mathnet-numerics/src/Numerics/Distributions/Continuous/Beta.cs

// <copyright file="Beta.cs" company="Math.NET">
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namespace MathNet.Numerics.Distributions
{
    using System;
    using System.Collections.Generic;
    using Properties;

    /// <summary>
    /// Implements the Beta distribution. For details about this distribution, see 
    /// <a href="http://en.wikipedia.org/wiki/Beta_distribution">Wikipedia - Beta distribution</a>.
    /// </summary>
    /// <remarks><para>The distribution will use the <see cref="System.Random"/> by default. 
    /// Users can get/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 false, all parameter checks can be turned off.</para></remarks>
    public class Beta : IContinuousDistribution
    {
        /// <summary>
        /// Beta shape parameter a.
        /// </summary>
        private double _shapeA;

        /// <summary>
        /// Beta shape parameter b.
        /// </summary>
        private double _shapeB;

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

        /// <summary>
        /// Initializes a new instance of the Beta class.
        /// </summary>
        /// <param name="a">The a shape parameter of the Beta distribution.</param>
        /// <param name="b">The b shape parameter of the Beta distribution.</param>
        /// <exception cref="ArgumentOutOfRangeException">If any of the Beta parameters are negative.</exception>
        public Beta(double a, double b)
        {
            SetParameters(a, b);
            RandomSource = new Random();
        }

        /// <summary>
        /// A string representation of the distribution.
        /// </summary>
        /// <returns>A string representation of the Beta distribution.</returns>
        public override string ToString()
        {
            return "Beta(A = " + _shapeA + ", B = " + _shapeB + ")";
        }

        /// <summary>
        /// Checks whether the parameters of the distribution are valid. 
        /// </summary>
        /// <param name="a">The a shape parameter of the Beta distribution.</param>
        /// <param name="b">The b shape parameter of the Beta distribution.</param>
        /// <returns>True when the parameters are valid, false otherwise.</returns>
        private static bool IsValidParameterSet(double a, double b)
        {
            if (a < 0.0 || b < 0.0 || Double.IsNaN(a) || Double.IsNaN(b))
            {
                return false;
            }

            return true;
        }

        /// <summary>
        /// Sets the parameters of the distribution after checking their validity.
        /// </summary>
        /// <param name="a">The a shape parameter of the Beta distribution.</param>
        /// <param name="b">The b shape parameter of the Beta distribution.</param>
        /// <exception cref="ArgumentOutOfRangeException">When the parameters don't pass the <see cref="IsValidParameterSet"/> function.</exception>
        private void SetParameters(double a, double b)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(a, b))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            _shapeA = a;
            _shapeB = b;
        }

        /// <summary>
        /// Gets or sets the A shape parameter of the Beta distribution.
        /// </summary>
        public double A
        {
            get { return _shapeA; }
            set { SetParameters(value, _shapeB); }
        }

        /// <summary>
        /// Gets or sets the B shape parameter of the Beta distribution.
        /// </summary>
        public double B
        {
            get { return _shapeB; }
            set { SetParameters(_shapeA, value); }
        }

        #region IDistribution implementation

        /// <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>
        /// Gets the mean of the Beta distribution.
        /// </summary>
        public double Mean
        {
            get
            {
                if(_shapeA == 0.0 && _shapeB == 0.0)
                {
                    return 0.5;
                }
                else if(_shapeA == 0.0)
                {
                    return 0.0;
                }
                else if(_shapeB == 0.0)
                {
                    return 1.0;
                }
                else if(Double.IsPositiveInfinity(_shapeA) && Double.IsPositiveInfinity(_shapeB))
                {
                    return 0.5;
                }
                else if (Double.IsPositiveInfinity(_shapeA))
                {
                    return 1.0;
                }
                else if (Double.IsPositiveInfinity(_shapeB))
                {
                    return 0.0;
                }
                else
                {
                    return _shapeA / (_shapeA + _shapeB);
                }
            }
        }

        /// <summary>
        /// Gets the variance of the Beta distribution.
        /// </summary>
        public double Variance
        {
            get
            {
                return (_shapeA * _shapeB) / ((_shapeA + _shapeB) * (_shapeA + _shapeB) * (_shapeA + _shapeB + 1.0));
            }
        }

        /// <summary>
        /// Gets the standard deviation of the Beta distribution.
        /// </summary>
        public double StdDev
        {
            get { return Math.Sqrt((_shapeA * _shapeB) / ((_shapeA + _shapeB) * (_shapeA + _shapeB) * (_shapeA + _shapeB + 1.0))); }
        }

        /// <summary>
        /// Gets the entropy of the Beta distribution.
        /// </summary>
        public double Entropy
        {
            get
            {
                return SpecialFunctions.BetaLn(_shapeA, _shapeB)
                       - ((_shapeA - 1.0) * SpecialFunctions.DiGamma(_shapeA))
                       - ((_shapeB - 1.0) * SpecialFunctions.DiGamma(_shapeB))
                       + ((_shapeA + _shapeB - 2.0) * SpecialFunctions.DiGamma(_shapeA + _shapeB));
            }
        }

        /// <summary>
        /// Gets the skewness of the Beta distribution.
        /// </summary>
        public double Skewness
        {
            get
            {
                if (Double.IsPositiveInfinity(_shapeA) && Double.IsPositiveInfinity(_shapeB))
                {
                    return 0.0;
                }
                else if (Double.IsPositiveInfinity(_shapeA))
                {
                    return -2.0;
                }
                else if (Double.IsPositiveInfinity(_shapeB))
                {
                    return 2.0;
                }
                else if (_shapeA == 0.0 && _shapeB == 0.0)
                {
                    return 0.0;
                }
                else if (_shapeA == 0.0)
                {
                    return 2.0;
                }
                else if (_shapeB == 0.0)
                {
                    return -2.0;
                }
                else
                {
                    return 2.0 * (_shapeB - _shapeA) * Math.Sqrt(_shapeA + _shapeB + 1.0)
                       / ((_shapeA + _shapeB + 2.0) * Math.Sqrt(_shapeA * _shapeB));
                }
            }
        }
        #endregion

        #region IContinuousDistribution implementation

        /// <summary>
        /// Gets the mode of the Beta distribution; when there are multiple answers, this routine will return 0.5.
        /// </summary>
        public double Mode
        {
            get
            {
                if (_shapeA == 0.0 && _shapeB == 0.0)
                {
                    return 0.5;
                }
                else if (_shapeA == 0.0)
                {
                    return 0.0;
                }
                else if (_shapeB == 0.0)
                {
                    return 1.0;
                }
                else if (Double.IsPositiveInfinity(_shapeA) && Double.IsPositiveInfinity(_shapeB))
                {
                    return 0.5;
                }
                else if (Double.IsPositiveInfinity(_shapeA))
                {
                    return 1.0;
                }
                else if (Double.IsPositiveInfinity(_shapeB))
                {
                    return 0.0;
                }
                else if(_shapeA == 1.0 && _shapeB == 1.0)
                {
                    return 0.5;
                }
                else
                {
                    return (_shapeA - 1) / (_shapeA + _shapeB - 2);
                }
            }
        }

        /// <summary>
        /// Gets the median of the Beta distribution.
        /// </summary>
        public double Median
        {
            get { throw new NotSupportedException(); }
        }

        /// <summary>
        /// Gets the minimum of the Beta distribution.
        /// </summary>
        public double Minimum
        {
            get { return 0.0; }
        }

        /// <summary>
        /// Gets the maximum of the Beta distribution.
        /// </summary>
        public double Maximum
        {
            get { return 1.0; }
        }

        /// <summary>
        /// Computes the density of the Beta distribution.
        /// </summary>
        /// <param name="x">The location at which to compute the density.</param>
        /// <returns>the density at <paramref name="x"/>.</returns>
        public double Density(double x)
        {
            if (x < 0.0 || x > 1.0)
            {
                return 0.0;
            }

            if (Double.IsPositiveInfinity(_shapeA) && Double.IsPositiveInfinity(_shapeB))
            {
                if (x == 0.5)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (Double.IsPositiveInfinity(_shapeA))
            {
                if (x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (Double.IsPositiveInfinity(_shapeB))
            {
                if (x == 0.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (_shapeA == 0.0 && _shapeB == 0.0)
            {
                if (x == 0.0 || x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (_shapeA == 0.0)
            {
                if (x == 0.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (_shapeB == 0.0)
            {
                if (x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (_shapeA == 1.0 && _shapeB == 1.0)
            {
                return 1.0;
            }
            else
            {
                double b = SpecialFunctions.Gamma(_shapeA + _shapeB) / (SpecialFunctions.Gamma(_shapeA) * SpecialFunctions.Gamma(_shapeB));
                return b * Math.Pow(x, _shapeA - 1.0) * Math.Pow(1.0 - x, _shapeB - 1.0);
            }
        }

        /// <summary>
        /// Computes the log density of the Beta distribution.
        /// </summary>
        /// <param name="x">The location at which to compute the log density.</param>
        /// <returns>the log density at <paramref name="x"/>.</returns>
        public double DensityLn(double x)
        {
            if (x < 0.0 || x > 1.0)
            {
                return Double.NegativeInfinity;
            }

            if (Double.IsPositiveInfinity(_shapeA) && Double.IsPositiveInfinity(_shapeB))
            {
                if (x == 0.5)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (Double.IsPositiveInfinity(_shapeA))
            {
                if (x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (Double.IsPositiveInfinity(_shapeB))
            {
                if (x == 0.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (_shapeA == 0.0 && _shapeB == 0.0)
            {
                if (x == 0.0 || x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (_shapeA == 0.0)
            {
                if (x == 0.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (_shapeB == 0.0)
            {
                if (x == 1.0)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if (_shapeA == 1.0 && _shapeB == 1.0)
            {
                return 0.0;
            }
            else
            {
                double a = SpecialFunctions.GammaLn(_shapeA + _shapeB) - SpecialFunctions.GammaLn(_shapeA) - SpecialFunctions.GammaLn(_shapeB);
                double b = x == 0.0 ? (_shapeA == 1.0 ? 0.0 : Double.NegativeInfinity) : (_shapeA - 1.0) * Math.Log(x);
                double c = x == 1.0 ? (_shapeB == 1.0 ? 0.0 : Double.NegativeInfinity) : (_shapeB - 1.0) * Math.Log(1.0 - x);
                return a + b + c;
            }
        }

        /// <summary>
        /// Computes the cumulative distribution function of the Beta distribution.
        /// </summary>
        /// <param name="x">The location at which to compute the cumulative density.</param>
        /// <returns>the cumulative density at <paramref name="x"/>.</returns>
        public double CumulativeDistribution(double x)
        {
            return SpecialFunctions.BetaRegularized(_shapeA, _shapeB, x);
        }

        /// <summary>
        /// Generates a sample from the Beta distribution.
        /// </summary>
        /// <returns>a sample from the distribution.</returns>
        public double Sample()
        {
            return SampleBeta(RandomSource, _shapeA, _shapeB);
        }

        /// <summary>
        /// Generates a sequence of samples from the Beta distribution.
        /// </summary>
        /// <returns>a sequence of samples from the distribution.</returns>
        public IEnumerable<double> Samples()
        {
            while (true)
            {
                yield return SampleBeta(RandomSource, _shapeA, _shapeB);
            }
        }
        #endregion

        /// <summary>
        /// Generates a sample from the normal distribution using the <i>Box-Muller</i> algorithm.
        /// </summary>
        /// <param name="rng">The random number generator to use.</param>
        /// <param name="a">The a shape parameter of the Beta distribution.</param>
        /// <param name="b">The b shape parameter of the Beta distribution.</param>
        /// <returns>a sample from the distribution.</returns>
        public static double Sample(Random rng, double a, double b)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(a, b))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            return SampleBeta(rng, a, b);
        }

        /// <summary>
        /// Generates a sequence of samples from the normal distribution using the <i>Box-Muller</i> algorithm.
        /// </summary>
        /// <param name="rng">The random number generator to use.</param>
        /// <param name="a">The a shape parameter of the Beta distribution.</param>
        /// <param name="b">The b shape parameter of the Beta distribution.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static IEnumerable<double> Samples(Random rng, double a, double b)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(a, b))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            while (true)
            {
                yield return SampleBeta(rng, a, b);
            }
        }

        /// <summary>
        /// Samples Beta distributed random variables by sampling two Gamma variables and normalizing.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="a">The A shape parameter.</param>
        /// <param name="b">The B shape parameter.</param>
        /// <returns>a random number from the Beta distribution.</returns>
        internal static double SampleBeta(Random rnd, double a, double b)
        {
            double x = Gamma.SampleGamma(rnd, a, 1.0);
            double y = Gamma.SampleGamma(rnd, b, 1.0);
            return x / (x + y);
        }
    }
}