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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 Gumbel 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 }