// <copyright file="BetaScaled.cs" company="Math.NET">
|
// Math.NET Numerics, part of the Math.NET Project
|
// http://numerics.mathdotnet.com
|
// http://github.com/mathnet/mathnet-numerics
|
//
|
// Copyright (c) 2009-2015 Math.NET
|
//
|
// Permission is hereby granted, free of charge, to any person
|
// obtaining a copy of this software and associated documentation
|
// files (the "Software"), to deal in the Software without
|
// restriction, including without limitation the rights to use,
|
// copy, modify, merge, publish, distribute, sublicense, and/or sell
|
// copies of the Software, and to permit persons to whom the
|
// Software is furnished to do so, subject to the following
|
// conditions:
|
//
|
// The above copyright notice and this permission notice shall be
|
// included in all copies or substantial portions of the Software.
|
//
|
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
|
// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
|
// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
|
// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
|
// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
|
// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
|
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
|
// OTHER DEALINGS IN THE SOFTWARE.
|
// </copyright>
|
|
using System;
|
using System.Collections.Generic;
|
using IStation.Numerics.Random;
|
using IStation.Numerics.Threading;
|
|
namespace IStation.Numerics.Distributions
|
{
|
public class BetaScaled : IContinuousDistribution
|
{
|
System.Random _random;
|
|
readonly double _shapeA;
|
readonly double _shapeB;
|
readonly double _location;
|
readonly double _scale;
|
|
/// <summary>
|
/// Initializes a new instance of the BetaScaled class.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
public BetaScaled(double a, double b, double location, double scale)
|
{
|
if (!IsValidParameterSet(a, b, location, scale))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = SystemRandomSource.Default;
|
_shapeA = a;
|
_shapeB = b;
|
_location = location;
|
_scale = scale;
|
}
|
|
/// <summary>
|
/// Initializes a new instance of the BetaScaled class.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
|
public BetaScaled(double a, double b, double location, double scale, System.Random randomSource)
|
{
|
if (!IsValidParameterSet(a, b, location, scale))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = randomSource ?? SystemRandomSource.Default;
|
_shapeA = a;
|
_shapeB = b;
|
_location = location;
|
_scale = scale;
|
}
|
|
/// <summary>
|
/// Create a Beta PERT distribution, used in risk analysis and other domains where an expert forecast
|
/// is used to construct an underlying beta distribution.
|
/// </summary>
|
/// <param name="min">The minimum value.</param>
|
/// <param name="max">The maximum value.</param>
|
/// <param name="likely">The most likely value (mode).</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
|
/// <returns>The Beta distribution derived from the PERT parameters.</returns>
|
public static BetaScaled PERT(double min, double max, double likely, System.Random randomSource = null)
|
{
|
if (min > max || likely > max || likely < min)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
// specified to make the formulas match the literature;
|
// traditionally set to 4 so that the range between min and max
|
// represents six standard deviations (sometimes called
|
// "the six-sigma assumption").
|
const double lambda = 4;
|
|
// calculate the mean
|
double mean = (min + max + lambda * likely) / (lambda + 2);
|
|
// derive the shape parameters a and b
|
double a;
|
|
// special case where mean and mode are identical
|
if (mean == likely)
|
{
|
a = (lambda / 2) + 1;
|
}
|
else
|
{
|
a = ((mean - min) * (2 * likely - min - max)) / ((likely - mean) * (max - min));
|
}
|
|
double b = (a * (max - mean)) / (mean - min);
|
|
return new BetaScaled(a, b, min, max - min, randomSource);
|
}
|
|
/// <summary>
|
/// A string representation of the distribution.
|
/// </summary>
|
/// <returns>A string representation of the BetaScaled distribution.</returns>
|
public override string ToString()
|
{
|
return $"BetaScaled(α = {_shapeA}, β = {_shapeB}, μ = {_location}, σ = {_scale})";
|
}
|
|
/// <summary>
|
/// Tests whether the provided values are valid parameters for this distribution.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
public static bool IsValidParameterSet(double a, double b, double location, double scale)
|
{
|
return a > 0.0 && b > 0.0 && scale > 0.0 && !double.IsNaN(location);
|
}
|
|
/// <summary>
|
/// Gets the α shape parameter of the BetaScaled distribution. Range: α > 0.
|
/// </summary>
|
public double A => _shapeA;
|
|
/// <summary>
|
/// Gets the β shape parameter of the BetaScaled distribution. Range: β > 0.
|
/// </summary>
|
public double B => _shapeB;
|
|
/// <summary>
|
/// Gets the location (μ) of the BetaScaled distribution.
|
/// </summary>
|
public double Location => _location;
|
|
/// <summary>
|
/// Gets the scale (σ) of the BetaScaled distribution. Range: σ > 0.
|
/// </summary>
|
public double Scale => _scale;
|
|
/// <summary>
|
/// Gets or sets the random number generator which is used to draw random samples.
|
/// </summary>
|
public System.Random RandomSource
|
{
|
get => _random;
|
set => _random = value ?? SystemRandomSource.Default;
|
}
|
|
/// <summary>
|
/// Gets the mean of the BetaScaled distribution.
|
/// </summary>
|
public double Mean
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_shapeA) && double.IsPositiveInfinity(_shapeB))
|
{
|
return _location + 0.5 * _scale;
|
}
|
|
if (double.IsPositiveInfinity(_shapeA))
|
{
|
return _location + _scale;
|
}
|
|
if (double.IsPositiveInfinity(_shapeB))
|
{
|
return _location;
|
}
|
|
return (_shapeB*_location + _shapeA*(_location + _scale))/(_shapeA + _shapeB);
|
}
|
}
|
|
/// <summary>
|
/// Gets the variance of the BetaScaled distribution.
|
/// </summary>
|
public double Variance
|
{
|
get
|
{
|
double sum = _shapeA + _shapeB;
|
return (_shapeA*_shapeB*_scale*_scale)/(sum*sum*(1.0 + sum));
|
}
|
}
|
|
/// <summary>
|
/// Gets the standard deviation of the BetaScaled distribution.
|
/// </summary>
|
public double StdDev => Math.Sqrt(Variance);
|
|
/// <summary>
|
/// Gets the entropy of the BetaScaled distribution.
|
/// </summary>
|
public double Entropy => throw new NotSupportedException();
|
|
/// <summary>
|
/// Gets the skewness of the BetaScaled distribution.
|
/// </summary>
|
public double Skewness
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_shapeA) && double.IsPositiveInfinity(_shapeB))
|
{
|
return 0.0;
|
}
|
|
if (double.IsPositiveInfinity(_shapeA))
|
{
|
return -2.0*_scale/Math.Sqrt(_shapeB*_scale*_scale);
|
}
|
|
if (double.IsPositiveInfinity(_shapeB))
|
{
|
return 2.0*_scale/Math.Sqrt(_shapeA*_scale*_scale);
|
}
|
|
double sum = _shapeA + _shapeB;
|
double variance = (_shapeA * _shapeB * _scale * _scale) / (sum * sum * (1.0 + sum));
|
return 2.0*(_shapeB - _shapeA)*_scale/(sum*(2.0 + sum)*Math.Sqrt(variance));
|
}
|
}
|
|
/// <summary>
|
/// Gets the mode of the BetaScaled distribution; when there are multiple answers, this routine will return 0.5.
|
/// </summary>
|
public double Mode
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_shapeA) && double.IsPositiveInfinity(_shapeB))
|
{
|
return _location + 0.5 * _scale;
|
}
|
|
if (double.IsPositiveInfinity(_shapeA))
|
{
|
return _location + _scale;
|
}
|
|
if (double.IsPositiveInfinity(_shapeB))
|
{
|
return _location;
|
}
|
|
if (_shapeA == 1.0 && _shapeB == 1.0)
|
{
|
return _location + 0.5 * _scale;
|
}
|
|
return ((_shapeA - 1)/(_shapeA + _shapeB - 2))*_scale + _location;
|
}
|
}
|
|
/// <summary>
|
/// Gets the median of the BetaScaled distribution.
|
/// </summary>
|
public double Median => throw new NotSupportedException();
|
|
/// <summary>
|
/// Gets the minimum of the BetaScaled distribution.
|
/// </summary>
|
public double Minimum => _location;
|
|
/// <summary>
|
/// Gets the maximum of the BetaScaled distribution.
|
/// </summary>
|
public double Maximum => _location + _scale;
|
|
/// <summary>
|
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
|
/// </summary>
|
/// <param name="x">The location at which to compute the density.</param>
|
/// <returns>the density at <paramref name="x"/>.</returns>
|
/// <seealso cref="PDF"/>
|
public double Density(double x)
|
{
|
return PDF(_shapeA, _shapeB, _location, _scale, x);
|
}
|
|
/// <summary>
|
/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
|
/// </summary>
|
/// <param name="x">The location at which to compute the log density.</param>
|
/// <returns>the log density at <paramref name="x"/>.</returns>
|
/// <seealso cref="PDFLn"/>
|
public double DensityLn(double x)
|
{
|
return PDFLn(_shapeA, _shapeB, _location, _scale, x);
|
}
|
|
/// <summary>
|
/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
|
/// </summary>
|
/// <param name="x">The location at which to compute the cumulative distribution function.</param>
|
/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
|
/// <seealso cref="CDF"/>
|
public double CumulativeDistribution(double x)
|
{
|
return CDF(_shapeA, _shapeB, _location, _scale, x);
|
}
|
|
/// <summary>
|
/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
|
/// at the given probability. This is also known as the quantile or percent point function.
|
/// </summary>
|
/// <param name="p">The location at which to compute the inverse cumulative density.</param>
|
/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
|
/// <seealso cref="InvCDF"/>
|
/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
|
public double InverseCumulativeDistribution(double p)
|
{
|
return InvCDF(_shapeA, _shapeB, _location, _scale, p);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <returns>a sample from the distribution.</returns>
|
public double Sample()
|
{
|
return SampleUnchecked(_random, _shapeA, _shapeB, _location, _scale);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
public void Samples(double[] values)
|
{
|
SamplesUnchecked(_random, values, _shapeA, _shapeB, _location, _scale);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public IEnumerable<double> Samples()
|
{
|
return SamplesUnchecked(_random, _shapeA, _shapeB, _location, _scale);
|
}
|
|
static double SampleUnchecked(System.Random rnd, double a, double b, double location, double scale)
|
{
|
return Beta.SampleUnchecked(rnd, a, b)*scale + location;
|
}
|
|
static void SamplesUnchecked(System.Random rnd, double[] values, double a, double b, double location, double scale)
|
{
|
Beta.SamplesUnchecked(rnd, values, a, b);
|
CommonParallel.For(0, values.Length, 4096, (aa, bb) =>
|
{
|
for (int i = aa; i < bb; i++)
|
{
|
values[i] = values[i]*scale + location;
|
}
|
});
|
}
|
|
static IEnumerable<double> SamplesUnchecked(System.Random rnd, double a, double b, double location, double scale)
|
{
|
while (true)
|
{
|
yield return SampleUnchecked(rnd, a, b, location, scale);
|
}
|
}
|
|
/// <summary>
|
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="x">The location at which to compute the density.</param>
|
/// <returns>the density at <paramref name="x"/>.</returns>
|
/// <seealso cref="Density"/>
|
public static double PDF(double a, double b, double location, double scale, double x)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return Beta.PDF(a, b, (x - location)/scale)/Math.Abs(scale);
|
}
|
|
/// <summary>
|
/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="x">The location at which to compute the density.</param>
|
/// <returns>the log density at <paramref name="x"/>.</returns>
|
/// <seealso cref="DensityLn"/>
|
public static double PDFLn(double a, double b, double location, double scale, double x)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return Beta.PDFLn(a, b, (x - location)/scale) - Math.Log(Math.Abs(scale));
|
}
|
|
/// <summary>
|
/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="x">The location at which to compute the cumulative distribution function.</param>
|
/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
|
/// <seealso cref="CumulativeDistribution"/>
|
public static double CDF(double a, double b, double location, double scale, double x)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return Beta.CDF(a, b, (x - location) / scale);
|
}
|
|
/// <summary>
|
/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
|
/// at the given probability. This is also known as the quantile or percent point function.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <param name="p">The location at which to compute the inverse cumulative density.</param>
|
/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
|
/// <seealso cref="InverseCumulativeDistribution"/>
|
/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
|
public static double InvCDF(double a, double b, double location, double scale, double p)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return Beta.InvCDF(a, b, p)*scale + location;
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(System.Random rnd, double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(rnd, a, b, location, scale);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(System.Random rnd, double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(rnd, a, b, location, scale);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(System.Random rnd, double[] values, double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(rnd, values, a, b, location, scale);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SampleUnchecked(SystemRandomSource.Default, a, b, location, scale);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
return SamplesUnchecked(SystemRandomSource.Default, a, b, location, scale);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="a">The α shape parameter of the BetaScaled distribution. Range: α > 0.</param>
|
/// <param name="b">The β shape parameter of the BetaScaled distribution. Range: β > 0.</param>
|
/// <param name="location">The location (μ) of the distribution.</param>
|
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(double[] values, double a, double b, double location, double scale)
|
{
|
if (!(a > 0.0 && b > 0.0 && scale > 0.0) || double.IsNaN(location))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
SamplesUnchecked(SystemRandomSource.Default, values, a, b, location, scale);
|
}
|
}
|
}
|
