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stat: implement Pareto (#309)
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kczimm
122
stat/distuv/pareto.go
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122
stat/distuv/pareto.go
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// 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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// (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.Rand
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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 1 - p.Survival(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 0
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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.Xm * math.Pow(2, 1/p.Alpha)
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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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// 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 = p.Src.ExpFloat64()
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}
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return math.Exp(math.Log(p.Xm) + 1/p.Alpha*rnd)
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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.Exp(p.Alpha * (math.Log(p.Xm) - math.Log(x)))
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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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162
stat/distuv/pareto_test.go
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162
stat/distuv/pareto_test.go
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@@ -0,0 +1,162 @@
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// 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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"sort"
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"testing"
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"golang.org/x/exp/rand"
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"gonum.org/v1/gonum/floats"
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)
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func TestParetoProb(t *testing.T) {
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for _, test := range []struct {
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x, xm, alpha, want float64
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}{
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{0, 1, 1, 0},
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{0.5, 1, 1, 0},
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{1, 1, 1, 1.0},
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{1.5, 1, 1, 0.444444444444444},
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{2, 1, 1, 0.25},
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{2.5, 1, 1, 0.16},
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{3, 1, 1, 0.1111111111111111},
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{3.5, 1, 1, 0.081632653061224},
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{4, 1, 1, 0.0625},
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{4.5, 1, 1, 0.049382716049383},
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{5, 1, 1, 0.04},
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{0, 1, 2, 0},
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{0.5, 1, 2, 0},
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{1, 1, 2, 2},
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{1.5, 1, 2, 0.592592592592593},
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{2, 1, 2, 0.25},
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{2.5, 1, 2, 0.128},
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{3, 1, 2, 0.074074074074074},
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{3.5, 1, 2, 0.046647230320700},
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{4, 1, 2, 0.03125},
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{4.5, 1, 2, 0.021947873799726},
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{5, 1, 2, 0.016},
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{0, 1, 3, 0},
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{0.5, 1, 3, 0},
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{1, 1, 3, 3.0},
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{1.5, 1, 3, 0.592592592592593},
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{2, 1, 3, 0.1875},
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{2.5, 1, 3, 0.0768},
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{3, 1, 3, 0.037037037037037},
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{3.5, 1, 3, 0.019991670137443},
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{4, 1, 3, 0.011718750000000},
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{4.5, 1, 3, 0.007315957933242},
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{5, 1, 3, 0.0048},
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} {
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pdf := Pareto{test.xm, test.alpha, nil}.Prob(test.x)
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if !floats.EqualWithinAbsOrRel(pdf, test.want, 1e-10, 1e-10) {
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t.Errorf("Pdf mismatch, x = %v, xm = %v, alpha = %v. Got %v, want %v", test.x, test.xm, test.alpha, pdf, test.want)
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}
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}
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}
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func TestParetoCDF(t *testing.T) {
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for _, test := range []struct {
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x, xm, alpha, want float64
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}{
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{0, 1, 1, 0},
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{0.5, 1, 1, 0},
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{1, 1, 1, 0},
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{1.5, 1, 1, 0.333333333333333},
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{2, 1, 1, 0.5},
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{2.5, 1, 1, 0.6},
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{3, 1, 1, 0.666666666666667},
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{3.5, 1, 1, 0.714285714285714},
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{4, 1, 1, 0.75},
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{4.5, 1, 1, 0.777777777777778},
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{5, 1, 1, 0.80},
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{5.5, 1, 1, 0.818181818181818},
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{6, 1, 1, 0.833333333333333},
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{6.5, 1, 1, 0.846153846153846},
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{7, 1, 1, 0.857142857142857},
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{7.5, 1, 1, 0.866666666666667},
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{8, 1, 1, 0.875},
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{8.5, 1, 1, 0.882352941176471},
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{9, 1, 1, 0.888888888888889},
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{9.5, 1, 1, 0.894736842105263},
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{10, 1, 1, 0.90},
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{0, 1, 2, 0},
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{0.5, 1, 2, 0},
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{1, 1, 2, 0},
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{1.5, 1, 2, 0.555555555555556},
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{2, 1, 2, 0.75},
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{2.5, 1, 2, 0.84},
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{3, 1, 2, 0.888888888888889},
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{3.5, 1, 2, 0.918367346938776},
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{4, 1, 2, 0.9375},
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{4.5, 1, 2, 0.950617283950617},
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{5, 1, 2, 0.96},
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{5.5, 1, 2, 0.966942148760331},
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{6, 1, 2, 0.972222222222222},
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{6.5, 1, 2, 0.976331360946746},
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{7, 1, 2, 0.979591836734694},
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{7.5, 1, 2, 0.982222222222222},
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{8, 1, 2, 0.984375000000000},
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{8.5, 1, 2, 0.986159169550173},
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{9, 1, 2, 0.987654320987654},
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{9.5, 1, 2, 0.988919667590028},
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{10, 1, 2, 0.99},
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{0, 1, 3, 0},
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{0.5, 1, 3, 0},
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{1, 1, 3, 0},
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{1.5, 1, 3, 0.703703703703704},
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{2, 1, 3, 0.875},
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{2.5, 1, 3, 0.936},
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{3, 1, 3, 0.962962962962963},
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{3.5, 1, 3, 0.976676384839650},
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{4, 1, 3, 0.984375000000000},
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{4.5, 1, 3, 0.989026063100137},
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{5, 1, 3, 0.992},
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{5.5, 1, 3, 0.993989481592787},
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{6, 1, 3, 0.995370370370370},
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{6.5, 1, 3, 0.996358670914884},
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{7, 1, 3, 0.997084548104956},
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{7.5, 1, 3, 0.997629629629630},
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{8, 1, 3, 0.998046875000000},
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{8.5, 1, 3, 0.998371667005903},
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{9, 1, 3, 0.998628257887517},
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{9.5, 1, 3, 0.998833649220003},
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{10, 1, 3, 0.999},
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} {
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cdf := Pareto{test.xm, test.alpha, nil}.CDF(test.x)
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if !floats.EqualWithinAbsOrRel(cdf, test.want, 1e-10, 1e-10) {
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t.Errorf("CDF mismatch, x = %v, xm = %v, alpha = %v. Got %v, want %v", test.x, test.xm, test.alpha, cdf, test.want)
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}
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}
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}
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func TestPareto(t *testing.T) {
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src := rand.New(rand.NewSource(1))
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for i, p := range []Pareto{
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{1, 10, src},
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{1, 20, src},
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} {
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testPareto(t, p, i)
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}
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}
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func testPareto(t *testing.T, p Pareto, i int) {
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tol := 1e-2
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const n = 1e6
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const bins = 50
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x := make([]float64, n)
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generateSamples(x, p)
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sort.Float64s(x)
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testRandLogProbContinuous(t, i, 0, x, p, tol, bins)
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checkMean(t, i, x, p, tol)
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checkVarAndStd(t, i, x, p, tol)
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checkExKurtosis(t, i, x, p, 7e-2)
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checkProbContinuous(t, i, x, p, 1e-3)
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}
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