mathnet-numerics/src/Numerics/Distributions/Multivariate/MatrixNormal.cs

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using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Double;
using MathNet.Numerics.LinearAlgebra.Factorization;
using MathNet.Numerics.Properties;

namespace MathNet.Numerics.Distributions
{
    using System;

    /// <summary>
    /// This class implements functionality for matrix valued normal distributions. The distribution
    /// is parameterized by a mean matrix (M), a covariance matrix for the rows (V) and a covariance matrix
    /// 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.
    /// <a href="http://en.wikipedia.org/wiki/Matrix_normal_distribution">Wikipedia - MatrixNormal distribution</a>.
    /// </summary>
    /// <remarks><para>The distribution will use the <see cref="System.Random"/> by default. 
    /// Users can set the random number generator by using the <see cref="RandomSource"/> property.</para>
    /// <para>The statistics classes will check all the incoming parameters whether they are in the allowed
    /// range. This might involve heavy computation. Optionally, by setting Control.CheckDistributionParameters
    /// to <c>false</c>, all parameter checks can be turned off.</para></remarks>
    public class MatrixNormal
    {
        /// <summary>
        /// The mean of the matrix normal distribution.        
        /// </summary>
        Matrix<double> _m;

        /// <summary>
        /// The covariance matrix for the rows.
        /// </summary>
        Matrix<double> _v;

        /// <summary>
        /// The covariance matrix for the columns.
        /// </summary>
        Matrix<double> _k;

        /// <summary>
        /// The distribution's random number generator.
        /// </summary>
        Random _random;

        /// <summary>
        /// Initializes a new instance of the <see cref="MatrixNormal"/> class. 
        /// </summary>
        /// <param name="m">The mean of the matrix normal.</param>
        /// <param name="v">The covariance matrix for the rows.</param>
        /// <param name="k">The covariance matrix for the columns.</param>
        /// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
        public MatrixNormal(Matrix<double> m, Matrix<double> v, Matrix<double> k)
        {
            _random = new Random();
            SetParameters(m, v, k);
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="MatrixNormal"/> class. 
        /// </summary>
        /// <param name="m">The mean of the matrix normal.</param>
        /// <param name="v">The covariance matrix for the rows.</param>
        /// <param name="k">The covariance matrix for the columns.</param>
        /// <param name="randomSource">The random number generator which is used to draw random samples.</param>
        /// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
        public MatrixNormal(Matrix<double> m, Matrix<double> v, Matrix<double> k, Random randomSource)
        {
            _random = randomSource ?? new Random();
            SetParameters(m, v, k);
        }

        /// <summary>
        /// Returns a <see cref="System.String"/> that represents this instance.
        /// </summary>
        /// <returns>
        /// A <see cref="System.String"/> that represents this instance.
        /// </returns>
        public override string ToString()
        {
            return "MatrixNormal(Rows = " + _m.RowCount + ", Columns = " + _m.ColumnCount + ")";
        }

        /// <summary>
        /// Gets or sets the mean. (M)
        /// </summary>
        /// <value>The mean of the distribution.</value>
        public Matrix<double> Mean
        {
            get { return _m; }
            set { SetParameters(value, _v, _k); }
        }

        /// <summary>
        /// Gets or sets the row covariance. (V)
        /// </summary>
        /// <value>The row covariance.</value>
        public Matrix<double> RowCovariance
        {
            get { return _v; }
            set { SetParameters(_m, value, _k); }
        }

        /// <summary>
        /// Gets or sets the column covariance. (K)
        /// </summary>
        /// <value>The column covariance.</value>
        public Matrix<double> ColumnCovariance
        {
            get { return _k; }
            set { SetParameters(_m, _v, value); }
        }

        /// <summary>
        /// Sets the parameters of the distribution after checking their validity.
        /// </summary>
        /// <param name="m">The mean of the matrix normal.</param>
        /// <param name="v">The covariance matrix for the rows.</param>
        /// <param name="k">The covariance matrix for the columns.</param>
        /// <exception cref="ArgumentOutOfRangeException">When the parameters don't pass the <see cref="IsValidParameterSet"/> function.</exception>
        void SetParameters(Matrix<double> m, Matrix<double> v, Matrix<double> k)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(m, v, k))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            _m = m;
            _v = v;
            _k = k;
        }

        /// <summary>
        /// Checks whether the parameters of the distribution are valid. 
        /// </summary>
        /// <param name="m">The mean of the matrix normal.</param>
        /// <param name="v">The covariance matrix for the rows.</param>
        /// <param name="k">The covariance matrix for the columns.</param>
        /// <returns><c>true</c> when the parameters are valid, <c>false</c> otherwise.</returns>
        static bool IsValidParameterSet(Matrix<double> m, Matrix<double> v, Matrix<double> k)
        {
            var n = m.RowCount;
            var p = m.ColumnCount;
            if (v.ColumnCount != n || v.RowCount != n)
            {
                return false;
            }

            if (k.ColumnCount != p || k.RowCount != p)
            {
                return false;
            }

            for (var i = 0; i < v.RowCount; i++)
            {
                if (v.At(i, i) <= 0)
                {
                    return false;
                }
            }

            for (var i = 0; i < k.RowCount; i++)
            {
                if (k.At(i, i) <= 0)
                {
                    return false;
                }
            }

            return true;
        }

        /// <summary>
        /// Gets or sets the random number generator which is used to draw random samples.
        /// </summary>
        public Random RandomSource
        {
            get { return _random; }
            set
            {
                if (value == null)
                {
                    throw new ArgumentNullException();
                }

                _random = value;
            }
        }

        /// <summary>
        /// Evaluates the probability density function for the matrix normal distribution.
        /// </summary>
        /// <param name="x">The matrix at which to evaluate the density at.</param>
        /// <returns>the density at <paramref name="x"/></returns>
        /// <exception cref="ArgumentOutOfRangeException">If the argument does not have the correct dimensions.</exception>
        public double Density(Matrix<double> x)
        {
            if (x.RowCount != _m.RowCount || x.ColumnCount != _m.ColumnCount)
            {
                throw Matrix.DimensionsDontMatch<ArgumentOutOfRangeException>(x, _m, "x");
            }

            var a = x - _m;
            var cholV = Cholesky<double>.Create(_v);
            var cholK = Cholesky<double>.Create(_k);

            return Math.Exp(-0.5*cholV.Solve(a.Transpose()*cholK.Solve(a)).Trace())
                /Math.Pow(2.0*Constants.Pi, x.RowCount*x.ColumnCount/2.0)
                /Math.Pow(cholV.Determinant, x.RowCount/2.0)
                /Math.Pow(cholK.Determinant, x.ColumnCount/2.0);
        }

        /// <summary>
        /// Samples a matrix normal distributed random variable.
        /// </summary>
        /// <returns>A random number from this distribution.</returns>
        public Matrix<double> Sample()
        {
            return Sample(RandomSource, _m, _v, _k);
        }

        /// <summary>
        /// Samples a matrix normal distributed random variable.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="m">The mean of the matrix normal.</param>
        /// <param name="v">The covariance matrix for the rows.</param>
        /// <param name="k">The covariance matrix for the columns.</param>
        /// <exception cref="ArgumentOutOfRangeException">If the dimensions of the mean and two covariance matrices don't match.</exception>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static Matrix<double> Sample(Random rnd, Matrix<double> m, Matrix<double> v, Matrix<double> k)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(m, v, k))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            var n = m.RowCount;
            var p = m.ColumnCount;

            // Compute the Kronecker product of V and K, this is the covariance matrix for the stacked matrix.
            var vki = v.KroneckerProduct(k.Inverse());

            // Sample a vector valued random variable with VKi as the covariance.
            var vector = SampleVectorNormal(rnd, new DenseVector(n*p), vki);

            // Unstack the vector v and add the mean.
            var r = m.Clone();
            for (var i = 0; i < n; i++)
            {
                for (var j = 0; j < p; j++)
                {
                    r.At(i, j, r.At(i, j) + vector[(j*n) + i]);
                }
            }

            return r;
        }

        /// <summary>
        /// Samples a vector normal distributed random variable.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="mean">The mean of the vector normal distribution.</param>
        /// <param name="covariance">The covariance matrix of the vector normal distribution.</param>
        /// <returns>a sequence of samples from defined distribution.</returns>
        static Vector<double> SampleVectorNormal(Random rnd, Vector<double> mean, Matrix<double> covariance)
        {
            var chol = Cholesky<double>.Create(covariance);
            return SampleVectorNormal(rnd, mean, chol);
        }

        /// <summary>
        /// Samples a vector normal distributed random variable.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="mean">The mean of the vector normal distribution.</param>
        /// <param name="cholesky">The Cholesky factorization of the covariance matrix.</param>
        /// <returns>a sequence of samples from defined distribution.</returns>
        static Vector<double> SampleVectorNormal(Random rnd, Vector<double> mean, Cholesky<double> cholesky)
        {
            var count = mean.Count;

            // Sample a standard normal variable.
            var v = new DenseVector(count);
            for (var d = 0; d < count; d += 2)
            {
                var sample = Normal.SampleUncheckedBoxMuller(rnd);
                v[d] = sample.Item1;
                if (d + 1 < count)
                {
                    v[d + 1] = sample.Item2;
                }
            }

            // Return the transformed variable.
            return mean + (cholesky.Factor*v);
        }
    }
}