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// Copyright ©2018 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package distuv
import (
"math"
"golang.org/x/exp/rand"
)
// GumbelRight implements the right-skewed Gumbel distribution, a two-parameter
// continuous distribution with support over the real numbers. The right-skewed
// Gumbel distribution is also sometimes known as the Extreme Value distribution.
//
// The right-skewed Gumbel distribution has density function
// 1/beta * exp(-(z + exp(-z)))
// z = (x - mu)/beta
// Beta must be greater than 0.
//
// For more information, see https://en.wikipedia.org/wiki/Gumbel_distribution.
type GumbelRight struct {
Mu float64
Beta float64
Src rand.Source
}
func (g GumbelRight) z(x float64) float64 {
return (x - g.Mu) / g.Beta
}
// CDF computes the value of the cumulative density function at x.
func (g GumbelRight) CDF(x float64) float64 {
z := g.z(x)
return math.Exp(-math.Exp(-z))
}
// Entropy returns the differential entropy of the distribution.
func (g GumbelRight) Entropy() float64 {
return math.Log(g.Beta) + eulerMascheroni + 1
}
// ExKurtosis returns the excess kurtosis of the distribution.
func (g GumbelRight) ExKurtosis() float64 {
return 12.0 / 5
}
// LogProb computes the natural logarithm of the value of the probability density function at x.
func (g GumbelRight) LogProb(x float64) float64 {
z := g.z(x)
return -math.Log(g.Beta) - z - math.Exp(-z)
}
// Mean returns the mean of the probability distribution.
func (g GumbelRight) Mean() float64 {
return g.Mu + g.Beta*eulerMascheroni
}
// Median returns the median of the normal distribution.
func (g GumbelRight) Median() float64 {
return g.Mu - g.Beta*math.Log(math.Ln2)
}
// Mode returns the mode of the normal distribution.
func (g GumbelRight) Mode() float64 {
return g.Mu
}
// NumParameters returns the number of parameters in the distribution.
func (GumbelRight) NumParameters() int {
return 2
}
// Prob computes the value of the probability density function at x.
func (g GumbelRight) Prob(x float64) float64 {
return math.Exp(g.LogProb(x))
}
// Quantile returns the inverse of the cumulative probability distribution.
func (g GumbelRight) Quantile(p float64) float64 {
if p < 0 || 1 < p {
panic(badPercentile)
}
return g.Mu - g.Beta*math.Log(-math.Log(p))
}
// Rand returns a random sample drawn from the distribution.
func (g GumbelRight) Rand() float64 {
var rnd float64
if g.Src == nil {
rnd = rand.ExpFloat64()
} else {
rnd = rand.New(g.Src).ExpFloat64()
}
return g.Mu - g.Beta*math.Log(rnd)
}
// Skewness returns the skewness of the distribution.
func (GumbelRight) Skewness() float64 {
return 12 * math.Sqrt(6) * apery / (math.Pi * math.Pi * math.Pi)
}
// StdDev returns the standard deviation of the probability distribution.
func (g GumbelRight) StdDev() float64 {
return (math.Pi / math.Sqrt(6)) * g.Beta
}
// Survival returns the survival function (complementary CDF) at x.
func (g GumbelRight) Survival(x float64) float64 {
return 1 - g.CDF(x)
}
// Variance returns the variance of the probability distribution.
func (g GumbelRight) Variance() float64 {
return math.Pi * math.Pi * g.Beta * g.Beta / 6
}
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