This document discusses clustering financial time series data using distances between dependent random variables. It notes that traditional clustering based only on correlation can lead to spurious clusters, as correlation does not fully capture dependence. The paper proposes a distance measure that combines information about both the correlation and distribution of random variables. It tests this distance measure on synthetic data from a hierarchical block model and real credit default swap market data, finding it performs better than distances based only on correlation or distribution individually. Some open questions are also discussed, such as how to select the optimal weighting of correlation vs distribution information.

1. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
On clustering financial time series
A need for distances between dependent random variables
Gautier Marti, Frank Nielsen, Philippe Very, Philippe Donnat
24 September 2015
Gautier Marti, Frank Nielsen On clustering financial time series

2. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
1 Introduction
2 Dependence and Distribution
3 Toward an extension to the multivariate case
Gautier Marti, Frank Nielsen On clustering financial time series

3. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Motivations: Why clustering?
Motivations:
Mathematical finance: Use of variance-covariance matrices
(e.g., Markowitz, Value-at-Risk)
Stylized fact: Empirical
variance-covariance matrices
estimated on financial time
series are very noisy
(Random Matrix Theory,
Noise Dressing of Financial
Correlation Matrices, Laloux
et al, 1999)
Figure: Marchenko-Pastur
distribution vs. eigenvalues of the
empirical correlation matrix
How to filter these variance-covariance matrices?
Gautier Marti, Frank Nielsen On clustering financial time series

4. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Information filtering? Clustering!
Mantegna (1999) et al’s work:
Limits: focus on ρij (Pearson correlation) which is not robust to
outliers / heavy tails → could lead to spurious clusters
Gautier Marti, Frank Nielsen On clustering financial time series

5. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Modelling
Asset i variations or returns follow random variable Xi
Assets variations or returns are ”correlated”
i.i.d. observations:
X1 : X1
1 , X2
1 , . . . , XT
1
X2 : X1
2 , X2
2 , . . . , XT
2
. . . , . . . , . . . , . . . , . . .
XN : X1
N, X2
N, . . . , XT
N
Which distances d(Xi , Xj ) between dependent random variables?
Gautier Marti, Frank Nielsen On clustering financial time series

6. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
1 Introduction
2 Dependence and Distribution
3 Toward an extension to the multivariate case
Gautier Marti, Frank Nielsen On clustering financial time series

7. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Pitfalls of a basic distance
Let (X, Y ) be a bivariate Gaussian vector, with X ∼ N(µX , σ2
X ),
Y ∼ N(µY , σ2
Y ) and whose correlation is ρ(X, Y ) ∈ [−1, 1].
E[(X − Y )2
] = (µX − µY )2
+ (σX − σY )2
+ 2σX σY (1 − ρ(X, Y ))
Now, consider the following values for correlation:
ρ(X, Y ) = 0, so E[(X − Y )2] = (µX − µY )2 + σ2
X + σ2
Y .
Assume µX = µY and σX = σY . For σX = σY 1, we
obtain E[(X − Y )2] 1 instead of the distance 0, expected
from comparing two equal Gaussians.
ρ(X, Y ) = 1, so E[(X − Y )2] = (µX − µY )2 + (σX − σY )2.
Gautier Marti, Frank Nielsen On clustering financial time series

8. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Pitfalls of a basic distance
(Marti, Nielsen, Very, Donnat, ICMLA 2015)
Gautier Marti, Frank Nielsen On clustering financial time series

9. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
The Financial Engineer Bias: Correlation
correlation patterns are blatant
Mantegna et al. aim at filtering information from the
correlation matrix using clustering
O(N2) (correlation) vs. O(N) (distribution) parameters
Gautier Marti, Frank Nielsen On clustering financial time series

10. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Information Geometry and its statistical distances
original poster: http://www.sonycsl.co.jp/person/nielsen/FrankNielsen-distances-figs.pdf
Gautier Marti, Frank Nielsen On clustering financial time series

11. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Sklar’s Theorem and the Copula Transform
Theorem (Sklar’s Theorem (1959))
For any random vector X = (X1, . . . , XN) having continuous
marginal cdfs Pi , 1 ≤ i ≤ N, its joint cumulative distribution P is
uniquely expressed as
P(X1, . . . , XN) = C(P1(X1), . . . , PN(XN)),
where C, the multivariate distribution of uniform marginals, is
known as the copula of X.
Gautier Marti, Frank Nielsen On clustering financial time series

12. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Sklar’s Theorem and the Copula Transform
Definition (The Copula Transform)
Let X = (X1, . . . , XN) be a random vector with continuous
marginal cumulative distribution functions (cdfs) Pi , 1 ≤ i ≤ N.
The random vector
U = (U1, . . . , UN) := P(X) = (P1(X1), . . . , PN(XN))
is known as the copula transform.
Ui , 1 ≤ i ≤ N, are uniformly distributed on [0, 1] (the probability
integral transform): for Pi the cdf of Xi , we have
x = Pi (Pi
−1
(x)) = Pr(Xi ≤ Pi
−1
(x)) = Pr(Pi (Xi ) ≤ x), thus
Pi (Xi ) ∼ U[0, 1].
Gautier Marti, Frank Nielsen On clustering financial time series

13. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Distance Design
d2
θ (Xi , Xj ) = θ3E |Pi (Xi ) − Pj (Xj )|2
+ (1 − θ)
1
2 R
dPi
dλ
−
dPj
dλ
2
dλ
Gautier Marti, Frank Nielsen On clustering financial time series

14. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Results: Data from Hierarchical Block Model
Adjusted Rand Index
Algo. Distance A B C
HC-AL
(1 − ρ)/2 0.00 ±0.01 0.99 ±0.01 0.56 ±0.01
E[(X − Y )2
] 0.00 ±0.00 0.09 ±0.12 0.55 ±0.05
GPR θ = 0 0.34 ±0.01 0.01 ±0.01 0.06 ±0.02
GPR θ = 1 0.00 ±0.01 0.99 ±0.01 0.56 ±0.01
GPR θ = .5 0.34 ±0.01 0.59 ±0.12 0.57 ±0.01
GNPR θ = 0 1 0.00 ±0.00 0.17 ±0.00
GNPR θ = 1 0.00 ±0.00 1 0.57 ±0.00
GNPR θ = .5 0.99 ±0.01 0.25 ±0.20 0.95 ±0.08
AP
(1 − ρ)/2 0.00 ±0.00 0.99 ±0.07 0.48 ±0.02
E[(X − Y )2
] 0.14 ±0.03 0.94 ±0.02 0.59 ±0.00
GPR θ = 0 0.25 ±0.08 0.01 ±0.01 0.05 ±0.02
GPR θ = 1 0.00 ±0.01 0.99 ±0.01 0.48 ±0.02
GPR θ = .5 0.06 ±0.00 0.80 ±0.10 0.52 ±0.02
GNPR θ = 0 1 0.00 ±0.00 0.18 ±0.01
GNPR θ = 1 0.00 ±0.01 1 0.59 ±0.00
GNPR θ = .5 0.39 ±0.02 0.39 ±0.11 1
Gautier Marti, Frank Nielsen On clustering financial time series

15. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Results: Data from CDS market
(Marti, Nielsen, Very, Donnat, ICMLA 2015)
Gautier Marti, Frank Nielsen On clustering financial time series

16. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Limits and questions
Why a convex combination? no a priori support from geometry
In practice:
no real control on the weight of correlation and on the weight
of distribution
stability methods are still prone to overfitting for selecting
parameters
θ actually depends on the convergence rate of the estimators:
correlation measures converge faster than distribution
estimation
Gautier Marti, Frank Nielsen On clustering financial time series

17. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
1 Introduction
2 Dependence and Distribution
3 Toward an extension to the multivariate case
Gautier Marti, Frank Nielsen On clustering financial time series

18. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Overview
Gautier Marti, Frank Nielsen On clustering financial time series

19. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Multivariate dependence
What is the state of the art on multivariate dependence?
multivariate mutual information: In information theory
there have been various attempts over the years to
extend the definition of mutual information to more than
two random variables. These attempts have met with a
great deal of confusion and a realization that interactions
among many random variables are poorly understood.
Gautier Marti, Frank Nielsen On clustering financial time series

20. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Optimal Copula Transport for intra-dependence
Dintra(X1, X2) := EMD(s1, s2),
EMD(s1, s2) := min
f
1≤i,j≤n
pi − qj fij
subject to fij ≥ 0, 1 ≤ i, j ≤ n,
n
j=1
fij ≤ wpi
, 1 ≤ i ≤ n,
n
i=1
fij ≤ wqj
, 1 ≤ j ≤ n,
n
i=1
n
j=1
fij = 1.
Gautier Marti, Frank Nielsen On clustering financial time series

21. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Optimal Copula Transport for inter-dependence
Gautier Marti, Frank Nielsen On clustering financial time series

22. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
Limits and questions
does not scale well with even moderate dimensionality:
density estimation
computing cost
full parametric approach?
how to connect with the (copula,margins) representation?
information geometry?
(approximate) optimal transport?
kernel embedding of distributions?
contact: gautier.marti@helleborecapital.com
Gautier Marti, Frank Nielsen On clustering financial time series

23. Introduction
Dependence and Distribution
Toward an extension to the multivariate case
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