// <copyright file="MatrixNormal.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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// 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.Random;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Multivariate Matrix-valued Normal distributions. The distribution
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/// is parameterized by a mean matrix (M), a covariance matrix for the rows (V) and a covariance matrix
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/// for the columns (K). If the dimension of M is d-by-m then V is d-by-d and K is m-by-m.
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/// <a href="http://en.wikipedia.org/wiki/Matrix_normal_distribution">Wikipedia - MatrixNormal distribution</a>.
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/// </summary>
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public class MatrixNormal : IDistribution
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{
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System.Random _random;
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/// <summary>
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/// The mean of the matrix normal distribution.
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/// </summary>
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readonly Matrix<double> _m;
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/// <summary>
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/// The covariance matrix for the rows.
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/// </summary>
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readonly Matrix<double> _v;
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/// <summary>
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/// The covariance matrix for the columns.
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/// </summary>
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readonly Matrix<double> _k;
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/// <summary>
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/// Initializes a new instance of the <see cref="MatrixNormal"/> class.
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/// </summary>
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/// <param name="m">The mean of the matrix normal.</param>
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/// <param name="v">The covariance matrix for the rows.</param>
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/// <param name="k">The covariance matrix for the columns.</param>
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/// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
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public MatrixNormal(Matrix<double> m, Matrix<double> v, Matrix<double> k)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(m, v, k))
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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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_m = m;
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_v = v;
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_k = k;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="MatrixNormal"/> class.
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/// </summary>
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/// <param name="m">The mean of the matrix normal.</param>
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/// <param name="v">The covariance matrix for the rows.</param>
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/// <param name="k">The covariance matrix for the columns.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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/// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
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public MatrixNormal(Matrix<double> m, Matrix<double> v, Matrix<double> k, System.Random randomSource)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(m, v, k))
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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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_m = m;
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_v = v;
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_k = k;
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}
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/// <summary>
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/// Returns a <see cref="System.String"/> that represents this instance.
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/// </summary>
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/// <returns>
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/// A <see cref="System.String"/> that represents this instance.
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/// </returns>
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public override string ToString()
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{
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return $"MatrixNormal(Rows = {_m.RowCount}, Columns = {_m.ColumnCount})";
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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="m">The mean of the matrix normal.</param>
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/// <param name="v">The covariance matrix for the rows.</param>
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/// <param name="k">The covariance matrix for the columns.</param>
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public static bool IsValidParameterSet(Matrix<double> m, Matrix<double> v, Matrix<double> k)
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{
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var n = m.RowCount;
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var p = m.ColumnCount;
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if (v.ColumnCount != n || v.RowCount != n)
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{
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return false;
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}
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if (k.ColumnCount != p || k.RowCount != p)
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{
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return false;
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}
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for (var i = 0; i < v.RowCount; i++)
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{
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if (v.At(i, i) <= 0)
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{
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return false;
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}
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}
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for (var i = 0; i < k.RowCount; i++)
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{
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if (k.At(i, i) <= 0)
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{
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return false;
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}
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}
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return true;
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}
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/// <summary>
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/// Gets the mean. (M)
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/// </summary>
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/// <value>The mean of the distribution.</value>
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public Matrix<double> Mean => _m;
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/// <summary>
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/// Gets the row covariance. (V)
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/// </summary>
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/// <value>The row covariance.</value>
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public Matrix<double> RowCovariance => _v;
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/// <summary>
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/// Gets the column covariance. (K)
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/// </summary>
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/// <value>The column covariance.</value>
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public Matrix<double> ColumnCovariance => _k;
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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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/// Evaluates the probability density function for the matrix normal 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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/// <returns>the density at <paramref name="x"/></returns>
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/// <exception cref="ArgumentOutOfRangeException">If the argument does not have the correct dimensions.</exception>
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public double Density(Matrix<double> x)
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{
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if (x.RowCount != _m.RowCount || x.ColumnCount != _m.ColumnCount)
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{
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throw Matrix.DimensionsDontMatch<ArgumentOutOfRangeException>(x, _m, "x");
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}
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var a = x - _m;
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var cholV = _v.Cholesky();
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var cholK = _k.Cholesky();
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return Math.Exp(-0.5*cholK.Solve(a.Transpose()*cholV.Solve(a)).Trace())
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/Math.Pow(2.0*Constants.Pi, x.RowCount*x.ColumnCount/2.0)
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/Math.Pow(cholK.Determinant, x.RowCount/2.0)
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/Math.Pow(cholV.Determinant, x.ColumnCount/2.0);
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}
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/// <summary>
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/// Samples a matrix normal distributed random variable.
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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 Sample(_random, _m, _v, _k);
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}
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/// <summary>
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/// Samples a matrix normal distributed random variable.
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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="m">The mean of the matrix normal.</param>
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/// <param name="v">The covariance matrix for the rows.</param>
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/// <param name="k">The covariance matrix for the columns.</param>
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/// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
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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, Matrix<double> m, Matrix<double> v, Matrix<double> k)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(m, v, k))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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var n = m.RowCount;
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var p = m.ColumnCount;
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// Compute the Kronecker product of V and K, this is the covariance matrix for the stacked matrix.
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var vki = v.KroneckerProduct(k.Inverse());
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// Sample a vector valued random variable with VKi as the covariance.
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var vector = SampleVectorNormal(rnd, new DenseVector(n*p), vki);
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// Unstack the vector v and add the mean.
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var r = m.Clone();
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for (var i = 0; i < n; i++)
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{
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for (var j = 0; j < p; j++)
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{
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r.At(i, j, r.At(i, j) + vector[(j*n) + i]);
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}
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}
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return r;
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}
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/// <summary>
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/// Samples a vector normal distributed random variable.
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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="mean">The mean of the vector normal distribution.</param>
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/// <param name="covariance">The covariance matrix of the vector normal distribution.</param>
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/// <returns>a sequence of samples from defined distribution.</returns>
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static Vector<double> SampleVectorNormal(System.Random rnd, Vector<double> mean, Matrix<double> covariance)
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{
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var chol = covariance.Cholesky();
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// Sample a standard normal variable.
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var v = Vector<double>.Build.Random(mean.Count, new Normal(rnd));
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// Return the transformed variable.
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return mean + (chol.Factor*v);
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}
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}
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}
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