Exploring and measuring non-linear correlationsG. Marti†?, S. Andler†‡, F. Nielsen?, P. Donnat† (presented by M. Binkowski†∗)

†Hellebore Capital Ltd, ?Ecole Polytechnique, ‡ENS de Lyon, ∗Imperial College London

Motivations

• Interpretability of pairwise dependence•Summary of associations between many variables•Find abnormal dependence patterns•Design robust and custom dependence coefficients•Query the dataset for specific associations•Realistic simulations of market variables

Copulas

Sklar’s Theorem

Let X = (Xi, Xj) be a random vector witha joint cumulative distribution function F , andhaving continuous marginal cumulative distribu-tion functions Fi, Fj respectively. Then, thereexists a unique distribution C such that

F (Xi, Xj) = C(Fi(Xi), Fj(Xj)).C, the copula of X , is the bivariate distributionof uniform marginals Ui, Uj := Fi(Xi), Fj(Xj).

Fréchet-Hoeffding copula bounds

0 0.5 1

ui

0

0.5

1

uj

w(ui, uj)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0 0.5 1

ui

0

0.5

1

uj

W(ui, uj)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1

ui

0

0.5

1

uj

π(ui, uj)

0.00036

0.00037

0.00038

0.00039

0.00040

0.00041

0.00042

0.00043

0.00044

0 0.5 1

ui

0

0.5

1

uj

Π(ui, uj)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1

ui

0

0.5

1

uj

m(ui, uj)

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0 0.5 1

ui

0

0.5

1

uj

M(ui, uj)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 1: Copulas measure (left column) and cumulative dis-tribution function (right column) heatmaps for negative de-pendence (first row), independence (second row), i.e. theuniform distribution over [0, 1]2, and positive dependence(third row)

The methodology - Clustering of copulas & custom dependence coefficients

The methodology leverages copulas for encoding depen-dence between two variables, state-of-the-art optimaltransport for providing a relevant geometry to the cop-ulas, and clustering for summarizing the main depen-dence patterns found between the variables. Some ofthe clusters centers can be used to parameterize a cus-tom dependence coefficient.

Target/Forget Dependence Coefficient: Let {C−l }l bethe set of forget-dependence copulas, and {C+

k }k be theset of target-dependence copulas. Let C be the copulaof (Xi, Xj).TFDC

(Xi, Xj; {C+

k }k, {C−l }l)

:=minl dM(C−l , C)

minl dM(C−l , C) + mink dM(C,C+k )∈ [0, 1].

Which geometry for copulas?

In [1], we detail the benefit of optimal transport overinformation divergences for clustering copulas.

Figure 2: Copulas C1, C2, C3 encoding a correlation of0.5, 0.99, 0.9999 respectively; Which pair of copulas is the near-est? For Fisher-Rao, Kullback-Leibler, Hellinger and related di-vergences: D(C1, C2) ≤ D(C2, C3); W2(C2, C3) ≤W2(C1, C2)

We use results from [2], [3] to compute faster thedistances and barycenters needed for the clustering.

0 0.5 10

0.5

1Bregman barycenter copula

0.0000

0.0008

0.0016

0.0024

0.0032

0.0040

0.0048

0.0056

0 0.5 10

0.5

1Wasserstein barycenter copula

0.0000

0.0004

0.0008

0.0012

0.0016

0.0020

0.0024

0.0028

0.0032

Figure 3: Barycenter for: (left) Bregman geometry (which in-cludes, for example, squared Euclidean and Kullback-Leibler dis-tances); (right) Wasserstein geometry.

Copulas of financial time series

We apply clustering to the(N2)bivariate copulas of

a financial time series dataset consisting in daily re-turns of stocks, credit default swaps and FX rates.

Figure 4: Credit default swaps: More mass in the top-rightcorner, i.e. upper tail dependence. Insurance cost against thedefault of companies tends to soar in distressed market.

Queries about dependence

(A) (B) (C) (D)Figure 5: Target copulas (simulated or handcrafted) and theirrespective nearest copulas which answer questions A,B,C,D

• (A) most Gaussian with ρ = 0.7?• (B) both positively and negatively correlated?• (C) extreme returns for one, small for the other?• (D) uncorrelated but correlated for small returns?

References

[1] G. Marti, S. Andler, F. Nielsen, P. Donnat, IEEEStatistical Signal Processing Workshop (2016), 1-5.

[2] M. Cuturi, Advances in Neural Information ProcessingSystems (2013), 2292-2300.

[3] M. Cuturi, A. Doucet, Proceedings of the 31thInternational Conference on Machine Learning (2014),685-693.

HELLEBORECAPITAL