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 // Copyright ©2017 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" ) // Pareto implements the Pareto (Type I) distribution, a one parameter distribution // with support above the scale parameter. // // The density function is given by // (α x_m^{α})/(x^{α+1}) for x >= x_m. // // For more information, see https://en.wikipedia.org/wiki/Pareto_distribution. type Pareto struct { // Xm is the scale parameter. // Xm must be greater than 0. Xm float64 // Alpha is the shape parameter. // Alpha must be greater than 0. Alpha float64 Src rand.Source } // CDF computes the value of the cumulative density function at x. func (p Pareto) CDF(x float64) float64 { if x < p.Xm { return 0 } return -math.Expm1(p.Alpha * math.Log(p.Xm/x)) } // Entropy returns the differential entropy of the distribution. func (p Pareto) Entropy() float64 { return math.Log(p.Xm) - math.Log(p.Alpha) + (1 + 1/p.Alpha) } // ExKurtosis returns the excess kurtosis of the distribution. func (p Pareto) ExKurtosis() float64 { if p.Alpha <= 4 { return math.NaN() } return 6 * (p.Alpha*p.Alpha*p.Alpha + p.Alpha*p.Alpha - 6*p.Alpha - 2) / (p.Alpha * (p.Alpha - 3) * (p.Alpha - 4)) } // LogProb computes the natural logarithm of the value of the probability // density function at x. func (p Pareto) LogProb(x float64) float64 { if x < p.Xm { return math.Inf(-1) } return math.Log(p.Alpha) + p.Alpha*math.Log(p.Xm) - (p.Alpha+1)*math.Log(x) } // Mean returns the mean of the probability distribution. func (p Pareto) Mean() float64 { if p.Alpha <= 1 { return math.Inf(1) } return p.Alpha * p.Xm / (p.Alpha - 1) } // Median returns the median of the pareto distribution. func (p Pareto) Median() float64 { return p.Quantile(0.5) } // Mode returns the mode of the distribution. func (p Pareto) Mode() float64 { return p.Xm } // NumParameters returns the number of parameters in the distribution. func (p Pareto) NumParameters() int { return 2 } // Prob computes the value of the probability density function at x. func (p Pareto) Prob(x float64) float64 { return math.Exp(p.LogProb(x)) } // Quantile returns the inverse of the cumulative probability distribution. func (p Pareto) Quantile(prob float64) float64 { if prob < 0 || 1 < prob { panic(badPercentile) } return p.Xm / math.Pow(1-prob, 1/p.Alpha) } // Rand returns a random sample drawn from the distribution. func (p Pareto) Rand() float64 { var rnd float64 if p.Src == nil { rnd = rand.ExpFloat64() } else { rnd = rand.New(p.Src).ExpFloat64() } return math.Exp(math.Log(p.Xm) + 1/p.Alpha*rnd) } // StdDev returns the standard deviation of the probability distribution. func (p Pareto) StdDev() float64 { return math.Sqrt(p.Variance()) } // Survival returns the survival function (complementary CDF) at x. func (p Pareto) Survival(x float64) float64 { if x < p.Xm { return 1 } return math.Pow(p.Xm/x, p.Alpha) } // Variance returns the variance of the probability distribution. func (p Pareto) Variance() float64 { if p.Alpha <= 2 { return math.Inf(1) } am1 := p.Alpha - 1 return p.Xm * p.Xm * p.Alpha / (am1 * am1 * (p.Alpha - 2)) }