// <copyright file="InverseWishart.cs" company="Math.NET">
|
// Math.NET Numerics, part of the Math.NET Project
|
// http://numerics.mathdotnet.com
|
// http://github.com/mathnet/mathnet-numerics
|
//
|
// Copyright (c) 2009-2013 Math.NET
|
//
|
// Permission is hereby granted, free of charge, to any person
|
// obtaining a copy of this software and associated documentation
|
// files (the "Software"), to deal in the Software without
|
// restriction, including without limitation the rights to use,
|
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
// copies of the Software, and to permit persons to whom the
|
// Software is furnished to do so, subject to the following
|
// conditions:
|
//
|
// The above copyright notice and this permission notice shall be
|
// included in all copies or substantial portions of the Software.
|
//
|
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
// OTHER DEALINGS IN THE SOFTWARE.
|
// </copyright>
|
|
using System;
|
using IStation.Numerics.LinearAlgebra;
|
using IStation.Numerics.LinearAlgebra.Factorization;
|
using IStation.Numerics.Random;
|
|
namespace IStation.Numerics.Distributions
|
{
|
/// <summary>
|
/// Multivariate Inverse Wishart distribution. This distribution is
|
/// parameterized by the degrees of freedom nu and the scale matrix S. The inverse Wishart distribution
|
/// is the conjugate prior for the covariance matrix of a multivariate normal distribution.
|
/// <a href="http://en.wikipedia.org/wiki/Inverse-Wishart_distribution">Wikipedia - Inverse-Wishart distribution</a>.
|
/// </summary>
|
public class InverseWishart : IDistribution
|
{
|
System.Random _random;
|
|
readonly double _freedom;
|
readonly Matrix<double> _scale;
|
|
/// <summary>
|
/// Caches the Cholesky factorization of the scale matrix.
|
/// </summary>
|
readonly Cholesky<double> _chol;
|
|
/// <summary>
|
/// Initializes a new instance of the <see cref="InverseWishart"/> class.
|
/// </summary>
|
/// <param name="degreesOfFreedom">The degree of freedom (ν) for the inverse Wishart distribution.</param>
|
/// <param name="scale">The scale matrix (Ψ) for the inverse Wishart distribution.</param>
|
public InverseWishart(double degreesOfFreedom, Matrix<double> scale)
|
{
|
if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = SystemRandomSource.Default;
|
_freedom = degreesOfFreedom;
|
_scale = scale;
|
_chol = _scale.Cholesky();
|
}
|
|
/// <summary>
|
/// Initializes a new instance of the <see cref="InverseWishart"/> class.
|
/// </summary>
|
/// <param name="degreesOfFreedom">The degree of freedom (ν) for the inverse Wishart distribution.</param>
|
/// <param name="scale">The scale matrix (Ψ) for the inverse Wishart distribution.</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
|
public InverseWishart(double degreesOfFreedom, Matrix<double> scale, System.Random randomSource)
|
{
|
if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = randomSource ?? SystemRandomSource.Default;
|
_freedom = degreesOfFreedom;
|
_scale = scale;
|
_chol = _scale.Cholesky();
|
}
|
|
/// <summary>
|
/// A string representation of the distribution.
|
/// </summary>
|
/// <returns>a string representation of the distribution.</returns>
|
public override string ToString()
|
{
|
return $"InverseWishart(ν = {_freedom}, Rows = {_scale.RowCount}, Columns = {_scale.ColumnCount})";
|
}
|
|
/// <summary>
|
/// Tests whether the provided values are valid parameters for this distribution.
|
/// </summary>
|
/// <param name="degreesOfFreedom">The degree of freedom (ν) for the inverse Wishart distribution.</param>
|
/// <param name="scale">The scale matrix (Ψ) for the inverse Wishart distribution.</param>
|
public static bool IsValidParameterSet(double degreesOfFreedom, Matrix<double> scale)
|
{
|
if (scale.RowCount != scale.ColumnCount)
|
{
|
return false;
|
}
|
|
for (var i = 0; i < scale.RowCount; i++)
|
{
|
if (scale.At(i, i) <= 0.0)
|
{
|
return false;
|
}
|
}
|
|
return degreesOfFreedom > 0.0;
|
}
|
|
/// <summary>
|
/// Gets or sets the degree of freedom (ν) for the inverse Wishart distribution.
|
/// </summary>
|
public double DegreesOfFreedom => _freedom;
|
|
/// <summary>
|
/// Gets or sets the scale matrix (Ψ) for the inverse Wishart distribution.
|
/// </summary>
|
public Matrix<double> Scale => _scale;
|
|
/// <summary>
|
/// Gets or sets the random number generator which is used to draw random samples.
|
/// </summary>
|
public System.Random RandomSource
|
{
|
get => _random;
|
set => _random = value ?? SystemRandomSource.Default;
|
}
|
|
/// <summary>
|
/// Gets the mean.
|
/// </summary>
|
/// <value>The mean of the distribution.</value>
|
public Matrix<double> Mean => _scale*(1.0/(_freedom - _scale.RowCount - 1.0));
|
|
/// <summary>
|
/// Gets the mode of the distribution.
|
/// </summary>
|
/// <value>The mode of the distribution.</value>
|
/// <remarks>A. O'Hagan, and J. J. Forster (2004). Kendall's Advanced Theory of Statistics: Bayesian Inference. 2B (2 ed.). Arnold. ISBN 0-340-80752-0.</remarks>
|
public Matrix<double> Mode => _scale*(1.0/(_freedom + _scale.RowCount + 1.0));
|
|
/// <summary>
|
/// Gets the variance of the distribution.
|
/// </summary>
|
/// <value>The variance of the distribution.</value>
|
/// <remarks>Kanti V. Mardia, J. T. Kent and J. M. Bibby (1979). Multivariate Analysis.</remarks>
|
public Matrix<double> Variance
|
{
|
get
|
{
|
return Matrix<double>.Build.Dense(_scale.RowCount, _scale.ColumnCount, (i, j) =>
|
{
|
var num1 = ((_freedom - _scale.RowCount + 1)*_scale.At(i, j)*_scale.At(i, j)) + ((_freedom - _scale.RowCount - 1)*_scale.At(i, i)*_scale.At(j, j));
|
var num2 = (_freedom - _scale.RowCount)*(_freedom - _scale.RowCount - 1)*(_freedom - _scale.RowCount - 1)*(_freedom - _scale.RowCount - 3);
|
return num1/num2;
|
});
|
}
|
}
|
|
/// <summary>
|
/// Evaluates the probability density function for the inverse Wishart distribution.
|
/// </summary>
|
/// <param name="x">The matrix at which to evaluate the density at.</param>
|
/// <exception cref="ArgumentOutOfRangeException">If the argument does not have the same dimensions as the scale matrix.</exception>
|
/// <returns>the density at <paramref name="x"/>.</returns>
|
public double Density(Matrix<double> x)
|
{
|
var p = _scale.RowCount;
|
|
if (x.RowCount != p || x.ColumnCount != p)
|
{
|
throw new ArgumentOutOfRangeException(nameof(x), "Matrix dimensions must agree.");
|
}
|
|
var chol = x.Cholesky();
|
var dX = chol.Determinant;
|
var sXi = chol.Solve(Scale);
|
|
// Compute the multivariate Gamma function.
|
var gp = Math.Pow(Constants.Pi, p*(p - 1.0)/4.0);
|
for (var j = 1; j <= p; j++)
|
{
|
gp *= SpecialFunctions.Gamma((_freedom + 1.0 - j)/2.0);
|
}
|
|
return Math.Pow(dX, -(_freedom + p + 1.0)/2.0)
|
*Math.Exp(-0.5*sXi.Trace())
|
*Math.Pow(_chol.Determinant, _freedom/2.0)
|
/Math.Pow(2.0, _freedom*p/2.0)
|
/gp;
|
}
|
|
/// <summary>
|
/// Samples an inverse Wishart distributed random variable by sampling
|
/// a Wishart random variable and inverting the matrix.
|
/// </summary>
|
/// <returns>a sample from the distribution.</returns>
|
public Matrix<double> Sample()
|
{
|
return Sample(_random, _freedom, _scale);
|
}
|
|
/// <summary>
|
/// Samples an inverse Wishart distributed random variable by sampling
|
/// a Wishart random variable and inverting the matrix.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="degreesOfFreedom">The degree of freedom (ν) for the inverse Wishart distribution.</param>
|
/// <param name="scale">The scale matrix (Ψ) for the inverse Wishart distribution.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static Matrix<double> Sample(System.Random rnd, double degreesOfFreedom, Matrix<double> scale)
|
{
|
if (Control.CheckDistributionParameters && !IsValidParameterSet(degreesOfFreedom, scale))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
var r = Wishart.Sample(rnd, degreesOfFreedom, scale.Inverse());
|
return r.PseudoInverse();
|
}
|
}
|
}
|
