mathnet-numerics/src/Numerics/Distributions/Multinomial.cs

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using System;
using System.Collections.Generic;
using System.Linq;
using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Double;
using MathNet.Numerics.Properties;
using MathNet.Numerics.Random;
using MathNet.Numerics.Statistics;

namespace MathNet.Numerics.Distributions
{
    /// <summary>
    /// Multivariate Multinomial distribution. For details about this distribution, see
    /// <a href="http://en.wikipedia.org/wiki/Multinomial_distribution">Wikipedia - Multinomial distribution</a>.
    /// </summary>
    /// <remarks>
    /// The distribution is parameterized by a vector of ratios: in other words, the parameter
    /// does not have to be normalized and sum to 1. The reason is that some vectors can't be exactly normalized
    /// to sum to 1 in floating point representation.
    /// </remarks>
    public class Multinomial : IDistribution
    {
        System.Random _random;

        /// <summary>
        /// Stores the normalized multinomial probabilities.
        /// </summary>
        double[] _p;

        /// <summary>
        /// The number of trials.
        /// </summary>
        int _trials;

        /// <summary>
        /// Initializes a new instance of the Multinomial class.
        /// </summary>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of trials.</param>
        /// <exception cref="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
        public Multinomial(double[] p, int n)
        {
            _random = SystemRandomSource.Default;
            SetParameters(p, n);
        }

        /// <summary>
        /// Initializes a new instance of the Multinomial class.
        /// </summary>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of trials.</param>
        /// <param name="randomSource">The random number generator which is used to draw random samples.</param>
        /// <exception cref="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
        public Multinomial(double[] p, int n, System.Random randomSource)
        {
            _random = randomSource ?? SystemRandomSource.Default;
            SetParameters(p, n);
        }

        /// <summary>
        /// Initializes a new instance of the Multinomial class from histogram <paramref name="h"/>. The distribution will
        /// not be automatically updated when the histogram changes.
        /// </summary>
        /// <param name="h">Histogram instance</param>
        /// <param name="n">The number of trials.</param>
        /// <exception cref="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
        /// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
        public Multinomial(Histogram h, int n)
        {
            if (h == null)
            {
                throw new ArgumentNullException("h");
            }

            // The probability distribution vector.
            var p = new double[h.BucketCount];

            // Fill in the distribution vector.
            for (var i = 0; i < h.BucketCount; i++)
            {
                p[i] = h[i].Count;
            }

            SetParameters(p, n);
            RandomSource = SystemRandomSource.Default;
        }

        /// <summary>
        /// A string representation of the distribution.
        /// </summary>
        /// <returns>a string representation of the distribution.</returns>
        public override string ToString()
        {
            return "Multinomial(Dimension = " + _p.Length + ", Number of Trails = " + _trials + ")";
        }

        /// <summary>
        /// Checks whether the parameters of the distribution are valid.
        /// </summary>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of trials.</param>
        /// <returns>If any of the probabilities are negative returns <c>false</c>,
        /// if the sum of parameters is 0.0, or if the number of trials is negative; otherwise <c>true</c>.</returns>
        static bool IsValidParameterSet(IEnumerable<double> p, int n)
        {
            var sum = 0.0;
            foreach (var t in p)
            {
                if (t < 0.0 || Double.IsNaN(t))
                {
                    return false;
                }

                sum += t;
            }

            if (sum == 0.0)
            {
                return false;
            }

            return n >= 0;
        }

        /// <summary>
        /// Sets the parameters of the distribution after checking their validity.
        /// </summary>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of trials.</param>
        /// <exception cref="ArgumentOutOfRangeException">When the parameters are out of range.</exception>
        void SetParameters(double[] p, int n)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            _p = (double[]) p.Clone();
            _trials = n;
        }

        /// <summary>
        /// Gets or sets the proportion of ratios.
        /// </summary>
        public double[] P
        {
            get { return (double[])_p.Clone(); }
            set { SetParameters(value, _trials); }
        }

        /// <summary>
        /// Gets or sets the number of trials.
        /// </summary>
        public int N
        {
            get { return _trials; }
            set { SetParameters(_p, value); }
        }

        /// <summary>
        /// Gets or sets the random number generator which is used to draw random samples.
        /// </summary>
        public System.Random RandomSource
        {
            get { return _random; }
            set { _random = value ?? SystemRandomSource.Default; }
        }

        /// <summary>
        /// Gets the mean of the distribution.
        /// </summary>
        public Vector<double> Mean
        {
            get { return _trials*(DenseVector) P; }
        }

        /// <summary>
        /// Gets the variance of the distribution.
        /// </summary>
        public Vector<double> Variance
        {
            get
            {
                // Do not use _p, because operations below will modify _p array. Use P or _p.Clone().
                var res = (DenseVector) P;
                for (var i = 0; i < res.Count; i++)
                {
                    res[i] *= _trials*(1 - res[i]);
                }

                return res;
            }
        }

        /// <summary>
        /// Gets the skewness of the distribution.
        /// </summary>
        public Vector<double> Skewness
        {
            get
            {
                // Do not use _p, because operations below will modify _p array. Use P or _p.Clone().
                var res = (DenseVector) P;
                for (var i = 0; i < res.Count; i++)
                {
                    res[i] = (1.0 - (2.0*res[i]))/Math.Sqrt(_trials*(1.0 - res[i])*res[i]);
                }

                return res;
            }
        }

        /// <summary>
        /// Computes values of the probability mass function.
        /// </summary>
        /// <param name="x">Non-negative integers x1, ..., xk</param>
        /// <returns>The probability mass at location <paramref name="x"/>.</returns>
        /// <exception cref="ArgumentNullException">When <paramref name="x"/> is null.</exception>
        /// <exception cref="ArgumentException">When length of <paramref name="x"/> is not equal to event probabilities count.</exception>
        public double Probability(int[] x)
        {
            if (null == x)
            {
                throw new ArgumentNullException("x");
            }

            if (x.Length != _p.Length)
            {
                throw new ArgumentException(Resources.ArgumentVectorsSameLength, "x");
            }

            if (x.Sum() == _trials)
            {
                var coef = SpecialFunctions.Multinomial(_trials, x);
                var num = 1.0;
                for (var i = 0; i < x.Length; i++)
                {
                    num *= Math.Pow(_p[i], x[i]);
                }

                return coef*num;
            }

            return 0.0;
        }

        /// <summary>
        /// Computes values of the log probability mass function.
        /// </summary>
        /// <param name="x">Non-negative integers x1, ..., xk</param>
        /// <returns>The log probability mass at location <paramref name="x"/>.</returns>
        /// <exception cref="ArgumentNullException">When <paramref name="x"/> is null.</exception>
        /// <exception cref="ArgumentException">When length of <paramref name="x"/> is not equal to event probabilities count.</exception>
        public double ProbabilityLn(int[] x)
        {
            if (null == x)
            {
                throw new ArgumentNullException("x");
            }

            if (x.Length != _p.Length)
            {
                throw new ArgumentException(Resources.ArgumentVectorsSameLength, "x");
            }

            if (x.Sum() == _trials)
            {
                var coef = Math.Log(SpecialFunctions.Multinomial(_trials, x));
                var num = x.Select((t, i) => t*Math.Log(_p[i])).Sum();
                return coef + num;
            }

            return 0.0;
        }

        /// <summary>
        /// Samples one multinomial distributed random variable.
        /// </summary>
        /// <returns>the counts for each of the different possible values.</returns>
        public int[] Sample()
        {
            return Sample(_random, _p, _trials);
        }

        /// <summary>
        /// Samples a sequence multinomially distributed random variables.
        /// </summary>
        /// <returns>a sequence of counts for each of the different possible values.</returns>
        public IEnumerable<int[]> Samples()
        {
            while (true)
            {
                yield return Sample(_random, _p, _trials);
            }
        }

        /// <summary>
        /// Samples one multinomial distributed random variable.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of trials.</param>
        /// <returns>the counts for each of the different possible values.</returns>
        public static int[] Sample(System.Random rnd, double[] p, int n)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            // The cumulative density of p.
            var cp = Categorical.ProbabilityMassToCumulativeDistribution(p);

            // The variable that stores the counts.
            var ret = new int[p.Length];

            for (var i = 0; i < n; i++)
            {
                ret[Categorical.SampleUnchecked(rnd, cp)]++;
            }

            return ret;
        }

        /// <summary>
        /// Samples a multinomially distributed random variable.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
        /// as this is often impossible using floating point arithmetic.</param>
        /// <param name="n">The number of variables needed.</param>
        /// <returns>a sequence of counts for each of the different possible values.</returns>
        public static IEnumerable<int[]> Samples(System.Random rnd, double[] p, int n)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            // The cumulative density of p.
            var cp = Categorical.ProbabilityMassToCumulativeDistribution(p);

            while (true)
            {
                // The variable that stores the counts.
                var ret = new int[p.Length];

                for (var i = 0; i < n; i++)
                {
                    ret[Categorical.SampleUnchecked(rnd, cp)]++;
                }

                yield return ret;
            }
        }
    }
}