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// Copyright ©2018 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"
"gonum.org/v1/gonum/mathext"
"gonum.org/v1/gonum/stat/combin"
)
// Binomial implements the binomial distribution, a discrete probability distribution
// that expresses the probability of a given number of successful Bernoulli trials
// out of a total of n, each with success probability p.
// The binomial distribution has the density function:
//
// f(k) = (n choose k) p^k (1-p)^(n-k)
//
// For more information, see https://en.wikipedia.org/wiki/Binomial_distribution.
type Binomial struct {
// N is the total number of Bernoulli trials. N must be greater than 0.
N float64
// P is the probability of success in any given trial. P must be in [0, 1].
P float64
Src rand.Source
}
// CDF computes the value of the cumulative distribution function at x.
func (b Binomial) CDF(x float64) float64 {
if x < 0 {
return 0
}
if x >= b.N {
return 1
}
x = math.Floor(x)
return mathext.RegIncBeta(b.N-x, x+1, 1-b.P)
}
// ExKurtosis returns the excess kurtosis of the distribution.
func (b Binomial) ExKurtosis() float64 {
v := b.P * (1 - b.P)
return (1 - 6*v) / (b.N * v)
}
// LogProb computes the natural logarithm of the value of the probability
// density function at x.
func (b Binomial) LogProb(x float64) float64 {
if x < 0 || x > b.N || math.Floor(x) != x {
return math.Inf(-1)
}
lb := combin.LogGeneralizedBinomial(b.N, x)
return lb + x*math.Log(b.P) + (b.N-x)*math.Log(1-b.P)
}
// Mean returns the mean of the probability distribution.
func (b Binomial) Mean() float64 {
return b.N * b.P
}
// NumParameters returns the number of parameters in the distribution.
func (Binomial) NumParameters() int {
return 2
}
// Prob computes the value of the probability density function at x.
func (b Binomial) Prob(x float64) float64 {
return math.Exp(b.LogProb(x))
}
// Rand returns a random sample drawn from the distribution.
func (b Binomial) Rand() float64 {
// NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)
// p. 295-6
// http://www.aip.de/groups/soe/local/numres/bookcpdf/c7-3.pdf
runif := rand.Float64
rexp := rand.ExpFloat64
if b.Src != nil {
rnd := rand.New(b.Src)
runif = rnd.Float64
rexp = rnd.ExpFloat64
}
p := b.P
if p > 0.5 {
p = 1 - p
}
am := b.N * p
if b.N < 25 {
// Use direct method.
bnl := 0.0
for i := 0; i < int(b.N); i++ {
if runif() < p {
bnl++
}
}
if p != b.P {
return b.N - bnl
}
return bnl
}
if am < 1 {
// Use rejection method with Poisson proposal.
const logM = 2.6e-2 // constant for rejection sampling (https://en.wikipedia.org/wiki/Rejection_sampling)
var bnl float64
z := -p
pclog := (1 + 0.5*z) * z / (1 + (1+1.0/6*z)*z) // Padé approximant of log(1 + x)
for {
bnl = 0.0
t := 0.0
for i := 0; i < int(b.N); i++ {
t += rexp()
if t >= am {
break
}
bnl++
}
bnlc := b.N - bnl
z = -bnl / b.N
log1p := (1 + 0.5*z) * z / (1 + (1+1.0/6*z)*z)
t = (bnlc+0.5)*log1p + bnl - bnlc*pclog + 1/(12*bnlc) - am + logM // Uses Stirling's expansion of log(n!)
if rexp() >= t {
break
}
}
if p != b.P {
return b.N - bnl
}
return bnl
}
// Original algorithm samples from a Poisson distribution with the
// appropriate expected value. However, the Poisson approximation is
// asymptotic such that the absolute deviation in probability is O(1/n).
// Rejection sampling produces exact variates with at worst less than 3%
// rejection with minimal additional computation.
// Use rejection method with Cauchy proposal.
g, _ := math.Lgamma(b.N + 1)
plog := math.Log(p)
pclog := math.Log1p(-p)
sq := math.Sqrt(2 * am * (1 - p))
for {
var em, y float64
for {
y = math.Tan(math.Pi * runif())
em = sq*y + am
if em >= 0 && em < b.N+1 {
break
}
}
em = math.Floor(em)
lg1, _ := math.Lgamma(em + 1)
lg2, _ := math.Lgamma(b.N - em + 1)
t := 1.2 * sq * (1 + y*y) * math.Exp(g-lg1-lg2+em*plog+(b.N-em)*pclog)
if runif() <= t {
if p != b.P {
return b.N - em
}
return em
}
}
}
// Skewness returns the skewness of the distribution.
func (b Binomial) Skewness() float64 {
return (1 - 2*b.P) / b.StdDev()
}
// StdDev returns the standard deviation of the probability distribution.
func (b Binomial) StdDev() float64 {
return math.Sqrt(b.Variance())
}
// Survival returns the survival function (complementary CDF) at x.
func (b Binomial) Survival(x float64) float64 {
return 1 - b.CDF(x)
}
// Variance returns the variance of the probability distribution.
func (b Binomial) Variance() float64 {
return b.N * b.P * (1 - b.P)
}
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