// Inverse of complementary error function. Returns x such that erfc(x) = p for argument p.
func InvErfc(p float64) float64 {
// From page 265 of numerical recipes
if p >= 2.0 {
return -100
}
if p <= 0.0 {
return 100
}
var pp float64
if p < 1.0 {
pp = p
} else {
pp = 2 - p
}
t := math.Sqrt(-2 * math.Log(pp/2.0)) // Initial guess
x := -0.70711 * ((2.30753+t*0.27061)/(1.0+t*(0.99229+t*0.04481)) - t)
for j := 0; j < 2; j++ {
err := math.Erfc(x) - pp
x += err / (1.12837916709551257*math.Exp(-(x*x)) - x*err) // Halley
}
if p < 1.0 {
return x
}
return -x
}
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
}
// Erfc returns the complementary error function of x (from math.Erfc).
func Erfc(x float64) float64 {
return math.Erfc(x)
}
// Erfc returns the complementary error function of x.
//
// Special cases are:
// Erfc(+Inf) = 0
// Erfc(-Inf) = 2
// Erfc(NaN) = NaN
func Erfc(x float32) float32 {
return float32(math.Erfc(float64(x)))
}
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) CDF(x float64) float64 {
return math.Erfc(-(x-n.Mu)/(n.Sigma*math.Sqrt2)) / 2
}
