// <copyright file="Pareto.cs" company="Math.NET">
// Math.NET Numerics, part of the Math.NET Project
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using System;
using System.Collections.Generic;
using System.Linq;
using IStation.Numerics.Random;
using IStation.Numerics.Threading;

namespace IStation.Numerics.Distributions
{
    /// <summary>
    /// Continuous Univariate Pareto distribution.
    /// The Pareto distribution is a power law probability distribution that coincides with social,
    /// scientific, geophysical, actuarial, and many other types of observable phenomena.
    /// For details about this distribution, see
    /// <a href="http://en.wikipedia.org/wiki/Pareto_distribution">Wikipedia - Pareto distribution</a>.
    /// </summary>
    public class Pareto : IContinuousDistribution
    {
        System.Random _random;

        readonly double _scale;
        readonly double _shape;

        /// <summary>
        /// Initializes a new instance of the <see cref="Pareto"/> class.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <exception cref="ArgumentException">If <paramref name="scale"/> or <paramref name="shape"/> are negative.</exception>
        public Pareto(double scale, double shape)
        {
            if (!IsValidParameterSet(scale, shape))
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            _random = SystemRandomSource.Default;
            _scale = scale;
            _shape = shape;
        }

        /// <summary>
        /// Initializes a new instance of the <see cref="Pareto"/> class.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <param name="randomSource">The random number generator which is used to draw random samples.</param>
        /// <exception cref="ArgumentException">If <paramref name="scale"/> or <paramref name="shape"/> are negative.</exception>
        public Pareto(double scale, double shape, System.Random randomSource)
        {
            if (!IsValidParameterSet(scale, shape))
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            _random = randomSource ?? SystemRandomSource.Default;
            _scale = scale;
            _shape = shape;
        }

        /// <summary>
        /// A string representation of the distribution.
        /// </summary>
        /// <returns>a string representation of the distribution.</returns>
        public override string ToString()
        {
            return $"Pareto(xm = {_scale}, α = {_shape})";
        }

        /// <summary>
        /// Tests whether the provided values are valid parameters for this distribution.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        public static bool IsValidParameterSet(double scale, double shape)
        {
            return scale > 0.0 && shape > 0.0;
        }

        /// <summary>
        /// Gets the scale (xm) of the distribution. Range: xm > 0.
        /// </summary>
        public double Scale => _scale;

        /// <summary>
        /// Gets the shape (α) of the distribution. Range: α > 0.
        /// </summary>
        public double Shape => _shape;

        /// <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 (_shape <= 1)
                {
                    throw new NotSupportedException();
                }

                return _shape*_scale/(_shape - 1.0);
            }
        }

        /// <summary>
        /// Gets the variance of the distribution.
        /// </summary>
        public double Variance
        {
            get
            {
                if (_shape <= 2.0)
                {
                    return double.PositiveInfinity;
                }

                return _scale*_scale*_shape/((_shape - 1.0)*(_shape - 1.0)*(_shape - 2.0));
            }
        }

        /// <summary>
        /// Gets the standard deviation of the distribution.
        /// </summary>
        public double StdDev => (_scale*Math.Sqrt(_shape))/(Math.Abs(_shape - 1.0)*Math.Sqrt(_shape - 2.0));

        /// <summary>
        /// Gets the entropy of the distribution.
        /// </summary>
        public double Entropy => Math.Log(_shape/_scale) - (1.0/_shape) - 1.0;

        /// <summary>
        /// Gets the skewness of the distribution.
        /// </summary>
        public double Skewness => (2.0*(_shape + 1.0)/(_shape - 3.0))*Math.Sqrt((_shape - 2.0)/_shape);

        /// <summary>
        /// Gets the mode of the distribution.
        /// </summary>
        public double Mode => _scale;

        /// <summary>
        /// Gets the median of the distribution.
        /// </summary>
        public double Median => _scale*Math.Pow(2.0, 1.0/_shape);

        /// <summary>
        /// Gets the minimum of the distribution.
        /// </summary>
        public double Minimum => _scale;

        /// <summary>
        /// Gets the maximum of the distribution.
        /// </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"/>
        public double Density(double x)
        {
            return _shape*Math.Pow(_scale, _shape)/Math.Pow(x, _shape + 1.0);
        }

        /// <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 Math.Log(_shape) + _shape*Math.Log(_scale) - (_shape + 1.0)*Math.Log(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 1.0 - Math.Pow(_scale/x, _shape);
        }

        /// <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"/>
        public double InverseCumulativeDistribution(double p)
        {
            return _scale*Math.Pow(1.0 - p, -1.0/_shape);
        }

        /// <summary>
        /// Draws a random sample from the distribution.
        /// </summary>
        /// <returns>A random number from this distribution.</returns>
        public double Sample()
        {
            return SampleUnchecked(_random, _scale, _shape);
        }

        /// <summary>
        /// Fills an array with samples generated from the distribution.
        /// </summary>
        public void Samples(double[] values)
        {
            SamplesUnchecked(_random, values, _scale, _shape);
        }

        /// <summary>
        /// Generates a sequence of samples from the Pareto distribution.
        /// </summary>
        /// <returns>a sequence of samples from the distribution.</returns>
        public IEnumerable<double> Samples()
        {
            return SamplesUnchecked(_random, _scale, _shape);
        }

        static double SampleUnchecked(System.Random rnd, double scale, double shape)
        {
            return scale*Math.Pow(rnd.NextDouble(), -1.0/shape);
        }

        static IEnumerable<double> SamplesUnchecked(System.Random rnd, double scale, double shape)
        {
            var power = -1.0/shape;
            return rnd.NextDoubleSequence().Select(x => scale*Math.Pow(x, power));
        }

        static void SamplesUnchecked(System.Random rnd, double[] values, double scale, double shape)
        {
            var power = -1.0/shape;
            rnd.NextDoubles(values);
            CommonParallel.For(0, values.Length, 4096, (a, b) =>
            {
                for (int i = a; i < b; i++)
                {
                    values[i] = scale*Math.Pow(values[i], power);
                }
            });
        }

        /// <summary>
        /// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double x)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return shape*Math.Pow(scale, shape)/Math.Pow(x, shape + 1.0);
        }

        /// <summary>
        /// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 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 scale, double shape, double x)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return Math.Log(shape) + shape*Math.Log(scale) - (shape + 1.0)*Math.Log(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>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
        /// <seealso cref="CumulativeDistribution"/>
        public static double CDF(double scale, double shape, double x)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return 1.0 - Math.Pow(scale/x, shape);
        }

        /// <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>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
        /// <seealso cref="InverseCumulativeDistribution"/>
        public static double InvCDF(double scale, double shape, double p)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return scale*Math.Pow(1.0 - p, -1.0/shape);
        }

        /// <summary>
        /// Generates a sample from the distribution.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sample from the distribution.</returns>
        public static double Sample(System.Random rnd, double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return scale*Math.Pow(rnd.NextDouble(), -1.0/shape);
        }

        /// <summary>
        /// Generates a sequence of samples from the distribution.
        /// </summary>
        /// <param name="rnd">The random number generator to use.</param>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static IEnumerable<double> Samples(System.Random rnd, double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return SamplesUnchecked(rnd, scale, shape);
        }

        /// <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="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static void Samples(System.Random rnd, double[] values, double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            SamplesUnchecked(rnd, values, scale, shape);
        }

        /// <summary>
        /// Generates a sample from the distribution.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sample from the distribution.</returns>
        public static double Sample(double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return SampleUnchecked(SystemRandomSource.Default, scale, shape);
        }

        /// <summary>
        /// Generates a sequence of samples from the distribution.
        /// </summary>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static IEnumerable<double> Samples(double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            return SamplesUnchecked(SystemRandomSource.Default, scale, shape);
        }

        /// <summary>
        /// Fills an array with samples generated from the distribution.
        /// </summary>
        /// <param name="values">The array to fill with the samples.</param>
        /// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
        /// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
        /// <returns>a sequence of samples from the distribution.</returns>
        public static void Samples(double[] values, double scale, double shape)
        {
            if (scale <= 0.0 || shape <= 0.0)
            {
                throw new ArgumentException("Invalid parametrization for the distribution.");
            }

            SamplesUnchecked(SystemRandomSource.Default, values, scale, shape);
        }
    }
}