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

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namespace MathNet.Numerics.Distributions
{
    using System;
    using System.Collections.Generic;
    using Properties;

    /// <summary>
    /// Implements the univariate Gamma distribution. For details about this distribution, see 
    /// <a href="http://en.wikipedia.org/wiki/Gamma_distribution">Wikipedia - Gamma distribution</a>.
    /// </summary>
    /// <remarks>
    /// <para>The Gamma distribution is parametrized by a shape and inverse scale parameter. When we want
    /// to specify a Gamma distribution which is a point distribution we set the shape parameter to be the
    /// location of the point distribution and the inverse scale as positive infinity. The distribution
    /// with shape and inverse scale both zero is undefined.</para>
    /// <para> Random number generation for the Gamma distribution is based on the algorithm in:
    /// "A Simple Method for Generating Gamma Variables" - Marsaglia &amp; Tsang
    /// ACM Transactions on Mathematical Software, Vol. 26, No. 3, September 2000, Pages 363�372.</para>
    /// <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 Gamma : IContinuousDistribution
    {
        /// <summary>
        /// Gamma shape parameter.
        /// </summary>
        private double _shape;

        /// <summary>
        /// Gamma inverse scale parameter.
        /// </summary>
        private double _invScale;

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

        /// <summary>
        /// Initializes a new instance of the Gamma class.
        /// </summary>
        /// <param name="shape">The shape of the Gamma distribution.</param>
        /// <param name="invScale">The inverse scale of the Gamma distribution.</param>
        public Gamma(double shape, double invScale)
        {
            SetParameters(shape, invScale);
            RandomSource = new Random();
        }

        /// <summary>
        /// Constructs a Gamma distribution from a shape and scale parameter. The distribution will
        /// be initialized with the default <seealso cref="System.Random"/> random number generator.
        /// </summary>
        /// <param name="shape">The shape of the Gamma distribution.</param>
        /// <param name="scale">The scale of the Gamma distribution.</param>
        /// <returns>a normal distribution.</returns>
        public static Gamma WithShapeScale(double shape, double scale)
        {
            return new Gamma(shape, 1.0/scale);
        }

        /// <summary>
        /// Constructs a Gamma distribution from a shape and inverse scale parameter. The distribution will
        /// be initialized with the default <seealso cref="System.Random"/> random number generator.
        /// </summary>
        /// <param name="shape">The shape of the Gamma distribution.</param>
        /// <param name="invScale">The inverse scale of the Gamma distribution.</param>
        /// <returns>a normal distribution.</returns>
        public static Gamma WithShapeInvScale(double shape, double invScale)
        {
            return new Gamma(shape, invScale);
        }

        /// <summary>
        /// A string representation of the distribution.
        /// </summary>
        /// <returns>a string representation of the distribution.</returns>
        public override string ToString()
        {
            return "Gamma(Shape = " + _shape + ", Inverse Scale = " + _invScale + ")";
        }

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

            return true;
        }

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

            _shape = shape;
            _invScale = invScale;
        }

        /// <summary>
        /// Gets or sets the shape of the Gamma distribution.
        /// </summary>
        public double Shape
        {
            get
            {
                return _shape;
            }

            set
            {
                SetParameters(value, _invScale);
            }
        }

        /// <summary>
        /// Gets or sets the scale of the Gamma distribution.
        /// </summary>
        public double Scale
        {
            get
            {
                return 1.0 / _invScale;
            }

            set
            {
                double invScale = 1.0/value;

                if(Double.IsNegativeInfinity(invScale))
                {
                    invScale = - invScale;
                }

                SetParameters(_shape, invScale);
            }
        }

        /// <summary>
        /// Gets or sets the inverse scale of the Gamma distribution.
        /// </summary>
        public double InvScale
        {
            get
            {
                return _invScale;
            }

            set
            {
                SetParameters(_shape, 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 Gamma distribution.
        /// </summary>
        public double Mean
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return _shape;
                }
                else if(_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return _shape / _invScale;
                }
            }
        }

        /// <summary>
        /// Gets the variance of the Gamma distribution.
        /// </summary>
        public double Variance
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return 0.0;
                }
                else if (_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return _shape / (_invScale * _invScale);
                }
            }
        }

        /// <summary>
        /// Gets the standard deviation of the Gamma distribution.
        /// </summary>
        public double StdDev
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return 0.0;
                }
                else if (_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return Math.Sqrt(_shape / (_invScale * _invScale));
                }
            }
        }

        /// <summary>
        /// Gets the entropy of the Gamma distribution.
        /// </summary>
        public double Entropy
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return 0.0;
                }
                else if (_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return _shape - Math.Log(_invScale) + SpecialFunctions.GammaLn(_shape) + ((1.0 - _shape) * SpecialFunctions.DiGamma(_shape));
                }
            }
        }

        /// <summary>
        /// Gets the skewness of the Gamma distribution.
        /// </summary>
        public double Skewness
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return 0.0;
                }
                else if (_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return 2.0 / Math.Sqrt(_shape);
                }
            }
        }
        #endregion

        #region IContinuousDistribution implementation

        /// <summary>
        /// Gets the mode of the Gamma distribution.
        /// </summary>
        public double Mode
        {
            get
            {
                if (Double.IsPositiveInfinity(_invScale))
                {
                    return _shape;
                }
                else if (_invScale == 0.0 && _shape == 0.0)
                {
                    return Double.NaN;
                }
                else
                {
                    return (_shape - 1.0) / _invScale;
                }
            }
        }

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

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

        /// <summary>
        /// Gets the maximum of the Gamma distribution.
        /// </summary>
        public double Maximum
        {
            get { return Double.PositiveInfinity; }
        }

        /// <summary>
        /// Computes the density of the Gamma 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 (Double.IsPositiveInfinity(_invScale))
            {
                if (x == _shape)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return 0.0;
                }
            }
            else if (_shape == 0.0 && _invScale == 0.0)
            {
                return 0.0;
            }
            else if (_shape == 1.0)
            {
                return _invScale * Math.Exp(- _invScale * x);
            }
            else
            {
                return Math.Pow(_invScale, _shape) * Math.Pow(x, _shape - 1.0) * Math.Exp(-_invScale * x) / SpecialFunctions.Gamma(_shape);
            } 
        }

        /// <summary>
        /// Computes the log density of the Gamma 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 (Double.IsPositiveInfinity(_invScale))
            {
                if (x == _shape)
                {
                    return Double.PositiveInfinity;
                }
                else
                {
                    return Double.NegativeInfinity;
                }
            }
            else if(_shape == 0.0 && _invScale == 0.0)
            {
                return Double.NegativeInfinity;
            }
            else if(_shape == 1.0)
            {
                return Math.Log(_invScale) - (_invScale * x);
            }
            else
            {
                return (_shape * Math.Log(_invScale)) + ((_shape - 1.0) * Math.Log(x)) - (_invScale * x) - SpecialFunctions.GammaLn(_shape);
            }
        }

        /// <summary>
        /// Computes the cumulative distribution function of the Gamma 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)
        {
            if (Double.IsPositiveInfinity(_invScale))
            {
                if (x >= _shape)
                {
                    return 1.0;
                }
                else
                {
                    return 0.0;
                }
            }
            else
            {
                return SpecialFunctions.IncompleteGamma(_shape, x * _invScale, true);
            } 
        }

        /// <summary>
        /// Generates a sample from the Gamma distribution.
        /// </summary>
        /// <returns>a sample from the distribution.</returns>
        public double Sample()
        {
            return SampleGamma(RandomSource, _shape, _invScale);
        }

        /// <summary>
        /// Generates a sequence of samples from the Gamma distribution.
        /// </summary>
        /// <returns>a sequence of samples from the distribution.</returns>
        public IEnumerable<double> Samples()
        {
            while (true)
            {
                yield return SampleGamma(RandomSource, _shape, _invScale);
            }
        }
        #endregion

        /// <summary>
        /// Generates a sample from the Gamma distribution.
        /// </summary>
        /// <param name="rng">The random number generator to use.</param>
        /// <param name="shape">The shape of the Gamma distribution from which to generate samples.</param>
        /// <param name="invScale">The inverse scale of the Gamma distribution from which to generate samples.</param>
        /// <returns>a sample from the distribution.</returns>
        public static double Sample(Random rng, double shape, double invScale)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(shape, invScale))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            return SampleGamma(rng, shape, invScale);
        }

        /// <summary>
        /// Generates a sequence of samples from the Gamma distribution.
        /// </summary>
        /// <param name="rng">The random number generator to use.</param>
        /// <param name="shape">The shape of the Gamma distribution from which to generate samples.</param>
        /// <param name="invScale">The inverse scale of the Gamma distribution from which to generate samples.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static IEnumerable<double> Samples(Random rng, double shape, double invScale)
        {
            if (Control.CheckDistributionParameters && !IsValidParameterSet(shape, invScale))
            {
                throw new ArgumentOutOfRangeException(Resources.InvalidDistributionParameters);
            }

            while (true)
            {
                yield return SampleGamma(rng, shape, invScale);
            }
        }

        /// <summary>
        /// Sampling implementation based on:
        /// "A Simple Method for Generating Gamma Variables" - Marsaglia &amp; Tsang
        /// ACM Transactions on Mathematical Software, Vol. 26, No. 3, September 2000, Pages 363�372.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="shape">The shape of the Gamma distribution.</param>
        /// <param name="invScale">The inverse scale of the Gamma distribution.</param>
        /// <returns>A sample from a Gamma distributed random variable.</returns>
        internal static double SampleGamma(System.Random rnd, double shape, double invScale)
        {
            if (Double.IsPositiveInfinity(invScale))
            {
                return shape;
            }
            else
            {
                double a = shape;
                double alphafix = 1.0;

                // Fix when alpha is less than one.
                if (shape < 1.0)
                {
                    a = shape + 1.0;
                    alphafix = System.Math.Pow(rnd.NextDouble(), 1.0 / shape);
                }

                double d = a - (1.0 / 3.0);
                double c = 1.0 / System.Math.Sqrt(9.0 * d);
                while (true)
                {
                    double x = Normal.Sample(rnd, 0.0, 1.0);
                    double v = 1.0 + (c * x);
                    while (v <= 0.0)
                    {
                        x = Normal.Sample(rnd, 0.0, 1.0);
                        v = 1.0 + (c * x);
                    }

                    v = v * v * v;
                    double u = rnd.NextDouble();
                    x = x * x;
                    if (u < 1.0 - (0.0331 * x * x))
                    {
                        return alphafix * d * v / invScale;
                    }

                    if (System.Math.Log(u) < (0.5 * x) + (d * (1.0 - v + System.Math.Log(v))))
                    {
                        return alphafix * d * v / invScale;
                    }
                }
            }
        }
    }
}