N IME ERIES - Data Science Summer School · max )( min) ; where max min = 1 + N=T 2 p N=T , and 2 [...
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O N C LUSTERING F INANCIAL T IME S ERIES G AUTIER M ARTI ,P HILIPPE D ONNAT AND F RANK N IELSEN N OISY CORRELATION MATRICES Let 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 T XX > . We can compute the empirical density of its eigenvalues ρ(λ)= 1 N dn(λ) dλ , where n(λ) counts the number of eigenvalues of C less than λ. From random matrix theory, the Marchenko- Pastur distribution gives the limit distribution as N →∞, T →∞ and T /N fixed. It reads: ρ(λ)= T /N 2π p (λ max - λ)(λ - λ min ) λ , where λ max min = 1+ N/T ± 2 p N/T , and λ ∈ [λ min ,λ max ]. Figure 1: Marchenko-Pastur density vs. empirical den- sity of the correlation matrix eigenvalues Notice that the Marchenko-Pastur density fits well the empirical density meaning that most of the information contained in the empirical corre- lation matrix amounts to noise: only 26 eigenval- ues are greater than λ max . The highest eigenvalue corresponds to the ‘mar- ket’, the 25 others can be associated to ‘industrial sectors’. C LUSTERING TIME SERIES Given a correlation matrix of the returns, 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, Figure 3: The same noisy correlation matrix re-ordered by a hierarchical clustering algorithm and finally filter the noise according to the corre- lation pattern: Figure 4: The resulting filtered correlation matrix B EYOND CORRELATION Sklar’s Theorem. For any random vector X =(X 1 ,...,X N ) having continuous marginal cumulative distribution functions F i , its joint cumulative distribution F is uniquely expressed as F (X 1 ,...,X N )= C (F 1 (X 1 ),...,F N (X N )), 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 their heavy-tailed exponential distribution. Let θ ∈ [0, 1]. Let (X, Y ) ∈V 2 . Let G =(G X ,G Y ), where G X and G Y are respectively X and Y marginal cdf. We define the following distance d 2 θ (X, Y )= θd 2 1 (G X (X ),G Y (Y )) + (1 - θ )d 2 0 (G X ,G Y ), where d 2 1 (G X (X ),G Y (Y )) = 3E[|G X (X ) - G Y (Y )| 2 ], and d 2 0 (G X ,G Y )= 1 2 R R q dG X dλ - q dG Y dλ 2 dλ. C LUSTERING R ESULTS &S TABILITY Figure 6: (Top) The returns correlation structure ap- pears more clearly using rank correlation; (Bottom) Clusters of returns distributions can be partly described by the returns volatility Figure 7: Stability test on Odd/Even trading days sub- sampling: our approach (GNPR) yields more stable clusters with respect to this perturbation than standard approaches (using Pearson correlation or L 2 distances).
Transcript of N IME ERIES - Data Science Summer School · max )( min) ; where max min = 1 + N=T 2 p N=T , and 2 [...
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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
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Standard Deviation in basis points0
5
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30
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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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