// <copyright file="Erlang.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) 2009-2014 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;
|
|
namespace IStation.Numerics.Distributions
|
{
|
/// <summary>
|
/// Continuous Univariate Erlang distribution.
|
/// This distribution is a continuous probability distribution with wide applicability primarily due to its
|
/// relation to the exponential and Gamma distributions.
|
/// <a href="http://en.wikipedia.org/wiki/Erlang_distribution">Wikipedia - Erlang distribution</a>.
|
/// </summary>
|
public class Erlang : IContinuousDistribution
|
{
|
System.Random _random;
|
|
readonly int _shape;
|
readonly double _rate;
|
|
/// <summary>
|
/// Initializes a new instance of the <see cref="Erlang"/> class.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
public Erlang(int shape, double rate)
|
{
|
if (!IsValidParameterSet(shape, rate))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = SystemRandomSource.Default;
|
_shape = shape;
|
_rate = rate;
|
}
|
|
/// <summary>
|
/// Initializes a new instance of the <see cref="Erlang"/> class.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
|
public Erlang(int shape, double rate, System.Random randomSource)
|
{
|
if (!IsValidParameterSet(shape, rate))
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
_random = randomSource ?? SystemRandomSource.Default;
|
_shape = shape;
|
_rate = rate;
|
}
|
|
/// <summary>
|
/// Constructs a Erlang distribution from a shape and scale parameter. The distribution will
|
/// be initialized with the default <seealso cref="System.Random"/> random number generator.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="scale">The scale (μ) of the Erlang distribution. Range: μ ≥ 0.</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples. Optional, can be null.</param>
|
public static Erlang WithShapeScale(int shape, double scale, System.Random randomSource = null)
|
{
|
return new Erlang(shape, 1.0/scale, randomSource);
|
}
|
|
/// <summary>
|
/// Constructs a Erlang distribution from a shape and inverse scale parameter. The distribution will
|
/// be initialized with the default <seealso cref="System.Random"/> random number generator.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <param name="randomSource">The random number generator which is used to draw random samples. Optional, can be null.</param>
|
public static Erlang WithShapeRate(int shape, double rate, System.Random randomSource = null)
|
{
|
return new Erlang(shape, rate, randomSource);
|
}
|
|
/// <summary>
|
/// A string representation of the distribution.
|
/// </summary>
|
/// <returns>a string representation of the distribution.</returns>
|
public override string ToString()
|
{
|
return $"Erlang(k = {_shape}, λ = {_rate})";
|
}
|
|
/// <summary>
|
/// Tests whether the provided values are valid parameters for this distribution.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
public static bool IsValidParameterSet(int shape, double rate)
|
{
|
return shape >= 0 && rate >= 0.0;
|
}
|
|
/// <summary>
|
/// Gets the shape (k) of the Erlang distribution. Range: k ≥ 0.
|
/// </summary>
|
public int Shape => _shape;
|
|
/// <summary>
|
/// Gets the rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.
|
/// </summary>
|
public double Rate => _rate;
|
|
/// <summary>
|
/// Gets the scale of the Erlang distribution.
|
/// </summary>
|
public double Scale => 1.0/_rate;
|
|
/// <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 distribution.
|
/// </summary>
|
public double Mean
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return _shape;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return _shape/_rate;
|
}
|
}
|
|
/// <summary>
|
/// Gets the variance of the distribution.
|
/// </summary>
|
public double Variance
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return 0.0;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return _shape/(_rate*_rate);
|
}
|
}
|
|
/// <summary>
|
/// Gets the standard deviation of the distribution.
|
/// </summary>
|
public double StdDev
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return 0.0;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return Math.Sqrt(_shape)/_rate;
|
}
|
}
|
|
/// <summary>
|
/// Gets the entropy of the distribution.
|
/// </summary>
|
public double Entropy
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return 0.0;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return _shape - Math.Log(_rate) + SpecialFunctions.GammaLn(_shape) + ((1.0 - _shape)*SpecialFunctions.DiGamma(_shape));
|
}
|
}
|
|
/// <summary>
|
/// Gets the skewness of the distribution.
|
/// </summary>
|
public double Skewness
|
{
|
get
|
{
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return 0.0;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return 2.0/Math.Sqrt(_shape);
|
}
|
}
|
|
/// <summary>
|
/// Gets the mode of the distribution.
|
/// </summary>
|
public double Mode
|
{
|
get
|
{
|
if (_shape < 1)
|
{
|
throw new NotSupportedException();
|
}
|
|
if (double.IsPositiveInfinity(_rate))
|
{
|
return _shape;
|
}
|
|
if (_rate == 0.0 && _shape == 0.0)
|
{
|
return double.NaN;
|
}
|
|
return (_shape - 1.0)/_rate;
|
}
|
}
|
|
/// <summary>
|
/// Gets the median of the distribution.
|
/// </summary>
|
public double Median => throw new NotSupportedException();
|
|
/// <summary>
|
/// Gets the minimum value.
|
/// </summary>
|
public double Minimum => 0.0;
|
|
/// <summary>
|
/// Gets the Maximum value.
|
/// </summary>
|
public double Maximum => double.PositiveInfinity;
|
|
/// <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(int, double, double)"/>
|
public double Density(double x)
|
{
|
return PDF(_shape, _rate, 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(int, double, double)"/>
|
public double DensityLn(double x)
|
{
|
return PDFLn(_shape, _rate, 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(int, double, double)"/>
|
public double CumulativeDistribution(double x)
|
{
|
return CDF(_shape, _rate, x);
|
}
|
|
/// <summary>
|
/// Generates a sample from the Erlang distribution.
|
/// </summary>
|
/// <returns>a sample from the distribution.</returns>
|
public double Sample()
|
{
|
return Gamma.SampleUnchecked(_random, _shape, _rate);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
public void Samples(double[] values)
|
{
|
Gamma.SamplesUnchecked(_random, values, _shape, _rate);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the Erlang distribution.
|
/// </summary>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public IEnumerable<double> Samples()
|
{
|
while (true)
|
{
|
yield return Gamma.SampleUnchecked(_random, _shape, _rate);
|
}
|
}
|
|
/// <summary>
|
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 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(int shape, double rate, double x)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
if (double.IsPositiveInfinity(rate))
|
{
|
return x == shape ? double.PositiveInfinity : 0.0;
|
}
|
|
if (shape == 0.0 && rate == 0.0)
|
{
|
return 0.0;
|
}
|
|
if (shape == 1.0)
|
{
|
return rate*Math.Exp(-rate*x);
|
}
|
|
if (shape > 160.0)
|
{
|
return Math.Exp(PDFLn(shape, rate, x));
|
}
|
|
return Math.Pow(rate, shape)*Math.Pow(x, shape - 1.0)*Math.Exp(-rate*x)/SpecialFunctions.Gamma(shape);
|
}
|
|
/// <summary>
|
/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 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(int shape, double rate, double x)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
if (double.IsPositiveInfinity(rate))
|
{
|
return x == shape ? double.PositiveInfinity : double.NegativeInfinity;
|
}
|
|
if (shape == 0.0 && rate == 0.0)
|
{
|
return double.NegativeInfinity;
|
}
|
|
if (shape == 1.0)
|
{
|
return Math.Log(rate) - (rate*x);
|
}
|
|
return (shape*Math.Log(rate)) + ((shape - 1.0)*Math.Log(x)) - (rate*x) - SpecialFunctions.GammaLn(shape);
|
}
|
|
/// <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>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
|
/// <seealso cref="CumulativeDistribution"/>
|
public static double CDF(int shape, double rate, double x)
|
{
|
if (shape < 0.0 || rate < 0.0)
|
{
|
throw new ArgumentException("Invalid parametrization for the distribution.");
|
}
|
|
if (double.IsPositiveInfinity(rate))
|
{
|
return x >= shape ? 1.0 : 0.0;
|
}
|
|
if (shape == 0.0 && rate == 0.0)
|
{
|
return 0.0;
|
}
|
|
return SpecialFunctions.GammaLowerRegularized(shape, x*rate);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(System.Random rnd, int shape, double rate)
|
{
|
return Gamma.Sample(rnd, shape, rate);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="rnd">The random number generator to use.</param>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(System.Random rnd, int shape, double rate)
|
{
|
return Gamma.Samples(rnd, shape, rate);
|
}
|
|
/// <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="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(System.Random rnd, double[] values, int shape, double rate)
|
{
|
Gamma.Samples(rnd, values, shape, rate);
|
}
|
|
/// <summary>
|
/// Generates a sample from the distribution.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sample from the distribution.</returns>
|
public static double Sample(int shape, double rate)
|
{
|
return Gamma.Sample(shape, rate);
|
}
|
|
/// <summary>
|
/// Generates a sequence of samples from the distribution.
|
/// </summary>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static IEnumerable<double> Samples(int shape, double rate)
|
{
|
return Gamma.Samples(shape, rate);
|
}
|
|
/// <summary>
|
/// Fills an array with samples generated from the distribution.
|
/// </summary>
|
/// <param name="values">The array to fill with the samples.</param>
|
/// <param name="shape">The shape (k) of the Erlang distribution. Range: k ≥ 0.</param>
|
/// <param name="rate">The rate or inverse scale (λ) of the Erlang distribution. Range: λ ≥ 0.</param>
|
/// <returns>a sequence of samples from the distribution.</returns>
|
public static void Samples(double[] values, int shape, double rate)
|
{
|
Gamma.Samples(values, shape, rate);
|
}
|
}
|
}
|
