// <copyright file="LogNormal.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) 2009-2013 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 System.Linq;
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using IStation.Numerics.Random;
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using IStation.Numerics.Statistics;
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using IStation.Numerics.Threading;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Continuous Univariate Log-Normal distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Log-normal_distribution">Wikipedia - Log-Normal distribution</a>.
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/// </summary>
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public class LogNormal : IContinuousDistribution
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{
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System.Random _random;
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readonly double _mu;
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readonly double _sigma;
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/// <summary>
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/// Initializes a new instance of the <see cref="LogNormal"/> class.
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/// The distribution will be initialized with the default <seealso cref="System.Random"/>
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/// random number generator.
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/// </summary>
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/// <param name="mu">The log-scale (μ) of the logarithm of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the logarithm of the distribution. Range: σ ≥ 0.</param>
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public LogNormal(double mu, double sigma)
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{
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if (!IsValidParameterSet(mu, sigma))
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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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_mu = mu;
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_sigma = sigma;
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}
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/// <summary>
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/// Initializes a new instance of the <see cref="LogNormal"/> class.
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/// The distribution will be initialized with the default <seealso cref="System.Random"/>
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/// random number generator.
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/// </summary>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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 LogNormal(double mu, double sigma, System.Random randomSource)
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{
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if (!IsValidParameterSet(mu, sigma))
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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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_mu = mu;
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_sigma = sigma;
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}
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/// <summary>
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/// Constructs a log-normal distribution with the desired mu and sigma parameters.
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/// </summary>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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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/// <returns>A log-normal distribution.</returns>
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public static LogNormal WithMuSigma(double mu, double sigma, System.Random randomSource = null)
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{
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return new LogNormal(mu, sigma, randomSource);
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}
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/// <summary>
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/// Constructs a log-normal distribution with the desired mean and variance.
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/// </summary>
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/// <param name="mean">The mean of the log-normal distribution.</param>
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/// <param name="var">The variance of the log-normal distribution.</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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/// <returns>A log-normal distribution.</returns>
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public static LogNormal WithMeanVariance(double mean, double var, System.Random randomSource = null)
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{
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var sigma2 = Math.Log(var/(mean*mean) + 1.0);
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return new LogNormal(Math.Log(mean) - sigma2/2.0, Math.Sqrt(sigma2), randomSource);
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}
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/// <summary>
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/// Estimates the log-normal distribution parameters from sample data with maximum-likelihood.
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/// </summary>
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/// <param name="samples">The samples to estimate the distribution parameters from.</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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/// <returns>A log-normal distribution.</returns>
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/// <remarks>MATLAB: lognfit</remarks>
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public static LogNormal Estimate(IEnumerable<double> samples, System.Random randomSource = null)
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{
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var muSigma = samples.Select(s => Math.Log(s)).MeanStandardDeviation();
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return new LogNormal(muSigma.Item1, muSigma.Item2, randomSource);
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}
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/// <summary>
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/// A string representation of the distribution.
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/// </summary>
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/// <returns>a string representation of the distribution.</returns>
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public override string ToString()
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{
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return $"LogNormal(μ = {_mu}, σ = {_sigma})";
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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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</param>
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public static bool IsValidParameterSet(double mu, double sigma)
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{
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return sigma >= 0.0 && !double.IsNaN(mu);
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}
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/// <summary>
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/// Gets the log-scale (μ) (mean of the logarithm) of the distribution.
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/// </summary>
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public double Mu => _mu;
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/// <summary>
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/// Gets the shape (σ) (standard deviation of the logarithm) of the distribution. Range: σ ≥ 0.
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/// </summary>
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public double Sigma => _sigma;
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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 mu of the log-normal distribution.
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/// </summary>
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public double Mean => Math.Exp(_mu + (_sigma*_sigma/2.0));
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/// <summary>
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/// Gets the variance of the log-normal 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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var sigma2 = _sigma*_sigma;
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return (Math.Exp(sigma2) - 1.0)*Math.Exp(_mu + _mu + sigma2);
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}
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}
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/// <summary>
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/// Gets the standard deviation of the log-normal 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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var sigma2 = _sigma*_sigma;
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return Math.Sqrt((Math.Exp(sigma2) - 1.0)*Math.Exp(_mu + _mu + sigma2));
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}
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}
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/// <summary>
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/// Gets the entropy of the log-normal distribution.
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/// </summary>
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public double Entropy => 0.5 + Math.Log(_sigma) + _mu + Constants.LogSqrt2Pi;
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/// <summary>
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/// Gets the skewness of the log-normal 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 expsigma2 = Math.Exp(_sigma*_sigma);
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return (expsigma2 + 2.0)*Math.Sqrt(expsigma2 - 1);
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}
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}
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/// <summary>
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/// Gets the mode of the log-normal distribution.
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/// </summary>
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public double Mode => Math.Exp(_mu - (_sigma*_sigma));
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/// <summary>
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/// Gets the median of the log-normal distribution.
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/// </summary>
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public double Median => Math.Exp(_mu);
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/// <summary>
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/// Gets the minimum of the log-normal 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 log-normal distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDF"/>
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public double Density(double x)
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{
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if (x < 0.0)
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{
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return 0.0;
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}
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var a = (Math.Log(x) - _mu)/_sigma;
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return Math.Exp(-0.5*a*a)/(x*_sigma*Constants.Sqrt2Pi);
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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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if (x < 0.0)
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{
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return double.NegativeInfinity;
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}
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var a = (Math.Log(x) - _mu)/_sigma;
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return (-0.5*a*a) - Math.Log(x*_sigma) - Constants.LogSqrt2Pi;
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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 x < 0.0 ? 0.0
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: 0.5*SpecialFunctions.Erfc((_mu - Math.Log(x))/(_sigma*Constants.Sqrt2));
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}
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/// <summary>
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/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
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/// at the given probability. This is also known as the quantile or percent point function.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative density.</param>
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/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
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/// <seealso cref="InvCDF"/>
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public double InverseCumulativeDistribution(double p)
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{
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return p <= 0.0 ? 0.0 : p >= 1.0 ? double.PositiveInfinity
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: Math.Exp(_mu - _sigma*Constants.Sqrt2*SpecialFunctions.ErfcInv(2.0*p));
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}
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/// <summary>
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/// Generates a sample from the log-normal distribution using the <i>Box-Muller</i> algorithm.
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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, _mu, _sigma);
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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, _mu, _sigma);
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}
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/// <summary>
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/// Generates a sequence of samples from the log-normal distribution using the <i>Box-Muller</i> algorithm.
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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, _mu, _sigma);
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}
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static double SampleUnchecked(System.Random rnd, double mu, double sigma)
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{
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return Math.Exp(Normal.SampleUnchecked(rnd, mu, sigma));
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double mu, double sigma)
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{
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return Normal.SamplesUnchecked(rnd, mu, sigma).Select(Math.Exp);
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double mu, double sigma)
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{
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Normal.SamplesUnchecked(rnd, values, mu, sigma);
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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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values[i] = Math.Exp(values[i]);
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}
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});
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}
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="Density"/>
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/// <remarks>MATLAB: lognpdf</remarks>
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public static double PDF(double mu, double sigma, double x)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (x < 0.0)
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{
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return 0.0;
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}
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var a = (Math.Log(x) - mu)/sigma;
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return Math.Exp(-0.5*a*a)/(x*sigma*Constants.Sqrt2Pi);
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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 density.</param>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</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 mu, double sigma, double x)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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if (x < 0.0)
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{
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return double.NegativeInfinity;
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}
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var a = (Math.Log(x) - mu)/sigma;
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return (-0.5*a*a) - Math.Log(x*sigma) - Constants.LogSqrt2Pi;
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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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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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/// <remarks>MATLAB: logncdf</remarks>
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public static double CDF(double mu, double sigma, double x)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return x < 0.0 ? 0.0
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: 0.5*(1.0 + SpecialFunctions.Erf((Math.Log(x) - mu)/(sigma*Constants.Sqrt2)));
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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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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>MATLAB: logninv</remarks>
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public static double InvCDF(double mu, double sigma, double p)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return p <= 0.0 ? 0.0 : p >= 1.0 ? double.PositiveInfinity
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: Math.Exp(mu - sigma*Constants.Sqrt2*SpecialFunctions.ErfcInv(2.0*p));
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}
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/// <summary>
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/// Generates a sample from the log-normal 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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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 mu, double sigma)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(rnd, mu, sigma);
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}
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/// <summary>
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/// Generates a sequence of samples from the log-normal 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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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 mu, double sigma)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(rnd, mu, sigma);
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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="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static void Samples(System.Random rnd, double[] values, double mu, double sigma)
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{
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if (sigma < 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(rnd, values, mu, sigma);
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}
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/// <summary>
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/// Generates a sample from the log-normal distribution using the <i>Box-Muller</i> algorithm.
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/// </summary>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(double mu, double sigma)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(SystemRandomSource.Default, mu, sigma);
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}
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/// <summary>
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/// Generates a sequence of samples from the log-normal distribution using the <i>Box-Muller</i> algorithm.
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/// </summary>
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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 0.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(double mu, double sigma)
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{
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if (sigma < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(SystemRandomSource.Default, mu, sigma);
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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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/// <param name="mu">The log-scale (μ) of the distribution.</param>
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/// <param name="sigma">The shape (σ) of the distribution. Range: σ ≥ 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 mu, double sigma)
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{
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if (sigma < 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, mu, sigma);
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}
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}
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}
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