This poster describes the Target Dependence Coefficient. This coefficient is designed to measure the dependence between random variables. Unlike other dependence measures which simply estimate the distance between the joint distribution and the distribution of independence (i.e. the product of marginals), this coefficient can target specific dependence relationships or forget others. To do so, dependence is measured as the relative distance from the forget-dependence (it can be independence) to the target-dependence (it can be comonotonicity, as usually considered). Earth Mover Distance (discrete optimal transport) is used to evaluate the distance between the different copulas encoding the data-dependence, target-dependence, forget-dependence. Finally, we use this methodology for clustering financial time series (5-year credit default swaps) and observe that it yields more stable clusters than using Pearson, Spearman or Kendall correlation.

1. Optimal Copula Transport for Clustering Time Series
Gautier Marti1,2
, Frank Nielsen2
, Philippe Donnat1
1
Hellebore Capital Limited & 2
Ecole Polytechnique
Clustering Time Series
Which Dependence Measure?
For Which Dependence?
Many bivariate dependence measures are avail-
able. Usually, they aim at measuring:
• any deviation from independence,
• any deviation from co/counter-monotonicity.
Motivation: What if
• we aim at specific dependence,
• and try to “ignore” some others?
Dependence to detect (ρij := 1)
Dependence to ignore (ρij := 0)
Problem: A dependence measure powerful
enough to detect y = f(x2
) will also detect
y = g(x), f increasing, g decreasing.
Copulas & Dependence
• Sklar’s Theorem:
F(xi, xj) = Cij(Fi(xi), Fj(xj))
• Cij, the copula, encodes the dependence structure
• Fréchet-Hoeffding bounds:
max{ui + uj − 1, 0} ≤ Cij(ui, uj) ≤ min{ui, uj}
• Bivariate dependence measures:
• deviation from lower and upper bounds
• Spearman’s ρS, Gini’s γ
• deviation from independence uiuj
• Spearman, Copula MMD, Schweizer-Wolff’s σ, Hoeffding’s Φ2
Figure 1: (left) lower-bound copula, (mid) independence copula,
(right) upper-bound copula
Optimal Transport
Wasserstein metrics:
Wp
p (µ, ν) := inf
γ∈Γ(µ,ν) M×M
d(x, y)p
dγ(x, y)
In practice, the distance W1 is estimated on discrete
data by solving the following linear program with
the Hungarian algorithm:
EMD(s1, s2) := min
f 1≤k,l≤n
pk − ql fkl
subject to fkl ≥ 0, 1 ≤ k, l ≤ n,
n
l=1
fkl ≤ wpk
, 1 ≤ k ≤ n,
n
k=1
fkl ≤ wql
, 1 ≤ l ≤ n,
n
k=1
n
l=1
fkl = 1.
It is called the Earth Mover Distance (EMD) in the
CS literature.
A target-oriented dependence
coefficient
• Build the independence copula Cind
• Build the target-dependence copulas {Ck}k
• Compute the empirical copula Cij from xi, xj
TDC(Cij) =
EMD(Cind, Cij)
EMD(Cind, Cij) + mink EMD(Cij, Ck)
Figure 2: Dependence is measured as the relative distance from
independence to the nearest target-dependence
EMD between Copulas
• Probability integral transform of a variable xi:
FT (xk
i ) =
1
T
T
t=1
I(xt
i ≤ xk
i ),
i.e. computing the ranks of the realizations, and
normalizing them into [0,1]
Why the Earth Mover Distance?
Figure 3: 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 re-
lated divergences: D(C1, C2) ≤ D(C2, C3); EMD(C2, C3) ≤
EMD(C1, C2)
Benchmark: Power of Estimators
Our coefficient can robustly target complex depen-
dence patterns such as the ones displayed in Fig. 4.
• x-axis measures the noise added to the sample
• y-axis measures the frequency the coefficient is
able to discern between the dependent sample
and the independent one
• Basic check: no coefficient can discern between
the “dependent” sample (with no dependence)
and the independent sample.
0.00.40.8
xvals
power.cor[typ,]
xvals
power.cor[typ,]
0.00.40.8
xvals
power.cor[typ,]
xvals
power.cor[typ,]
cor
dCor
MIC
ACE
RDC
TDC
0.00.40.8
xvals
power.cor[typ,]
xvals
power.cor[typ,]
0 20 40 60 80 100
0.00.40.8
xvals
power.cor[typ,]
0 20 40 60 80 100
xvals
power.cor[typ,]
Noise Level
Power
Figure 4: Dependence estimators power as a function of the
noise for several deterministic patterns + noise. Their power is
the percentage of times that they are able to distinguish between
dependent and independent samples.
Clustering of Credit Default Swaps
• We use the two targets from Fig. 2
• Clustering distance: Dij = (1 − TDC(Cij))/2
Figure 5: Impact of different measures on clusters
Conclusion
The methodology presented is
• non-parametric, robust, deterministic.
It has some scalability issues:
• in dimension, non-parametric density estimation;
• in time, EMD is costly to compute.
Approximation schemes or parametric modelling
can alleviate these issues.
Information
• Web: www.datagrapple.com
• Email: gautier.marti@helleborecapital.com