132 lines
3.2 KiB
Go
132 lines
3.2 KiB
Go
// Copyright ©2017 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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)
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// Pareto implements the Pareto (Type I) distribution, a one parameter distribution
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// with support above the scale parameter.
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//
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// The density function is given by
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//
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// (α x_m^{α})/(x^{α+1}) for x >= x_m.
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//
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// For more information, see https://en.wikipedia.org/wiki/Pareto_distribution.
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type Pareto struct {
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// Xm is the scale parameter.
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// Xm must be greater than 0.
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Xm float64
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// Alpha is the shape parameter.
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// Alpha must be greater than 0.
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Alpha 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 (p Pareto) CDF(x float64) float64 {
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if x < p.Xm {
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return 0
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}
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return -math.Expm1(p.Alpha * math.Log(p.Xm/x))
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}
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// Entropy returns the differential entropy of the distribution.
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func (p Pareto) Entropy() float64 {
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return math.Log(p.Xm) - math.Log(p.Alpha) + (1 + 1/p.Alpha)
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}
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// ExKurtosis returns the excess kurtosis of the distribution.
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func (p Pareto) ExKurtosis() float64 {
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if p.Alpha <= 4 {
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return math.NaN()
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}
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return 6 * (p.Alpha*p.Alpha*p.Alpha + p.Alpha*p.Alpha - 6*p.Alpha - 2) / (p.Alpha * (p.Alpha - 3) * (p.Alpha - 4))
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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 (p Pareto) LogProb(x float64) float64 {
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if x < p.Xm {
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return math.Inf(-1)
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}
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return math.Log(p.Alpha) + p.Alpha*math.Log(p.Xm) - (p.Alpha+1)*math.Log(x)
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}
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// Mean returns the mean of the probability distribution.
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func (p Pareto) Mean() float64 {
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if p.Alpha <= 1 {
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return math.Inf(1)
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}
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return p.Alpha * p.Xm / (p.Alpha - 1)
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}
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// Median returns the median of the pareto distribution.
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func (p Pareto) Median() float64 {
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return p.Quantile(0.5)
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}
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// Mode returns the mode of the distribution.
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func (p Pareto) Mode() float64 {
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return p.Xm
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}
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// NumParameters returns the number of parameters in the distribution.
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func (p Pareto) NumParameters() int {
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return 2
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}
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// Prob computes the value of the probability density function at x.
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func (p Pareto) Prob(x float64) float64 {
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return math.Exp(p.LogProb(x))
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}
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// Quantile returns the inverse of the cumulative probability distribution.
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func (p Pareto) Quantile(prob float64) float64 {
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if prob < 0 || 1 < prob {
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panic(badPercentile)
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}
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return p.Xm / math.Pow(1-prob, 1/p.Alpha)
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}
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// Rand returns a random sample drawn from the distribution.
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func (p Pareto) Rand() float64 {
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var rnd float64
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if p.Src == nil {
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rnd = rand.ExpFloat64()
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} else {
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rnd = rand.New(p.Src).ExpFloat64()
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}
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return p.Xm * math.Exp(rnd/p.Alpha)
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}
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// StdDev returns the standard deviation of the probability distribution.
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func (p Pareto) StdDev() float64 {
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return math.Sqrt(p.Variance())
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}
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// Survival returns the survival function (complementary CDF) at x.
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func (p Pareto) Survival(x float64) float64 {
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if x < p.Xm {
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return 1
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}
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return math.Pow(p.Xm/x, p.Alpha)
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}
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// Variance returns the variance of the probability distribution.
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func (p Pareto) Variance() float64 {
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if p.Alpha <= 2 {
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return math.Inf(1)
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}
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am1 := p.Alpha - 1
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return p.Xm * p.Xm * p.Alpha / (am1 * am1 * (p.Alpha - 2))
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}
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