// <copyright file="Weibull.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.Threading;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Continuous Univariate Weibull distribution.
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/// For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Weibull_distribution">Wikipedia - Weibull distribution</a>.
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/// </summary>
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/// <remarks>
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/// The Weibull distribution is parametrized by a shape and scale parameter.
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/// </remarks>
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public class Weibull : IContinuousDistribution
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{
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System.Random _random;
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readonly double _shape;
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readonly double _scale;
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/// <summary>
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/// Reusable intermediate result 1 / (_scale ^ _shape)
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/// </summary>
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/// <remarks>
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/// By caching this parameter we can get slightly better numerics precision
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/// in certain constellations without any additional computations.
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/// </remarks>
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readonly double _scalePowShapeInv;
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/// <summary>
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/// Initializes a new instance of the Weibull class.
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/// </summary>
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/// <param name="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 0.</param>
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public Weibull(double shape, double scale)
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{
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if (!IsValidParameterSet(shape, scale))
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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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_shape = shape;
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_scale = scale;
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_scalePowShapeInv = Math.Pow(scale, -shape);
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}
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/// <summary>
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/// Initializes a new instance of the Weibull class.
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/// </summary>
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/// <param name="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 Weibull(double shape, double scale, System.Random randomSource)
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{
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if (!IsValidParameterSet(shape, scale))
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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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_shape = shape;
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_scale = scale;
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_scalePowShapeInv = Math.Pow(scale, -shape);
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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 $"Weibull(k = {_shape}, λ = {_scale})";
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 0.</param>
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public static bool IsValidParameterSet(double shape, double scale)
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{
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return shape > 0.0 && scale > 0.0;
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}
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/// <summary>
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/// Gets the shape (k) of the Weibull distribution. Range: k > 0.
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/// </summary>
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public double Shape => _shape;
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/// <summary>
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/// Gets the scale (λ) of the Weibull distribution. Range: λ > 0.
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/// </summary>
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public double Scale => _scale;
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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 Weibull distribution.
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/// </summary>
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public double Mean => _scale*SpecialFunctions.Gamma(1.0 + (1.0/_shape));
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/// <summary>
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/// Gets the variance of the Weibull distribution.
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/// </summary>
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public double Variance => (_scale*_scale*SpecialFunctions.Gamma(1.0 + (2.0/_shape))) - (Mean*Mean);
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/// <summary>
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/// Gets the standard deviation of the Weibull 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 Weibull distribution.
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/// </summary>
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public double Entropy => (Constants.EulerMascheroni*(1.0 - (1.0/_shape))) + Math.Log(_scale/_shape) + 1.0;
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/// <summary>
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/// Gets the skewness of the Weibull 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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double mu = Mean;
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double sigma = StdDev;
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double sigma2 = sigma*sigma;
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double sigma3 = sigma2*sigma;
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return ((_scale*_scale*_scale*SpecialFunctions.Gamma(1.0 + (3.0/_shape))) - (3.0*sigma2*mu) - (mu*mu*mu))/sigma3;
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}
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}
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/// <summary>
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/// Gets the mode of the Weibull distribution.
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/// </summary>
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public double Mode
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{
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get
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{
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if (_shape <= 1.0)
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{
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return 0.0;
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}
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return _scale*Math.Pow((_shape - 1.0)/_shape, 1.0/_shape);
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}
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}
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/// <summary>
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/// Gets the median of the Weibull distribution.
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/// </summary>
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public double Median => _scale*Math.Pow(Constants.Ln2, 1.0/_shape);
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/// <summary>
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/// Gets the minimum of the Weibull 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 Weibull distribution.
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/// </summary>
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public double Maximum => double.PositiveInfinity;
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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public double Density(double x)
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{
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if (x >= 0.0)
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{
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if (x == 0.0 && _shape == 1.0)
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{
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return _shape/_scale;
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}
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return _shape*Math.Pow(x/_scale, _shape - 1.0)*Math.Exp(-Math.Pow(x, _shape)*_scalePowShapeInv)/_scale;
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}
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return 0.0;
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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public double DensityLn(double x)
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{
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if (x >= 0.0)
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{
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if (x == 0.0 && _shape == 1.0)
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{
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return Math.Log(_shape) - Math.Log(_scale);
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}
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return Math.Log(_shape) + ((_shape - 1.0)*Math.Log(x/_scale)) - (Math.Pow(x, _shape)*_scalePowShapeInv) - Math.Log(_scale);
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}
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return double.NegativeInfinity;
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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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if (x < 0.0)
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{
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return 0.0;
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}
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return -SpecialFunctions.ExponentialMinusOne(-Math.Pow(x, _shape)*_scalePowShapeInv);
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}
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/// <summary>
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/// Generates a sample from the Weibull distribution.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public double Sample()
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{
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return SampleUnchecked(_random, _shape, _scale);
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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, _shape, _scale);
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}
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/// <summary>
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/// Generates a sequence of samples from the Weibull distribution.
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/// </summary>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<double> Samples()
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{
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return SamplesUnchecked(_random, _shape, _scale);
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}
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static double SampleUnchecked(System.Random rnd, double shape, double scale)
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{
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var x = rnd.NextDouble();
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return scale*Math.Pow(-Math.Log(x), 1.0/shape);
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double shape, double scale)
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{
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var exponent = 1.0/shape;
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return rnd.NextDoubleSequence().Select(x => scale*Math.Pow(-Math.Log(x), exponent));
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double shape, double scale)
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{
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var exponent = 1.0/shape;
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rnd.NextDoubles(values);
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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] = scale*Math.Pow(-Math.Log(values[i]), exponent);
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 0.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="Density"/>
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public static double PDF(double shape, double scale, double x)
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{
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if (shape <= 0.0 || scale <= 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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if (x == 0.0 && shape == 1.0)
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{
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return shape/scale;
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}
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return shape
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*Math.Pow(x/scale, shape - 1.0)
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*Math.Exp(-Math.Pow(x, shape)*Math.Pow(scale, -shape))
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/scale;
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}
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return 0.0;
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale, double x)
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{
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if (shape <= 0.0 || scale <= 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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if (x == 0.0 && shape == 1.0)
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{
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return Math.Log(shape) - Math.Log(scale);
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}
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return Math.Log(shape)
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+ ((shape - 1.0)*Math.Log(x/scale))
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- (Math.Pow(x, shape)*Math.Pow(scale, -shape))
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- Math.Log(scale);
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}
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return double.NegativeInfinity;
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale, double x)
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{
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if (shape <= 0.0 || scale <= 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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return -SpecialFunctions.ExponentialMinusOne(-Math.Pow(x, shape)*Math.Pow(scale, -shape));
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}
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/// <summary>
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/// Implemented according to: Parameter estimation of the Weibull probability distribution, 1994, Hongzhu Qiao, Chris P. Tsokos
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/// </summary>
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/// <param name="samples"></param>
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/// <param name="randomSource"></param>
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/// <returns>Returns a Weibull distribution.</returns>
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public static Weibull Estimate(IEnumerable<double> samples, System.Random randomSource = null)
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{
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var samp = samples as double[] ?? samples.ToArray();
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double n = samp.Length, s1 = 0, s2 = 0, s3 = 0, previousC = Int32.MinValue, QofC = 0;
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if (n <= 1) throw new Exception("Observations not sufficient");
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// Start values
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double c = 10; double b = 0;
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while (Math.Abs(c - previousC) >= 0.0001)
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{
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s1 = s2 = s3 = 0;
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foreach (double x in samp)
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{
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if (x > 0)
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{
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s1 += Math.Log(x);
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s2 += Math.Pow(x, c);
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s3 += Math.Pow(x, c) * Math.Log(x);
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}
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}
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QofC = n * s2 / (n * s3 - s1 * s2);
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previousC = c;
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c = (c + QofC) / 2;
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}
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foreach (double x in samp)
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{
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if (x > 0)
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{
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b += Math.Pow(x, c);
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}
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}
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b = Math.Pow(b / n, 1 / c);
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return new Weibull(c, b, randomSource);
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}
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/// <summary>
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/// Generates a sample from the Weibull 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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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}
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/// <summary>
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/// Generates a sequence of samples from the Weibull 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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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}
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/// <summary>
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/// Generates a sample from the Weibull distribution.
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/// </summary>
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/// <param name="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 0.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(double shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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}
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/// <summary>
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/// Generates a sequence of samples from the Weibull distribution.
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/// </summary>
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/// <param name="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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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="shape">The shape (k) of the Weibull distribution. Range: k > 0.</param>
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/// <param name="scale">The scale (λ) of the Weibull distribution. Range: λ > 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 shape, double scale)
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{
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if (shape <= 0.0 || scale <= 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, shape, scale);
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}
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}
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}
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