// <copyright file="Triangular.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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/// Triangular distribution.
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/// For details, see <a href="https://en.wikipedia.org/wiki/Triangular_distribution">Wikipedia - Triangular distribution</a>.
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/// </summary>
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/// <remarks><para>The distribution will use the <see cref="System.Random"/> by default.
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/// Users can get/set the random number generator by using the <see cref="RandomSource"/> property.</para>
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/// <para>The statistics classes will check whether all the incoming parameters are in the allowed range. This might involve heavy computation. Optionally, by setting Control.CheckDistributionParameters
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/// to <c>false</c>, all parameter checks can be turned off.</para></remarks>
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public class Triangular : IContinuousDistribution
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{
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System.Random _random;
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readonly double _lower;
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readonly double _upper;
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readonly double _mode;
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/// <summary>
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/// Initializes a new instance of the Triangular class with the given lower bound, upper bound and mode.
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/// </summary>
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/// <param name="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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/// <exception cref="ArgumentException">If the upper bound is smaller than the mode or if the mode is smaller than the lower bound.</exception>
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public Triangular(double lower, double upper, double mode)
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{
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if (!IsValidParameterSet(lower, upper, mode))
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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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_lower = lower;
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_upper = upper;
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_mode = mode;
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}
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/// <summary>
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/// Initializes a new instance of the Triangular class with the given lower bound, upper bound and mode.
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/// </summary>
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/// <param name="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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/// <exception cref="ArgumentException">If the upper bound is smaller than the mode or if the mode is smaller than the lower bound.</exception>
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public Triangular(double lower, double upper, double mode, System.Random randomSource)
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{
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if (!IsValidParameterSet(lower, upper, mode))
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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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_lower = lower;
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_upper = upper;
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_mode = mode;
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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 $"Triangular(Lower = {_lower}, Upper = {_upper}, Mode = {_mode})";
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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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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public static bool IsValidParameterSet(double lower, double upper, double mode)
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{
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return upper >= mode && mode >= lower && !double.IsInfinity(upper) && !double.IsInfinity(lower) && !double.IsInfinity(mode);
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}
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/// <summary>
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/// Gets the lower bound of the distribution.
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/// </summary>
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public double LowerBound => _lower;
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/// <summary>
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/// Gets the upper bound of the distribution.
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/// </summary>
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public double UpperBound => _upper;
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/// <summary>
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/// Gets or sets the random number generator which is used to draw random samples.
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/// </summary>
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public System.Random RandomSource
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{
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get => _random;
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set => _random = value ?? SystemRandomSource.Default;
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}
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/// <summary>
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/// Gets the mean of the distribution.
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/// </summary>
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public double Mean => (_lower + _upper + _mode)/3.0;
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/// <summary>
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/// Gets the variance of the distribution.
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/// </summary>
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public double Variance
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{
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get
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{
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var a = _lower;
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var b = _upper;
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var c = _mode;
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return (a*a + b*b + c*c - a*b - a*c - b*c)/18.0;
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}
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}
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/// <summary>
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/// Gets the standard deviation of the distribution.
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/// </summary>
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public double StdDev => Math.Sqrt(Variance);
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/// <summary>
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/// Gets the entropy of the distribution.
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/// </summary>
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/// <value></value>
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public double Entropy => 0.5 + Math.Log((_upper - _lower)/2);
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/// <summary>
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/// Gets the skewness of the distribution.
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/// </summary>
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public double Skewness
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{
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get
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{
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var a = _lower;
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var b = _upper;
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var c = _mode;
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var q = Math.Sqrt(2)*(a + b - 2*c)*(2*a - b - c)*(a - 2*b + c);
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var d = 5*Math.Pow(a*a + b*b + c*c - a*b - a*c - b*c, 3.0/2);
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return q/d;
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}
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}
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/// <summary>
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/// Gets or sets the mode of the distribution.
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/// </summary>
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public double Mode => _mode;
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/// <summary>
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/// Gets the median of the distribution.
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/// </summary>
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/// <value></value>
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public double Median
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{
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get
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{
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var a = _lower;
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var b = _upper;
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var c = _mode;
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return c >= (a + b)/2
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? a + Math.Sqrt((b - a)*(c - a)/2)
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: b - Math.Sqrt((b - a)*(b - c)/2);
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}
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}
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/// <summary>
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/// Gets the minimum of the distribution.
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/// </summary>
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public double Minimum => _lower;
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/// <summary>
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/// Gets the maximum of the distribution.
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/// </summary>
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public double Maximum => _upper;
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDF"/>
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public double Density(double x)
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{
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return PDF(_lower, _upper, _mode, x);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDFLn"/>
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public double DensityLn(double x)
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{
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return PDFLn(_lower, _upper, _mode, x);
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CDF"/>
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public double CumulativeDistribution(double x)
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{
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return CDF(_lower, _upper, _mode, x);
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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 InvCDF(_lower, _upper, _mode, p);
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}
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/// <summary>
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/// Generates a sample from the <c>Triangular</c> 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, _lower, _upper, _mode);
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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, _lower, _upper, _mode);
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}
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/// <summary>
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/// Generates a sequence of samples from the <c>Triangular</c> 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, _lower, _upper, _mode);
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}
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static double SampleUnchecked(System.Random rnd, double lower, double upper, double mode)
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{
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var u = rnd.NextDouble();
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return u < (mode - lower)/(upper - lower)
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? lower + Math.Sqrt(u*(upper - lower)*(mode - lower))
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: upper - Math.Sqrt((1 - u)*(upper - lower)*(upper - mode));
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}
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static IEnumerable<double> SamplesUnchecked(System.Random rnd, double lower, double upper, double mode)
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{
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double ml = mode - lower, ul = upper - lower, um = upper - mode;
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double u = ml/ul, v = ul*ml, w = ul*um;
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return rnd.NextDoubleSequence().Select(x => x < u ? lower + Math.Sqrt(x*v) : upper - Math.Sqrt((1 - x)*w));
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}
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static void SamplesUnchecked(System.Random rnd, double[] values, double lower, double upper, double mode)
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{
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double ml = mode - lower, ul = upper - lower, um = upper - mode;
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double u = ml/ul, v = ul*ml, w = ul*um;
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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] = values[i] < u
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? lower + Math.Sqrt(values[i]*v)
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: upper - Math.Sqrt((1 - values[i])*w);
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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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode, double x)
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{
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if (!(upper >= mode && mode >= lower))
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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 a = lower;
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var b = upper;
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var c = mode;
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if (a <= x && x <= c)
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{
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return 2*(x - a)/((b - a)*(c - a));
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}
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if (c < x & x <= b)
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{
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return 2*(b - x)/((b - a)*(b - c));
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}
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return 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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode, double x)
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{
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return Math.Log(PDF(lower, upper, mode, x));
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <param name="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode, double x)
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{
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if (!(upper >= mode && mode >= lower))
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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 a = lower;
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var b = upper;
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var c = mode;
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if (x < a)
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{
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return 0;
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}
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if (a <= x && x <= c)
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{
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return (x - a)*(x - a)/((b - a)*(c - a));
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}
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if (c < x & x <= b)
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{
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return 1 - (b - x)*(b - x)/((b - a)*(b - c));
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}
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return 1;
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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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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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public static double InvCDF(double lower, double upper, double mode, double p)
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{
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if (!(upper >= mode && mode >= lower))
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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 a = lower;
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var b = upper;
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var c = mode;
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if (p <= 0)
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{
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return lower;
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}
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// Taken from http://www.ntrand.com/triangular-distribution/
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if (p < (c - a)/(b - a))
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{
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return a + Math.Sqrt(p*(c - a)*(b - a));
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}
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if (p < 1)
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{
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return b - Math.Sqrt((1 - p)*(b - c)*(b - a));
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}
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return upper;
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}
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/// <summary>
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/// Generates a sample from the <c>Triangular</c> 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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, double lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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}
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/// <summary>
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/// Generates a sequence of samples from the <c>Triangular</c> 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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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}
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/// <summary>
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/// Generates a sample from the <c>Triangular</c> distribution.
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/// </summary>
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/// <param name="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(double lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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}
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/// <summary>
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/// Generates a sequence of samples from the <c>Triangular</c> distribution.
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/// </summary>
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/// <param name="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(double lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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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="lower">Lower bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="upper">Upper bound. Range: lower ≤ mode ≤ upper</param>
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/// <param name="mode">Mode (most frequent value). Range: lower ≤ mode ≤ upper</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 lower, double upper, double mode)
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{
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if (!(upper >= mode && mode >= lower))
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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, lower, upper, mode);
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}
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}
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}
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