// <copyright file="Wishart.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2013 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using IStation.Numerics.LinearAlgebra;
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using IStation.Numerics.LinearAlgebra.Double;
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using IStation.Numerics.LinearAlgebra.Factorization;
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using IStation.Numerics.Random;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Multivariate Wishart distribution. This distribution is
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/// parameterized by the degrees of freedom nu and the scale matrix S. The Wishart distribution
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/// is the conjugate prior for the precision (inverse covariance) matrix of the multivariate
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/// normal distribution.
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/// <a href="http://en.wikipedia.org/wiki/Wishart_distribution">Wikipedia - Wishart distribution</a>.
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/// </summary>
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public class Wishart : IDistribution
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{
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System.Random _random;
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/// <summary>
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/// The degrees of freedom for the Wishart distribution.
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/// </summary>
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readonly double _degreesOfFreedom;
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/// <summary>
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/// The scale matrix for the Wishart distribution.
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/// </summary>
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readonly Matrix<double> _scale;
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/// <summary>
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/// Caches the Cholesky factorization of the scale matrix.
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/// </summary>
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readonly Cholesky<double> _chol;
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/// <summary>
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/// Initializes a new instance of the <see cref="Wishart"/> class.
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/// </summary>
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/// <param name="degreesOfFreedom">The degrees of freedom (n) for the Wishart distribution.</param>
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/// <param name="scale">The scale matrix (V) for the Wishart distribution.</param>
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public Wishart(double degreesOfFreedom, Matrix<double> scale)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = SystemRandomSource.Default;
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_degreesOfFreedom = degreesOfFreedom;
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_scale = scale;
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_chol = _scale.Cholesky();
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="Wishart"/> class.
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/// </summary>
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/// <param name="degreesOfFreedom">The degrees of freedom (n) for the Wishart distribution.</param>
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/// <param name="scale">The scale matrix (V) for the Wishart distribution.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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public Wishart(double degreesOfFreedom, Matrix<double> scale, System.Random randomSource)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = randomSource ?? SystemRandomSource.Default;
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_degreesOfFreedom = degreesOfFreedom;
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_scale = scale;
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_chol = _scale.Cholesky();
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}
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/// <summary>
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/// Tests whether the provided values are valid parameters for this distribution.
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/// </summary>
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/// <param name="degreesOfFreedom">The degrees of freedom (n) for the Wishart distribution.</param>
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/// <param name="scale">The scale matrix (V) for the Wishart distribution.</param>
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public static bool IsValidParameterSet(double degreesOfFreedom, Matrix<double> scale)
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{
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if (scale.RowCount != scale.ColumnCount)
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{
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return false;
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}
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for (var i = 0; i < scale.RowCount; i++)
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{
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if (scale.At(i, i) <= 0.0)
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{
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return false;
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}
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}
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if (degreesOfFreedom <= 0.0 || double.IsNaN(degreesOfFreedom))
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{
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return false;
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}
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return true;
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}
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/// <summary>
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/// Gets or sets the degrees of freedom (n) for the Wishart distribution.
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/// </summary>
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public double DegreesOfFreedom => _degreesOfFreedom;
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/// <summary>
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/// Gets or sets the scale matrix (V) for the Wishart distribution.
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/// </summary>
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public Matrix<double> Scale => _scale;
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/// <summary>
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/// A string representation of the distribution.
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/// </summary>
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/// <returns>a string representation of the distribution.</returns>
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public override string ToString()
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{
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return $"Wishart(DegreesOfFreedom = {_degreesOfFreedom}, Rows = {_scale.RowCount}, Columns = {_scale.ColumnCount})";
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}
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/// <summary>
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/// Gets or sets the random number generator which is used to draw random samples.
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/// </summary>
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public System.Random RandomSource
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{
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get => _random;
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set => _random = value ?? SystemRandomSource.Default;
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}
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/// <summary>
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/// Gets the mean of the distribution.
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/// </summary>
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/// <value>The mean of the distribution.</value>
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public Matrix<double> Mean => _degreesOfFreedom*_scale;
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/// <summary>
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/// Gets the mode of the distribution.
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/// </summary>
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/// <value>The mode of the distribution.</value>
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public Matrix<double> Mode => (_degreesOfFreedom - _scale.RowCount - 1.0)*_scale;
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/// <summary>
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/// Gets the variance of the distribution.
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/// </summary>
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/// <value>The variance of the distribution.</value>
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public Matrix<double> Variance
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{
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get
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{
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return Matrix<double>.Build.Dense(_scale.RowCount, _scale.ColumnCount,
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(i, j) => _degreesOfFreedom*((_scale.At(i, j)*_scale.At(i, j)) + (_scale.At(i, i)*_scale.At(j, j))));
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}
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}
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/// <summary>
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/// Evaluates the probability density function for the Wishart distribution.
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/// </summary>
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/// <param name="x">The matrix at which to evaluate the density at.</param>
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/// <exception cref="ArgumentOutOfRangeException">If the argument does not have the same dimensions as the scale matrix.</exception>
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/// <returns>the density at <paramref name="x"/>.</returns>
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public double Density(Matrix<double> x)
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{
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var p = _scale.RowCount;
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if (x.RowCount != p || x.ColumnCount != p)
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{
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throw Matrix.DimensionsDontMatch<ArgumentOutOfRangeException>(x, _scale, "x");
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}
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var dX = x.Determinant();
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var siX = _chol.Solve(x);
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// Compute the multivariate Gamma function.
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var gp = Math.Pow(Constants.Pi, p*(p - 1.0)/4.0);
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for (var j = 1; j <= p; j++)
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{
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gp *= SpecialFunctions.Gamma((_degreesOfFreedom + 1.0 - j)/2.0);
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}
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return Math.Pow(dX, (_degreesOfFreedom - p - 1.0)/2.0)
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*Math.Exp(-0.5*siX.Trace())
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/Math.Pow(2.0, _degreesOfFreedom*p/2.0)
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/Math.Pow(_chol.Determinant, _degreesOfFreedom/2.0)
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/gp;
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}
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/// <summary>
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/// Samples a Wishart distributed random variable using the method
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/// Algorithm AS 53: Wishart Variate Generator
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/// W. B. Smith and R. R. Hocking
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/// Applied Statistics, Vol. 21, No. 3 (1972), pp. 341-345
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/// </summary>
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/// <returns>A random number from this distribution.</returns>
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public Matrix<double> Sample()
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{
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return DoSample(RandomSource, _degreesOfFreedom, _scale, _chol);
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}
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/// <summary>
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/// Samples a Wishart distributed random variable using the method
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/// Algorithm AS 53: Wishart Variate Generator
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/// W. B. Smith and R. R. Hocking
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/// Applied Statistics, Vol. 21, No. 3 (1972), pp. 341-345
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="degreesOfFreedom">The degrees of freedom (n) for the Wishart distribution.</param>
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/// <param name="scale">The scale matrix (V) for the Wishart distribution.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static Matrix<double> Sample(System.Random rnd, double degreesOfFreedom, Matrix<double> scale)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return DoSample(rnd, degreesOfFreedom, scale, scale.Cholesky());
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}
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/// <summary>
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/// Samples the distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="degreesOfFreedom">The degrees of freedom (n) for the Wishart distribution.</param>
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/// <param name="scale">The scale matrix (V) for the Wishart distribution.</param>
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/// <param name="chol">The cholesky decomposition to use.</param>
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/// <returns>a random number from the distribution.</returns>
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static Matrix<double> DoSample(System.Random rnd, double degreesOfFreedom, Matrix<double> scale, Cholesky<double> chol)
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{
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var count = scale.RowCount;
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// First generate a lower triangular matrix with Sqrt(Chi-Squares) on the diagonal
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// and normal distributed variables in the lower triangle.
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var a = new DenseMatrix(count, count);
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for (var d = 0; d < count; d++)
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{
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a.At(d, d, Math.Sqrt(Gamma.Sample(rnd, (degreesOfFreedom - d)/2.0, 0.5)));
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}
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for (var i = 1; i < count; i++)
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{
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for (var j = 0; j < i; j++)
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{
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a.At(i, j, Normal.Sample(rnd, 0.0, 1.0));
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}
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}
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var factor = chol.Factor;
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return factor*a*a.Transpose()*factor.Transpose();
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}
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}
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}
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