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125 lines
3.2 KiB
Go
125 lines
3.2 KiB
Go
// Copyright ©2021 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package distuv
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import (
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"math"
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"golang.org/x/exp/rand"
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"gonum.org/v1/gonum/mathext"
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)
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// Chi implements the χ distribution, a one parameter distribution
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// with support on the positive numbers.
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//
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// The density function is given by
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// 1/(2^{k/2-1} * Γ(k/2)) * x^{k - 1} * e^{-x^2/2}
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//
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// For more information, see https://en.wikipedia.org/wiki/Chi_distribution.
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type Chi struct {
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// K is the shape parameter, corresponding to the degrees of freedom. Must
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// be greater than 0.
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K float64
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Src rand.Source
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}
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// CDF computes the value of the cumulative density function at x.
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func (c Chi) CDF(x float64) float64 {
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return mathext.GammaIncReg(c.K/2, (x*x)/2)
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}
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// Entropy returns the differential entropy of the distribution.
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func (c Chi) Entropy() float64 {
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lg, _ := math.Lgamma(c.K / 2)
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return lg + 0.5*(c.K-math.Ln2-(c.K-1)*mathext.Digamma(c.K/2))
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}
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// ExKurtosis returns the excess kurtosis of the distribution.
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func (c Chi) ExKurtosis() float64 {
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v := c.Variance()
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s := math.Sqrt(v)
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return 2 / v * (1 - c.Mean()*s*c.Skewness() - v)
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}
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// LogProb computes the natural logarithm of the value of the probability
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// density function at x.
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func (c Chi) LogProb(x float64) float64 {
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if x < 0 {
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return math.Inf(-1)
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}
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lg, _ := math.Lgamma(c.K / 2)
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return (c.K-1)*math.Log(x) - (x*x)/2 - (c.K/2-1)*math.Ln2 - lg
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}
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// Mean returns the mean of the probability distribution.
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func (c Chi) Mean() float64 {
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lg1, _ := math.Lgamma((c.K + 1) / 2)
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lg, _ := math.Lgamma(c.K / 2)
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return math.Sqrt2 * math.Exp(lg1-lg)
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}
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// Median returns the median of the distribution.
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func (c Chi) Median() float64 {
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return c.Quantile(0.5)
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}
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// Mode returns the mode of the distribution.
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//
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// Mode returns NaN if K is less than one.
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func (c Chi) Mode() float64 {
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return math.Sqrt(c.K - 1)
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}
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// NumParameters returns the number of parameters in the distribution.
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func (c Chi) NumParameters() int {
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return 1
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}
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// Prob computes the value of the probability density function at x.
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func (c Chi) Prob(x float64) float64 {
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return math.Exp(c.LogProb(x))
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}
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// Rand returns a random sample drawn from the distribution.
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func (c Chi) Rand() float64 {
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return math.Sqrt(Gamma{c.K / 2, 0.5, c.Src}.Rand())
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}
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// Quantile returns the inverse of the cumulative distribution function.
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func (c Chi) Quantile(p float64) float64 {
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if p < 0 || 1 < p {
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panic(badPercentile)
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}
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return math.Sqrt(2 * mathext.GammaIncRegInv(0.5*c.K, p))
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}
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// Skewness returns the skewness of the distribution.
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func (c Chi) Skewness() float64 {
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v := c.Variance()
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s := math.Sqrt(v)
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return c.Mean() / (s * v) * (1 - 2*v)
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}
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// StdDev returns the standard deviation of the probability distribution.
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func (c Chi) StdDev() float64 {
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return math.Sqrt(c.Variance())
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}
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// Survival returns the survival function (complementary CDF) at x.
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func (c Chi) Survival(x float64) float64 {
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if x < 0 {
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return 1
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}
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return mathext.GammaIncRegComp(0.5*c.K, 0.5*(x*x))
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}
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// Variance returns the variance of the probability distribution.
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func (c Chi) Variance() float64 {
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m := c.Mean()
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return math.Max(0, c.K-m*m)
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}
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