// <copyright file="Beta.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) 2009-2015 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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using IStation.Numerics.RootFinding;
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using IStation.Numerics.Threading;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Continuous Univariate Beta distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Beta_distribution">Wikipedia - Beta distribution</a>.
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/// </summary>
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/// <remarks>
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/// There are a few special cases for the parameterization of the Beta distribution. When both
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/// shape parameters are positive infinity, the Beta distribution degenerates to a point distribution
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/// at 0.5. When one of the shape parameters is positive infinity, the distribution degenerates to a point
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/// distribution at the positive infinity. When both shape parameters are 0.0, the Beta distribution
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/// degenerates to a Bernoulli distribution with parameter 0.5. When one shape parameter is 0.0, the
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/// distribution degenerates to a point distribution at the non-zero shape parameter.
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/// </remarks>
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public class Beta : IContinuousDistribution
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{
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System.Random _random;
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readonly double _shapeA;
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readonly double _shapeB;
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/// <summary>
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/// Initializes a new instance of the Beta class.
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/// </summary>
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/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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public Beta(double a, double b)
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{
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if (!IsValidParameterSet(a, b))
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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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_shapeA = a;
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_shapeB = b;
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}
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/// <summary>
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/// Initializes a new instance of the Beta class.
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/// </summary>
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/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta 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 Beta(double a, double b, System.Random randomSource)
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{
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if (!IsValidParameterSet(a, b))
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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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_shapeA = a;
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_shapeB = b;
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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 Beta distribution.</returns>
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public override string ToString()
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{
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return $"Beta(α = {_shapeA}, β = {_shapeB})";
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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public static bool IsValidParameterSet(double a, double b)
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{
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return a >= 0.0 && b >= 0.0;
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}
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/// <summary>
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/// Gets the α shape parameter of the Beta distribution. Range: α ≥ 0.
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/// </summary>
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public double A => _shapeA;
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/// <summary>
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/// Gets the β shape parameter of the Beta distribution. Range: β ≥ 0.
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/// </summary>
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public double B => _shapeB;
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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 Beta 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 (_shapeA == 0.0 && _shapeB == 0.0)
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{
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return 0.5;
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}
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if (_shapeA == 0.0)
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{
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return 0.0;
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}
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if (_shapeB == 0.0)
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{
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return 1.0;
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}
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if (double.IsPositiveInfinity(_shapeA) && double.IsPositiveInfinity(_shapeB))
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{
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return 0.5;
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}
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if (double.IsPositiveInfinity(_shapeA))
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{
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return 1.0;
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}
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if (double.IsPositiveInfinity(_shapeB))
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{
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return 0.0;
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}
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return _shapeA/(_shapeA + _shapeB);
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}
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}
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/// <summary>
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/// Gets the variance of the Beta distribution.
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/// </summary>
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public double Variance => (_shapeA*_shapeB)/((_shapeA + _shapeB)*(_shapeA + _shapeB)*(_shapeA + _shapeB + 1.0));
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/// <summary>
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/// Gets the standard deviation of the Beta distribution.
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/// </summary>
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public double StdDev => Math.Sqrt((_shapeA*_shapeB)/((_shapeA + _shapeB)*(_shapeA + _shapeB)*(_shapeA + _shapeB + 1.0)));
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/// <summary>
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/// Gets the entropy of the Beta 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(_shapeA) || double.IsPositiveInfinity(_shapeB))
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{
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return 0.0;
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}
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if (_shapeA == 0.0 && _shapeB == 0.0)
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{
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return -Math.Log(0.5);
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}
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if (_shapeA == 0.0 || _shapeB == 0.0)
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{
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return 0.0;
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}
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return SpecialFunctions.BetaLn(_shapeA, _shapeB)
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- ((_shapeA - 1.0)*SpecialFunctions.DiGamma(_shapeA))
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- ((_shapeB - 1.0)*SpecialFunctions.DiGamma(_shapeB))
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+ ((_shapeA + _shapeB - 2.0)*SpecialFunctions.DiGamma(_shapeA + _shapeB));
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}
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}
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/// <summary>
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/// Gets the skewness of the Beta 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(_shapeA) && double.IsPositiveInfinity(_shapeB))
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{
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return 0.0;
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}
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if (double.IsPositiveInfinity(_shapeA))
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{
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return -2.0;
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}
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if (double.IsPositiveInfinity(_shapeB))
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{
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return 2.0;
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}
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if (_shapeA == 0.0 && _shapeB == 0.0)
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{
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return 0.0;
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}
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if (_shapeA == 0.0)
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{
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return 2.0;
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}
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if (_shapeB == 0.0)
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{
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return -2.0;
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}
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return 2.0*(_shapeB - _shapeA)*Math.Sqrt(_shapeA + _shapeB + 1.0)
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/((_shapeA + _shapeB + 2.0)*Math.Sqrt(_shapeA*_shapeB));
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}
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}
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/// <summary>
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/// Gets the mode of the Beta distribution; when there are multiple answers, this routine will return 0.5.
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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 (_shapeA == 0.0 && _shapeB == 0.0)
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{
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return 0.5;
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}
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if (_shapeA == 0.0)
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{
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return 0.0;
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}
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if (_shapeB == 0.0)
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{
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return 1.0;
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}
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if (double.IsPositiveInfinity(_shapeA) && double.IsPositiveInfinity(_shapeB))
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{
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return 0.5;
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}
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if (double.IsPositiveInfinity(_shapeA))
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{
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return 1.0;
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}
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if (double.IsPositiveInfinity(_shapeB))
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{
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return 0.0;
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}
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if (_shapeA == 1.0 && _shapeB == 1.0)
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{
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return 0.5;
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}
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return (_shapeA - 1)/(_shapeA + _shapeB - 2);
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}
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}
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/// <summary>
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/// Gets the median of the Beta 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 Beta 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 Beta distribution.
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/// </summary>
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public double Maximum => 1.0;
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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(_shapeA, _shapeB, 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(_shapeA, _shapeB, 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(_shapeA, _shapeB, 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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/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
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public double InverseCumulativeDistribution(double p)
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{
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return InvCDF(_shapeA, _shapeB, p);
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}
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/// <summary>
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/// Generates a sample from the Beta 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, _shapeA, _shapeB);
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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, _shapeA, _shapeB);
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}
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/// <summary>
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/// Generates a sequence of samples from the Beta 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, _shapeA, _shapeB);
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}
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/// <summary>
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/// Samples Beta distributed random variables by sampling two Gamma variables and normalizing.
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
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/// <returns>a random number from the Beta distribution.</returns>
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internal static double SampleUnchecked(System.Random rnd, double a, double b)
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{
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var x = Gamma.SampleUnchecked(rnd, a, 1.0);
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var y = Gamma.SampleUnchecked(rnd, b, 1.0);
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return x/(x + y);
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double a, double b)
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{
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var y = new double[values.Length];
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Gamma.SamplesUnchecked(rnd, values, a, 1.0);
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Gamma.SamplesUnchecked(rnd, y, b, 1.0);
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CommonParallel.For(0, values.Length, 4096, (aa, bb) =>
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{
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for (int i = aa; i < bb; i++)
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{
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values[i] = values[i]/(values[i] + y[i]);
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}
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});
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double a, double b)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, a, b);
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta 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 a, double b, double x)
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{
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if (a < 0.0 || b < 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 (x < 0.0 || x > 1.0)
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{
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return 0.0;
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}
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if (double.IsPositiveInfinity(a) && double.IsPositiveInfinity(b))
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{
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return x == 0.5 ? double.PositiveInfinity : 0.0;
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}
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if (double.IsPositiveInfinity(a))
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{
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return x == 1.0 ? double.PositiveInfinity : 0.0;
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}
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if (double.IsPositiveInfinity(b))
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{
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return x == 0.0 ? double.PositiveInfinity : 0.0;
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}
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if (a == 0.0 && b == 0.0)
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{
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if (x == 0.0 || x == 1.0)
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{
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return double.PositiveInfinity;
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}
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return 0.0;
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}
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if (a == 0.0)
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{
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return x == 0.0 ? double.PositiveInfinity : 0.0;
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}
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if (b == 0.0)
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{
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return x == 1.0 ? double.PositiveInfinity : 0.0;
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}
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if (a == 1.0 && b == 1.0)
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{
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return 1.0;
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}
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if (a > 80.0 || b > 80.0)
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{
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return Math.Exp(PDFLn(a, b, x));
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}
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var bb = SpecialFunctions.Gamma(a + b)/(SpecialFunctions.Gamma(a)*SpecialFunctions.Gamma(b));
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return bb*Math.Pow(x, a - 1.0)*Math.Pow(1.0 - x, b - 1.0);
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta 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 a, double b, double x)
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{
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if (a < 0.0 || b < 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 (x < 0.0 || x > 1.0)
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{
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return double.NegativeInfinity;
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}
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if (double.IsPositiveInfinity(a) && double.IsPositiveInfinity(b))
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{
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return x == 0.5 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (double.IsPositiveInfinity(a))
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{
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return x == 1.0 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (double.IsPositiveInfinity(b))
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{
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return x == 0.0 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (a == 0.0 && b == 0.0)
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{
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return x == 0.0 || x == 1.0 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (a == 0.0)
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{
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return x == 0.0 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (b == 0.0)
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{
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return x == 1.0 ? double.PositiveInfinity : double.NegativeInfinity;
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}
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if (a == 1.0 && b == 1.0)
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{
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return 0.0;
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}
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var aa = SpecialFunctions.GammaLn(a + b) - SpecialFunctions.GammaLn(a) - SpecialFunctions.GammaLn(b);
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var bb = x == 0.0 ? (a == 1.0 ? 0.0 : double.NegativeInfinity) : (a - 1.0)*Math.Log(x);
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var cc = x == 1.0 ? (b == 1.0 ? 0.0 : double.NegativeInfinity) : (b - 1.0)*Math.Log(1.0 - x);
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return aa + bb + cc;
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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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
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/// <param name="b">The β shape parameter of the Beta 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 a, double b, double x)
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{
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if (a < 0.0 || b < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
|
|
if (x < 0.0)
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{
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return 0.0;
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}
|
|
if (x >= 1.0)
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{
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return 1.0;
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}
|
|
if (double.IsPositiveInfinity(a) && double.IsPositiveInfinity(b))
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{
|
return x < 0.5 ? 0.0 : 1.0;
|
}
|
|
if (double.IsPositiveInfinity(a))
|
{
|
return x < 1.0 ? 0.0 : 1.0;
|
}
|
|
if (double.IsPositiveInfinity(b))
|
{
|
return x >= 0.0 ? 1.0 : 0.0;
|
}
|
|
if (a == 0.0 && b == 0.0)
|
{
|
if (x >= 0.0 && x < 1.0)
|
{
|
return 0.5;
|
}
|
|
return 1.0;
|
}
|
|
if (a == 0.0)
|
{
|
return 1.0;
|
}
|
|
if (b == 0.0)
|
{
|
return x >= 1.0 ? 1.0 : 0.0;
|
}
|
|
if (a == 1.0 && b == 1.0)
|
{
|
return x;
|
}
|
|
return SpecialFunctions.BetaRegularized(a, b, x);
|
}
|
|
/// <summary>
|
/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
|
/// at the given probability. This is also known as the quantile or percent point function.
|
/// </summary>
|
/// <param name="p">The location at which to compute the inverse cumulative density.</param>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
|
/// <seealso cref="InverseCumulativeDistribution"/>
|
/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
|
public static double InvCDF(double a, double b, double p)
|
{
|
if (a < 0.0 || b < 0.0 || p < 0.0 || p > 1.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return Brent.FindRoot(x => SpecialFunctions.BetaRegularized(a, b, x) - p, 0.0, 1.0, accuracy: 1e-12);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(System.Random rnd, double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(rnd, a, b);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(System.Random rnd, double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(rnd, a, b);
|
}
|
|
/// <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="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(System.Random rnd, double[] values, double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(rnd, values, a, b);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(SystemRandomSource.Default, a, b);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(SystemRandomSource.Default, a, b);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="a">The α shape parameter of the Beta distribution. Range: α ≥ 0.</param>
|
/// <param name="b">The β shape parameter of the Beta distribution. Range: β ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(double[] values, double a, double b)
|
{
|
if (a < 0.0 || b < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(SystemRandomSource.Default, values, a, b);
|
}
|
}
|
}
|
