stat/distuv/alphastable.go

// Copyright ©2020 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"
)

// AlphaStable represents an α-stable distribution with four parameters.
// See https://en.wikipedia.org/wiki/Stable_distribution for more information.
type AlphaStable struct {
	// Alpha is the stability parameter.
	// It is valid within the range 0 < α ≤ 2.
	Alpha float64
	// Beta is the skewness parameter.
	// It is valid within the range -1 ≤ β ≤ 1.
	Beta float64
	// C is the scale parameter.
	// It is valid when positive.
	C float64
	// Mu is the location parameter.
	Mu  float64
	Src rand.Source
}

// ExKurtosis returns the excess kurtosis of the distribution.
// ExKurtosis returns NaN when Alpha != 2.
func (a AlphaStable) ExKurtosis() float64 {
	if a.Alpha == 2 {
		return 0
	}
	return math.NaN()
}

// Mean returns the mean of the probability distribution.
// Mean returns NaN when Alpha <= 1.
func (a AlphaStable) Mean() float64 {
	if a.Alpha > 1 {
		return a.Mu
	}
	return math.NaN()
}

// Median returns the median of the distribution.
// Median panics when Beta != 0, because then the mode is not analytically
// expressible.
func (a AlphaStable) Median() float64 {
	if a.Beta == 0 {
		return a.Mu
	}
	panic("distuv: cannot compute Median for Beta != 0")
}

// Mode returns the mode of the distribution.
// Mode panics when Beta != 0, because then the mode is not analytically
// expressible.
func (a AlphaStable) Mode() float64 {
	if a.Beta == 0 {
		return a.Mu
	}
	panic("distuv: cannot compute Mode for Beta != 0")
}

// NumParameters returns the number of parameters in the distribution.
func (a AlphaStable) NumParameters() int {
	return 4
}

// Rand returns a random sample drawn from the distribution.
func (a AlphaStable) Rand() float64 {
	// From https://en.wikipedia.org/wiki/Stable_distribution#Simulation_of_stable_variables
	const halfPi = math.Pi / 2
	u := Uniform{-halfPi, halfPi, a.Src}.Rand()
	w := Exponential{1, a.Src}.Rand()
	if a.Alpha == 1 {
		f := halfPi + a.Beta*u
		x := (f*math.Tan(u) - a.Beta*math.Log(halfPi*w*math.Cos(u)/f)) / halfPi
		return a.C*(x+a.Beta*math.Log(a.C)/halfPi) + a.Mu
	}
	zeta := -a.Beta * math.Tan(halfPi*a.Alpha)
	xi := math.Atan(-zeta) / a.Alpha
	f := a.Alpha * (u + xi)
	g := math.Sqrt(1+zeta*zeta) * math.Pow(math.Cos(u-f)/w, 1-a.Alpha) / math.Cos(u)
	x := math.Pow(g, 1/a.Alpha) * math.Sin(f)
	return a.C*x + a.Mu
}

// Skewness returns the skewness of the distribution.
// Skewness returns NaN when Alpha != 2.
func (a AlphaStable) Skewness() float64 {
	if a.Alpha == 2 {
		return 0
	}
	return math.NaN()
}

// StdDev returns the standard deviation of the probability distribution.
func (a AlphaStable) StdDev() float64 {
	return math.Sqrt(a.Variance())
}

// Variance returns the variance of the probability distribution.
// Variance returns +Inf when Alpha != 2.
func (a AlphaStable) Variance() float64 {
	if a.Alpha == 2 {
		return 2 * a.C * a.C
	}
	return math.Inf(1)
}