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// Copyright ©2016 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 distmv
import (
"math"
"golang.org/x/exp/rand"
"gonum.org/v1/gonum/floats"
"gonum.org/v1/gonum/mat"
"gonum.org/v1/gonum/stat/distuv"
)
// Dirichlet implements the Dirichlet probability distribution.
//
// The Dirichlet distribution is a continuous probability distribution that
// generates elements over the probability simplex, i.e. ||x||_1 = 1. The Dirichlet
// distribution is the conjugate prior to the categorical distribution and the
// multivariate version of the beta distribution. The probability of a point x is
//
// 1/Beta(α) \prod_i x_i^(α_i - 1)
//
// where Beta(α) is the multivariate Beta function (see the mathext package).
//
// For more information see https://en.wikipedia.org/wiki/Dirichlet_distribution
type Dirichlet struct {
alpha []float64
dim int
src rand.Source
lbeta float64
sumAlpha float64
}
// NewDirichlet creates a new dirichlet distribution with the given parameters alpha.
// NewDirichlet will panic if len(alpha) == 0, or if any alpha is <= 0.
func NewDirichlet(alpha []float64, src rand.Source) *Dirichlet {
dim := len(alpha)
if dim == 0 {
panic(badZeroDimension)
}
for _, v := range alpha {
if v <= 0 {
panic("dirichlet: non-positive alpha")
}
}
a := make([]float64, len(alpha))
copy(a, alpha)
d := &Dirichlet{
alpha: a,
dim: dim,
src: src,
}
d.lbeta, d.sumAlpha = d.genLBeta(a)
return d
}
// CovarianceMatrix calculates the covariance matrix of the distribution,
// storing the result in dst. Upon return, the value at element {i, j} of the
// covariance matrix is equal to the covariance of the i^th and j^th variables.
//
// covariance(i, j) = E[(x_i - E[x_i])(x_j - E[x_j])]
//
// If the dst matrix is empty it will be resized to the correct dimensions,
// otherwise dst must match the dimension of the receiver or CovarianceMatrix
// will panic.
func (d *Dirichlet) CovarianceMatrix(dst *mat.SymDense) {
if dst.IsEmpty() {
*dst = *(dst.GrowSym(d.dim).(*mat.SymDense))
} else if dst.SymmetricDim() != d.dim {
panic("dirichelet: input matrix size mismatch")
}
scale := 1 / (d.sumAlpha * d.sumAlpha * (d.sumAlpha + 1))
for i := 0; i < d.dim; i++ {
ai := d.alpha[i]
v := ai * (d.sumAlpha - ai) * scale
dst.SetSym(i, i, v)
for j := i + 1; j < d.dim; j++ {
aj := d.alpha[j]
v := -ai * aj * scale
dst.SetSym(i, j, v)
}
}
}
// genLBeta computes the generalized LBeta function.
func (d *Dirichlet) genLBeta(alpha []float64) (lbeta, sumAlpha float64) {
for _, alpha := range d.alpha {
lg, _ := math.Lgamma(alpha)
lbeta += lg
sumAlpha += alpha
}
lg, _ := math.Lgamma(sumAlpha)
return lbeta - lg, sumAlpha
}
// Dim returns the dimension of the distribution.
func (d *Dirichlet) Dim() int {
return d.dim
}
// LogProb computes the log of the pdf of the point x.
//
// It does not check that ||x||_1 = 1.
func (d *Dirichlet) LogProb(x []float64) float64 {
dim := d.dim
if len(x) != dim {
panic(badSizeMismatch)
}
var lprob float64
for i, x := range x {
lprob += (d.alpha[i] - 1) * math.Log(x)
}
lprob -= d.lbeta
return lprob
}
// Mean returns the mean of the probability distribution.
//
// If dst is not nil, the mean will be stored in-place into dst and returned,
// otherwise a new slice will be allocated first. If dst is not nil, it must
// have length equal to the dimension of the distribution.
func (d *Dirichlet) Mean(dst []float64) []float64 {
dst = reuseAs(dst, d.dim)
floats.ScaleTo(dst, 1/d.sumAlpha, d.alpha)
return dst
}
// Prob computes the value of the probability density function at x.
func (d *Dirichlet) Prob(x []float64) float64 {
return math.Exp(d.LogProb(x))
}
// Rand generates a random number according to the distributon.
//
// If dst is not nil, the sample will be stored in-place into dst and returned,
// otherwise a new slice will be allocated first. If dst is not nil, it must
// have length equal to the dimension of the distribution.
func (d *Dirichlet) Rand(dst []float64) []float64 {
dst = reuseAs(dst, d.dim)
for i, alpha := range d.alpha {
dst[i] = distuv.Gamma{Alpha: alpha, Beta: 1, Src: d.src}.Rand()
}
sum := floats.Sum(dst)
floats.Scale(1/sum, dst)
return dst
}
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