// <copyright file="StudentT.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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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// Software is furnished to do so, subject to the following
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// conditions:
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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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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Continuous Univariate Student's T-distribution.
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/// Implements the univariate Student t-distribution. For details about this
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/// distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Student%27s_t-distribution">
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/// Wikipedia - Student's t-distribution</a>.
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/// </summary>
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/// <remarks><para>We use a slightly generalized version (compared to
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/// Wikipedia) of the Student t-distribution. Namely, one which also
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/// parameterizes the location and scale. See the book "Bayesian Data
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/// Analysis" by Gelman et al. for more details.</para>
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/// <para>The density of the Student t-distribution p(x|mu,scale,dof) =
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/// Gamma((dof+1)/2) (1 + (x - mu)^2 / (scale * scale * dof))^(-(dof+1)/2) /
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/// (Gamma(dof/2)*Sqrt(dof*pi*scale)).</para>
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/// <para>The distribution will use the <see cref="System.Random"/> by
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/// default. Users can get/set the random number generator by using the
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/// <see cref="RandomSource"/> property.</para>
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/// <para>The statistics classes will check all the incoming parameters
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/// whether they are in the allowed range. This might involve heavy
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/// computation. Optionally, by setting Control.CheckDistributionParameters
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/// to <c>false</c>, all parameter checks can be turned off.</para></remarks>
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public class StudentT : IContinuousDistribution
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{
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System.Random _random;
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readonly double _location;
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readonly double _scale;
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readonly double _freedom;
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/// <summary>
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/// Initializes a new instance of the StudentT class. This is a Student t-distribution with location 0.0
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/// scale 1.0 and degrees of freedom 1.
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/// </summary>
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public StudentT()
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{
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_random = SystemRandomSource.Default;
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_location = 0.0;
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_scale = 1.0;
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_freedom = 1.0;
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}
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/// <summary>
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/// Initializes a new instance of the StudentT class with a particular location, scale and degrees of
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/// freedom.
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/// </summary>
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/// <param name="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
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public StudentT(double location, double scale, double freedom)
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{
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if (!IsValidParameterSet(location, scale, freedom))
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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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_location = location;
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_scale = scale;
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_freedom = freedom;
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}
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/// <summary>
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/// Initializes a new instance of the StudentT class with a particular location, scale and degrees of
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/// freedom.
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/// </summary>
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/// <param name="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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 StudentT(double location, double scale, double freedom, System.Random randomSource)
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{
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if (!IsValidParameterSet(location, scale, freedom))
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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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_location = location;
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_scale = scale;
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_freedom = freedom;
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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 $"StudentT(μ = {_location}, σ = {_scale}, ν = {_freedom})";
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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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
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public static bool IsValidParameterSet(double location, double scale, double freedom)
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{
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return scale > 0.0 && freedom > 0.0 && !double.IsNaN(location);
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}
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/// <summary>
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/// Gets the location (μ) of the Student t-distribution.
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/// </summary>
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public double Location => _location;
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/// <summary>
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/// Gets the scale (σ) of the Student t-distribution. Range: σ > 0.
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/// </summary>
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public double Scale => _scale;
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/// <summary>
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/// Gets the degrees of freedom (ν) of the Student t-distribution. Range: ν > 0.
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/// </summary>
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public double DegreesOfFreedom => _freedom;
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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 Student t-distribution.
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/// </summary>
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public double Mean => _freedom > 1.0 ? _location : double.NaN;
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/// <summary>
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/// Gets the variance of the Student t-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(_freedom))
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{
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return _scale*_scale;
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}
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if (_freedom > 2.0)
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{
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return _freedom*_scale*_scale/(_freedom - 2.0);
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}
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return _freedom > 1.0 ? double.PositiveInfinity : double.NaN;
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}
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}
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/// <summary>
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/// Gets the standard deviation of the Student t-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(_freedom))
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{
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return Math.Sqrt(_scale*_scale);
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}
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if (_freedom > 2.0)
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{
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return Math.Sqrt(_freedom*_scale*_scale/(_freedom - 2.0));
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}
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return _freedom > 1.0 ? double.PositiveInfinity : double.NaN;
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}
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}
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/// <summary>
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/// Gets the entropy of the Student t-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 (_location != 0 || _scale != 1.0)
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{
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throw new NotSupportedException();
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}
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return (((_freedom + 1.0)/2.0)*(SpecialFunctions.DiGamma((1.0 + _freedom)/2.0) - SpecialFunctions.DiGamma(_freedom/2.0)))
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+ Math.Log(Math.Sqrt(_freedom)*SpecialFunctions.Beta(_freedom/2.0, 1.0/2.0));
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}
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}
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/// <summary>
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/// Gets the skewness of the Student t-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 (_freedom <= 3)
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{
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throw new NotSupportedException();
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}
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return 0.0;
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}
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}
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/// <summary>
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/// Gets the mode of the Student t-distribution.
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/// </summary>
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public double Mode => _location;
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/// <summary>
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/// Gets the median of the Student t-distribution.
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/// </summary>
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public double Median => _location;
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/// <summary>
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/// Gets the minimum of the Student t-distribution.
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/// </summary>
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public double Minimum => double.NegativeInfinity;
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/// <summary>
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/// Gets the maximum of the Student t-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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public double Density(double x)
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{
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return PDF(_location, _scale, _freedom, 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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public double DensityLn(double x)
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{
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return PDFLn(_location, _scale, _freedom, 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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public double CumulativeDistribution(double x)
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{
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return CDF(_location, _scale, _freedom, 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(_location, _scale, _freedom, p);
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}
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/// <summary>
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/// Samples student-t distributed random variables.
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/// </summary>
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/// <remarks>The algorithm is method 2 in section 5, chapter 9
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/// in L. Devroye's "Non-Uniform Random Variate Generation"</remarks>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
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/// <returns>a random number from the standard student-t distribution.</returns>
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static double SampleUnchecked(System.Random rnd, double location, double scale, double freedom)
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{
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var gamma = Gamma.SampleUnchecked(rnd, 0.5*freedom, 0.5);
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return Normal.Sample(rnd, location, scale*Math.Sqrt(freedom/gamma));
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double location, double scale, double freedom)
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{
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Gamma.SamplesUnchecked(rnd, values, 0.5*freedom, 0.5);
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for (int i = 0; i < values.Length; i++)
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{
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values[i] = Normal.Sample(rnd, location, scale*Math.Sqrt(freedom/values[i]));
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}
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double location, double scale, double freedom)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, location, scale, freedom);
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}
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}
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/// <summary>
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/// Generates a sample from the Student t-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, _location, _scale, _freedom);
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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, _location, _scale, _freedom);
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}
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/// <summary>
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/// Generates a sequence of samples from the Student t-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, _location, _scale, _freedom);
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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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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 location, double scale, double freedom, double x)
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{
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if (scale <= 0.0 || freedom <= 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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// TODO JVG we can probably do a better job for Cauchy special case
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if (freedom >= 1e+8d)
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{
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return Normal.PDF(location, scale, x);
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}
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var d = (x - location)/scale;
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return Math.Exp(SpecialFunctions.GammaLn((freedom + 1.0)/2.0) - SpecialFunctions.GammaLn(freedom/2.0))
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*Math.Pow(1.0 + (d*d/freedom), -0.5*(freedom + 1.0))
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/Math.Sqrt(freedom*Math.PI)
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/scale;
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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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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 location, double scale, double freedom, double x)
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{
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if (scale <= 0.0 || freedom <= 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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// TODO JVG we can probably do a better job for Cauchy special case
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if (freedom >= 1e+8d)
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{
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return Normal.PDFLn(location, scale, x);
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}
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var d = (x - location)/scale;
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return SpecialFunctions.GammaLn((freedom + 1.0)/2.0)
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- (0.5*((freedom + 1.0)*Math.Log(1.0 + (d*d/freedom))))
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- SpecialFunctions.GammaLn(freedom/2.0)
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- (0.5*Math.Log(freedom*Math.PI)) - Math.Log(scale);
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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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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 location, double scale, double freedom, double x)
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{
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if (scale <= 0.0 || freedom <= 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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// TODO JVG we can probably do a better job for Cauchy special case
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if (double.IsPositiveInfinity(freedom))
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{
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return Normal.CDF(location, scale, x);
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}
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var k = (x - location)/scale;
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var h = freedom/(freedom + (k*k));
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var ib = 0.5*SpecialFunctions.BetaRegularized(freedom/2.0, 0.5, h);
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return x <= location ? ib : 1.0 - ib;
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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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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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/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
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public static double InvCDF(double location, double scale, double freedom, double p)
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{
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if (scale <= 0.0 || freedom <= 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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// TODO JVG we can probably do a better job for Cauchy special case
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if (double.IsPositiveInfinity(freedom))
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{
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return Normal.InvCDF(location, scale, p);
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}
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if (p == 0.5d)
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{
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return location;
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}
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// TODO PERF: We must implement this explicitly instead of solving for CDF^-1
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return Brent.FindRoot(x =>
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{
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var k = (x - location)/scale;
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var h = freedom/(freedom + (k*k));
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var ib = 0.5*SpecialFunctions.BetaRegularized(freedom/2.0, 0.5, h);
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return x <= location ? ib - p : 1.0 - ib - p;
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}, -800, 800, accuracy: 1e-12);
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}
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/// <summary>
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/// Generates a sample from the Student t-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="location">The location (μ) of the distribution.</param>
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/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, double location, double scale, double freedom)
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{
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if (scale <= 0.0 || freedom <= 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
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return SampleUnchecked(rnd, location, scale, freedom);
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}
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/// <summary>
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/// Generates a sequence of samples from the Student t-distribution using the <i>Box-Muller</i> algorithm.
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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="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(System.Random rnd, double location, double scale, double freedom)
|
{
|
if (scale <= 0.0 || freedom <= 0.0)
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{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(rnd, location, scale, freedom);
|
}
|
|
/// <summary>
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/// 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="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(System.Random rnd, double[] values, double location, double scale, double freedom)
|
{
|
if (scale <= 0.0 || freedom <= 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(rnd, values, location, scale, freedom);
|
}
|
|
/// <summary>
|
/// Generates a sample from the Student t-distribution.
|
/// </summary>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(double location, double scale, double freedom)
|
{
|
if (scale <= 0.0 || freedom <= 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(SystemRandomSource.Default, location, scale, freedom);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the Student t-distribution using the <i>Box-Muller</i> algorithm.
|
/// </summary>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(double location, double scale, double freedom)
|
{
|
if (scale <= 0.0 || freedom <= 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(SystemRandomSource.Default, location, scale, freedom);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
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/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 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 location, double scale, double freedom)
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{
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if (scale <= 0.0 || freedom <= 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, location, scale, freedom);
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}
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}
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}
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