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stat/distuv: add binomial distribution
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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. | ||
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package distuv | ||
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import ( | ||
"math" | ||
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"golang.org/x/exp/rand" | ||
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"gonum.org/v1/gonum/mathext" | ||
"gonum.org/v1/gonum/stat/combin" | ||
) | ||
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// 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 successs 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 probablity of success in any given trial. P must be in [0, 1]. | ||
P float64 | ||
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Src rand.Source | ||
} | ||
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// 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) | ||
} | ||
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// 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) | ||
} | ||
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// 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) | ||
} | ||
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// Mean returns the mean of the probability distribution. | ||
func (b Binomial) Mean() float64 { | ||
return b.N * b.P | ||
} | ||
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// NumParameters returns the number of parameters in the distribution. | ||
func (Binomial) NumParameters() int { | ||
return 2 | ||
} | ||
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// Prob computes the value of the probability density function at x. | ||
func (b Binomial) Prob(x float64) float64 { | ||
return math.Exp(b.LogProb(x)) | ||
} | ||
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// 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 | ||
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runif := rand.Float64 | ||
rexp := rand.ExpFloat64 | ||
if b.Src != nil { | ||
rnd := rand.New(b.Src) | ||
runif = rnd.Float64 | ||
rexp = rnd.ExpFloat64 | ||
} | ||
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p := b.P | ||
if p > 0.5 { | ||
p = 1 - p | ||
} | ||
am := b.N * p | ||
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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 | ||
} | ||
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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 miminal additional computation. | ||
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// 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 | ||
} | ||
} | ||
} | ||
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// Skewness returns the skewness of the distribution. | ||
func (b Binomial) Skewness() float64 { | ||
return (1 - 2*b.P) / b.StdDev() | ||
} | ||
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// StdDev returns the standard deviation of the probability distribution. | ||
func (b Binomial) StdDev() float64 { | ||
return math.Sqrt(b.Variance()) | ||
} | ||
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// Survival returns the survival function (complementary CDF) at x. | ||
func (b Binomial) Survival(x float64) float64 { | ||
return 1 - b.CDF(x) | ||
} | ||
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// 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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// 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. | ||
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package distuv | ||
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import ( | ||
"sort" | ||
"testing" | ||
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"golang.org/x/exp/rand" | ||
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"gonum.org/v1/gonum/floats" | ||
) | ||
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func TestBinomialProb(t *testing.T) { | ||
const tol = 1e-10 | ||
for i, tt := range []struct { | ||
k float64 | ||
n float64 | ||
p float64 | ||
want float64 | ||
}{ | ||
// Probabilities computed with Wolfram|Alpha (http://wwww.wolframalpha.com) | ||
{0, 10, 0.5, 0.0009765625}, | ||
{1, 10, 0.5, 0.009765625}, | ||
{2, 10, 0.5, 0.0439453125}, | ||
{3, 10, 0.5, 0.1171875}, | ||
{4, 10, 0.5, 0.205078125}, | ||
{5, 10, 0.75, 5.839920043945313e-02}, | ||
{6, 10, 0.75, 0.1459980010986328}, | ||
{7, 10, 0.75, 0.2502822875976563}, | ||
{8, 10, 0.75, 0.2815675735473633}, | ||
{9, 10, 0.75, 0.1877117156982422}, | ||
{10, 10, 0.75, 5.6313514709472656e-02}, | ||
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{0, 25, 0.25, 7.525434581650003e-04}, | ||
{2, 25, 0.25, 2.508478193883334e-02}, | ||
{5, 25, 0.25, 0.1645375881987921}, | ||
{7, 25, 0.25, 0.1654081574485211}, | ||
{10, 25, 0.25, 4.165835076481272e-02}, | ||
{12, 25, 0.01, 4.563372575901533e-18}, | ||
{15, 25, 0.01, 2.956207951505780e-24}, | ||
{17, 25, 0.01, 9.980175928758777e-29}, | ||
{20, 25, 0.99, 4.345539559454088e-06}, | ||
{22, 25, 0.99, 1.843750355939806e-03}, | ||
{25, 25, 0.99, 0.7778213593991468}, | ||
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{0.5, 25, 0.5, 0}, | ||
{1.5, 25, 0.5, 0}, | ||
{2.5, 25, 0.5, 0}, | ||
{3.5, 25, 0.5, 0}, | ||
{4.5, 25, 0.5, 0}, | ||
{5.5, 25, 0.5, 0}, | ||
{6.5, 25, 0.5, 0}, | ||
{7.5, 25, 0.5, 0}, | ||
{8.5, 25, 0.5, 0}, | ||
{9.5, 25, 0.5, 0}, | ||
} { | ||
b := Binomial{N: tt.n, P: tt.p} | ||
got := b.Prob(tt.k) | ||
if !floats.EqualWithinRel(got, tt.want, tol) { | ||
t.Errorf("test-%d: got=%e. want=%e\n", i, got, tt.want) | ||
} | ||
} | ||
} | ||
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func TestBinomialCDF(t *testing.T) { | ||
const tol = 1e-10 | ||
for i, tt := range []struct { | ||
k float64 | ||
n float64 | ||
p float64 | ||
want float64 | ||
}{ | ||
// Cumulative probabilities computed with SciPy | ||
{0, 10, 0.5, 9.765625e-04}, | ||
{1, 10, 0.5, 1.0742187499999998e-02}, | ||
{2, 10, 0.5, 5.468749999999999e-02}, | ||
{3, 10, 0.5, 1.7187499999999994e-01}, | ||
{4, 10, 0.5, 3.769531249999999e-01}, | ||
{5, 10, 0.25, 9.802722930908203e-01}, | ||
{6, 10, 0.25, 9.964942932128906e-01}, | ||
{7, 10, 0.25, 9.995841979980469e-01}, | ||
{8, 10, 0.25, 9.999704360961914e-01}, | ||
{9, 10, 0.25, 9.999990463256836e-01}, | ||
{10, 10, 0.25, 1.0}, | ||
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{0, 25, 0.75, 8.881784197001252e-16}, | ||
{2.5, 25, 0.75, 2.4655832930875472e-12}, | ||
{5, 25, 0.75, 1.243460090449844e-08}, | ||
{7.5, 25, 0.75, 1.060837565347583e-06}, | ||
{10, 25, 0.75, 2.1451240486669576e-04}, | ||
{12.5, 25, 0.01, 9.999999999999999e-01}, | ||
{15, 25, 0.01, 9.999999999999999e-01}, | ||
{17.5, 25, 0.01, 9.999999999999999e-01}, | ||
{20, 25, 0.99, 4.495958469027147e-06}, | ||
{22.5, 25, 0.99, 1.9506768897388268e-03}, | ||
{25, 25, 0.99, 1.0}, | ||
} { | ||
b := Binomial{N: tt.n, P: tt.p} | ||
got := b.CDF(tt.k) | ||
if !floats.EqualWithinRel(got, tt.want, tol) { | ||
t.Errorf("test-%d: got=%e. want=%e\n", i, got, tt.want) | ||
} | ||
} | ||
} | ||
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func TestBinomial(t *testing.T) { | ||
src := rand.New(rand.NewSource(1)) | ||
for i, b := range []Binomial{ | ||
{100, 0.5, src}, | ||
{15, 0.25, src}, | ||
{10, 0.75, src}, | ||
{9000, 0.102, src}, | ||
{1e6, 0.001, src}, | ||
{25, 0.02, src}, | ||
{3, 0.8, src}, | ||
} { | ||
testBinomial(t, b, i) | ||
} | ||
} | ||
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func testBinomial(t *testing.T, b Binomial, i int) { | ||
const tol = 1e-2 | ||
const n = 1e6 | ||
x := make([]float64, n) | ||
generateSamples(x, b) | ||
sort.Float64s(x) | ||
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checkMean(t, i, x, b, tol) | ||
checkVarAndStd(t, i, x, b, tol) | ||
checkExKurtosis(t, i, x, b, 7e-2) | ||
} |