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// Copyright ©2014 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"
)
// Weibull distribution. Valid range for x is [0,+∞).
type Weibull struct {
// Shape parameter of the distribution. A value of 1 represents
// the exponential distribution. A value of 2 represents the
// Rayleigh distribution. Valid range is (0,+∞).
K float64
// Scale parameter of the distribution. Valid range is (0,+∞).
Lambda float64
// Source of random numbers
Src rand.Source
}
// CDF computes the value of the cumulative density function at x.
func (w Weibull) CDF(x float64) float64 {
if x < 0 {
return 0
}
return -math.Expm1(-math.Pow(x/w.Lambda, w.K))
}
// Entropy returns the entropy of the distribution.
func (w Weibull) Entropy() float64 {
return eulerGamma*(1-1/w.K) + math.Log(w.Lambda/w.K) + 1
}
// ExKurtosis returns the excess kurtosis of the distribution.
func (w Weibull) ExKurtosis() float64 {
return (-6*w.gammaIPow(1, 4) + 12*w.gammaIPow(1, 2)*math.Gamma(1+2/w.K) - 3*w.gammaIPow(2, 2) - 4*math.Gamma(1+1/w.K)*math.Gamma(1+3/w.K) + math.Gamma(1+4/w.K)) / math.Pow(math.Gamma(1+2/w.K)-w.gammaIPow(1, 2), 2)
}
// gammIPow is a shortcut for computing the gamma function to a power.
func (w Weibull) gammaIPow(i, pow float64) float64 {
return math.Pow(math.Gamma(1+i/w.K), pow)
}
// LogProb computes the natural logarithm of the value of the probability
// density function at x. -Inf is returned if x is less than zero.
//
// Special cases occur when x == 0, and the result depends on the shape
// parameter as follows:
//
// If 0 < K < 1, LogProb returns +Inf.
// If K == 1, LogProb returns 0.
// If K > 1, LogProb returns -Inf.
func (w Weibull) LogProb(x float64) float64 {
if x < 0 {
return math.Inf(-1)
}
if x == 0 && w.K == 1 {
return 0
}
return math.Log(w.K) - math.Log(w.Lambda) + (w.K-1)*(math.Log(x)-math.Log(w.Lambda)) - math.Pow(x/w.Lambda, w.K)
}
// LogSurvival returns the log of the survival function (complementary CDF) at x.
func (w Weibull) LogSurvival(x float64) float64 {
if x < 0 {
return 0
}
return -math.Pow(x/w.Lambda, w.K)
}
// Mean returns the mean of the probability distribution.
func (w Weibull) Mean() float64 {
return w.Lambda * math.Gamma(1+1/w.K)
}
// Median returns the median of the normal distribution.
func (w Weibull) Median() float64 {
return w.Lambda * math.Pow(ln2, 1/w.K)
}
// Mode returns the mode of the normal distribution.
//
// The mode is NaN in the special case where the K (shape) parameter
// is less than 1.
func (w Weibull) Mode() float64 {
if w.K > 1 {
return w.Lambda * math.Pow((w.K-1)/w.K, 1/w.K)
}
return 0
}
// NumParameters returns the number of parameters in the distribution.
func (Weibull) NumParameters() int {
return 2
}
// Prob computes the value of the probability density function at x.
func (w Weibull) Prob(x float64) float64 {
if x < 0 {
return 0
}
return math.Exp(w.LogProb(x))
}
// Quantile returns the inverse of the cumulative probability distribution.
func (w Weibull) Quantile(p float64) float64 {
if p < 0 || p > 1 {
panic(badPercentile)
}
return w.Lambda * math.Pow(-math.Log(1-p), 1/w.K)
}
// Rand returns a random sample drawn from the distribution.
func (w Weibull) Rand() float64 {
var rnd float64
if w.Src == nil {
rnd = rand.Float64()
} else {
rnd = rand.New(w.Src).Float64()
}
return w.Quantile(rnd)
}
// Score returns the score function with respect to the parameters of the
// distribution at the input location x. The score function is the derivative
// of the log-likelihood at x with respect to the parameters
//
// (∂/∂θ) log(p(x;θ))
//
// If deriv is non-nil, len(deriv) must equal the number of parameters otherwise
// Score will panic, and the derivative is stored in-place into deriv. If deriv
// is nil a new slice will be allocated and returned.
//
// The order is [∂LogProb / ∂K, ∂LogProb / ∂λ].
//
// For more information, see https://en.wikipedia.org/wiki/Score_%28statistics%29.
//
// Special cases:
//
// Score(x) = [NaN, NaN] for x <= 0
func (w Weibull) Score(deriv []float64, x float64) []float64 {
if deriv == nil {
deriv = make([]float64, w.NumParameters())
}
if len(deriv) != w.NumParameters() {
panic(badLength)
}
if x > 0 {
deriv[0] = 1/w.K + math.Log(x) - math.Log(w.Lambda) - (math.Log(x)-math.Log(w.Lambda))*math.Pow(x/w.Lambda, w.K)
deriv[1] = (w.K * (math.Pow(x/w.Lambda, w.K) - 1)) / w.Lambda
return deriv
}
deriv[0] = math.NaN()
deriv[1] = math.NaN()
return deriv
}
// ScoreInput returns the score function with respect to the input of the
// distribution at the input location specified by x. The score function is the
// derivative of the log-likelihood
//
// (d/dx) log(p(x)) .
//
// Special cases:
//
// ScoreInput(x) = NaN for x <= 0
func (w Weibull) ScoreInput(x float64) float64 {
if x > 0 {
return (-w.K*math.Pow(x/w.Lambda, w.K) + w.K - 1) / x
}
return math.NaN()
}
// Skewness returns the skewness of the distribution.
func (w Weibull) Skewness() float64 {
stdDev := w.StdDev()
firstGamma, firstGammaSign := math.Lgamma(1 + 3/w.K)
logFirst := firstGamma + 3*(math.Log(w.Lambda)-math.Log(stdDev))
logSecond := math.Log(3) + math.Log(w.Mean()) + 2*math.Log(stdDev) - 3*math.Log(stdDev)
logThird := 3 * (math.Log(w.Mean()) - math.Log(stdDev))
return float64(firstGammaSign)*math.Exp(logFirst) - math.Exp(logSecond) - math.Exp(logThird)
}
// StdDev returns the standard deviation of the probability distribution.
func (w Weibull) StdDev() float64 {
return math.Sqrt(w.Variance())
}
// Survival returns the survival function (complementary CDF) at x.
func (w Weibull) Survival(x float64) float64 {
return math.Exp(w.LogSurvival(x))
}
// setParameters modifies the parameters of the distribution.
func (w *Weibull) setParameters(p []Parameter) {
if len(p) != w.NumParameters() {
panic("weibull: incorrect number of parameters to set")
}
if p[0].Name != "K" {
panic("weibull: " + panicNameMismatch)
}
if p[1].Name != "λ" {
panic("weibull: " + panicNameMismatch)
}
w.K = p[0].Value
w.Lambda = p[1].Value
}
// Variance returns the variance of the probability distribution.
func (w Weibull) Variance() float64 {
return math.Pow(w.Lambda, 2) * (math.Gamma(1+2/w.K) - w.gammaIPow(1, 2))
}
// parameters returns the parameters of the distribution.
func (w Weibull) parameters(p []Parameter) []Parameter {
nParam := w.NumParameters()
if p == nil {
p = make([]Parameter, nParam)
} else if len(p) != nParam {
panic("weibull: improper parameter length")
}
p[0].Name = "K"
p[0].Value = w.K
p[1].Name = "λ"
p[1].Value = w.Lambda
return p
}
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