gonum/stat/distmv/statdist.go

// 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"

	"gonum.org/v1/gonum/floats"
	"gonum.org/v1/gonum/mat"
	"gonum.org/v1/gonum/mathext"
	"gonum.org/v1/gonum/stat"
)

// Bhattacharyya is a type for computing the Bhattacharyya distance between
// probability distributions.
//
// The Bhattacharyya distance is defined as
//  D_B = -ln(BC(l,r))
//  BC = \int_-∞^∞ (p(x)q(x))^(1/2) dx
// Where BC is known as the Bhattacharyya coefficient.
// The Bhattacharyya distance is related to the Hellinger distance by
//  H(l,r) = sqrt(1-BC(l,r))
// For more information, see
//  https://en.wikipedia.org/wiki/Bhattacharyya_distance
type Bhattacharyya struct{}

// DistNormal computes the Bhattacharyya distance between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
//
// For Normal distributions, the Bhattacharyya distance is
//  Σ = (Σ_l + Σ_r)/2
//  D_B = (1/8)*(μ_l - μ_r)^T*Σ^-1*(μ_l - μ_r) + (1/2)*ln(det(Σ)/(det(Σ_l)*det(Σ_r))^(1/2))
func (Bhattacharyya) DistNormal(l, r *Normal) float64 {
	dim := l.Dim()
	if dim != r.Dim() {
		panic(badSizeMismatch)
	}

	var sigma mat.SymDense
	sigma.AddSym(&l.sigma, &r.sigma)
	sigma.ScaleSym(0.5, &sigma)

	var chol mat.Cholesky
	chol.Factorize(&sigma)

	mahalanobis := stat.Mahalanobis(mat.NewVecDense(dim, l.mu), mat.NewVecDense(dim, r.mu), &chol)
	mahalanobisSq := mahalanobis * mahalanobis

	dl := l.chol.LogDet()
	dr := r.chol.LogDet()
	ds := chol.LogDet()

	return 0.125*mahalanobisSq + 0.5*ds - 0.25*dl - 0.25*dr
}

// DistUniform computes the Bhattacharyya distance between uniform distributions l and r.
// The dimensions of the input distributions must match or DistUniform will panic.
func (Bhattacharyya) DistUniform(l, r *Uniform) float64 {
	if len(l.bounds) != len(r.bounds) {
		panic(badSizeMismatch)
	}
	// BC = \int \sqrt(p(x)q(x)), which for uniform distributions is a constant
	// over the volume where both distributions have positive probability.
	// Compute the overlap and the value of sqrt(p(x)q(x)). The entropy is the
	// negative log probability of the distribution (use instead of LogProb so
	// it is not necessary to construct an x value).
	//
	// BC = volume * sqrt(p(x)q(x))
	// logBC = log(volume) + 0.5*(logP + logQ)
	// D_B = -logBC
	return -unifLogVolOverlap(l.bounds, r.bounds) + 0.5*(l.Entropy()+r.Entropy())
}

// unifLogVolOverlap computes the log of the volume of the hyper-rectangle where
// both uniform distributions have positive probability.
func unifLogVolOverlap(b1, b2 []Bound) float64 {
	var logVolOverlap float64
	for dim, v1 := range b1 {
		v2 := b2[dim]
		// If the surfaces don't overlap, then the volume is 0
		if v1.Max <= v2.Min || v2.Max <= v1.Min {
			return math.Inf(-1)
		}
		vol := math.Min(v1.Max, v2.Max) - math.Max(v1.Min, v2.Min)
		logVolOverlap += math.Log(vol)
	}
	return logVolOverlap
}

// CrossEntropy is a type for computing the cross-entropy between probability
// distributions.
//
// The cross-entropy is defined as
//  - \int_x l(x) log(r(x)) dx = KL(l || r) + H(l)
// where KL is the Kullback-Leibler divergence and H is the entropy.
// For more information, see
//  https://en.wikipedia.org/wiki/Cross_entropy
type CrossEntropy struct{}

// DistNormal returns the cross-entropy between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
func (CrossEntropy) DistNormal(l, r *Normal) float64 {
	if l.Dim() != r.Dim() {
		panic(badSizeMismatch)
	}
	kl := KullbackLeibler{}.DistNormal(l, r)
	return kl + l.Entropy()
}

// Hellinger is a type for computing the Hellinger distance between probability
// distributions.
//
// The Hellinger distance is defined as
//  H^2(l,r) = 1/2 * int_x (\sqrt(l(x)) - \sqrt(r(x)))^2 dx
// and is bounded between 0 and 1. Note the above formula defines the squared
// Hellinger distance, while this returns the Hellinger distance itself.
// The Hellinger distance is related to the Bhattacharyya distance by
//  H^2 = 1 - exp(-D_B)
// For more information, see
//  https://en.wikipedia.org/wiki/Hellinger_distance
type Hellinger struct{}

// DistNormal returns the Hellinger distance between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
//
// See the documentation of Bhattacharyya.DistNormal for the formula for Normal
// distributions.
func (Hellinger) DistNormal(l, r *Normal) float64 {
	if l.Dim() != r.Dim() {
		panic(badSizeMismatch)
	}
	db := Bhattacharyya{}.DistNormal(l, r)
	bc := math.Exp(-db)
	return math.Sqrt(1 - bc)
}

// KullbackLeibler is a type for computing the Kullback-Leibler divergence from l to r.
//
// The Kullback-Leibler divergence is defined as
//  D_KL(l || r ) = \int_x p(x) log(p(x)/q(x)) dx
// Note that the Kullback-Leibler divergence is not symmetric with respect to
// the order of the input arguments.
type KullbackLeibler struct{}

// DistDirichlet returns the Kullback-Leibler divergence between Dirichlet
// distributions l and r. The dimensions of the input distributions must match
// or DistDirichlet will panic.
//
// For two Dirichlet distributions, the KL divergence is computed as
//   D_KL(l || r) = log Γ(α_0_l) - \sum_i log Γ(α_i_l) - log Γ(α_0_r) + \sum_i log Γ(α_i_r)
//                  + \sum_i (α_i_l - α_i_r)(ψ(α_i_l)- ψ(α_0_l))
// Where Γ is the gamma function, ψ is the digamma function, and α_0 is the
// sum of the Dirichlet parameters.
func (KullbackLeibler) DistDirichlet(l, r *Dirichlet) float64 {
	// http://bariskurt.com/kullback-leibler-divergence-between-two-dirichlet-and-beta-distributions/
	if l.Dim() != r.Dim() {
		panic(badSizeMismatch)
	}
	l0, _ := math.Lgamma(l.sumAlpha)
	r0, _ := math.Lgamma(r.sumAlpha)
	dl := mathext.Digamma(l.sumAlpha)

	var l1, r1, c float64
	for i, al := range l.alpha {
		ar := r.alpha[i]
		vl, _ := math.Lgamma(al)
		l1 += vl
		vr, _ := math.Lgamma(ar)
		r1 += vr
		c += (al - ar) * (mathext.Digamma(al) - dl)
	}
	return l0 - l1 - r0 + r1 + c
}

// DistNormal returns the KullbackLeibler divergence between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
//
// For two normal distributions, the KL divergence is computed as
//   D_KL(l || r) = 0.5*[ln(|Σ_r|) - ln(|Σ_l|) + (μ_l - μ_r)^T*Σ_r^-1*(μ_l - μ_r) + tr(Σ_r^-1*Σ_l)-d]
func (KullbackLeibler) DistNormal(l, r *Normal) float64 {
	dim := l.Dim()
	if dim != r.Dim() {
		panic(badSizeMismatch)
	}

	mahalanobis := stat.Mahalanobis(mat.NewVecDense(dim, l.mu), mat.NewVecDense(dim, r.mu), &r.chol)
	mahalanobisSq := mahalanobis * mahalanobis

	// TODO(btracey): Optimize where there is a SolveCholeskySym
	// TODO(btracey): There may be a more efficient way to just compute the trace
	// Compute tr(Σ_r^-1*Σ_l) using the fact that Σ_l = U^T * U
	var u mat.TriDense
	l.chol.UTo(&u)
	var m mat.Dense
	err := r.chol.Solve(&m, u.T())
	if err != nil {
		return math.NaN()
	}
	m.Mul(&m, &u)
	tr := mat.Trace(&m)

	return r.logSqrtDet - l.logSqrtDet + 0.5*(mahalanobisSq+tr-float64(l.dim))
}

// DistUniform returns the KullbackLeibler divergence between uniform distributions
// l and r. The dimensions of the input distributions must match or DistUniform
// will panic.
func (KullbackLeibler) DistUniform(l, r *Uniform) float64 {
	bl := l.Bounds(nil)
	br := r.Bounds(nil)
	if len(bl) != len(br) {
		panic(badSizeMismatch)
	}

	// The KL is ∞ if l is not completely contained within r, because then
	// r(x) is zero when l(x) is non-zero for some x.
	contained := true
	for i, v := range bl {
		if v.Min < br[i].Min || br[i].Max < v.Max {
			contained = false
			break
		}
	}
	if !contained {
		return math.Inf(1)
	}

	// The KL divergence is finite.
	//
	// KL defines 0*ln(0) = 0, so there is no contribution to KL where l(x) = 0.
	// Inside the region, l(x) and r(x) are constant (uniform distribution), and
	// this constant is integrated over l(x), which integrates out to one.
	// The entropy is -log(p(x)).
	logPx := -l.Entropy()
	logQx := -r.Entropy()
	return logPx - logQx
}

// Renyi is a type for computing the Rényi divergence of order α from l to r.
//
// The Rényi divergence with α > 0, α ≠ 1 is defined as
//  D_α(l || r) = 1/(α-1) log(\int_-∞^∞ l(x)^α r(x)^(1-α)dx)
// The Rényi divergence has special forms for α = 0 and α = 1. This type does
// not implement α = ∞. For α = 0,
//  D_0(l || r) = -log \int_-∞^∞ r(x)1{p(x)>0} dx
// that is, the negative log probability under r(x) that l(x) > 0.
// When α = 1, the Rényi divergence is equal to the Kullback-Leibler divergence.
// The Rényi divergence is also equal to half the Bhattacharyya distance when α = 0.5.
//
// The parameter α must be in 0 ≤ α < ∞ or the distance functions will panic.
type Renyi struct {
	Alpha float64
}

// DistNormal returns the Rényi divergence between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
//
// For two normal distributions, the Rényi divergence is computed as
//  Σ_α = (1-α) Σ_l + αΣ_r
//  D_α(l||r) = α/2 * (μ_l - μ_r)'*Σ_α^-1*(μ_l - μ_r) + 1/(2(α-1))*ln(|Σ_λ|/(|Σ_l|^(1-α)*|Σ_r|^α))
//
// For a more nicely formatted version of the formula, see Eq. 15 of
//  Kolchinsky, Artemy, and Brendan D. Tracey. "Estimating Mixture Entropy
//  with Pairwise Distances." arXiv preprint arXiv:1706.02419 (2017).
// Note that the this formula is for Chernoff divergence, which differs from
// Rényi divergence by a factor of 1-α. Also be aware that most sources in
// the literature report this formula incorrectly.
func (renyi Renyi) DistNormal(l, r *Normal) float64 {
	if renyi.Alpha < 0 {
		panic("renyi: alpha < 0")
	}
	dim := l.Dim()
	if dim != r.Dim() {
		panic(badSizeMismatch)
	}
	if renyi.Alpha == 0 {
		return 0
	}
	if renyi.Alpha == 1 {
		return KullbackLeibler{}.DistNormal(l, r)
	}

	logDetL := l.chol.LogDet()
	logDetR := r.chol.LogDet()

	// Σ_α = (1-α)Σ_l + αΣ_r.
	sigA := mat.NewSymDense(dim, nil)
	for i := 0; i < dim; i++ {
		for j := i; j < dim; j++ {
			v := (1-renyi.Alpha)*l.sigma.At(i, j) + renyi.Alpha*r.sigma.At(i, j)
			sigA.SetSym(i, j, v)
		}
	}

	var chol mat.Cholesky
	ok := chol.Factorize(sigA)
	if !ok {
		return math.NaN()
	}
	logDetA := chol.LogDet()

	mahalanobis := stat.Mahalanobis(mat.NewVecDense(dim, l.mu), mat.NewVecDense(dim, r.mu), &chol)
	mahalanobisSq := mahalanobis * mahalanobis

	return (renyi.Alpha/2)*mahalanobisSq + 1/(2*(1-renyi.Alpha))*(logDetA-(1-renyi.Alpha)*logDetL-renyi.Alpha*logDetR)
}

// Wasserstein is a type for computing the Wasserstein distance between two
// probability distributions.
//
// The Wasserstein distance is defined as
//  W(l,r) := inf 𝔼(||X-Y||_2^2)^1/2
// For more information, see
//  https://en.wikipedia.org/wiki/Wasserstein_metric
type Wasserstein struct{}

// DistNormal returns the Wasserstein distance between normal distributions l and r.
// The dimensions of the input distributions must match or DistNormal will panic.
//
// The Wasserstein distance for Normal distributions is
//  d^2 = ||m_l - m_r||_2^2 + Tr(Σ_l + Σ_r - 2(Σ_l^(1/2)*Σ_r*Σ_l^(1/2))^(1/2))
// For more information, see
//  http://djalil.chafai.net/blog/2010/04/30/wasserstein-distance-between-two-gaussians/
func (Wasserstein) DistNormal(l, r *Normal) float64 {
	dim := l.Dim()
	if dim != r.Dim() {
		panic(badSizeMismatch)
	}

	d := floats.Distance(l.mu, r.mu, 2)
	d = d * d

	// Compute Σ_l^(1/2)
	var ssl mat.SymDense
	ssl.PowPSD(&l.sigma, 0.5)
	// Compute Σ_l^(1/2)*Σ_r*Σ_l^(1/2)
	var mean mat.Dense
	mean.Mul(&ssl, &r.sigma)
	mean.Mul(&mean, &ssl)

	// Reinterpret as symdense, and take Σ^(1/2)
	meanSym := mat.NewSymDense(dim, mean.RawMatrix().Data)
	ssl.PowPSD(meanSym, 0.5)

	tr := mat.Trace(&r.sigma)
	tl := mat.Trace(&l.sigma)
	tm := mat.Trace(&ssl)

	return d + tl + tr - 2*tm
}