Some contributions to the clustering of financial time series - Applications to credit default swaps

Introduction to credit default swaps
About the consistency of clustering financial time series
Design of distances and alternative dependence coefficients
Summary and open questions
Some contributions to the clustering
of financial time series
and its applications to credit default swaps
Gautier Marti
November 10, 2017
Gautier Marti Some contributions to the clustering of financial time series

Context of the PhD thesis
PhD studies in hedge funds:
Hellebore Capital Management,
63 Avenue des Champs-Elys´ees, Paris, France
(April 1, 2014 - February 29, 2016)
Hellebore Capital Limited,
81 Fulham Road, London, United Kingdom
(March 1, 2016 - September 20, 2017)
AXA IM Chorus,
18 Westlands Road, Hong Kong
(October 1, 2017 - present)

Outline
1. Introduction to the credit default swap dataset
Contributions:
2. About the consistency of clustering financial time series
3. Improving standard distances between financial time series
i) a simple correlation + distribution distance
ii) a geometrical approach to define dependence coefficients from
copulas
4. Perspectives

Some movation for using clustering
Statistical modelling is difficult as the time series
are non stationary, e.g. economic regimes are changing, it
can be misleading to use data from a too distant past
are near efficient, i.e. behaving nearly like random walks (cf.
the efficient-market hypothesis (Fama, 1970) [5])
have a low signal-to-noise ratio, i.e. measure artifacts hide
information in random fluctuations
are in an unfavorable statistical setting, too few relevant
observations (length) wrt the number of variables (time series)

Some motivation for using clustering
Clustering helps to reduce dimensionality, and thus can be used as
a preprocessing for:
Risk management, e.g. ﬁltering covariances, performance
and risk attribution
Investment, e.g. portfolio design, statarb, beta neutralization
Data analysis, e.g. outliers detection and missing values
imputation, exploration (www.datagrapple.com)

1 Introduction to credit default swaps
2 About the consistency of clustering financial time series
3 Design of distances and alternative dependence coefficients
Alternative representation and correlation+distribution distance
Copula-based dependence coefficients
4 Summary and open questions

Introduction to the credit default swap

Introduction to the CDS market

Introduction to the CDS raw dataset
Putting Self-Supervised Token Embedding on the Tables [15]
(ICMLA 2017)

A ‘tick-by-tick’ dataset
Autoregressive Convolutional Neural Networks for Asynchronous
Time Series [1] (ICML Time Series WS 2017)

Historical daily time series of spread
From the received ‘tick-by-tick’ prices, a synthetic order book is built. At
5pm London time, we save the mid-price of the best bid and best oﬀer in
the order book for each entity. N ≈ 800 liquid CDSs, with T ≈ 3000.

Clustering of Financial Time Series
Stylized fact I: Financial time series correlations have a strong
hierarchical block diagonal structure (Mantegna, 1999) [6]
https://gmarti.gitlab.io/ml/2017/09/07/how-to-sort-distance-matrix.html
Stylized fact II: Most correlations are spurious (Bouchaud,
1999) [3]
Motivation for clustering ﬁnancial time series using correlation as a
similarity measure:
dimensionality reduction ≡ ﬁltering noisy correlations

Challenge for the statistical practitioner
The dilemma:
the longer the time interval, the more precise the correlation
estimates, but also
the longer the time interval, the more unrealistic the
stationarity hypothesis for these time series.
Question: How does the clustering behave with statistical errors
of the correlation estimates?

A first theoretical approach - simplified setting
We consider the following framework:
financial time series ≡ random walks
they follow a joint elliptical distribution (e.g. Gaussian,
Student) parameterized by a correlation matrix
the correlation matrix has a hierarchical block structure:

Hierarchical clustering algorithms - A taxonomy
We consider Hierarchical Agglomerative Clustering algorithms.
Such as single linkage, average linkage, Ward.
Space contracting vs. Space conserving vs. Space dilating [2]
D(t+1)
C
(t)
i
∪ C
(t)
j
, C
(t)
k
≤ min D
(t)
ik
, D
(t)
jk
D(t+1)
C
(t)
i
∪ C
(t)
j
, C
(t)
k
∈
min D
(t)
ik
, D
(t)
jk
, max D
(t)
ik
, D
(t)
jk
D(t+1)
C
(t)
i
∪ C
(t)
j
, C
(t)
k
≥ max D
(t)
ik
, D
(t)
jk

Simulations in the simpliﬁed setting
Some inﬂuential parameters:
clustering algorithm
number of observations T
number of variables N relative to T
contrast between the correlations, and their values
correlation estimator (e.g. Pearson, Spearman)
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Single Linkage
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Average Linkage
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
100 200 300 400 500
Sample size
0.0
0.2
0.4
0.6
0.8
1.0
Score
Empirical rates of convergence for Ward
Gaussian - Pearson
Gaussian - Spearman
Student - Pearson
Student - Spearman
Ratio of the number of correct clustering obtained over the
number of trials as a function of T

A consistency proof & first convergence bounds
A 2-step proof. First step:
Which geometrical configurations lead to the true clustering?
For space-conserving algorithms (e.g. Single, Complete, Average
Linkage), a sufficient separability condition reads
max Dintra := max
1≤i,j≤N
C(i)=C(j)
d(Xi , Xj ) < min
1≤i,j≤N
C(i)=C(j)
d(Xi , Xj ) =: min Dinter

A consistency proof & first convergence bounds
A 2-step proof. Second step:
How long does it take for the estimates of the correlation
coefficients to be precise enough to be with high probability in
a good configuration for the clustering algorithm?
Answer: Concentration inequalities for correlation coefficients.

Convergence bounds
Combining both steps, we get the following convergence rate:
Convergence rate
The probability of the clustering algorithm making an error is
O
log N
T
.

Proof. Step 1 - A bit more details
By induction.
Let’s assume the separability condition is satisfied at step t,
then
min D
(t)
intra ≤ max D
(t)
intra < min D
(t)
inter ≤ max D
(t)
inter
From the space-conserving property, we get:
D
(t+1)
intra ∈ min D
(t)
intra, max D
(t)
intra and D
(t+1)
inter ∈ min D
(t)
inter, max D
(t)
inter .
separability condition is satisfied at t+1,
the clustering algorithm has not linked points from two
different clusters between step t and step t + 1.

Proof. Step 2 - A bit more details
Maximum statistical error - (Marti and Andler, IJCAI 2016)
For space conserving algorithm the separability condition is met if
ˆΣ − Σ ∞ <
minρi ,ρj
|ρi − ρj |
2
,
where C(i) = C(j).
This means that the statistical error has to be below the minimum
correlation ‘contrast’ between the clusters.
Weaker the ‘contrast’, more precise the correlation estimates have to be.
N.B. From Cram´er–Rao lower bound, we get for Pearson correlation
estimator:
var(ˆρ) ≥
1
I(ρ)
=
(ρ − 1)2
(ρ + 1)2
3(ρ2 + 1)
.
When correlation is high, it is easier to estimate. (Marti, SSP 2016)

Correlation estimates concentration bounds
number of variables N, observations T, minimum separation d
Concentration bounds [4]
If Σ and ˆΣ are the population and empirical Spearman correlation
matrices respectively, then for N ≥ 24
log T + 2, we have with
probability at least 1 − 1
T2 ,
ˆΣ − Σ ∞ ≤ 24
log N
T
.
P(“correct clustering”) ≥ 1 − 2N2
e−Td2/24
Not sharp enough for reasonable values of N, T, d.
For example, for N = 500, T = 2500, d = 0.2, we obtain
≈ −7750.

Future developments & open questions
Bounds are not sharp enough. We can try to refine them using:
(theoretical) Intrinsic dimension of the HCBM model [16];
(theoretical Use PSD-ness to refine the bounds for the matrix
(theoretical/empirical) A distance between dendrograms
(instead of correct/incorrect) for a finer analysis;
(empirical) A study of ‘correctness’ isoquants:
Precise convergence rates of clustering methodologies can provide
a useful model selection criterion for practitioners!

1 Introduction to credit default swaps
2 About the consistency of clustering ﬁnancial time series
3 Design of distances and alternative dependence coeﬃcients
4 Summary and open questions

Motivations
Not only correlation!
Diﬀerent kinds of returns distributions may exist in the data.
We may want to reﬁne ‘correlation’ clusters with this information.

A (too) naive distance and its pitfalls
Applying L2 directly on the time series mixes correlation and
volatility.
We are looking for a better representation so that a L2 is
meaningful.

A (too) naive distance and its pitfalls

Revisting Sklar theorem (1959)...

and Deheuvels empirical copulas (1981)

A novel representation for time series
Basically, we transform each time series of returns to a (normalized
ranks, square root of marginal density) vector.
Applying a L2 between two of these vectors is now equivalent to a
distance in Spearman correlation + Hellinger between the densities.
cf. (Marti et al., 2016) [13] (Pattern Recognition Letters) for more
details on this representation and the associated distance.

Analysing the diﬀerences - Using the Sankey diagram
cf. (Marti et al., 2015) [14] (ICMLA 2015) for guidelines on how to
compare several clustering methodologies for ﬁnancial time series.

Exploring and improving dependence coeﬃcients –
Motivations
We want to use dependence measures which are:
robust to noise (not Pearson then) and preserve as much
information as possible, so that clusterings are more stable;
can be tuned to look for speciﬁc dependencies, e.g.
tail-dependence or more exotic ones.
As copulas are a convenient way to capture all the dependence
between two variables, we aim at leveraging them.

Minimum, Independence, Maximum copulas

Relation to existing dependence coefficients
Some dependence coefficients can be readily expressed as:
deviation from Fréchet-Hoeffding bounds
Spearman’s ρS = 1 − 6 [0,1]2 (ui − uj )2
dC(ui , uj ),
Gini’s γ,
Kendall distribution distance,
deviation from independence ui uj
Spearman ρS = 12 [0,1]2 (C(ui , uj ) − ui uj )dui
duj
,
Copula MMD,
Schweizer-Wolff’s σ,
Hoeffding’s φ2

Idea: Relative position of empirical copula wrt ‘targets’
and ‘forgets’
In a classical setting, we choose the positive and negative
dependence copulas as ‘targets’, and the independence one as a
‘forget’ dependence.

Optimal Transport between empirical copulas
cf. (Marti et al., 2016) [12] (ICASSP 2016)

Why choosing Optimal Transport over f -divergences?
Distances between Gaussian copulas C1, C2, C3:
cf. (Marti et al., 2016) [7] (SSP 2016)

Standard setting: TFDC vs. Spearman
cf. (Marti et al., 2017) [8] (NIPS Time Series WS 2016)

Power of TFDC and state-of-the-art dependence measures

Some applications of the Target/Forget Dependence
Coeﬃcient
Applications in non-standard settings: We can look for particular
associations between random variables.
cf. (Marti et al., 2017) [8] (NIPS Time Series 2016)

Impact of different coefficients on clustering
Different results... Stability and empirical convergence rates may
help for choosing one over the others.

Summary of contributions
The contribution of the PhD thesis:
bring a greater focus on statistical reliability (convergence
rates and consistency) [9] (IJCAI 2016)
consider alternative representation and distances [13]
(Pattern Recognition Letters), [8] (NIPS Time Series 2016)
visualizations [10] and a framework to test for clustering
stability [14] (ICMLA 2015)
an extensive and regularly updated survey of the literature:
https://arxiv.org/pdf/1703.00485.pdf [11] (350+
references)

Perspectives and open questions
“How many clusters?”
What is multivariate correlation? How to use it for
hierarchical clustering?
Using several time series representing a given entity, and
dependence between random vectors?
Riemannian geometry of correlation matrices (not a totally
geodesic submanifold of the well-explored manifold of
covariances)
Entities switching clusters: noise or signal?
More precise results for (empirical) convergence rates?

Mikolaj Binkowski, Gautier Marti, and Philippe Donnat.
Autoregressive convolutional neural networks for asynchronous
time series.
arXiv preprint arXiv:1703.04122, 2017.
Zhenmin Chen and John W Van Ness.
Space-conserving agglomerative algorithms.
Journal of classification, 13(1):157–168, 1996.
Laurent Laloux, Pierre Cizeau, Marc Potters, and
Jean-Philippe Bouchaud.
Random matrix theory and financial correlations.
International Journal of Theoretical and Applied Finance,
3(03):391–397, 2000.
Han Liu, Fang Han, Ming Yuan, John Lafferty, Larry
Wasserman, et al.

High-dimensional semiparametric gaussian copula graphical
models.
The Annals of Statistics, 40(4):2293–2326, 2012.
Burton G Malkiel and Eugene F Fama.
Efficient capital markets: A review of theory and empirical
work.
The journal of Finance, 25(2):383–417, 1970.
Rosario N Mantegna.
Hierarchical structure in financial markets.
The European Physical Journal B-Condensed Matter and
Complex Systems, 11(1):193–197, 1999.
Gautier Marti, Sébastien Andler, Frank Nielsen, and Philippe
Donnat.
Optimal transport vs. fisher-rao distance between copulas for
clustering multivariate time series.

In Statistical Signal Processing Workshop (SSP), 2016 IEEE,
pages 1–5. IEEE, 2016.
Donnat.
Exploring and measuring non-linear correlations: Copulas,
lightspeed transportation and clustering.
In NIPS 2016 Time Series Workshop, pages 59–69, 2017.
Donnat.
Clustering ﬁnancial time series: How long is enough?
2016.
Gautier Marti, Philippe Donnat, Frank Nielsen, and Philippe
Very.

HCMapper: An interactive visualization tool to compare
partition-based flat clustering extracted from pairs of
dendrograms.
Gautier Marti, Frank Nielsen, Mikolaj Bińkowski, and Philippe
Donnat.
A review of two decades of correlations, hierarchies, networks
and clustering in financial markets.
Gautier Marti, Frank Nielsen, and Philippe Donnat.
Optimal copula transport for clustering multivariate time
series.
In Acoustics, Speech and Signal Processing (ICASSP), 2016
IEEE International Conference on, pages 2379–2383. IEEE,
2016.

Gautier Marti, Philippe Very, and Philippe Donnat.
Toward a generic representation of random variables for
machine learning.
Gautier Marti, Philippe Very, Philippe Donnat, and Frank
Nielsen.
A proposal of a methodological framework with experimental
guidelines to investigate clustering stability on ﬁnancial time
series.
In 14th IEEE International Conference on Machine Learning
and Applications, ICMLA 2015, Miami, FL, USA, December
9-11, 2015, pages 32–37, 2015.
Marc Szafraniec, Gautier Marti, and Philippe Donnat.
Putting self-supervised token embedding on the tables.

Joel A Tropp.
An introduction to matrix concentration inequalities.

Appendix - CRLB for correlation

Appendix - CRLB for correlation - Proof

Appendix - Fisher-Rao geodesic distance

Appendix - Optimal Transport distances
Other transportation distances: regularized discrete optimal
transport, Sinkhorn distances, etc.

Appendix - Geometry of covariances

Appendix - The standard methodology: Pearson + MST

Appendix - The Target/Forget Dependence Coeﬃcient

Appendix - The Copula Transform

Appendix - The correlation + distribution distance

Appendix - Pearson correlation

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