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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) }