// <copyright file="Normal.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2015 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// 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.Statistics;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Continuous Univariate Normal distribution, also known as Gaussian distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Normal_distribution">Wikipedia - Normal distribution</a>.
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/// </summary>
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public class Normal : IContinuousDistribution
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{
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System.Random _random;
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readonly double _mean;
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readonly double _stdDev;
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/// <summary>
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/// Initializes a new instance of the Normal class. This is a normal distribution with mean 0.0
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/// and standard deviation 1.0. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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public Normal()
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: this(0.0, 1.0)
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{
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}
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/// <summary>
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/// Initializes a new instance of the Normal class. This is a normal distribution with mean 0.0
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/// and standard deviation 1.0. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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public Normal(System.Random randomSource)
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: this(0.0, 1.0, randomSource)
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{
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}
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/// <summary>
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/// Initializes a new instance of the Normal class with a particular mean and standard deviation. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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/// <param name="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
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public Normal(double mean, double stddev)
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{
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if (!IsValidParameterSet(mean, stddev))
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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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_mean = mean;
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_stdDev = stddev;
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}
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/// <summary>
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/// Initializes a new instance of the Normal class with a particular mean and standard deviation. The distribution will
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/// be initialized with the default <seealso cref="System.Random"/> random number generator.
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/// </summary>
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/// <param name="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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 Normal(double mean, double stddev, System.Random randomSource)
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{
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if (!IsValidParameterSet(mean, stddev))
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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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_mean = mean;
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_stdDev = stddev;
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}
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/// <summary>
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/// Constructs a normal distribution from a mean and standard deviation.
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/// </summary>
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/// <param name="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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 normal distribution.</returns>
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public static Normal WithMeanStdDev(double mean, double stddev, System.Random randomSource = null)
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{
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return new Normal(mean, stddev, randomSource);
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}
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/// <summary>
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/// Constructs a normal distribution from a mean and variance.
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/// </summary>
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/// <param name="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="var">The variance (σ^2) of the 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 normal distribution.</returns>
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public static Normal WithMeanVariance(double mean, double var, System.Random randomSource = null)
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{
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return new Normal(mean, Math.Sqrt(var), randomSource);
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}
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/// <summary>
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/// Constructs a normal distribution from a mean and precision.
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/// </summary>
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/// <param name="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="precision">The precision of the 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 normal distribution.</returns>
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public static Normal WithMeanPrecision(double mean, double precision, System.Random randomSource = null)
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{
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return new Normal(mean, 1.0/Math.Sqrt(precision), randomSource);
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}
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/// <summary>
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/// Estimates the 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 normal distribution.</returns>
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/// <remarks>MATLAB: normfit</remarks>
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public static Normal Estimate(IEnumerable<double> samples, System.Random randomSource = null)
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{
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var meanStdDev = samples.MeanStandardDeviation();
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return new Normal(meanStdDev.Item1, meanStdDev.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 $"Normal(μ = {_mean}, σ = {_stdDev})";
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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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
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public static bool IsValidParameterSet(double mean, double stddev)
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{
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return stddev >= 0.0 && !double.IsNaN(mean);
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}
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/// <summary>
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/// Gets the mean (μ) of the normal distribution.
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/// </summary>
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public double Mean => _mean;
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/// <summary>
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/// Gets the standard deviation (σ) of the normal distribution. Range: σ ≥ 0.
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/// </summary>
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public double StdDev => _stdDev;
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/// <summary>
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/// Gets the variance of the normal distribution.
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/// </summary>
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public double Variance => _stdDev*_stdDev;
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/// <summary>
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/// Gets the precision of the normal distribution.
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/// </summary>
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public double Precision => 1.0/(_stdDev*_stdDev);
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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 entropy of the normal distribution.
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/// </summary>
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public double Entropy => Math.Log(_stdDev) + Constants.LogSqrt2PiE;
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/// <summary>
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/// Gets the skewness of the normal distribution.
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/// </summary>
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public double Skewness => 0.0;
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/// <summary>
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/// Gets the mode of the normal distribution.
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/// </summary>
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public double Mode => _mean;
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/// <summary>
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/// Gets the median of the normal distribution.
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/// </summary>
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public double Median => _mean;
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/// <summary>
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/// Gets the minimum of the normal distribution.
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/// </summary>
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public double Minimum => double.NegativeInfinity;
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/// <summary>
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/// Gets the maximum of the 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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var d = (x - _mean)/_stdDev;
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return Math.Exp(-0.5*d*d)/(Constants.Sqrt2Pi*_stdDev);
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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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var d = (x - _mean)/_stdDev;
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return (-0.5*d*d) - Math.Log(_stdDev) - 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 0.5*SpecialFunctions.Erfc((_mean - x)/(_stdDev*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 _mean - (_stdDev*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 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, _mean, _stdDev);
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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, _mean, _stdDev);
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}
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/// <summary>
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/// Generates a sequence of samples from the 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, _mean, _stdDev);
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}
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internal static double SampleUnchecked(System.Random rnd, double mean, double stddev)
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{
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double x, y;
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while (!PolarTransform(rnd.NextDouble(), rnd.NextDouble(), out x, out y))
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{
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}
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return mean + (stddev*x);
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}
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internal static IEnumerable<double> SamplesUnchecked(System.Random rnd, double mean, double stddev)
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{
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while (true)
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{
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double x, y;
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if (!PolarTransform(rnd.NextDouble(), rnd.NextDouble(), out x, out y))
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{
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continue;
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}
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yield return mean + (stddev*x);
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yield return mean + (stddev*y);
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}
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double mean, double stddev)
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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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// Since we only accept points within the unit circle
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// we need to generate roughly 4/pi=1.27 times the numbers needed.
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int n = (int)Math.Ceiling(values.Length*4*Constants.InvPi);
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if (n.IsOdd())
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{
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n++;
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}
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var uniform = rnd.NextDoubles(n);
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// Polar transform
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double x, y;
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int index = 0;
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for (int i = 0; i < uniform.Length && index < values.Length; i += 2)
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{
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if (!PolarTransform(uniform[i], uniform[i + 1], out x, out y))
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{
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continue;
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}
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values[index++] = mean + stddev*x;
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if (index == values.Length)
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{
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return;
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}
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values[index++] = mean + stddev*y;
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if (index == values.Length)
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{
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return;
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}
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}
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// remaining, if any
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while (index < values.Length)
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{
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if (!PolarTransform(rnd.NextDouble(), rnd.NextDouble(), out x, out y))
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{
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continue;
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}
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values[index++] = mean + stddev*x;
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if (index == values.Length)
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{
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return;
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}
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values[index++] = mean + stddev*y;
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if (index == values.Length)
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{
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return;
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}
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}
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}
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static bool PolarTransform(double a, double b, out double x, out double y)
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{
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var v1 = (2.0*a) - 1.0;
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var v2 = (2.0*b) - 1.0;
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var r = (v1*v1) + (v2*v2);
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if (r >= 1.0 || r == 0.0)
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{
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x = 0;
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y = 0;
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return false;
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}
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var fac = Math.Sqrt(-2.0*Math.Log(r)/r);
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x = v1*fac;
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y = v2*fac;
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return true;
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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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 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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/// <remarks>MATLAB: normpdf</remarks>
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public static double PDF(double mean, double stddev, double x)
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{
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if (stddev < 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 d = (x - mean)/stddev;
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return Math.Exp(-0.5*d*d)/(Constants.Sqrt2Pi*stddev);
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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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="DensityLn"/>
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public static double PDFLn(double mean, double stddev, double x)
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{
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if (stddev < 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 d = (x - mean)/stddev;
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return (-0.5*d*d) - Math.Log(stddev) - 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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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: normcdf</remarks>
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public static double CDF(double mean, double stddev, double x)
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{
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if (stddev < 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 0.5*SpecialFunctions.Erfc((mean - x)/(stddev*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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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: norminv</remarks>
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public static double InvCDF(double mean, double stddev, double p)
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{
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if (stddev < 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 mean - (stddev*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 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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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 mean, double stddev)
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{
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if (stddev < 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, mean, stddev);
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}
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/// <summary>
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/// Generates a sequence of samples from the 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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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 mean, double stddev)
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{
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if (stddev < 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, mean, stddev);
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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="mean">The mean (μ) of the normal distribution.</param>
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/// <param name="stddev">The standard deviation (σ) of the normal 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 mean, double stddev)
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{
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if (stddev < 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, mean, stddev);
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}
|
|
/// <summary>
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/// Generates a sample from the normal distribution using the <i>Box-Muller</i> algorithm.
|
/// </summary>
|
/// <param name="mean">The mean (μ) of the normal distribution.</param>
|
/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
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public static double Sample(double mean, double stddev)
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{
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if (stddev < 0.0)
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
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return SampleUnchecked(SystemRandomSource.Default, mean, stddev);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the normal distribution using the <i>Box-Muller</i> algorithm.
|
/// </summary>
|
/// <param name="mean">The mean (μ) of the normal distribution.</param>
|
/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(double mean, double stddev)
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{
|
if (stddev < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(SystemRandomSource.Default, mean, stddev);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="mean">The mean (μ) of the normal distribution.</param>
|
/// <param name="stddev">The standard deviation (σ) of the normal distribution. Range: σ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(double[] values, double mean, double stddev)
|
{
|
if (stddev < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(SystemRandomSource.Default, values, mean, stddev);
|
}
|
}
|
}
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