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// Copyright ©2015 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"
)
// LogNormal represents a random variable whose log is normally distributed.
// The probability density function is given by
//
// 1/(x σ √2π) exp(-(ln(x)-μ)^2)/(2σ^2))
type LogNormal struct {
Mu float64
Sigma float64
Src rand.Source
}
// CDF computes the value of the cumulative density function at x.
func (l LogNormal) CDF(x float64) float64 {
return 0.5 * math.Erfc(-(math.Log(x)-l.Mu)/(math.Sqrt2*l.Sigma))
}
// Entropy returns the differential entropy of the distribution.
func (l LogNormal) Entropy() float64 {
return 0.5 + 0.5*math.Log(2*math.Pi*l.Sigma*l.Sigma) + l.Mu
}
// ExKurtosis returns the excess kurtosis of the distribution.
func (l LogNormal) ExKurtosis() float64 {
s2 := l.Sigma * l.Sigma
return math.Exp(4*s2) + 2*math.Exp(3*s2) + 3*math.Exp(2*s2) - 6
}
// LogProb computes the natural logarithm of the value of the probability density function at x.
func (l LogNormal) LogProb(x float64) float64 {
if x < 0 {
return math.Inf(-1)
}
logx := math.Log(x)
normdiff := (logx - l.Mu) / l.Sigma
return -0.5*normdiff*normdiff - logx - math.Log(l.Sigma) - logRoot2Pi
}
// Mean returns the mean of the probability distribution.
func (l LogNormal) Mean() float64 {
return math.Exp(l.Mu + 0.5*l.Sigma*l.Sigma)
}
// Median returns the median of the probability distribution.
func (l LogNormal) Median() float64 {
return math.Exp(l.Mu)
}
// Mode returns the mode of the probability distribution.
func (l LogNormal) Mode() float64 {
return math.Exp(l.Mu - l.Sigma*l.Sigma)
}
// NumParameters returns the number of parameters in the distribution.
func (LogNormal) NumParameters() int {
return 2
}
// Prob computes the value of the probability density function at x.
func (l LogNormal) Prob(x float64) float64 {
return math.Exp(l.LogProb(x))
}
// Quantile returns the inverse of the cumulative probability distribution.
func (l LogNormal) Quantile(p float64) float64 {
if p < 0 || p > 1 {
panic(badPercentile)
}
// Formula from http://www.math.uah.edu/stat/special/LogNormal.html.
return math.Exp(l.Mu + l.Sigma*UnitNormal.Quantile(p))
}
// Rand returns a random sample drawn from the distribution.
func (l LogNormal) Rand() float64 {
var rnd float64
if l.Src == nil {
rnd = rand.NormFloat64()
} else {
rnd = rand.New(l.Src).NormFloat64()
}
return math.Exp(rnd*l.Sigma + l.Mu)
}
// Skewness returns the skewness of the distribution.
func (l LogNormal) Skewness() float64 {
s2 := l.Sigma * l.Sigma
return (math.Exp(s2) + 2) * math.Sqrt(math.Exp(s2)-1)
}
// StdDev returns the standard deviation of the probability distribution.
func (l LogNormal) StdDev() float64 {
return math.Sqrt(l.Variance())
}
// Survival returns the survival function (complementary CDF) at x.
func (l LogNormal) Survival(x float64) float64 {
return 0.5 * (1 - math.Erf((math.Log(x)-l.Mu)/(math.Sqrt2*l.Sigma)))
}
// Variance returns the variance of the probability distribution.
func (l LogNormal) Variance() float64 {
s2 := l.Sigma * l.Sigma
return (math.Exp(s2) - 1) * math.Exp(2*l.Mu+s2)
}
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