// <copyright file="Multinomial.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.LinearAlgebra;
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using IStation.Numerics.LinearAlgebra.Double;
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using IStation.Numerics.Random;
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using IStation.Numerics.Statistics;
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namespace IStation.Numerics.Distributions
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{
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/// <summary>
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/// Multivariate Multinomial distribution. For details about this distribution, see
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/// <a href="http://en.wikipedia.org/wiki/Multinomial_distribution">Wikipedia - Multinomial distribution</a>.
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/// </summary>
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/// <remarks>
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/// The distribution is parameterized by a vector of ratios: in other words, the parameter
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/// does not have to be normalized and sum to 1. The reason is that some vectors can't be exactly normalized
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/// to sum to 1 in floating point representation.
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/// </remarks>
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public class Multinomial : IDistribution
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{
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System.Random _random;
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/// <summary>
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/// Stores the normalized multinomial probabilities.
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/// </summary>
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readonly double[] _p;
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/// <summary>
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/// The number of trials.
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/// </summary>
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readonly int _trials;
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/// <summary>
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/// Initializes a new instance of the Multinomial class.
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/// </summary>
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/// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
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/// as this is often impossible using floating point arithmetic.</param>
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/// <param name="n">The number of trials.</param>
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/// <exception cref="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
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public Multinomial(double[] p, int n)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
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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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_p = (double[])p.Clone();
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_trials = n;
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}
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/// <summary>
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/// Initializes a new instance of the Multinomial class.
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/// </summary>
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/// <param name="p">An array of nonnegative ratios: this array does not need to be normalized
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/// as this is often impossible using floating point arithmetic.</param>
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/// <param name="n">The number of trials.</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="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
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public Multinomial(double[] p, int n, System.Random randomSource)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
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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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_p = (double[])p.Clone();
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_trials = n;
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}
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/// <summary>
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/// Initializes a new instance of the Multinomial class from histogram <paramref name="h"/>. The distribution will
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/// not be automatically updated when the histogram changes.
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/// </summary>
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/// <param name="h">Histogram instance</param>
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/// <param name="n">The number of trials.</param>
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/// <exception cref="ArgumentOutOfRangeException">If any of the probabilities are negative or do not sum to one.</exception>
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/// <exception cref="ArgumentOutOfRangeException">If <paramref name="n"/> is negative.</exception>
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public Multinomial(Histogram h, int n)
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{
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if (h == null)
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{
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throw new ArgumentNullException(nameof(h));
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}
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// The probability distribution vector.
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var p = new double[h.BucketCount];
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// Fill in the distribution vector.
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for (var i = 0; i < h.BucketCount; i++)
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{
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p[i] = h[i].Count;
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}
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if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_p = (double[])p.Clone();
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_trials = n;
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RandomSource = SystemRandomSource.Default;
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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 $"Multinomial(Dimension = {_p.Length}, Number of Trails = {_trials})";
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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="p">An array of nonnegative ratios: this array does not need to be normalized
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/// as this is often impossible using floating point arithmetic.</param>
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/// <param name="n">The number of trials.</param>
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/// <returns>If any of the probabilities are negative returns <c>false</c>,
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/// if the sum of parameters is 0.0, or if the number of trials is negative; otherwise <c>true</c>.</returns>
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public static bool IsValidParameterSet(IEnumerable<double> p, int n)
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{
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var sum = 0.0;
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foreach (var t in p)
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{
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if (t < 0.0 || double.IsNaN(t))
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{
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return false;
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}
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sum += t;
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}
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if (sum == 0.0)
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{
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return false;
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}
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return n >= 0;
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}
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/// <summary>
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/// Gets the proportion of ratios.
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/// </summary>
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public double[] P => (double[])_p.Clone();
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/// <summary>
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/// Gets the number of trials.
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/// </summary>
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public int N => _trials;
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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 Vector<double> Mean => _trials*(DenseVector)P;
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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 Vector<double> Variance
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{
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get
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{
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// Do not use _p, because operations below will modify _p array. Use P or _p.Clone().
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var res = (DenseVector)P;
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for (var i = 0; i < res.Count; i++)
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{
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res[i] *= _trials*(1 - res[i]);
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}
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return res;
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}
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}
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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 Vector<double> Skewness
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{
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get
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{
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// Do not use _p, because operations below will modify _p array. Use P or _p.Clone().
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var res = (DenseVector)P;
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for (var i = 0; i < res.Count; i++)
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{
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res[i] = (1.0 - (2.0*res[i]))/Math.Sqrt(_trials*(1.0 - res[i])*res[i]);
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}
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return res;
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}
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}
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/// <summary>
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/// Computes values of the probability mass function.
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/// </summary>
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/// <param name="x">Non-negative integers x1, ..., xk</param>
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/// <returns>The probability mass at location <paramref name="x"/>.</returns>
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/// <exception cref="ArgumentNullException">When <paramref name="x"/> is null.</exception>
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/// <exception cref="ArgumentException">When length of <paramref name="x"/> is not equal to event probabilities count.</exception>
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public double Probability(int[] x)
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{
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if (null == x)
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{
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throw new ArgumentNullException(nameof(x));
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}
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if (x.Length != _p.Length)
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{
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throw new ArgumentException("All vectors must have the same dimensionality.", nameof(x));
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}
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if (x.Sum() == _trials)
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{
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var coef = SpecialFunctions.Multinomial(_trials, x);
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var num = 1.0;
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for (var i = 0; i < x.Length; i++)
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{
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num *= Math.Pow(_p[i], x[i]);
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}
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return coef*num;
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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 values of the log probability mass function.
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/// </summary>
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/// <param name="x">Non-negative integers x1, ..., xk</param>
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/// <returns>The log probability mass at location <paramref name="x"/>.</returns>
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/// <exception cref="ArgumentNullException">When <paramref name="x"/> is null.</exception>
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/// <exception cref="ArgumentException">When length of <paramref name="x"/> is not equal to event probabilities count.</exception>
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public double ProbabilityLn(int[] x)
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{
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if (null == x)
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{
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throw new ArgumentNullException(nameof(x));
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}
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if (x.Length != _p.Length)
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{
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throw new ArgumentException("All vectors must have the same dimensionality.", nameof(x));
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}
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if (x.Sum() == _trials)
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{
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var coef = Math.Log(SpecialFunctions.Multinomial(_trials, x));
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var num = x.Select((t, i) => t*Math.Log(_p[i])).Sum();
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return coef + num;
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}
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return 0.0;
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}
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/// <summary>
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/// Samples one multinomial distributed random variable.
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/// </summary>
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/// <returns>the counts for each of the different possible values.</returns>
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public int[] Sample()
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{
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return Sample(_random, _p, _trials);
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}
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/// <summary>
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/// Samples a sequence multinomially distributed random variables.
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/// </summary>
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/// <returns>a sequence of counts for each of the different possible values.</returns>
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public IEnumerable<int[]> Samples()
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{
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while (true)
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{
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yield return Sample(_random, _p, _trials);
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}
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}
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/// <summary>
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/// Samples one multinomial distributed random variable.
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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="p">An array of nonnegative ratios: this array does not need to be normalized
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/// as this is often impossible using floating point arithmetic.</param>
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/// <param name="n">The number of trials.</param>
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/// <returns>the counts for each of the different possible values.</returns>
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public static int[] Sample(System.Random rnd, double[] p, int n)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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// The cumulative density of p.
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var cp = Categorical.ProbabilityMassToCumulativeDistribution(p);
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// The variable that stores the counts.
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var ret = new int[p.Length];
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for (var i = 0; i < n; i++)
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{
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ret[Categorical.SampleUnchecked(rnd, cp)]++;
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}
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return ret;
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}
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/// <summary>
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/// Samples a multinomially distributed random variable.
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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="p">An array of nonnegative ratios: this array does not need to be normalized
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/// as this is often impossible using floating point arithmetic.</param>
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/// <param name="n">The number of variables needed.</param>
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/// <returns>a sequence of counts for each of the different possible values.</returns>
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public static IEnumerable<int[]> Samples(System.Random rnd, double[] p, int n)
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{
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if (Control.CheckDistributionParameters && !IsValidParameterSet(p, n))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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// The cumulative density of p.
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var cp = Categorical.ProbabilityMassToCumulativeDistribution(p);
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while (true)
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{
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// The variable that stores the counts.
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var ret = new int[p.Length];
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for (var i = 0; i < n; i++)
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{
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ret[Categorical.SampleUnchecked(rnd, cp)]++;
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}
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yield return ret;
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}
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}
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}
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}
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