Construction and estimation of high dimensional copulas

Transcription

1 Construction and estimation of high dimensional copulas Gildas Mazo PhD work supervised by S. Girard and F. Forbes Mistis, Inria and laboratoire Jean Kuntzmann, Grenoble, France Séminaire Statistiques, Strasbourg, 14 octobre / 51

2 Copulas: why and what? Multivariate vector (X 1,..., X d ) F F is often nongaussian / tail dependent Example flood {X 1 > q,..., X d > q} portfolio of financial assets X X d Two steps modeling of F : 1. Margins F 1,..., F d 2. Dependence Copula: d.f. of (F 1 (X 1 ),..., F d (X d )) takes care of dependence 2 / 51

3 Challenge: modeling in high dimension with copulas Two issues: Construction of high-dimensional copulas: difficult to describe the interplay between many variables Estimation of copulas: we have the sample of F not that of C (X (k) 1,..., X (k) ), k = 1,..., n d (F 1 (X (k) 1 ),..., F d (X (k) )), k = 1,..., n d 3 / 51

4 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results 4 / 51

5 Definition and Sklar s theorem Definition (d-dimensional copula) d.f. of (U 1,..., U d ), U i U([0, 1]) C(u 1,..., u d ) = P[U 1 u 1,..., U d u d ] Theorem (Sklar, 1959) If C copula and F 1,..., F d are univariate d.f. then F (x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )) (1) defines a multivariate d.f. Every F (X 1,..., X d ) can be written as (1) C is the d.f. of (F 1 (X 1 ),..., F d (X d )) 5 / 51

6 Quantifying dependence (X (1) 1, X (1) 2 ), (X (2) 1, X (2) 2 ) i.i.d. (X 1, X 2 ) with margins F 1, F 2 Kendall s tau: [ ] τ =P (X (1) 1 X (2) 1 )(X (1) 2 X (2) 2 ) > 0 [ ] P (X (1) 1 X (2) 1 )(X (1) 2 X (2) 2 ) < 0 Spearman s rho: ρ =cor(f 1 (X 1 ), F 2 (X 2 )) Property: τ = 4 CdC 1, [0,1] 2 ρ = 12 CdΠ 3 [0,1] 2 6 / 51

7 Tail dependence coefficients Upper tail dependence coefficient: 0.6 λ(u) = lim P [F2 (X2 ) > u F1 (X1 ) > u] u Lower tail dependence coefficient: 0.6 λ(l) = lim P [F2 (X2 ) u F1 (X1 ) u] u 0 Property: λ(l) = lim u 0 C (u, u), u λ(u) = lim u 1 1 2u + C (u, u) 1 u 7 / 51

8 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results 8 / 51

9 Important copulas in high dimension tractability flexibility tail asymmetry / 51

10 Important copulas in high dimension tractability flexibility Archimedean tail asymmetry / 51

11 Important copulas in high dimension tractability flexibility Archimedean Vines tail asymmetry / 51

12 Important copulas in high dimension tractability Elliptical flexibility Archimedean Vines tail asymmetry / 51

13 Archimedean copulas [Nelsen 1996], [Joe 2001], [McNeil Neslehova 2009] C(u 1,..., u d ) = ψ(ψ 1 (u 1 ) + + ψ 1 (u d )) ψ univariate function, called the generator of C Kendall s tau: 1 ψ 1 (t) τ = (ψ 1 ) (t) dt Examples: ψ 1 (t) = ( log t) θ, τ = (θ 1)/θ, θ 1 (Gumbel) ψ 1 (t) = (t θ 1)/θ, τ = θ/(θ + 2), θ > 0 (Clayton) Not flexible because exchangeable! τ 12 = τ 13 = τ 37 = / 51

14 Vines [Joe 2001], [Bedford Cook 2002], [Aas et al 2009] Decomposition of the density into (conditional) bivariate densities Drawable vines: d 1 d j j=1 i=1 c(u 1,..., u d ) = c i,i+j i+1,...,i+j 1 {F (u i u i+1,..., u i+j 1 ), F (u i+j u i+1,..., u i+j 1 )} Canonical vines: d 1 d j c j,j+1 1,...,j 1 {F (u j u 1,..., u j 1 ), F (u j+i u 1,..., u j 1 )} j=1 i=1 For each of above, 60 decompositions with d = 5, >1 million with d = 10, etc... Overfitting issues Flexible, but not tractable 14 / 51

15 Elliptical copulas [e.g., Frahm et al 2003] Are the copulas of elliptical distributions: density f writes x R d, g density generator, P = (θ ij ) correlation matrix. Examples: Kendall s tau: f (x) = P 1/2 g(x T P 1 x), g(t) exp( t/2), t 0, (Gaussian) ( g(t) 1 + t ) (ν+d)/2, t 0, ν > 2, (Student) ν τ ij = 2 π arcsin(θ ij) Elliptical copulas are tail symmetric: λ (U) ij = λ (L) ij 15 / 51

16 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results Existing copulas have drawbacks Wish to construct copulas with all of tractability flexibility tail asymmetry 16 / 51

17 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results Product of Bivariate Copulas: build high-dimensional copulas by making products of bivariate copulas 17 / 51

18 Generalized product of copulas (Liebscher) is unhandy [Liebscher 2009]: C(u 1,..., u d ) = C e (g e1 (u 1 ),..., g ed (u d )) e constraints: g ei (u) = u, i = 1,..., d, u [0, 1] e Almost not been used in practice (because of the constraints?) Can we get a tractable/flexible model out of Liebscher? 18 / 51

19 Product of Bivariate Copulas (PBC) Let ({U 1,..., U d }, E) be a graph Assumption Set g ei = 1 if e / E, does not depend on e otherwise Theorem Liebscher reduces to PBC C(u 1,..., u d ) = {ij} E C ij (u 1/n i i, u 1/n j j ), (n i number of neighbors of the i-th variable) No constraints on the parameters anymore! 19 / 51

20 Example C(u 1, u 2, u 3, u 4, u 5 ) = C 12 (u 1, u 1/3 2 ) C 23 (u 1/3 2, u 1/2 3 ) C 24 (u 1/3 2, u 4 ) C 35 (u 1/2 3, u 5 ) 20 / 51

21 Properties and Limitations Properties One parameter per edge in the graph Benefits from the rich literature of bivariate copulas Limitations Marginal independence: if {ij} does not belong to E, then C ij Π Dependence weakens as graph gets more connected: if n i, then C ik Π coefficient ρ kl τ kl λ kl (n k, n l ) (1, 2) [ 0.60, 0.60] [ 0.50, 0.50] [0.00, 0.50] (2, 2) [ 0.30, 0.43] [ 0.21, 0.33] [0.00, 0.50] (1, 3) [ 0.43, 0.43] [ 0.33, 0.33] [0.00, 0.33] (2, 3) [ 0.19, 0.33] [ 0.13, 0.25] [0.00, 0.33] (3, 3) [ 0.12, 0.27] [ 0.08, 0.20] [0.00, 0.33] 21 / 51

22 Triangle position Can we do better? if the data do not exhibit too much dependence: tractability flexibility PBC tail asymmetry 22 / 51

23 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results one-factor copulas with Durante Generators: Factor property for tractability Durante class for flexibility 23 / 51

24 Conditional independence property [Krupskii Joe 2013] U 1,..., U d independent given a latent factor U 0 Linking copulas: C(u 1,..., u d ) = 1 0 d i=1 (U 0, U i ) C 0i C 0i (u 0, u i ) u 0 du 0 Can we calculate the integral for suitable linking copulas? (without removing flexibility) 24 / 51

25 Durante class of bivariate copulas [Durante 2006] (Nonparametric) class D f : f generator Coefficients: Idea: ρ = 12 C f (u, v) = min(u, v)f (max(u, v)), 1 0 x 2 f (x)dx 3, τ = 4 1 λ (L) = f (0), and λ (U) = 1 f (1) 0 xf (x) 2 dx 1, C 0i C fi D fi 25 / 51

26 Expression of FDG copulas We can calculate the integral: C(u 1,..., u d ) = d u (1) + j=2 d k 1 k=3 j=2 u (j) u (j) 1 d u (d) j=1 d j (x)dx + f (1) (u (2) ) f (j) (u (j) ) f d f (j) (u (j) ) j=k u(k) k 1 u (k 1) j=1 f j=2 (j) (x)dx, u (i) := u σ(i), f (i) := f σ(i), σ permutation of (1,..., d) such that u σ(1) u σ(d) Bivariate margins C ij D fij ( Durante ) with f ij (t) = f i (t)f j (t) + t 1 t f i (x)f j (x)dx 26 / 51

27 A remarkable tail dependence structure Both in the upper and lower tails: Examples: λ ( ) ij = λ ( ) i λ ( ) j generator tail dependence Type f i (t) θ i λ (L) ij λ (U) ij F (1 θ i )t + θ i [0, 1] θ i θ j θ i θ j CA t 1 θ i [0, 1] 0 θ i θ j sin(θ S i t) sin(θ i ) (0, π 2 ] 0 (1 θ i tan θ i )(1 ( ) E exp t θ i 1 θ i (0, ) exp( 1 θ i 1 θ j ) 0 F: [Fréchet 1958], CA: [Cuadras-Augé 1981] S: sinus, E: exponential [Durante 2006] θ j tan θ j ) 27 / 51

28 We can calculate the extreme-value copulas of FDG EVFDG(u 1,..., u d ) = lim FDG n (u 1/n 1,..., u 1/n n d ) = d i=1 u χ i (i), with i 1 χ i = (1 λ (j) ) λ (i) + 1 λ (i) j=1 λ i = 1 f i (1) Bivariate margins are Cuadras-Augé copulas [Cuadras-Augé 1981]: EVFDG #,ij (u i, u j ) = min(u i, u j ) max(u i, u j ) 1 λ i λ j 28 / 51

29 FDG copulas seem to have a good triangle position tractability flexibility FDG tail asymmetry / 51

30 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results 30 / 51

31 Two main methods for estimating copulas Setup (X 1,..., X d ) F with copula C C(θ), θ Θ R q for some integer q. We observe (X (k) 1,..., X (k) d ) for k = 1,..., n. Methods Likelihood maximization max θ n k=1 log c( F 1 (X (k) 1 ),..., F d (X (k) d ); θ) where F i some approximation of F i [Joe 2001], [Genest et al 1995] Method-of-moment, based on dependence coefficients [Favre Genest 2007], [Kluppelberg Kuhn 2009], [Oh Patton 2013] 31 / 51

32 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results 32 / 51

33 Estimating PBC copulas by maximizing the likelihood We want to maximize log L(θ) = where the density is and n k=1 log c( F 1 (X (k) 1 ),..., F d (X (k) d ); θ) c(u 1,..., u d ) = d C(u 1,..., u d ) u 1... u d C(u 1,..., u d ) = {ij} E C ij (u 1/n i i, u 1/n j j ). Numerical derivation unhandy in high-dimension How to compute the density? 33 / 51

34 A message-passing algorithm to compute the density [Huang Jojic 2010] u = (u 1,..., u d ), τ i e subtree rooted at variable i and containing edge e, τ j e subtree rooted at edge e and containing variable i. Write C(u) = e N(i) T τ i e (u). uv C(u) = ui,u V \i = ui e N(i) e N(i) T τ i e (u) T uτ i e \i τe i (u) }{{} µ e i (u) 34 / 51

35 A message-passing algorithm to compute the density µ e i (u) = T uτ i e \i τe i (u) = uj C e (u 1/n i i, u 1/n j j ) uτ e j \j T τ e j (u) } {{ } µ j e (u) Recursive algorithm 35 / 51

36 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results 36 / 51

37 Issue: FDG copulas are not differentiable Recall bivariate FDG: C ij (u, v) = min(u, v)f ij (uv/ min(u, v)) No likelihood No density No partial derivatives Attempt to calculate first partial derivative (Fréchet): C ij (u + h, u) C ij (u, u) lim = h 0 h { (1 θi θ j )u + θ i θ j if h < 0, (1 θ i θ j )u if h > / 51

38 Let s try method-of-moment instead Match empirical with parametric dependence coefficients Definition (Weighted Least-Squares estimator (WLS) based on dependence coefficients) where ˆθ = arg min (ˆr r(θ)) T Ŵ (ˆr r(θ)), θ Θ r(θ) = (r 1,2 (θ),..., r d 1,d (θ)) vector of pairwise dependence coefficients ˆr = (ˆr 1,2,..., ˆr d 1,d ) empirical counterpart Ŵ is a weight matrix 38 / 51

39 No asymptotic properties are derived in general case Particular cases: Single, real parameter [Favre Genest 2007] Elliptical copulas [Kluppelberg Kuhn 2009] Smooth copulas (i.e., with partial derivatives) [Oh Patton 2013] None of these references are suitable for FDG copulas 39 / 51

40 How to derive asymptotic properties for nonsmooth copulas? Use [Hoeffding 1948] n(ˆr r(θ0 )) N(0, Σ) and pass convergence to n(ˆθ θ 0 ) by homeomorphism condition: Assumption (main) Θ r(θ), θ r(θ) is one-to-one, continuous, and its inverse is continous as well Theorem n(ˆθ θ 0 ) N(0, Ξ) (even though C has no partial derivatives) r i,j (θ) = λ (U) i,j (θ) = λ (U) i (θ)λ (U) j (θ) verifies the Assumption for extreme-value copulas associated to FDG 40 / 51

41 A short review of copulas Definition and main properties/ideas High dimensional copulas Two novel high-dimensional copulas PBC copulas FDG copulas Estimation PBC copulas FDG copulas Numerical results Assess gain of efficiency for message-passing in PBC copulas estimation Check WLS estimator s performance for FDG copulas Analysis of extreme hydrological events 41 / 51

42 PBC: is maximizing the full likelihood worth it? 100 datasets of dimension d = 9 and size n = 500 Monitoring coefficients GE = 1 d 1 Var e=1 d 1 Var e=1 (ˆθ e FULL (ˆθ e PW ) ) MAE ρ = 1 d 1 ρ(θ e ) ρ(ˆθ e FULL ) d 1 e=1 MAE τ = 1 d 1 τ(θ e ) τ(ˆθ e FULL ) d 1 e=1 42 / 51

43 It depends on the families Copula GE MAE ρ MAE τ PBC AMH PBC FGM PBC Frank PBC Gumbel PBC Joe AMH: Ali-Mikhail-Haq FGM: Farlie-Gumbel-Morgenstern 43 / 51

44 FDG: how behaves WLS estimator on simulated data? 200 datasets of dimension d = 4, 50 and size n = 500 from FDG-CA, FDG-F, FDG-S, FDG-E Estimation by WLS estimator with Spearman s rho For each dataset and each model: MAE r = 1 r i,j r(ˆθ i, ˆθ j ) and RMAE = 1 p d i<j d i=1 ˆθ i θ 0i θ 0i, (averaged over the replications) 44 / 51

45 Performs well, even on high-dimensional data MAE r RMAE Copula d = 4 d = 50 d = 4 d = 50 FDG-CA FDG-F FDG-sinus FDG-exponential / 51

46 Hydrological application Assessment of flood critical levels 46 / 51

47 Critical levels Extreme event: {F 1 (X 1 ) > q,..., F d (X d ) > q} Return perdiod T and critical level q: T = 1 1 M(q), where M(q) = P(min(F 1(X 1 ),..., F d (X d )) q) A return period of T = 30 years and a critical level of q = 70% means that each X i exceeds its quantile of order 70% once every 30 years in average. 47 / 51

48 Estimated critical levels 48 / 51

49 Conclusion PBC Liebscher leads to PBC full likelihood of PBC can be computed via message-passing PBC (and, therefore, Liebscher?) are of limited use in practice FDG tractable, flexible, tail asymmetric theoretically well grounded estimation Estimation: derivation of asymptotic properties without assuming partial derivatives 49 / 51

50 Achievements Submitted papers: FDG copulas: A flexible and tractable class of one-factor copulas, WLS estimator: Weighted least-squares inference based on dependence coefficients for multivariate copulas, PBC: A class of multivariate copulas based on products of bivariate copulas, Softwares: PBC: Product of Bivariate Copulas (PBC), FDGcopulas: dealing with FDG copulas, out soon 50 / 51

51 Perspectives Do extreme-value copulas associated to FDG have the one-factor property? Derivation of EM algorithm for one-factor copulas Mixing different types of dependence coefficients in WLS estimator Extension of one-factor structure to spatial data (interpolation) Derivation of explicit / simple copula functionals (critical levels, value-at-risk, etc) from FDG copulas 51 / 51