clustering cds: algorithms, distances, stability and convergence rates

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IntroductionThe standard methodology

Exploring dependence between returnsCopula-based dependence coefficients (clustering distances)

Empirical convergence ratesBeyond dependence: a (copula,margins) representation

Clustering CDS: algorithms, distances,stability and convergence rates

CMStatistics 2016, University of Seville, Spain

Gautier Marti, Frank Nielsen, Philippe Donnat

HELLEBORECAPITALDecember 9, 2016

Gautier Marti Clustering CDS: algorithms, distances, stability and convergence rates

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1 Introduction

2 The standard methodology

3 Exploring dependence between returns

4 Copula-based dependence coefficients (clustering distances)

5 Empirical convergence rates

6 Beyond dependence: a (copula,margins) representation


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Introduction

Goal: Finding groups of ’homogeneous’ assets that can help to:

• build alternative measures of risk,

• elaborate trading strategies. . .

But, we need a high confidence in these clusters (networks).

So, we need appropriate AND fast converging methodologies [8]:

. to be consistent yet efficient (bias–variance tradeoff),

. to avoid non-stationarity of the time series (too large sample).

A good model selection criterion:Minimum sample size to reach a given ’accuracy’.


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1 Introduction







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The standard methodology - description

The methodology widely adopted in empirical studies: [7].

Let N be the number of assets.Let Pi (t) be the price at time t of asset i , 1 ≤ i ≤ N.Let ri (t) be the log-return at time t of asset i :

ri (t) = log Pi (t)− log Pi (t − 1).

For each pair i , j of assets, compute their correlation:

ρij =〈ri rj〉 − 〈ri 〉〈rj〉√

(〈r 2i 〉 − 〈ri 〉2)

(〈r 2

j 〉 − 〈rj〉2) .

Convert the correlation coefficients ρij into distances:

dij =√

2(1− ρij).


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The standard methodology - description

From all the distances dij , compute a minimum spanning tree:

Figure: A minimum spanning tree of stocks (from [1]); stocks from thesame industry (represented by color) tend to cluster together


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The standard methodology - limitations

• MST clustering equivalent to Single Linkage clustering:

• chaining phenomenon• not stable to noise / small perturbations [11]

• Use of the Pearson correlation:

• can take value 0 whereas variables are strongly dependent• not invariant to variable monotone transformations• not robust to outliers

Is it still useful for financial time series? stocks? CDS??!


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The standard methodology - limitations

• MST clustering equivalent to Single Linkage clustering:

• chaining phenomenon• not stable to noise / small perturbations [11]

• Use of the Pearson correlation:

• can take value 0 whereas variables are strongly dependent• not invariant to variables monotone transformations• not robust to outliers

Is it still useful for financial time series? stocks? CDS??!


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1 Introduction







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Copulas

Sklar’s Theorem [13]

For (Xi ,Xj) having continuous marginal cdfs FXi ,FXj , its joint cumulativedistribution F is uniquely expressed as

F (Xi ,Xj) = C (FXi (Xi ),FXj (Xj)),

where C is known as the copula of (Xi ,Xj).

Copula’s uniform marginals jointly encode all the dependence.


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From ranks to empirical copula

ri , rj are the rank statistics of Xi ,Xj respectively, i.e. r ti is the rank

of X ti in {X 1

i , . . . ,XTi }: r ti =

∑Tk=1 1{X k

i ≤ X ti }.

Deheuvels’ empirical copula [3]

Any copula C defined on the lattice L = {( tiT ,

tjT ) : ti , tj = 0, . . . ,T} by

C ( tiT ,

tjT ) = 1

T

∑Tt=1 1{r ti ≤ ti , r

tj ≤ tj} is an empirical copula.

C is a consistent estimator of C with uniform convergence [4].


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Clustering of bivariate empirical copulas

Generate the(N

2

)bivariate empirical copulas

Find clusters of copulas using optimal transport [10, 9]

Compute and display the clusters’ centroids [2]

Some code available at www.datagrapple.com/Tech.


www.datagrapple.com/Tech

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Copula-centers for stocks (CAC 40)

Figure: Stocks: More mass in the bottom-left corner, i.e. lower taildependence. Stock prices tend to plummet together.


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Copula-centers for Credit Default Swaps (XO index)

Figure: Credit default swaps: More mass in the top-right corner, i.e.upper tail dependence. Insurance cost against entities’ default tends tosoar in stressed market.


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1 Introduction







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Dependence as relative distances between copulas

C copula of (Xi ,Xj),|u − v |/

√2 distance between (u, v) to the diagonal

Spearman’s ρS :

ρS(Xi ,Xj) = 12

∫ 1

0

∫ 1

0(C (u, v)− uv)dudv

= 1− 6

∫ 1

0

∫ 1

0(u − v)2dC (u, v)

Many correlation coefficients can be expressed as distances to the

Frechet–Hoeffding bounds or the independence [6]. Some are explicitely

built this way (e.g. [12, 5, 9]).


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A metric space for copulas: Optimal Transport


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The Target/Forget Dependence Coefficient (TFDC)

Now, we can define our bespoke dependence coefficient:

Build the forget-dependence copulas {CFl }l

Build the target-dependence copulas {CTk }k

Compute the empirical copula Cij from xi , xj

TFDC(Cij) =minl D(CF

l ,Cij)

minl D(CFl ,Cij) + mink D(Cij ,CT

k )


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Spearman vs. TFDC

0.0 0.2 0.4 0.6 0.8 1.0

discontinuity position a

0.0

0.2

0.4

0.6

0.8

1.0

Est

imate

d p

osi

tive d

ependence

Spearman & TFDC values as a function of a

TFDC

Spearman

Figure: Empirical copulas for (X ,Y ) whereX = Z1{Z < a}+ εX1{Z > a},Y = Z1{Z < a + 0.25}+ εY 1{Z > a + 0.25}, a = 0, 0.05, . . . , 0.95, 1,and where Z is uniform on [0, 1] and εX , εY are independent noises (left).TFDC and Spearman coefficients estimated between X and Y as afunction of a (right).For a = 0.75, Spearman coefficient yields a negative value, yet X = Yover [0, a].


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1 Introduction







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Process: Recovering a simulated ground-truth [8]

A simulation & benchmark process that needs to be refined:

. Extract (using a large sample) a filtered correlation matrix R


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. Generate samples of size T = 10, . . . , 20, . . . from a relevantdistribution (parameterized by R)


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. Compute the ratio of the number of correct clusteringobtained over the number of trials as a function of T

100 200 300 400 500Sample size

0.0

0.2

0.4

0.6

0.8

1.0

Scor

e

Empirical rates of convergence for Single Linkage

Gaussian - PearsonGaussian - SpearmanStudent - PearsonStudent - Spearman

100 200 300 400 500Sample size

0.0

0.2

0.4

0.6

0.8

1.0

Scor

e

Empirical rates of convergence for Average Linkage


100 200 300 400 500Sample size

0.0

0.2

0.4

0.6

0.8

1.0

Scor

e

Empirical rates of convergence for Ward


A full comparative study will be posted online at www.datagrapple.com/Tech.


www.datagrapple.com/Tech

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1 Introduction







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ON CLUSTERING FINANCIAL TIME SERIESGAUTIER MARTI, PHILIPPE DONNAT AND FRANK NIELSEN

NOISY CORRELATION MATRICESLet X be the matrix storing the standardized re-turns of N = 560 assets (credit default swaps)over a period of T = 2500 trading days.

Then, the empirical correlation matrix of the re-turns is

C =1

TXX>.

We can compute the empirical density of itseigenvalues

ρ(λ) =1

N

dn(λ)

dλ,

where n(λ) counts the number of eigenvalues ofC less than λ.

From random matrix theory, the Marchenko-Pastur distribution gives the limit distribution asN →∞, T →∞ and T/N fixed. It reads:

ρ(λ) =T/N

2π

√(λmax − λ)(λ− λmin)

λ,

where λmaxmin = 1 + N/T ± 2

√N/T , and λ ∈

[λmin, λmax].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

λ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

ρ(λ

)

Figure 1: Marchenko-Pastur density vs. empirical den-sity of the correlation matrix eigenvalues

Notice that the Marchenko-Pastur density fitswell the empirical density meaning that most ofthe information contained in the empirical corre-lation matrix amounts to noise: only 26 eigenval-ues are greater than λmax.The highest eigenvalue corresponds to the ‘mar-ket’, the 25 others can be associated to ‘industrialsectors’.

CLUSTERING TIME SERIESGiven a correlation matrix of the returns,

0 100 200 300 400 5000

100

200

300

400

500

Figure 2: An empirical and noisy correlation matrix

one can re-order assets using a hierarchical clus-tering algorithm to make the hierarchical correla-tion pattern blatant,

0 100 200 300 400 5000

100

200

300

400

500

Figure 3: The same noisy correlation matrix re-orderedby a hierarchical clustering algorithm

and finally filter the noise according to the corre-lation pattern:

0 100 200 300 400 5000

100

200

300

400

500

Figure 4: The resulting filtered correlation matrix

BEYOND CORRELATIONSklar’s Theorem. For any random vector X = (X1, . . . , XN ) having continuous marginal cumulativedistribution functions Fi, its joint cumulative distribution F is uniquely expressed as

F (X1, . . . , XN ) = C(F1(X1), . . . , FN (XN )),

where C, the multivariate distribution of uniform marginals, is known as the copula of X .

Figure 5: ArcelorMittal and Société générale prices are projected on dependence ⊕ distribution space; notice theirheavy-tailed exponential distribution.

Let θ ∈ [0, 1]. Let (X,Y ) ∈ V2. Let G = (GX , GY ), where GX and GY are respectively X and Y marginalcdf. We define the following distance

d2θ(X,Y ) = θd21(GX(X), GY (Y )) + (1− θ)d20(GX , GY ),

where d21(GX(X), GY (Y )) = 3E[|GX(X)−GY (Y )|2], and d20(GX , GY ) =12

∫R

(√dGX

dλ −√

dGY

dλ

)2

dλ.

CLUSTERING RESULTS & STABILITY

0 5 10 15 20 25 30

Standard Deviation in basis points0

5

10

15

20

25

30

35

Num

ber

of

occ

urr

ence

s

Standard Deviations Histogram

Figure 6: (Top) The returns correlation structure ap-pears more clearly using rank correlation; (Bottom)Clusters of returns distributions can be partly describedby the returns volatility

Figure 7: Stability test on Odd/Even trading days sub-sampling: our approach (GNPR) yields more stableclusters with respect to this perturbation than standardapproaches (using Pearson correlation or L2 distances).


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Gautier Marti, Sebastien Andler, Frank Nielsen, and PhilippeDonnat.Clustering financial time series: How long is enough?Proceedings of the Twenty-Fifth International JointConference on Artificial Intelligence, IJCAI 2016, New York,NY, USA, 9-15 July 2016, pages 2583–2589, 2016.

Gautier Marti, Sebastien Andler, Frank Nielsen, and PhilippeDonnat.Exploring and measuring non-linear correlations: Copulas,lightspeed transportation and clustering.NIPS 2016 Time Series Workshop, 55, 2016.

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