, Previous: cauchy
– Cauchy distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
chi
– Chi distributionx^(nu1) * exp( x^2/2 )
1 / (2^((nu/2)1) * Gamma(nu/2))
0 <= x < infinity
No.  name  default  
[0]  nu  > 0  (shape) 
DEF
Ratio of Uniforms with shift (only for nu >= 1) [MJa87]
Next: exponential
– Exponential distribution, Previous: chi
– Chi distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
chisquare
– Chisquare distributionx^((nu/2)1) * exp( x/2 )
1 / (2^(nu/2) * Gamma(nu/2))
0 <= x < infinity
No.  name  default  
[0]  nu  > 0  (shape (degrees of freedom)) 
Next: extremeI
– Extreme value type I (Gumbeltype) distribution, Previous: chisquare
– Chisquare distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
exponential
– Exponential distributionexp( (xtheta)/sigma)
1/sigma
theta <= x < infinity
No.  name  default  
[0]  sigma  > 0  1  (scale) 
[1]  theta  0  (location) 
INV
Inversion method
Next: extremeII
– Extreme value type II (Frechettype) distribution, Previous: exponential
– Exponential distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
extremeI
– Extreme value type I (Gumbeltype) distributionexp( exp( (xzeta)/theta )  (xzeta)/theta )
1/theta
infinity < x <infinity
No.  name  default  
[0]  zeta  0  (location)  
[1]  theta  > 0  1  (scale) 
INV
Inversion method
Next: gamma
– Gamma distribution, Previous: extremeI
– Extreme value type I (Gumbeltype) distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
extremeII
– Extreme value type II (Frechettype) distributionexp( ((xzeta)/theta)^(k)) * ((xzeta)/theta)^(k1)
k/theta
zeta < x <infinity
No.  name  default  
[0]  k  > 0  (shape)  
[1]  zeta  0  (location)  
[2]  theta  > 0  1  (scale) 
INV
Inversion method
Next: gig
– Generalized Inverse Gaussian distribution, Previous: extremeII
– Extreme value type II (Frechettype) distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
gamma
– Gamma distribution((xgamma)/beta)^(alpha1) * exp( (xgamma)/beta )
1 / (beta * Gamma(alpha))
gamma < x < infinity
No.  name  default  
[0]  alpha  > 0  (shape)  
[1]  beta  > 0  1  (scale) 
[2]  gamma  0  (location) 
Next: gig2
– Generalized Inverse Gaussian distribution, Previous: gamma
– Gamma distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
gig
– Generalized Inverse Gaussian distributionx^(theta1) * exp( 1/2 * omega * (x/eta + eta/x))
not implemented!
not implemented!
0 < x <infinity
No.  name  default  
[0]  theta  (shape)  
[1]  omega  > 0  (scale)  
[2]  eta  > 0  1  (shape) 
DEF
RatioofUniforms method [Dag89]
Next: hyperbolic
– Hyperbolic distribution, Previous: gig
– Generalized Inverse Gaussian distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
gig2
– Generalized Inverse Gaussian distributionx^(theta1) * exp( 1/2 * (chi/x + psi*x))
not implemented!
not implemented!
0 < x <infinity
No.  name  default  
[0]  theta  (shape)  
[1]  psi  > 0  (shape)  
[2]  chi  > 0  (shape) 
Next: ig
– Inverse Gaussian distribution, Previous: gig2
– Generalized Inverse Gaussian distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
hyperbolic
– Hyperbolic distributionexp( alpha * sqrt(delta^2 + (x  mu)^2) + beta*(xmu) )
not implemented!
not implemented!
infinity < x <infinity
No.  name  default  
[0]  alpha  >beta  (shape (tail))  
[1]  beta  (shape (asymmetry))  
[2]  delta  > 0  (scale)  
[3]  mu  (location) 
Next: laplace
– Laplace distribution, Previous: hyperbolic
– Hyperbolic distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
ig
– Inverse Gaussian distributionsqrt( lambda/2*pi*x^3 ) * exp( (lambda*(xmu)^2)/2*mu^2*x )
1
not implemented!
0 < x <infinity
No.  name  default  
[0]  mu  > 0  (mean)  
[1]  lambda  > 0  (shape) 
Next: logistic
– Logistic distribution, Previous: ig
– Inverse Gaussian distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
laplace
– Laplace distributionexp( xtheta / phi )
1/(2 * phi)
infinity < x <infinity
No.  name  default  
[0]  theta  0  (location)  
[1]  phi  > 0  1  (scale) 
INV
Inversion method
Next: lognormal
– LogNormal distribution, Previous: laplace
– Laplace distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
logistic
– Logistic distributionexp((xalpha)/beta) * (1 + exp((xalpha)/beta))^(2)
1/beta
infinity < x <infinity
No.  name  default  
[0]  alpha  0  (location)  
[1]  beta  > 0  1  (scale) 
INV
Inversion method
Next: lomax
– Lomax distribution (Pareto distribution of second kind), Previous: logistic
– Logistic distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
lognormal
– LogNormal distribution1/(xtheta) * exp( (log(xtheta)zeta)^2/(2 sigma^2) )
1/(sigma * sqrt(2 pi))
theta <= x < infinity
No.  name  default  
[0]  zeta  (shape)  
[1]  sigma  > 0  (shape)  
[2]  theta  0  (location) 
Next: normal
– Normal distribution, Previous: lognormal
– LogNormal distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
lomax
– Lomax distribution (Pareto distribution of second kind)(x+C)^((a+1))
a * C^a
0 <= x < infinity
No.  name  default  
[0]  a  > 0  (shape)  
[1]  C  > 0  1  (scale) 
INV
Inversion method
Next: pareto
– Pareto distribution (of first kind), Previous: lomax
– Lomax distribution (Pareto distribution of second kind), Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
normal
– Normal distributionexp( 1/2 * ((xmu)/sigma)^2 )
1 / (sigma * sqrt(2 pi))
infinity < x < infinity
No.  name  default  
[0]  mu  0  (location)  
[1]  sigma  > 0  1  (scale) 
Next: powerexponential
– Powerexponential (Subbotin) distribution, Previous: normal
– Normal distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
pareto
– Pareto distribution (of first kind)x^((a+1))
a * k^a
k < x < infinity
No.  name  default  
[0]  k  > 0  (shape, location)  
[1]  a  > 0  (shape) 
INV
Inversion method
Next: rayleigh
– Rayleigh distribution, Previous: pareto
– Pareto distribution (of first kind), Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
powerexponential
– Powerexponential (Subbotin) distributionexp( x^tau )
1 / (2 * Gamma(1+1/tau))
infinity < x < infinity
No.  name  default  
[0]  tau  > 0  (shape) 
DEF
Transformed density rejection (only for tau >= 1) [DLa86]
Next: slash
– Slash distribution, Previous: powerexponential
– Powerexponential (Subbotin) distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
rayleigh
– Rayleigh distributionx * exp( 1/2 * (x/sigma)^2 )
1 / sigma^2
0 <= x < infinity
No.  name  default  
[0]  sigma  > 0  (scale) 
Next: student
– Student’s t distribution, Previous: rayleigh
– Rayleigh distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
slash
– Slash distribution(1  exp(x^2/2)) / x^2
1 / sqrt(2 pi)
not implemented!
infinity < x < infinity
DEF
Ratio of normal and uniform random variates
Next: triangular
– Triangular distribution, Previous: slash
– Slash distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
student
– Student’s t distribution(1+t^2/nu)^((nu+1)/2)
1 / (sqrt(nu) * B(1/2,nu/2))
not implemented!
infinity < x < infinity
No.  name  default  
[0]  nu  > 0  (shape) 
Next: uniform
– Uniform distribution, Previous: student
– Student’s t distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
triangular
– Triangular distribution2*x / H, for 0 <= x <= H
2*(1x) / (1H), for H <= x <= 1
1
0 <= x <= 1
No.  name  default  
[0]  H  0 <= H <= 1  1/2  (shape) 
INV
Inversion method
Next: weibull
– Weibull distribution, Previous: triangular
– Triangular distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
uniform
– Uniform distribution1 / (ba)
1
a < x < b
No.  name  default  
[0]  a  0  (location)  
[1]  b  > a  1  (location) 
INV
Inversion method
Previous: uniform
– Uniform distribution, Up: UNU.RAN Library of continuous univariate distributions [Contents][Index]
weibull
– Weibull distribution((xzeta)/alpha)^(c1) * exp( ((xzeta)/alpha)^c )
c / alpha
zeta < x < infinity
No.  name  default  
[0]  c  > 0  (shape)  
[1]  alpha  > 0  1  (scale) 
[2]  zeta  0  (location) 
INV
Inversion method
Next: UNU.RAN Library of discrete univariate distributions, Previous: UNU.RAN Library of continuous univariate distributions, Up: UNU.RAN Library of standard distributions [Contents][Index]
copula
– Copula (distribution with uniform marginals)multicauchy
– Multicauchy distributionmultiexponential
– Multiexponential distributionmultinormal
– Multinormal distributionmultistudent
– Multistudent distribution
Next: multicauchy
– Multicauchy distribution, Up: UNU.RAN Library of continuous multivariate distributions [Contents][Index]
copula
– Copula (distribution with uniform marginals)UNUR_DISTR *unur_distr_copula(int dim, const double *rankcorr)
creates a distribution object for a copula with dim components.
rankcorr is an array of size dimxdim and holds the
rank correlation matrix (Spearman’s correlation), where the rows of
the matrix are stored consecutively in this array. The NULL
pointer
can be used instead the identity matrix.
If covar is not a valid rank correlation matrix (i.e., not positive
definite) then no distribution object is created and NULL
is returned.
Next: multiexponential
– Multiexponential distribution, Previous: copula
– Copula (distribution with uniform marginals), Up: UNU.RAN Library of continuous multivariate distributions [Contents][Index]
multicauchy
– Multicauchy distributionf(x) = 1 / ( 1 + (xmu)^t . Sigma^(1) . (xmu) )^((dim+1)/2)
Gamma((dim+1)/2) / ( pi^((dim+1)/2) * sqrt(det(Sigma)) )
infinity^(dim) < x < infinity^(dim)
No.  name  default  
[0]  mu  (0,…,0)  (location)  
[1]  Sigma  Symm, Pos. def.  I  (shape) 
DEF
Cholesky factor
Next: multinormal
– Multinormal distribution, Previous: multicauchy
– Multicauchy distribution, Up: UNU.RAN Library of continuous multivariate distributions [Contents][Index]
multiexponential
– Multiexponential distributionf(x) = Prod_(i=0)^(i=dim1) exp((dimi) (x_(i)x_(i1)  (theta_itheta_(i1)) ) / sigma_i); with x_(1)=0 and theta_(i1)=0
Prod_(i=0)^(i=dim1) 1/sigma_i
0^(dim) <= x < infinity^(dim)
No.  name  default  
[0]  sigma  (1,…,1)  (shape)  
[1]  theta  (0,…,0)  (location) 
Next: multistudent
– Multistudent distribution, Previous: multiexponential
– Multiexponential distribution, Up: UNU.RAN Library of continuous multivariate distributions [Contents][Index]
multinormal
– Multinormal distributionf(x) = exp( 1/2 * (xmu)^t . Sigma^(1) . (xmu) )
1 / ( (2 pi)^(dim/2) * sqrt(det(Sigma)) )
infinity^(dim) < x < infinity^(dim)
No.  name  default  
[0]  mu  (0,…,0)  (location)  
[1]  Sigma  Symm, Pos. def.  I  (shape) 
Previous: multinormal
– Multinormal distribution, Up: UNU.RAN Library of continuous multivariate distributions [Contents][Index]
multistudent
– Multistudent distributionf(x) = 1 / ( 1 + (xmu)^t . Sigma^(1) . (xmu) / m)^((dim+m)/2) )
Gamma((dim+m)/2) / ( Gamma(m/2) (m*pi)^(dim/2) * sqrt(det(Sigma)) )
infinity^(dim) < x < infinity^(dim)
No.  name  default  
[0]  m  m>0  1  (location) 
[1]  mu  (0,…,0)  (location)  
[2]  Sigma  Symm, Pos. def.  I  (shape) 
Next: UNU.RAN Library of random matrices, Previous: UNU.RAN Library of continuous multivariate distributions, Up: UNU.RAN Library of standard distributions [Contents][Index]
At the moment there are no CDFs implemented for discrete distribution.
Thus
unur_distr_discr_upd_pmfsum
does not work properly for truncated
distribution.
binomial
– Binomial distributiongeometric
– Geometric distributionhypergeometric
– Hypergeometric distributionlogarithmic
– Logarithmic distributionnegativebinomial
– Negative Binomial distributionpoisson
– Poisson distribution
Next: geometric
– Geometric distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
binomial
– Binomial distribution(n \choose k) * p^k * (1p)^(nk)
1
0 <= k <= n
No.  name  default  
[0]  n  >= 1  (no. of elements)  
[1]  p  0 < p < 1  (shape) 
DEF
Ratio of Uniforms/Inversion [STa89]
Next: hypergeometric
– Hypergeometric distribution, Previous: binomial
– Binomial distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
geometric
– Geometric distributionp * (1p)^k
1
0 <= k < infinity
No.  name  default  
[0]  p  0 < p < 1  (shape) 
INV
Inversion method
Next: logarithmic
– Logarithmic distribution, Previous: geometric
– Geometric distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
hypergeometric
– Hypergeometric distribution(M \choose k) * (NM \choose nk) / (N \choose n)
1
max(0,nN+M) <= k <= min(n,M)
No.  name  default  
[0]  N  >= 1  (no. of elements)  
[1]  M  1 <= M <= N  (shape)  
[2]  n  1 <= n <= N  (shape) 
DEF
Ratio of Uniforms/Inversion [STa89]
Next: negativebinomial
– Negative Binomial distribution, Previous: hypergeometric
– Hypergeometric distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
logarithmic
– Logarithmic distributiontheta^k / k
 log( 1.theta);
1 <= k < infinity
No.  name  default  
[0]  theta  0 < theta < 1  (shape) 
DEF
Inversion/Transformation [KAa81]
Next: poisson
– Poisson distribution, Previous: logarithmic
– Logarithmic distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
negativebinomial
– Negative Binomial distribution(k+r1 \choose r1) * p^r * (1p)^k
1
0 <= k < infinity
No.  name  default  
[0]  p  0 < p < 1  (shape)  
[1]  r  > 0  (shape) 
Previous: negativebinomial
– Negative Binomial distribution, Up: UNU.RAN Library of discrete univariate distributions [Contents][Index]
poisson
– Poisson distributiontheta^k / k!
exp(theta)
0 <= k < infinity
No.  name  default  
[0]  theta  > 0  (shape) 
Previous: UNU.RAN Library of discrete univariate distributions, Up: UNU.RAN Library of standard distributions [Contents][Index]
correlation
– Random correlation matrixUNUR_DISTR *unur_distr_correlation( int n )
creates a distribution object for a random correlation matrix of
n rows and columns. It can be used with method MCORR
(see Random Correlation Matrix) to
generate random correlation matrices of the given size.
Next: Testing, Previous: UNU.RAN Library of standard distributions, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
UNU.RAN routines report an error whenever they cannot perform the requested task. Additionally it is possible to get information about the generated distribution of generator objects for debugging purposes. However, the latter must be enabled when compiling and installing the library. (It is disabled by default.) This chapter describes all necessary details:
Next: Debugging, Up: Error handling and Debugging [Contents][Index]
UNU.RAN uses a logfile for writing all error messages, warnings,
and debugging information onto an output stream. This stream can
be set at runtime by the
unur_set_stream
call.
If no such stream is given by the user a default stream is used
by the library: all messages are written into the file
unuran.log in the current working directory. The name of
this logfile is defined by the macro UNUR_LOG_FILE
in
unuran_config.h.
(If UNU.RAN fails to open this file for writing, stderr
is used instead.)
To destinguish between messages for different objects each of these has its own identifier which is composed by the name of the distribution obejct and generator type, resp., followed by a dot and three digits. (If there are more than 999 generators then the identifiers are not unique.)
Remark: Writting debugging information must be switched
on at compile time using the configure flag
enablelogging
, see Debugging.
This function sets a new file handler for the output stream, new_stream, for the UNU.RAN library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined file handler is stored in a static variable, so there can be only one output stream handler per program. This function should be not be used in multithreaded programs except to set up a programwide error handler from a master thread.
The NULL
pointer is not allowed.
(If you want to disable logging of debugging information use
unur_set_default_debug(UNUR_DEBUG_OFF) instead.
If you want to disable error messages at all use
unur_set_error_handler_off
.
)
Get the file handle for the current output stream. It can be used to allow applications to write additional information into the logfile.
Next: Error reporting, Previous: Output streams, Up: Error handling and Debugging [Contents][Index]
The UNU.RAN library has several debugging levels which
can be switched on/off by debugging flags. This debugging
feature must be enabled when building the library using the
enablelogging
configure flag.
The debugging levels range from print a short description of the created generator object to a detailed description of hat functions and tracing the sampling routines. The output is printed onto the debugging output stream (see Output streams).
The debugging flags can be set or changed by the respective calls
unur_set_debug
and
unur_chg_debug
independently for each
generator.
By default flag UNUR_DEBUG_INIT
(see
below) is used. This default flags is set by the macro
UNUR_DEBUGFLAG_DEFAULT
in unuran_config.h and can
be changed at runtime by a
unur_set_default_debug
call.
Off course these debugging flags depend on the chosen method. Since most of these are merely for debugging the library itself, a description of the flags are given in the corresponding source files of the method. Nevertheless, the following flags can be used with all methods.
Common debug flags:
UNUR_DEBUG_OFF
¶switch off all debuging information
UNUR_DEBUG_ALL
¶all avaivable information
UNUR_DEBUG_INIT
¶parameters of generator object after initialization
UNUR_DEBUG_SETUP
¶data created at setup
UNUR_DEBUG_ADAPT
¶data created during adaptive steps
UNUR_DEBUG_SAMPLE
¶trace sampling
Notice that these are flags which could be combined using the

operator.
Almost all routines check a given pointer before they read from or write
to the given address. This does not hold for timecritical routines
like all sampling routines. Thus you are responsible for checking a
pointer that is returned from a
unur_init
call.
However, it is possible to turn on checking for invalid NULL
pointers
even in such timecritical routines by building the library
using the enablecheckstruct
configure flag.
Another debugging tool used in the library are magic cookies that
validate a given pointer. It produces an error whenever a given
pointer points to an object that is invalid in the context.
The usage of magic cookies is also switched on by the
enablecheckstruct
configure flag.
Set debugging flags for generator.
Change debugging flags for generator.
Change default debugging flag.
Next: Error codes, Previous: Debugging, Up: Error handling and Debugging [Contents][Index]
UNU.RAN routines report an error whenever they cannot perform the requested task. For example, applying transformed density rejection to a distribution that violates the Tconcavity condition, or trying to set a parameter that is out of range, result in an error message. It might also happen that the setup fails for transformed density rejection for a Tconcave distribution with some extreme density function simply because of roundoff errors that makes the generation of a hat function numerically impossible. Situations like this may happen when using black box algorithms and you should check the return values of all routines.
All ..._set_...
, and ..._chg_...
calls
return UNUR_SUCCESS
if they could be executed
successfully. Otherwise, some error codes are returned if it was
not possible to set or change the desired parameters,
e.g. because the given values are out of range, or simply
because the set call does not work for the chosen method.
All routines that return a pointer to the requested object will
return a NULL
pointer in case of error.
(Thus you should always check the pointer to avoid possible
segmentation faults. Sampling routines usually do not check the
given pointer to the generator object.)
The library distinguishes between two major classes of error:
The library was not able to construct the requested object.
Some problems encounters while constructing a generator object. The routine has tried to solve the problem but the resulting object might not be what you want. For example, chosing a special variant of a method does not work and the initialization routine might switch to another variant. Then the generator produces random variates of the requested distribution but correlation induction is not possible. However, it also might happen that changing the domain of a distribution has failed. Then the generator produced random variates with too large/too small range, i.e. their distribution is not correct.
It is obvious from the example that this distinction between errors and warning is rather crude and sometimes arbitrary.
UNU.RAN routines use the global variable unur_errno to
report errors, completely analogously to errno in the ANSI
C standard library.
(However this approach is not threadsafe. There can
be only one instance of a global variable per program. Different
threads of execution may overwrite unur_errno
simultaneously).
Thus when an error occurs the caller of the routine can examine the
error code in unur_errno to get more details about the
reason why a routine failed. You get a short
description of the error by a
unur_get_strerror
call.
All the error code numbers have prefix UNUR_ERR_
and expand
to nonzero constant unsigned integer values.
Error codes are divided into six main groups,
see Error codes.
Alternatively, the variable unur_errno can also read by a
unur_get_errno
call and can be reset by the
unur_reset_errno
call (this is in particular required for the Windows version of the
library).
Additionally, there exists a error handler (see Error handlers) that is invoked in case of an error.
In addition to reporting errors by setting error codes in unur_errno, the library also has an error handler function. This function is called by other library functions when they report an error, just before they return to the caller (see Error handlers). The default behavior of the error handler is to print a short message:
AROU.004: [error] arou.c:1500  (generator) condition for method violated: AROU.004: ..> PDF not unimodal
The purpose of the error handler is to provide a function where a breakpoint can be set that will catch library errors when running under the debugger. It is not intended for use in production programs, which should handle any errors using the return codes.
Global variable for reporting diagnostics of error.
Get current value of global variable unur_errno.
Reset global variable unur_errno to UNUR_SUCCESS
(i.e., no errors occured).
Get a short description for error code value.
Next: Error handlers, Previous: Error reporting, Up: Error handling and Debugging [Contents][Index]
UNUR_SUCCESS (0x0u)
¶success (no error)
UNUR_ERR_DISTR_SET
¶set failed (invalid parameter).
UNUR_ERR_DISTR_GET
¶get failed (parameter not set).
UNUR_ERR_DISTR_NPARAMS
¶invalid number of parameters.
UNUR_ERR_DISTR_DOMAIN
¶parameter(s) out of domain.
UNUR_ERR_DISTR_GEN
¶invalid variant for special generator.
UNUR_ERR_DISTR_REQUIRED
¶incomplete distribution object, entry missing.
UNUR_ERR_DISTR_UNKNOWN
¶unknown distribution, cannot handle.
UNUR_ERR_DISTR_INVALID
¶invalid distribution object.
UNUR_ERR_DISTR_DATA
¶data are missing.
UNUR_ERR_DISTR_PROP
¶desired property does not exist
UNUR_ERR_GEN
¶error with generator object.
UNUR_ERR_GEN_DATA
¶(possibly) invalid data.
UNUR_ERR_GEN_CONDITION
¶condition for method violated.
UNUR_ERR_GEN_INVALID
¶invalid generator object.
UNUR_ERR_GEN_SAMPLING
¶sampling error.
UNUR_ERR_NO_REINIT
¶reinit routine not implemented.
UNUR_ERR_NO_QUANTILE
¶quantile routine not implemented.
UNUR_ERR_DOMAIN
¶argument out of domain.
UNUR_ERR_ROUNDOFF
¶(serious) roundoff error.
UNUR_ERR_MALLOC
¶virtual memory exhausted.
UNUR_ERR_NULL
¶invalid NULL
pointer.
UNUR_ERR_COOKIE
¶invalid cookie.
UNUR_ERR_GENERIC
¶generic error.
UNUR_ERR_SILENT
¶silent error (no error message).
UNUR_ERR_INF
¶infinity occured.
UNUR_ERR_NAN
¶NaN occured.
UNUR_ERR_COMPILE
¶Requested routine requires different compilation switches. Recompilation of library necessary.
UNUR_ERR_SHOULD_NOT_HAPPEN
¶Internal error, that should not happen. Please report this bug!
Previous: Error codes, Up: Error handling and Debugging [Contents][Index]
The default behavior of the UNU.RAN error handler is to print a short message onto the output stream, usually a logfile (see Output streams), e.g.,
AROU.004: [error] arou.c:1500  (generator) condition for method violated: AROU.004: ..> PDF not unimodal
This error handler can be switched off using the
unur_set_error_handler_off
call, or replace it by a new one.
Thus it allows to set a breakpoint that will catch library errors when
running under the debugger. It also can be used to redirect
error messages when UNU.RAN is included in general purpose
libraries or in interactive programming environments.
This is the type of UNU.RAN error handler functions. An error
handler will be passed six arguments which specify
the identifier of the object where the error occured (a string),
the name of the source file in which it occurred (also a string),
the line number in that file (an integer),
the type of error (a string: "error"
or "warning"
),
the error number (an integert), and
the reason for the error (a string).
The source file and line number are set at compile time
using the __FILE__
and __LINE__
directives in the
preprocessor.
The error number can be translated into a short description
using a
unur_get_strerror
call.
An error handler function returns type void
.
Error handler functions should be defined like this,
void my_handler( const char *objid, const char *file, int line, const char *errortype, int unur_errno, const char *reason )
To request the use of your own error handler you need the call
unur_set_error_handler
.
This function sets a new error handler, new_handler, for the UNU.RAN library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored in a static variable, so there can be only one error handler per program. This function should be not be used in multithreaded programs except to set up a programwide error handler from a master thread.
To use the default behavior set the error handler to NULL
.
This function turns off the error handler by defining an error handler which does nothing (except of setting unur_errno. The previous handler is returned (so that you can restore it later).
Next: Miscelleanous, Previous: Error handling and Debugging, Up: UNU.RAN – Universal NonUniform RANdom number generators [Contents][Index]
The following routines can be used to test the performance of the implemented generators and can be used to verify the implementions. They are declared in unuran_tests.h which has to be included.
Run a battery of tests.
The following tests are available (use 
to combine these
tests):
UNUR_TEST_ALL
run all possible tests.
UNUR_TEST_TIME
estimate generation times.
UNUR_TEST_N_URNG
count number of uniform random numbers
UNUR_TEST_N_PDF
count number of PDF calls
UNUR_TEST_CHI2
run chi^2 test for goodness of fit
UNUR_TEST_SAMPLE
print a small sample.
All these tests can be started individually (see below).
Print a small sample with n_rows rows and n_cols columns. out is the output stream to which all results are written.
Timing. parameters is an parameter object for which setup time and marginal generation times have to be measured. The results are written into time_setup and time_sample, respectively. log10_samplesize is the common logarithm of the sample size that is used for timing.
If verbosity is TRUE
then a small table is printed to
output stream out with setup time, marginal generation time and
average generation times for generating 10, 100, … random
variates. All times are given in micro seconds and relative to
the generation times for the underlying uniform random number
(using the UNIF interface) and an exponential distributed
random variate using the inversion method.
The created generator object is returned.
If a generator object could not be created successfully, then NULL
is returned.
If verbosity is TRUE
the result is written to the output
stream out.
Notice: All timing results are subject to heavy changes. Reruning timings usually results in different results. Minor changes in the source code can cause changes in such timings up to 25 percent.
Timing. parameters is an parameter object for which setup time and marginal generation times have to be measured. The results are written into time_setup and time_marginal, respectively. log10_samplesize is the common logarithm of the sample size that is used for timing.
Alternatively, one could provide the "parameter object" using
strings distrstr and methodstr as used for
unur_makegen_ssu
.
The results are more accurate than those of function
unur_test_timing
as the timings are computed using linear
regression with several timings for sample size 1 and
10^log10_samplesize.
For each sample size total generation time (including setup) is
measured 10 times. Since the these timings can be influenced by
external effects (like disc sync or handling of interupts) the 2
fastest and the 3 slowest timings are discarded.
Intercept and slope for simple linear regression are than stored
and
R^{2}
is returned.
In case of an error 100.
is returned.
Notice: All timing results are subject to heavy changes. Reruning timings usually results in different results. Minor changes in the source code can cause changes in such timings up to 25 percent.
Marginal generation times for the underlying uniform random number
(using the UNIF interface) and an exponential distributed
random variate using the inversion method. These times are used in
unur_test_timing
to compute the relative timings results.
Timing. parameters is an parameter object for which average times a sample of size samplesize (including setup) are estimated. Thus sampling is repeated and the median of these timings is returned (in micro seconds). The number of iterations is computed automatically such that the total amount of time necessary for the test ist approximately avg_duration (given in seconds). However, for very slow generator with expensive setup time the time necessary for this test may be (much) larger.
If an error occurs then 1
is returned.
Notice: All timing results are subject to heavy changes. Reruning timings usually results in different results. Minor changes in the source code can cause changes in such timings up to 25 percent.
Count used uniform random numbers. It returns the total number of
uniform random numbers required for a sample of nonuniform random
variates of size samplesize. In case of an error 1
is returned.
If verbosity is TRUE
the result is written to the output
stream out.
Notice: This test uses global variables to store counters. Thus it is not thread save.
Count evaluations of PDF and similar functions. It returns the
total number of evaluations of all such functions required for a
sample of nonuniform random variates of size samplesize.
If verbosity is TRUE
then a more detailed report is printed
to the output stream out.
In case of an error 1
is returned.
This test is run on a copy of the given generator object.
Notice: The printed numbers of evaluation should be interpreted with care. For example, methods either use the PDF or the logPDF; if only the logPDF is given, but a method needs the PDF then both the logPDF and the PDF (a wrapper around the logPDF) are called and thus one call to the PDF is counted twice.
Notice: This test uses global variables to store function pointers and counters. Thus it is not thread save.
Same as
unur_test_count_pdf
except that it is run on a parameter
object. Thus it also prints the number of function evaluations for
the setup. The temporary created generator object is destroyed
before the results are returned.
Run a Chi^2 test with the generator. The resulting pvalue is returned.
It works with discrete und continuous univariate distributions. For the latter the CDF of the distribution is required.
intervals is the number of intervals that is used for
continuous univariate distributions. samplesize is the size
of the sample that is used for testing. If it is set to 0
then a sample of size intervals^2 is used (bounded to some
upper bound).
classmin is the minimum number of expected entries per class. If a class has to few entries then some classes are joined.
verbosity controls the output of the routine. If it is set
to 1
then the result is written to the output stream
out. If it is set to 2
additionally the list of
expected and observed data is printed.
If it is set to 3
then all generated numbers are printed.
There is no output when it is set to 0
.
Notice: For multivariate distributions the generated points
are transformed by the inverse of the Cholesky factor of the
covariance matrix and the mean vectors (if given for the underlying
distribution). The marginal distributions of the transformed
vectors are then tested against the marginal distribution given by
a
unur_distr_cvec_set_marginals
or
unur_distr_cvec_set_marginal_array
call.
(Notice that these marginal distributions are never set by default
for any of the distributions provided by UNU.RAN.)
Then the Bonferroni corrected pvalue of all these tests is returned.
However, the test may not be performed correctly if the domain of the
underlying distribution is truncated by a
unur_distr_cvec_set_domain_rect
call and the components of the
distribution are correlated (i.e.
unur_distr_cvec_set_covar
is
called with the nonNULL argument). Then it almost surely will fail.
Computes the first n_moments central moments for a sample of
size samplesize. The result is stored into the array
moments.
n_moments must be an integer between 1
and 4
.
For multivariate distributions the moments are stored consecutively
for each dimension and the provided momentsarray must have
a length of at least (n_moments+1) * dim, where dim
is the dimension of the multivariate distribution.
The m’th moment for the d’th dimension (0<=d<dim)
is thus stored in the array element
moments[