// <copyright file="Burr.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) 2009-2019 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 IStation.Numerics.Random;
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using System;
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using System.Collections.Generic;
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namespace IStation.Numerics.Distributions
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{
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public class Burr : IContinuousDistribution
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{
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System.Random _random;
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/// <summary>
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/// Gets the scale (a) of the distribution. Range: a > 0.
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/// </summary>
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public double a { get; }
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/// <summary>
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/// Gets the first shape parameter (c) of the distribution. Range: c > 0.
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/// </summary>
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public double c { get; }
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/// <summary>
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/// Gets the second shape parameter (k) of the distribution. Range: k > 0.
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/// </summary>
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public double k { get; }
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/// <summary>
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/// Initializes a new instance of the Burr Type XII class.
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/// </summary>
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/// <param name="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 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 Burr(double a, double c, double k, System.Random randomSource = null)
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{
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if (!IsValidParameterSet(a, c, k))
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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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this.a = a;
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this.c = c;
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this.k = k;
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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 $"Burr(a = {a}, c = {c}, k = {k})";
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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 scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 0.</param>
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public static bool IsValidParameterSet(double a, double c, double k)
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{
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var allFinite = a.IsFinite() && c.IsFinite() && k.IsFinite();
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return allFinite && a > 0.0 && c > 0.0 && k > 0.0;
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}
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/// <summary>
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/// Gets 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 Burr distribution.
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/// </summary>
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public double Mean => (1 / SpecialFunctions.Gamma(k)) * a * SpecialFunctions.Gamma(1 + 1 / c) * SpecialFunctions.Gamma(k - 1 / c);
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/// <summary>
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/// Gets the variance of the Burr distribution.
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/// </summary>
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public double Variance =>
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(1 / SpecialFunctions.Gamma(k)) * Math.Pow(a, 2) * SpecialFunctions.Gamma(1 + 2 / c) * SpecialFunctions.Gamma(k - 2 / c)
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- Math.Pow((1 / SpecialFunctions.Gamma(k)) * a * SpecialFunctions.Gamma(1 + 1 / c) * SpecialFunctions.Gamma(k - 1 / c), 2);
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/// <summary>
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/// Gets the standard deviation of the Burr distribution.
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/// </summary>
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public double StdDev => Math.Sqrt(Variance);
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/// <summary>
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/// Gets the mode of the Burr distribution.
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/// </summary>
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public double Mode => a * Math.Pow((c - 1) / (c * k + 1), 1 / c);
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/// <summary>
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/// Gets the minimum of the Burr 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 Burr distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Gets the entropy of the Burr distribution (currently not supported).
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/// </summary>
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public double Entropy => throw new NotSupportedException();
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/// <summary>
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/// Gets the skewness of the Burr 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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var mean = Mean;
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var variance = Variance;
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var std = StdDev;
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return (GetMoment(3) - 3 * mean * variance - mean * mean * mean) / (std * std * std);
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}
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}
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/// <summary>
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/// Gets the median of the Burr distribution.
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/// </summary>
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public double Median => a * Math.Pow(Math.Pow(2, 1 / k) - 1, 1 / c);
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/// <summary>
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/// Generates a sample from the Burr 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, a, c, k);
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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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/// <param name="values">The array to fill with the samples.</param>
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public void Samples(double[] values)
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{
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SamplesUnchecked(_random, values, a, c, k);
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}
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/// <summary>
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/// Generates a sequence of samples from the Burr 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, a, c, k);
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}
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/// <summary>
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/// Generates a sample from the Burr 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="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 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 a, double c, double k)
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{
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if (!IsValidParameterSet(a, c, k))
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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 SampleUnchecked(rnd, a, c, k);
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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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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="values">The array to fill with the samples.</param>
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/// <param name="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 0.</param>
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public static void Samples(System.Random rnd, double[] values, double a, double c, double k)
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{
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if (!IsValidParameterSet(a, c, k))
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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(rnd, values, a, c, k);
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}
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/// <summary>
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/// Generates a sequence of samples from the Burr 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="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(System.Random rnd, double a, double c, double k)
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{
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if (!IsValidParameterSet(a, c, k))
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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 SamplesUnchecked(rnd, a, c, k);
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}
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internal static double SampleUnchecked(System.Random rnd, double a, double c, double k)
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{
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var k_inv = 1 / k;
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var c_inv = 1 / c;
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double u = rnd.NextDouble();
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return a * Math.Pow(Math.Pow(1 - u, -k_inv) - 1, c_inv);
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double a, double c, double k)
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{
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if (values.Length == 0)
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{
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return;
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}
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var k_inv = 1 / k;
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var c_inv = 1 / c;
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double[] u = rnd.NextDoubles(values.Length);
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for (var j = 0; j < values.Length; ++j)
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{
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values[j] = a * Math.Pow(Math.Pow(1 - u[j], -k_inv) - 1, c_inv);
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}
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}
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internal static IEnumerable<double> SamplesUnchecked(System.Random rnd, double a, double c, double k)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, a, c, k);
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}
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}
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/// <summary>
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/// Gets the n-th raw moment of the distribution.
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/// </summary>
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/// <param name="n">The order (n) of the moment. Range: n ≥ 1.</param>
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/// <returns>the n-th moment of the distribution.</returns>
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public double GetMoment(double n)
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{
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if (n > k * c)
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{
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throw new ArgumentException("The chosen parameter set is invalid (probably some value is out of range).");
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}
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var lambda_n = (n / c) * SpecialFunctions.Gamma(n / c) * SpecialFunctions.Gamma(k - n / c);
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return Math.Pow(a, n) * lambda_n / SpecialFunctions.Gamma(k);
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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="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 DensityImpl(a, c, k, 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 DensityLnImpl(a, c, k, 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 CumulativeDistributionImpl(a, c, k, x);
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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 scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 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 c, double k, double x)
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{
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if (!IsValidParameterSet(a, c, k))
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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 DensityImpl(a, c, k, 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="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 0.</param>
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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="DensityLn"/>
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public static double PDFLn(double a, double c, double k, double x)
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{
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if (!IsValidParameterSet(a, c, k))
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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 DensityLnImpl(a, c, k, 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="a">The scale parameter a of the Burr distribution. Range: a > 0.</param>
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/// <param name="c">The first shape parameter c of the Burr distribution. Range: c > 0.</param>
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/// <param name="k">The second shape parameter k of the Burr distribution. Range: k > 0.</param>
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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="CumulativeDistribution"/>
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public static double CDF(double a, double c, double k, double x)
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{
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if (!IsValidParameterSet(a, c, k))
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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 CumulativeDistributionImpl(a, c, k, x);
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}
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internal static double DensityImpl(double a, double c, double k, double x)
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{
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var numerator = (k * c / a) * Math.Pow(x / a, c - 1);
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var denominator = Math.Pow(1 + Math.Pow(x / a, c), k + 1);
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return numerator / denominator;
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}
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internal static double DensityLnImpl(double a, double c, double k, double x)
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{
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return Math.Log(DensityImpl(a, c, k, x));
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}
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internal static double CumulativeDistributionImpl(double a, double c, double k, double x)
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{
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var denominator = Math.Pow(1 + Math.Pow(x / a, c), k);
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return 1 - 1 / denominator;
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}
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}
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}
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