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// Copyright 2015 The Go 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 stats
import (
"math"
"math/rand"
)
// NormalDist is a normal (Gaussian) distribution with mean Mu and
// standard deviation Sigma.
type NormalDist struct {
Mu, Sigma float64
}
// StdNormal is the standard normal distribution (Mu = 0, Sigma = 1)
var StdNormal = NormalDist{0, 1}
// 1/sqrt(2 * pi)
const invSqrt2Pi = 0.39894228040143267793994605993438186847585863116493465766592583
func (n NormalDist) PDF(x float64) float64 {
z := x - n.Mu
return math.Exp(-z*z/(2*n.Sigma*n.Sigma)) * invSqrt2Pi / n.Sigma
}
func (n NormalDist) pdfEach(xs []float64) []float64 {
res := make([]float64, len(xs))
if n.Mu == 0 && n.Sigma == 1 {
// Standard normal fast path
for i, x := range xs {
res[i] = math.Exp(-x*x/2) * invSqrt2Pi
}
} else {
a := -1 / (2 * n.Sigma * n.Sigma)
b := invSqrt2Pi / n.Sigma
for i, x := range xs {
z := x - n.Mu
res[i] = math.Exp(z*z*a) * b
}
}
return res
}
func (n NormalDist) CDF(x float64) float64 {
return math.Erfc(-(x-n.Mu)/(n.Sigma*math.Sqrt2)) / 2
}
func (n NormalDist) cdfEach(xs []float64) []float64 {
res := make([]float64, len(xs))
a := 1 / (n.Sigma * math.Sqrt2)
for i, x := range xs {
res[i] = math.Erfc(-(x-n.Mu)*a) / 2
}
return res
}
func (n NormalDist) InvCDF(p float64) (x float64) {
// This is based on Peter John Acklam's inverse normal CDF
// algorithm: http://home.online.no/~pjacklam/notes/invnorm/
const (
a1 = -3.969683028665376e+01
a2 = 2.209460984245205e+02
a3 = -2.759285104469687e+02
a4 = 1.383577518672690e+02
a5 = -3.066479806614716e+01
a6 = 2.506628277459239e+00
b1 = -5.447609879822406e+01
b2 = 1.615858368580409e+02
b3 = -1.556989798598866e+02
b4 = 6.680131188771972e+01
b5 = -1.328068155288572e+01
c1 = -7.784894002430293e-03
c2 = -3.223964580411365e-01
c3 = -2.400758277161838e+00
c4 = -2.549732539343734e+00
c5 = 4.374664141464968e+00
c6 = 2.938163982698783e+00
d1 = 7.784695709041462e-03
d2 = 3.224671290700398e-01
d3 = 2.445134137142996e+00
d4 = 3.754408661907416e+00
plow = 0.02425
phigh = 1 - plow
)
if p < 0 || p > 1 {
return nan
} else if p == 0 {
return -inf
} else if p == 1 {
return inf
}
if p < plow {
// Rational approximation for lower region.
q := math.Sqrt(-2 * math.Log(p))
x = (((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q + c6) /
((((d1*q+d2)*q+d3)*q+d4)*q + 1)
} else if phigh < p {
// Rational approximation for upper region.
q := math.Sqrt(-2 * math.Log(1-p))
x = -(((((c1*q+c2)*q+c3)*q+c4)*q+c5)*q + c6) /
((((d1*q+d2)*q+d3)*q+d4)*q + 1)
} else {
// Rational approximation for central region.
q := p - 0.5
r := q * q
x = (((((a1*r+a2)*r+a3)*r+a4)*r+a5)*r + a6) * q /
(((((b1*r+b2)*r+b3)*r+b4)*r+b5)*r + 1)
}
// Refine approximation.
e := 0.5*math.Erfc(-x/math.Sqrt2) - p
u := e * math.Sqrt(2*math.Pi) * math.Exp(x*x/2)
x = x - u/(1+x*u/2)
// Adjust from standard normal.
return x*n.Sigma + n.Mu
}
func (n NormalDist) Rand(r *rand.Rand) float64 {
var x float64
if r == nil {
x = rand.NormFloat64()
} else {
x = r.NormFloat64()
}
return x*n.Sigma + n.Mu
}
func (n NormalDist) Bounds() (float64, float64) {
const stddevs = 3
return n.Mu - stddevs*n.Sigma, n.Mu + stddevs*n.Sigma
}
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