Merge pull request #159 from gonum/statdist

Add distance functions between probability distributions

distmv/statdist.go

@@ -0,0 +1,184 @@
+// 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"
+
+	"github.com/gonum/floats"
+	"github.com/gonum/matrix/mat64"
+	"github.com/gonum/stat"
+)
+
+// Bhattacharyya is a type for computing the Bhattacharyya distance between
+// probability distributions.
+//
+// The Battachara distance is defined as
+//  D_B = -ln(BC(l,r))
+//  BC = \int_x (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 = sqrt(1-BC)
+// 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 mat64.SymDense
+	sigma.AddSym(&l.sigma, &r.sigma)
+	sigma.ScaleSym(0.5, &sigma)
+
+	var chol mat64.Cholesky
+	chol.Factorize(&sigma)
+
+	mahalanobis := stat.Mahalanobis(mat64.NewVector(dim, l.mu), mat64.NewVector(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
+}
+
+// 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.
+// The Hellinger distance is related to the Bhattacharyya distance by
+//  H^2 = 1 - exp(-Db)
+// 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)
+}
+
+// KullbackLiebler is a type for computing the Kullback-Leibler divergence from l to r.
+// The dimensions of the input distributions must match or the function will panic.
+//
+// The Kullback-Liebler divergence is defined as
+//  D_KL(l || r ) = \int_x p(x) log(p(x)/q(x)) dx
+// Note that the Kullback-Liebler divergence is not symmetric with respect to
+// the order of the input arguments.
+type KullbackLeibler struct{}
+
+// DistNormal returns the KullbackLeibler distance 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(mat64.NewVector(dim, l.mu), mat64.NewVector(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 mat64.TriDense
+	u.UFromCholesky(&l.chol)
+	var m mat64.Dense
+	err := m.SolveCholesky(&r.chol, u.T())
+	if err != nil {
+		return math.NaN()
+	}
+	m.Mul(&m, &u)
+	tr := mat64.Trace(&m)
+
+	return r.logSqrtDet - l.logSqrtDet + 0.5*(mahalanobisSq+tr-float64(l.dim))
+}
+
+// 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 mat64.SymDense
+	ssl.PowPSD(&l.sigma, 0.5)
+	// Compute Σ_l^(1/2)*Σ_r*Σ_l^(1/2)
+	var mean mat64.Dense
+	mean.Mul(&ssl, &r.sigma)
+	mean.Mul(&mean, &ssl)
+
+	// Reinterpret as symdense, and take Σ^(1/2)
+	meanSym := mat64.NewSymDense(dim, mean.RawMatrix().Data)
+	ssl.PowPSD(meanSym, 0.5)
+
+	tr := mat64.Trace(&r.sigma)
+	tl := mat64.Trace(&l.sigma)
+	tm := mat64.Trace(&ssl)
+
+	return d + tl + tr - 2*tm
+}

distmv/statdist_test.go

@@ -0,0 +1,181 @@
+// 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"
+	"math/rand"
+	"testing"
+
+	"github.com/gonum/floats"
+	"github.com/gonum/matrix/mat64"
+)
+
+func TestBhattacharyyaNormal(t *testing.T) {
+	for cas, test := range []struct {
+		am, bm  []float64
+		ac, bc  *mat64.SymDense
+		samples int
+		tol     float64
+	}{
+		{
+			am:      []float64{2, 3},
+			ac:      mat64.NewSymDense(2, []float64{3, -1, -1, 2}),
+			bm:      []float64{-1, 1},
+			bc:      mat64.NewSymDense(2, []float64{1.5, 0.2, 0.2, 0.9}),
+			samples: 100000,
+			tol:     1e-2,
+		},
+	} {
+		rnd := rand.New(rand.NewSource(1))
+		a, ok := NewNormal(test.am, test.ac, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		b, ok := NewNormal(test.bm, test.bc, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		lBhatt := make([]float64, test.samples)
+		x := make([]float64, a.Dim())
+		for i := 0; i < test.samples; i++ {
+			// Do importance sampling over a: \int sqrt(a*b)/a * a dx
+			a.Rand(x)
+			pa := a.LogProb(x)
+			pb := b.LogProb(x)
+			lBhatt[i] = 0.5*pb - 0.5*pa
+		}
+		logBc := floats.LogSumExp(lBhatt) - math.Log(float64(test.samples))
+		db := -logBc
+		got := Bhattacharyya{}.DistNormal(a, b)
+		if math.Abs(db-got) > test.tol {
+			t.Errorf("Bhattacharyya mismatch, case %d: got %v, want %v", cas, got, db)
+		}
+	}
+}
+
+func TestCrossEntropyNormal(t *testing.T) {
+	for cas, test := range []struct {
+		am, bm  []float64
+		ac, bc  *mat64.SymDense
+		samples int
+		tol     float64
+	}{
+		{
+			am:      []float64{2, 3},
+			ac:      mat64.NewSymDense(2, []float64{3, -1, -1, 2}),
+			bm:      []float64{-1, 1},
+			bc:      mat64.NewSymDense(2, []float64{1.5, 0.2, 0.2, 0.9}),
+			samples: 100000,
+			tol:     1e-2,
+		},
+	} {
+		rnd := rand.New(rand.NewSource(1))
+		a, ok := NewNormal(test.am, test.ac, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		b, ok := NewNormal(test.bm, test.bc, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		var ce float64
+		x := make([]float64, a.Dim())
+		for i := 0; i < test.samples; i++ {
+			a.Rand(x)
+			ce -= b.LogProb(x)
+		}
+		ce /= float64(test.samples)
+		got := CrossEntropy{}.DistNormal(a, b)
+		if math.Abs(ce-got) > test.tol {
+			t.Errorf("CrossEntropy mismatch, case %d: got %v, want %v", cas, got, ce)
+		}
+	}
+}
+
+func TestHellingerNormal(t *testing.T) {
+	for cas, test := range []struct {
+		am, bm  []float64
+		ac, bc  *mat64.SymDense
+		samples int
+		tol     float64
+	}{
+		{
+			am:      []float64{2, 3},
+			ac:      mat64.NewSymDense(2, []float64{3, -1, -1, 2}),
+			bm:      []float64{-1, 1},
+			bc:      mat64.NewSymDense(2, []float64{1.5, 0.2, 0.2, 0.9}),
+			samples: 100000,
+			tol:     5e-1,
+		},
+	} {
+		rnd := rand.New(rand.NewSource(1))
+		a, ok := NewNormal(test.am, test.ac, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		b, ok := NewNormal(test.bm, test.bc, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		lAitchEDoubleHockeySticks := make([]float64, test.samples)
+		x := make([]float64, a.Dim())
+		for i := 0; i < test.samples; i++ {
+			// Do importance sampling over a: \int (\sqrt(a)-\sqrt(b))^2/a * a dx
+			a.Rand(x)
+			pa := a.LogProb(x)
+			pb := b.LogProb(x)
+			d := math.Exp(0.5*pa) - math.Exp(0.5*pb)
+			d = d * d
+			lAitchEDoubleHockeySticks[i] = math.Log(d) - pa
+		}
+		want := math.Sqrt(0.5 * math.Exp(floats.LogSumExp(lAitchEDoubleHockeySticks)-math.Log(float64(test.samples))))
+		got := Hellinger{}.DistNormal(a, b)
+		if math.Abs(want-got) > test.tol {
+			t.Errorf("Hellinger mismatch, case %d: got %v, want %v", cas, got, want)
+		}
+	}
+}
+
+func TestKullbackLieblerNormal(t *testing.T) {
+	for cas, test := range []struct {
+		am, bm  []float64
+		ac, bc  *mat64.SymDense
+		samples int
+		tol     float64
+	}{
+		{
+			am:      []float64{2, 3},
+			ac:      mat64.NewSymDense(2, []float64{3, -1, -1, 2}),
+			bm:      []float64{-1, 1},
+			bc:      mat64.NewSymDense(2, []float64{1.5, 0.2, 0.2, 0.9}),
+			samples: 10000,
+			tol:     1e-2,
+		},
+	} {
+		rnd := rand.New(rand.NewSource(1))
+		a, ok := NewNormal(test.am, test.ac, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		b, ok := NewNormal(test.bm, test.bc, rnd)
+		if !ok {
+			panic("bad test")
+		}
+		var klmc float64
+		x := make([]float64, a.Dim())
+		for i := 0; i < test.samples; i++ {
+			a.Rand(x)
+			pa := a.LogProb(x)
+			pb := b.LogProb(x)
+			klmc += pa - pb
+		}
+		klmc /= float64(test.samples)
+		kl := KullbackLeibler{}.DistNormal(a, b)
+		if !floats.EqualWithinAbsOrRel(kl, klmc, test.tol, test.tol) {
+			t.Errorf("Case %d, KL mismatch: got %v, want %v", cas, kl, klmc)
+		}
+	}
+}