// <copyright file="Gamma.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) 2009-2014 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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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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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 System.Collections.Generic;
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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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/// Continuous Univariate Gamma distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Gamma_distribution">Wikipedia - Gamma distribution</a>.
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/// </summary>
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/// <remarks>
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/// The Gamma distribution is parametrized by a shape and inverse scale parameter. When we want
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/// to specify a Gamma distribution which is a point distribution we set the shape parameter to be the
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/// location of the point distribution and the inverse scale as positive infinity. The distribution
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/// with shape and inverse scale both zero is undefined.
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///
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/// Random number generation for the Gamma distribution is based on the algorithm in:
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/// "A Simple Method for Generating Gamma Variables" - Marsaglia & Tsang
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/// ACM Transactions on Mathematical Software, Vol. 26, No. 3, September 2000, Pages 363–372.
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/// </remarks>
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public class Gamma : IContinuousDistribution
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{
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System.Random _random;
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readonly double _shape;
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readonly double _rate;
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/// <summary>
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/// Initializes a new instance of the Gamma class.
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/// </summary>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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public Gamma(double shape, double rate)
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{
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if (!IsValidParameterSet(shape, rate))
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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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_shape = shape;
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_rate = rate;
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}
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/// <summary>
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/// Initializes a new instance of the Gamma class.
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/// </summary>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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public Gamma(double shape, double rate, System.Random randomSource)
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{
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if (!IsValidParameterSet(shape, rate))
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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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_shape = shape;
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_rate = rate;
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}
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/// <summary>
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/// Constructs a Gamma distribution from a shape and scale parameter. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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/// <param name="shape">The shape (k) of the Gamma distribution. Range: k ≥ 0.</param>
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/// <param name="scale">The scale (θ) of the Gamma distribution. Range: θ ≥ 0</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples. Optional, can be null.</param>
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public static Gamma WithShapeScale(double shape, double scale, System.Random randomSource = null)
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{
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return new Gamma(shape, 1.0/scale, randomSource);
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}
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/// <summary>
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/// Constructs a Gamma distribution from a shape and inverse scale parameter. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples. Optional, can be null.</param>
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public static Gamma WithShapeRate(double shape, double rate, System.Random randomSource = null)
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{
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return new Gamma(shape, rate, randomSource);
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}
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/// <summary>
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/// A string representation of the distribution.
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/// </summary>
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/// <returns>a string representation of the distribution.</returns>
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public override string ToString()
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{
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return $"Gamma(α = {_shape}, β = {_rate})";
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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="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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public static bool IsValidParameterSet(double shape, double rate)
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{
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return shape >= 0.0 && rate >= 0.0;
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}
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/// <summary>
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/// Gets or sets the shape (k, α) of the Gamma distribution. Range: α ≥ 0.
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/// </summary>
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public double Shape => _shape;
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/// <summary>
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/// Gets or sets the rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.
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/// </summary>
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public double Rate => _rate;
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/// <summary>
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/// Gets or sets the scale (θ) of the Gamma distribution.
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/// </summary>
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public double Scale => 1.0/_rate;
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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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/// Gets the mean of the Gamma distribution.
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/// </summary>
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public double Mean
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return _shape;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return _shape/_rate;
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}
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}
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/// <summary>
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/// Gets the variance of the Gamma distribution.
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/// </summary>
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public double Variance
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return 0.0;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return _shape/(_rate*_rate);
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}
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}
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/// <summary>
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/// Gets the standard deviation of the Gamma distribution.
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/// </summary>
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public double StdDev
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return 0.0;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return Math.Sqrt(_shape/(_rate*_rate));
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}
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}
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/// <summary>
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/// Gets the entropy of the Gamma distribution.
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/// </summary>
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public double Entropy
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return 0.0;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return _shape - Math.Log(_rate) + SpecialFunctions.GammaLn(_shape) + ((1.0 - _shape)*SpecialFunctions.DiGamma(_shape));
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}
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}
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/// <summary>
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/// Gets the skewness of the Gamma distribution.
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/// </summary>
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public double Skewness
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return 0.0;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return 2.0/Math.Sqrt(_shape);
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}
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}
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/// <summary>
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/// Gets the mode of the Gamma distribution.
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/// </summary>
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public double Mode
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{
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get
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{
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if (double.IsPositiveInfinity(_rate))
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{
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return _shape;
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}
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if (_rate == 0.0 && _shape == 0.0)
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{
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return double.NaN;
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}
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return (_shape - 1.0)/_rate;
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}
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}
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/// <summary>
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/// Gets the median of the Gamma distribution.
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/// </summary>
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public double Median => throw new NotSupportedException();
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/// <summary>
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/// Gets the minimum of the Gamma distribution.
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/// </summary>
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public double Minimum => 0.0;
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/// <summary>
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/// Gets the maximum of the Gamma distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDF"/>
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public double Density(double x)
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{
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return PDF(_shape, _rate, x);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDFLn"/>
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public double DensityLn(double x)
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{
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return PDFLn(_shape, _rate, x);
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CDF"/>
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public double CumulativeDistribution(double x)
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{
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return CDF(_shape, _rate, x);
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}
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/// <summary>
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/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
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/// at the given probability. This is also known as the quantile or percent point function.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative density.</param>
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/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
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/// <seealso cref="InvCDF"/>
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public double InverseCumulativeDistribution(double p)
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{
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return InvCDF(_shape, _rate, p);
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}
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/// <summary>
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/// Generates a sample from the Gamma distribution.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public double Sample()
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{
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return SampleUnchecked(_random, _shape, _rate);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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public void Samples(double[] values)
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{
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SamplesUnchecked(_random, values, _shape, _rate);
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}
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/// <summary>
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/// Generates a sequence of samples from the Gamma distribution.
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/// </summary>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<double> Samples()
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{
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return SamplesUnchecked(_random, _shape, _rate);
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}
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/// <summary>
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/// <para>Sampling implementation based on:
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/// "A Simple Method for Generating Gamma Variables" - Marsaglia & Tsang
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/// ACM Transactions on Mathematical Software, Vol. 26, No. 3, September 2000, Pages 363–372.</para>
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/// <para>This method performs no parameter checks.</para>
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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="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <returns>A sample from a Gamma distributed random variable.</returns>
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internal static double SampleUnchecked(System.Random rnd, double shape, double rate)
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{
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if (double.IsPositiveInfinity(rate))
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{
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return shape;
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}
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var a = shape;
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var alphafix = 1.0;
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// Fix when alpha is less than one.
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if (shape < 1.0)
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{
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a = shape + 1.0;
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alphafix = Math.Pow(rnd.NextDouble(), 1.0/shape);
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}
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var d = a - (1.0/3.0);
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var c = 1.0/Math.Sqrt(9.0*d);
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while (true)
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{
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var x = Normal.Sample(rnd, 0.0, 1.0);
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var v = 1.0 + (c*x);
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while (v <= 0.0)
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{
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x = Normal.Sample(rnd, 0.0, 1.0);
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v = 1.0 + (c*x);
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}
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v = v*v*v;
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var u = rnd.NextDouble();
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x = x*x;
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if (u < 1.0 - (0.0331*x*x))
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{
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return alphafix*d*v/rate;
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}
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if (Math.Log(u) < (0.5*x) + (d*(1.0 - v + Math.Log(v))))
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{
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return alphafix*d*v/rate;
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}
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}
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double shape, double rate)
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{
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for (int i = 0; i < values.Length; i++)
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{
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values[i] = SampleUnchecked(rnd, shape, rate);
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}
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}
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internal static IEnumerable<double> SamplesUnchecked(System.Random rnd, double location, double scale)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, location, scale);
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}
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}
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="Density"/>
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public static double PDF(double shape, double rate, double x)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(rate))
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{
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return x == shape ? double.PositiveInfinity : 0.0;
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}
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if (shape == 0.0 && rate == 0.0)
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{
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return 0.0;
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}
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if (shape == 1.0)
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{
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return rate*Math.Exp(-rate*x);
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}
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if (shape > 160.0)
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{
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return Math.Exp(PDFLn(shape, rate, x));
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}
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return Math.Pow(rate, shape)*Math.Pow(x, shape - 1.0)*Math.Exp(-rate*x)/SpecialFunctions.Gamma(shape);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="DensityLn"/>
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public static double PDFLn(double shape, double rate, double x)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(rate))
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{
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return x == shape ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (shape == 0.0 && rate == 0.0)
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{
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return double.NegativeInfinity;
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}
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if (shape == 1.0)
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{
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return Math.Log(rate) - (rate*x);
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}
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return (shape*Math.Log(rate)) + ((shape - 1.0)*Math.Log(x)) - (rate*x) - SpecialFunctions.GammaLn(shape);
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CumulativeDistribution"/>
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public static double CDF(double shape, double rate, double x)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (double.IsPositiveInfinity(rate))
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{
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return x >= shape ? 1.0 : 0.0;
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}
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if (shape == 0.0 && rate == 0.0)
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{
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return 0.0;
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}
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return SpecialFunctions.GammaLowerRegularized(shape, x*rate);
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}
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/// <summary>
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/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
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/// at the given probability. This is also known as the quantile or percent point function.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative density.</param>
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/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
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/// <seealso cref="InverseCumulativeDistribution"/>
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public static double InvCDF(double shape, double rate, double p)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SpecialFunctions.GammaLowerRegularizedInv(shape, p)/rate;
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}
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/// <summary>
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/// Generates a sample from the Gamma distribution.
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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="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, double shape, double rate)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(rnd, shape, rate);
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}
|
|
/// <summary>
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/// Generates a sequence of samples from the Gamma distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(System.Random rnd, double shape, double rate)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(rnd, shape, rate);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(System.Random rnd, double[] values, double shape, double rate)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(rnd, values, shape, rate);
|
}
|
|
/// <summary>
|
/// Generates a sample from the Gamma distribution.
|
/// </summary>
|
/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(double shape, double rate)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(SystemRandomSource.Default, shape, rate);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the Gamma distribution.
|
/// </summary>
|
/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(double shape, double rate)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(SystemRandomSource.Default, shape, rate);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="shape">The shape (k, α) of the Gamma distribution. Range: α ≥ 0.</param>
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/// <param name="rate">The rate or inverse scale (β) of the Gamma distribution. Range: β ≥ 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static void Samples(double[] values, double shape, double rate)
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{
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if (shape < 0.0 || rate < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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SamplesUnchecked(SystemRandomSource.Default, values, shape, rate);
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}
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}
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}
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