// <copyright file="ConwayMaxwellPoisson.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2014 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using System;
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using System.Collections.Generic;
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using IStation.Numerics.Random;
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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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/// Discrete Univariate Conway-Maxwell-Poisson distribution.
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/// <para>The Conway-Maxwell-Poisson distribution is a generalization of the Poisson, Geometric and Bernoulli
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/// distributions. It is parameterized by two real numbers "lambda" and "nu". For
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/// <list>
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/// <item>nu = 0 the distribution reverts to a Geometric distribution</item>
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/// <item>nu = 1 the distribution reverts to the Poisson distribution</item>
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/// <item>nu -> infinity the distribution converges to a Bernoulli distribution</item>
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/// </list></para>
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/// This implementation will cache the value of the normalization constant.
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/// <a href="http://en.wikipedia.org/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution">Wikipedia - ConwayMaxwellPoisson distribution</a>.
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/// </summary>
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public class ConwayMaxwellPoisson : IDiscreteDistribution
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{
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System.Random _random;
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readonly double _lambda;
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readonly double _nu;
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/// <summary>
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/// The mean of the distribution.
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/// </summary>
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double _mean = double.MinValue;
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/// <summary>
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/// The variance of the distribution.
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/// </summary>
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double _variance = double.MinValue;
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/// <summary>
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/// Caches the value of the normalization constant.
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/// </summary>
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double _z = double.MinValue;
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/// <summary>
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/// Since many properties of the distribution can only be computed approximately, the tolerance
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/// level specifies how much error we accept.
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/// </summary>
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const double Tolerance = 1e-12;
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/// <summary>
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/// Initializes a new instance of the <see cref="ConwayMaxwellPoisson"/> class.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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public ConwayMaxwellPoisson(double lambda, double nu)
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{
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if (!IsValidParameterSet(lambda, nu))
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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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_lambda = lambda;
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_nu = nu;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="ConwayMaxwellPoisson"/> class.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. 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 ConwayMaxwellPoisson(double lambda, double nu, System.Random randomSource)
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{
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if (!IsValidParameterSet(lambda, nu))
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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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_lambda = lambda;
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_nu = nu;
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}
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/// <summary>
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/// Returns a <see cref="System.String"/> that represents this instance.
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/// </summary>
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/// <returns>A <see cref="System.String"/> that represents this instance.</returns>
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public override string ToString()
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{
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return $"ConwayMaxwellPoisson(λ = {_lambda}, ν = {_nu})";
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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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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public static bool IsValidParameterSet(double lambda, double nu)
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{
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return lambda > 0.0 && nu >= 0.0;
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}
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/// <summary>
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/// Gets the lambda (λ) parameter. Range: λ > 0.
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/// </summary>
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public double Lambda => _lambda;
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/// <summary>
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/// Gets the rate of decay (ν) parameter. Range: ν ≥ 0.
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/// </summary>
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public double Nu => _nu;
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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 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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// Special case requiring no computation.
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if (_lambda == 0)
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{
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return 0.0;
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}
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if (_mean != double.MinValue)
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{
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return _mean;
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}
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// The normalization constant for the distribution.
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var z = 1 + _lambda;
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// The probability of the next term.
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var a1 = _lambda*_lambda/Math.Pow(2, _nu);
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// The unnormalized mean.
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var zx = _lambda;
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// The contribution of the next term to the mean.
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var ax1 = 2*a1;
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for (var i = 3; i < 1000; i++)
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{
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var e = _lambda/Math.Pow(i, _nu);
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var ex = _lambda/Math.Pow(i, _nu - 1)/(i - 1);
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var a2 = a1*e;
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var ax2 = ax1*ex;
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if ((ax2 < ax1) && (a2 < a1))
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{
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var m = zx/z;
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var upper = (zx + (ax1/(1 - (ax2/ax1))))/z;
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var lower = zx/(z + (a1/(1 - (a2/a1))));
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var r = (upper - lower)/m;
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if (r < Tolerance)
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{
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break;
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}
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}
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z = z + a1;
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zx = zx + ax1;
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a1 = a2;
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ax1 = ax2;
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}
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_mean = zx/z;
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return _mean;
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}
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}
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/// <summary>
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/// Gets the variance of the 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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// Special case requiring no computation.
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if (_lambda == 0)
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{
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return 0.0;
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}
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if (_variance != double.MinValue)
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{
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return _variance;
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}
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// The normalization constant for the distribution.
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var z = 1 + _lambda;
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// The probability of the next term.
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var a1 = _lambda*_lambda/Math.Pow(2, _nu);
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// The unnormalized second moment.
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var zxx = _lambda;
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// The contribution of the next term to the second moment.
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var axx1 = 4*a1;
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for (var i = 3; i < 1000; i++)
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{
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var e = _lambda/Math.Pow(i, _nu);
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var exx = _lambda/Math.Pow(i, _nu - 2)/(i - 1)/(i - 1);
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var a2 = a1*e;
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var axx2 = axx1*exx;
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if ((axx2 < axx1) && (a2 < a1))
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{
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var m = zxx/z;
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var upper = (zxx + (axx1/(1 - (axx2/axx1))))/z;
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var lower = zxx/(z + (a1/(1 - (a2/a1))));
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var r = (upper - lower)/m;
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if (r < Tolerance)
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{
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break;
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}
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}
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z = z + a1;
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zxx = zxx + axx1;
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a1 = a2;
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axx1 = axx2;
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}
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var mean = Mean;
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_variance = (zxx/z) - (mean*mean);
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return _variance;
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}
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}
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/// <summary>
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/// Gets the standard deviation of the 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 entropy of the distribution.
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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 distribution.
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/// </summary>
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public double Skewness => throw new NotSupportedException();
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/// <summary>
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/// Gets the mode of the distribution
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/// </summary>
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public int Mode => throw new NotSupportedException();
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/// <summary>
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/// Gets the median of the 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 smallest element in the domain of the distributions which can be represented by an integer.
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/// </summary>
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public int Minimum => 0;
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/// <summary>
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/// Gets the largest element in the domain of the distributions which can be represented by an integer.
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/// </summary>
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public int Maximum => throw new NotSupportedException();
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/// <summary>
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/// Computes the probability mass (PMF) at k, i.e. P(X = k).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the probability mass function.</param>
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/// <returns>the probability mass at location <paramref name="k"/>.</returns>
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public double Probability(int k)
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{
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return Math.Pow(_lambda, k)/Math.Pow(SpecialFunctions.Factorial(k), _nu)/Z;
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}
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/// <summary>
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/// Computes the log probability mass (lnPMF) at k, i.e. ln(P(X = k)).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the log probability mass function.</param>
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/// <returns>the log probability mass at location <paramref name="k"/>.</returns>
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public double ProbabilityLn(int k)
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{
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return k*Math.Log(_lambda) - _nu*SpecialFunctions.FactorialLn(k) - Math.Log(Z);
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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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var z = Z;
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double sum = 0;
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for (var i = 0; i < x + 1; i++)
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{
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sum += Math.Pow(_lambda, i)/Math.Pow(SpecialFunctions.Factorial(i), _nu)/z;
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}
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return sum;
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}
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/// <summary>
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/// Computes the probability mass (PMF) at k, i.e. P(X = k).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the probability mass function.</param>
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/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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/// <returns>the probability mass at location <paramref name="k"/>.</returns>
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public static double PMF(double lambda, double nu, int k)
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{
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if (!(lambda > 0.0 && nu >= 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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var z = Normalization(lambda, nu);
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return Math.Pow(lambda, k)/Math.Pow(SpecialFunctions.Factorial(k), nu)/z;
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}
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/// <summary>
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/// Computes the log probability mass (lnPMF) at k, i.e. ln(P(X = k)).
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/// </summary>
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/// <param name="k">The location in the domain where we want to evaluate the log probability mass function.</param>
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/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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/// <returns>the log probability mass at location <paramref name="k"/>.</returns>
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public static double PMFLn(double lambda, double nu, int k)
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{
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if (!(lambda > 0.0 && nu >= 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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var z = Normalization(lambda, nu);
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return k*Math.Log(lambda) - nu*SpecialFunctions.FactorialLn(k) - Math.Log(z);
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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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. 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 lambda, double nu, double x)
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{
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if (!(lambda > 0.0 && nu >= 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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var z = Normalization(lambda, nu);
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double sum = 0;
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for (var i = 0; i < x + 1; i++)
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{
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sum += Math.Pow(lambda, i)/Math.Pow(SpecialFunctions.Factorial(i), nu)/z;
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}
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return sum;
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}
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/// <summary>
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/// Gets the normalization constant of the Conway-Maxwell-Poisson distribution.
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/// </summary>
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double Z
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{
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get
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{
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if (_z != double.MinValue)
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{
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return _z;
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}
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_z = Normalization(_lambda, _nu);
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return _z;
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}
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}
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/// <summary>
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/// Computes an approximate normalization constant for the CMP distribution.
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/// </summary>
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/// <param name="lambda">The lambda (λ) parameter for the CMP distribution.</param>
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/// <param name="nu">The rate of decay (ν) parameter for the CMP distribution.</param>
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/// <returns>
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/// an approximate normalization constant for the CMP distribution.
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/// </returns>
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static double Normalization(double lambda, double nu)
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{
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// Initialize Z with the first two terms.
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var z = 1.0 + lambda;
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// Remembers the last term added.
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var t = lambda;
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// Start adding more terms until convergence.
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for (var i = 2; i < 1000; i++)
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{
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// The new addition for term i.
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var e = lambda/Math.Pow(i, nu);
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// The new term.
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t = t*e;
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// The updated normalization constant.
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z = z + t;
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// The stopping criterion.
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if (e < 1)
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{
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if (t/(1 - e)/z < Tolerance)
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{
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break;
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}
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}
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}
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return z;
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}
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/// <summary>
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/// Returns one trials 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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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/// <param name="z">The z parameter.</param>
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/// <returns>
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/// One sample from the distribution implied by <paramref name="lambda"/>, <paramref name="nu"/>, and <paramref name="z"/>.
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/// </returns>
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static int SampleUnchecked(System.Random rnd, double lambda, double nu, double z)
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{
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var u = rnd.NextDouble();
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var p = 1.0/z;
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var cdf = p;
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var i = 0;
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while (u > cdf)
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{
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i++;
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p = p*lambda/Math.Pow(i, nu);
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cdf += p;
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}
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return i;
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}
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static void SamplesUnchecked(System.Random rnd, int[] values, double lambda, double nu, double z)
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{
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var uniform = rnd.NextDoubles(values.Length);
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CommonParallel.For(0, values.Length, 4096, (a, b) =>
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{
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for (int i = a; i < b; i++)
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{
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var u = uniform[i];
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var p = 1.0/z;
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var cdf = p;
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var k = 0;
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while (u > cdf)
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{
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k++;
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p = p*lambda/Math.Pow(k, nu);
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cdf += p;
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}
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values[i] = k;
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}
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});
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}
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static IEnumerable<int> SamplesUnchecked(System.Random rnd, double lambda, double nu, double z)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, lambda, nu, z);
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}
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}
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/// <summary>
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/// Samples a Conway-Maxwell-Poisson distributed random variable.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public int Sample()
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{
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return SampleUnchecked(_random, _lambda, _nu, Z);
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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(int[] values)
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{
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SamplesUnchecked(_random, values, _lambda, _nu, Z);
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}
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/// <summary>
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/// Samples a sequence of a Conway-Maxwell-Poisson distributed random variables.
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/// </summary>
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/// <returns>
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/// a sequence of samples from a Conway-Maxwell-Poisson distribution.
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/// </returns>
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public IEnumerable<int> Samples()
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{
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return SamplesUnchecked(_random, _lambda, _nu, Z);
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}
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/// <summary>
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/// Samples a random variable.
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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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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public static int Sample(System.Random rnd, double lambda, double nu)
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{
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if (!(lambda > 0.0 && nu >= 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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var z = Normalization(lambda, nu);
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return SampleUnchecked(rnd, lambda, nu, z);
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}
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/// <summary>
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/// Samples a sequence of this random variable.
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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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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public static IEnumerable<int> Samples(System.Random rnd, double lambda, double nu)
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{
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if (!(lambda > 0.0 && nu >= 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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var z = Normalization(lambda, nu);
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return SamplesUnchecked(rnd, lambda, nu, z);
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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="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
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/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
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public static void Samples(System.Random rnd, int[] values, double lambda, double nu)
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{
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if (!(lambda > 0.0 && nu >= 0.0))
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{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
var z = Normalization(lambda, nu);
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SamplesUnchecked(rnd, values, lambda, nu, z);
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}
|
|
/// <summary>
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/// Samples a random variable.
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/// </summary>
|
/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
|
/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
|
public static int Sample(double lambda, double nu)
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{
|
if (!(lambda > 0.0 && nu >= 0.0))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
var z = Normalization(lambda, nu);
|
return SampleUnchecked(SystemRandomSource.Default, lambda, nu, z);
|
}
|
|
/// <summary>
|
/// Samples a sequence of this random variable.
|
/// </summary>
|
/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
|
/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
|
public static IEnumerable<int> Samples(double lambda, double nu)
|
{
|
if (!(lambda > 0.0 && nu >= 0.0))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
var z = Normalization(lambda, nu);
|
return SamplesUnchecked(SystemRandomSource.Default, lambda, nu, z);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="lambda">The lambda (λ) parameter. Range: λ > 0.</param>
|
/// <param name="nu">The rate of decay (ν) parameter. Range: ν ≥ 0.</param>
|
public static void Samples(int[] values, double lambda, double nu)
|
{
|
if (!(lambda > 0.0 && nu >= 0.0))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
var z = Normalization(lambda, nu);
|
SamplesUnchecked(SystemRandomSource.Default, values, lambda, nu, z);
|
}
|
}
|
}
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