Optimal Transport vs. Fisher-Rao distance between Copulas

Introduction
Statistical distances
Optimal Transport vs. Fisher-Rao distance
between Copulas
IEEE SSP 2016
G. Marti, S. Andler, F. Nielsen, P. Donnat
June 28, 2016
Gautier Marti Optimal Transport vs. Fisher-Rao distance between Copulas

Introduction
Clustering of Time Series
We need a distance Dij between time series xi and xj
If we look for ‘correlation’, Dij is a decreasing function of ρij ,
a measure of ‘correlation’
Several choices are available for ρij . . .

Introduction
Copulas
Sklar’s Theorem:
F(xi , xj ) = Cij (Fi (xi ), Fj (xj ))
Cij , the copula, encodes the dependence structure
Fr´echet-Hoeﬀding bounds:
max{ui + uj − 1, 0} ≤ Cij (ui , uj ) ≤ min{ui , uj }
(left) lower-bound, (mid) independence, (right) upper-bound copulas

Introduction
Copulas - Gaussian Example
Gaussian copula: CGauss
R (ui , uj ) = ΦR(Φ−1(ui ), Φ−1(uj ))
The distribution is parametrized by a correlation matrix R.

Introduction
The Target/Forget (copula-based) Dependence Coeﬃcient
Dependence is measured as the relative distance from independence to
the nearest target-dependence: comonotonicity or counter-monotonicity
Which distances are appropriate between copulas for the task of
clustering (copulas and time series)?

Introduction
Definitions - Fisher-Rao geodesic distance
Metrization of the paramater space {θ ∈ Rd | p(X; θ)dx = 1}.
Consider the metric gjk(θ) = − ∂2 log p(x,θ)
∂θj ∂θk
p(x, θ)dx,
the infinitesimal length ds(θ) = ( θ) G(θ) θ,
the Fisher-Rao geodesic distance
FR(θ1, θ2) =
θ2
θ1
ds(θ).
f -divergences induce infinitesimal length proportional to
Fisher-Rao infinitesimal length:
Df (θ θ + dθ) =
1
2
( θ) G(θ) θ.
Thus, they have the same local behaviour [1].

Introduction
Deﬁnitions - Optimal Transport distances
Wasserstein metric
Wp(µ, ν)p
= inf
γ∈Γ(µ,ν) M×M
d(x, y)p
dγ(x, y)
Image from Optimal Transport for Image Processing, Papadakis
Other transportation distances: regularized discrete optimal
transport [3], Sinkhorn distances [2], . . .

Introduction
Geometry of covariances

Introduction
Distances between Gaussian copulas
Copulas C1, C2, C3 encoding a correlation of 0.5, 0.99, 0.9999 respectively;
Which pair of copulas is the nearest?
- For Fisher-Rao, Kullback-Leibler, Hellinger and related divergences:
D(C1, C2) ≤ D(C2, C3);
- For Wasserstein: W2(C2, C3) ≤ W2(C1, C2)

Introduction
Distances as a function of (ρ1, ρ2)
Distance heatmap and surface as a function of (ρ1, ρ2)
for Fisher-Rao for Wasserstein W2

Introduction
Distances impact on clustering
Datasets of bivariate time series are generated from six Gaussian copulas
with correlation .1, .2, .6, .7, .99, .9999
Distance heatmaps for Fisher-Rao (left), W2 (right); Using Ward
clustering, Fisher-Rao yields clusters of copulas with correlations
{.1, .2, .6, .7}, {.99}, {.9999}, W2 yields {.1, .2}, {.6, .7}, {.99, .9999}

Introduction
Fisher metric and the Cram´er–Rao lower bound
Cram´er–Rao lower bound (CRLB)
The variance of any unbiased estimator ˆθ of θ is bounded by the
reciprocal of the Fisher information G(θ):
var(ˆθ) ≥
1
G(θ)
.
In the bivariate Gaussian copula case,
var(ˆρ) ≥
(ρ − 1)2(ρ + 1)2
3(ρ2 + 1)
.

Introduction
We consider the set of 2 × 2 correlation matrices C =
1 θ
θ 1
parameterized by θ.
Let x =
x1
x2
∈ R2
.
f (x; θ) = 1
2π 1−θ2
exp − 1
2
x C−1
x = 1
2π 1−θ2
exp − 1
2(1−θ2)
(x2
1 + x2
2 − 2θx1x2)
log f (x; θ) = − log(2π 1 − θ2) − 1
2(1−θ2)
(x2
1 + x2
2 − 2θx1x2)
∂2 log f (x;θ)
∂θ2 = − θ2+1
(θ2−1)2 −
x2
1
2(θ+1)3 +
x2
1
2(θ−1)3 −
x2
2
2(θ+1)3 +
x2
2
2(θ−1)3 −
x1x2
(θ+1)3 −
x1x2
(θ−1)3
Then, we compute ∞
−∞
∂2 log f (x;θ)
∂θ2 f (x; θ)dx.
Since E[x1] = E[x2] = 0, E[x1x2] = θ, E[x2
1 ] = E[x2
2 ] = 1, we get
∞
−∞
∂2 log f (x;θ)
∂θ2 f (x; θ)dx =
− θ2+1
(θ2−1)2 − 1
2(θ+1)3 + 1
2(θ−1)3 − 1
2(θ+1)3 + 1
2(θ−1)3 − θ
(θ+1)3 − θ
(θ−1)3 = −
3(θ2+1)
(θ−1)2(θ+1)2
Thus,
G(θ) =
3(θ2
+ 1)
(θ − 1)2(θ + 1)2
.

Introduction
In the bivariate Gaussian copula case,
var(ˆρ) ≥
(ρ − 1)2(ρ + 1)2
3(ρ2 + 1)
.
Recall that locally Fisher-Rao and the f -divergences are a
quadratic form of the Fisher metric ( θ) G(θ) θ. So, the
discriminative power of these distances is well calibrated with
respect to statistical uncertainty. For this purpose, they induce the
appropriate curvature on the parameter space.

Introduction
Properties of these distances
In addition, for clustering we prefer OT since:
in a parametric setting:
Fisher-Rao and f -divergences are defined on density manifolds,
but some important copulas (such as the Fréchet-Hoeffding
upper bound) do not belong to these manifolds;
Thus, in case of closed-form formulas (such as in the Gaussian
case), they are ill-defined for these copulas (for perfect
dependence, covariance is not invertible)
in a non-parametric/empirical setting:
f -divergences are defined for absolutely continuous measures,
thus require a pre-processing KDE
they are not aware of the support geometry, thus badly handle
noise on the support

Introduction
Barycenters
OT is deﬁned for both discrete/empirical and continuous measures
and is support-geometry aware:
0 0.5 1
0
0.5
1
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
0.0120
0 0.5 1
0
0.5
1
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
0.0120
0 0.5 1
0
0.5
1
0.0000
0.0008
0.0016
0.0024
0.0032
0.0040
0.0048
0.0056
0 0.5 1
0
0.5
1
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
0.0120
0 0.5 1
0
0.5
1
0.0000
0.0015
0.0030
0.0045
0.0060
0.0075
0.0090
0.0105
0.0120
5 copulas describing the dependence between X ∼ U([0, 1]) and
Y ∼ (X ± i )2
, where i is a constant noise speciﬁc for each distribution
0 0.5 1
0
0.5
1 Wasserstein barycenter copula
0.0000
0.0004
0.0008
0.0012
0.0016
0.0020
0.0024
0.0028
0.0032
Barycenter of the 5 copulas for a divergence and OT

Introduction
Future Research
Develop further geometries of copulas
using Optimal Transport: show that dependence-clustering of
time series is improved over standard correlations
using f -divergences: detect efficiently dependence-regime
switching in multivariate time series (cf. Frédéric Barbaresco’s
work on radar signal processing)
Numerical experiments and code:
https://www.datagrapple.com/Tech/fisher-vs-ot.html

Introduction
Shun-ichi Amari and Andrzej Cichocki.
Information geometry of divergence functions.
Bulletin of the Polish Academy of Sciences: Technical
Sciences, 58(1):183–195, 2010.
Marco Cuturi.
Sinkhorn distances: Lightspeed computation of optimal
transport.
In Advances in Neural Information Processing Systems, pages
2292–2300, 2013.
Sira Ferradans, Nicolas Papadakis, Julien Rabin, Gabriel Peyr´e,
and Jean-Fran¸cois Aujol.
Regularized discrete optimal transport.
Springer, 2013.

Optimal Transport vs. Fisher-Rao distance between Copulas

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