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<html>
<head>
<title>
WISHART - Sample the Wishart Distribution for Random Covariance Matrices
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
WISHART <br> Sample the Wishart Distribution for Random Covariance Matrices
</h1>
<hr>
<p>
<b>WISHART</b>
is a MATLAB library which
produces sample matrices from the Wishart or Bartlett distributions,
useful for sampling random covariance matrices.
</p>
<p>
The Wishart distribution is a probability distribution for random
nonnegative-definite NxN matrices that can be used to select random
covariance matrices.
</p>
<p>
The objects of the distribution are NxN matrices which are the sum of
DF rank-one matrices X*X' constructed from N-vectors X, where the vectors
X have zero mean and covariance SIGMA. This implies that the expected
value of a Wishart matrix is DF * SIGMA.
</p>
<p>
A simplified version of the Wishart distribution assumes that SIGMA is
the identity matrix. We will call this the "unit Wishart distribution".
</p>
<p>
Because any Wishart matrix W is symmetric nonnegative definite,
there is an upper triangular factor T so that W = T' * T.
There is a corresponding Bartlett distribution of the matrices T,
so that one can alternatively sample the Bartlett distribution by
sampling the Bartlett distribution for T, and then forming W.
</p>
<p>
In order to generate the necessary random values, the library relies
on the PDFLIB and RNGLIB libraries.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>WISHART</b> is available in
<a href = "../../c_src/wishart/wishart.html">a C version</a> and
<a href = "../../cpp_src/wishart/wishart.html">a C++ version</a> and
<a href = "../../f77_src/wishart/wishart.html">a FORTRAN77 version</a> and
<a href = "../../f_src/wishart/wishart.html">a FORTRAN90 version</a> and
<a href = "../../m_src/wishart/wishart.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/asa053/asa053.html">
ASA053</a>,
a MATLAB library which
produces sample matrices from the Wishart distribution,
by William Smith and Ronald Hocking.
This is a version of Applied Statistics Algorithm 53.
</p>
<p>
<a href = "../../m_src/pdflib/pdflib.html">
PDFLIB</a>,
a MATLAB library which
evaluates Probability Density Functions (PDF's)
and produces random samples from them,
including beta, binomial, chi, exponential, gamma, inverse chi,
inverse gamma, multinomial, normal, scaled inverse chi, and uniform.
</p>
<p>
<a href = "../../m_src/ranlib/ranlib.html">
RANLIB</a>,
a MATLAB library which
produces random samples from Probability Density Functions (PDF's),
including Beta, Chi-square Exponential, F, Gamma, Multivariate normal,
Noncentral chi-square, Noncentral F, Univariate normal, random permutations,
Real uniform, Binomial, Negative Binomial, Multinomial, Poisson
and Integer uniform,
by Barry Brown and James Lovato.
</p>
<p>
<a href = "../../m_src/rnglib/rnglib.html">
RNGLIB</a>,
a MATLAB library which
implements a random number generator (RNG) with splitting facilities,
allowing multiple independent streams to be computed,
by L'Ecuyer and Cote.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ul>
<li>
Patrick Odell, Alan Feiveson,<br>
A numerical procedure to generate a sample covariance matrix,<br>
Journal of the American Statistical Association,<br>
Volume 61, Number 313, March 1966, pages 199-203.
</li>
<li>
Stanley Sawyer,<br>
Wishart Distributions and Inverse-Wishart Sampling,<br>
Washington University,<br>
30 April 2007, 12 pages.
</li>
</ul>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "bartlett_sample.m">bartlett_sample.m</a>,
returns a sample matrix from the Bartlett distribution.
</li>
<li>
<a href = "bartlett_unit_sample.m">bartlett_unit_sample.m</a>,
returns a sample matrix from the unit Bartlett distribution.
</li>
<li>
<a href = "r8mat_print.m">r8mat_print.m</a>,
prints an R8MAT.
</li>
<li>
<a href = "r8mat_print_some.m">r8mat_print_some.m</a>,
prints some of an R8MAT.
</li>
<li>
<a href = "r8ut_inverse.m">r8ut_inverse.m</a>,
returns the inverse of an upper triangular matrix.
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>,
prints an R8VEC.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp.
</li>
<li>
<a href = "wishart_sample.m">wishart_sample.m</a>,
returns a sample matrix from the Wishart distribution.
</li>
<li>
<a href = "wishart_sample_inverse.m">wishart_sample_inverse.m</a>,
returns the inverse of a sample matrix from the Wishart distribution.
</li>
<li>
<a href = "wishart_unit_sample.m">wishart_unit_sample.m</a>,
returns a sample matrix from the unit Wishart distribution.
</li>
<li>
<a href = "wishart_unit_sample_inverse.m">wishart_unit_sample_inverse.m</a>,
returns the inverse of a sample matrix from the unit Wishart distribution.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "wishart_test.m">wishart_test.m</a>,
calls all the tests.
</li>
<li>
<a href = "wishart_test_output.txt">wishart_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "wishart_test01.m">wishart_test01.m</a>
demonstrates the use of wishart_unit_sample();
</li>
<li>
<a href = "wishart_test02.m">wishart_test02.m</a>
demonstrates the use of bartlett_unit_sample();
</li>
<li>
<a href = "wishart_test03.m">wishart_test03.m</a>
checks a relationship between the unit Wishart and Bartlett distributions.
</li>
<li>
<a href = "wishart_test04.m">wishart_test04.m</a>
demonstrates the use of wishart_sample();
</li>
<li>
<a href = "wishart_test05.m">wishart_test05.m</a>
demonstrates the use of bartlett_sample();
</li>
<li>
<a href = "wishart_test06.m">wishart_test06.m</a>
checks a relationship between the Wishart and Bartlett distributions.
</li>
<li>
<a href = "wishart_test07.m">wishart_test07.m</a>
checks that the expected value of a Wishart matrix is the product
of the degrees of freedom times the distribution covariance matrix.
</li>
<li>
<a href = "wishart_test08.m">wishart_test08.m</a>
verifies that the inverse of a unit Wishart sample matrix has been
correctly computed.
</li>
<li>
<a href = "wishart_test09.m">wishart_test09.m</a>
verifies that the inverse of a Wishart sample matrix has been
correctly computed.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 10 October 2013.
</i>
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