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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 C++ 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 = "../../cpp_src/asa053/asa053.html">
ASA053</a>,
a C++ 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 = "../../cpp_src/pdflib/pdflib.html">
PDFLIB</a>,
a C++ 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 = "../../cpp_src/ranlib/ranlib.html">
RANLIB</a>,
a C++ 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 = "../../cpp_src/rnglib/rnglib.html">
RNGLIB</a>,
a C++ 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 = "wishart.cpp">wishart.cpp</a>, the source code.
</li>
<li>
<a href = "wishart.hpp">wishart.hpp</a>, the include file.
</li>
<li>
<a href = "wishart.sh">wishart.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "wishart_prb.cpp">wishart_prb.cpp</a>
a sample calling program.
</li>
<li>
<a href = "wishart_prb.sh">wishart_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "wishart_prb_output.txt">wishart_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BARTLETT_SAMPLE</b> samples the Bartlett distribution.
</li>
<li>
<b>BARTLETT_UNIT_SAMPLE</b> samples the unit Bartlett distribution.
</li>
<li>
<b>JACOBI_EIGENVALUE</b> carries out the Jacobi eigenvalue iteration.
</li>
<li>
<b>R8MAT_ADD</b> adds one R8MAT to another.
</li>
<li>
<b>R8MAT_CHOLESKY_FACTOR_UPPER</b> computes the upper Cholesky factor of a symmetric R8MAT.
</li>
<li>
<b>R8MAT_COPY_NEW</b> copies one R8MAT to a "new" R8MAT.
</li>
<li>
<b>R8MAT_DIAG_GET_VECTOR</b> gets the value of the diagonal of an R8MAT.
</li>
<li>
<b>R8MAT_DIAGONAL_NEW</b> returns a diagonal matrix.
</li>
<li>
<b>R8MAT_DIVIDE</b> divides an R8MAT by a scalar.
</li>
<li>
<b>R8MAT_IDENTITY</b> sets an R8MAT to the identity matrix.
</li>
<li>
<b>R8MAT_IDENTITY_NEW</b> returns an identity matrix.
</li>
<li>
<b>R8MAT_MM_NEW</b> multiplies two matrices.
</li>
<li>
<b>R8MAT_MTM_NEW</b> computes C = A' * B.
</li>
<li>
<b>R8MAT_NORM_FRO_AFFINE</b> returns the Frobenius norm of an R8MAT difference.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8MAT_ZERO_NEW</b> returns a new zeroed R8MAT.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>WISHART_SAMPLE</b> samples the Wishart distribution.
</li>
<li>
<b>WISHART_UNIT_SAMPLE</b> samples the unit Wishart distribution.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 02 August 2013.
</i>
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