1/25 The extremal elliptical model: Theoretical properties and statistical inference Thomas OPITZ Supervisors: Jean-Noel Bacro, Pierre Ribereau Institute of Mathematics and Modeling in Montpellier (I3M) University of Montpellier 2 23/04/2013
2/25 An extreme value framework asymptotic dependence in elliptically contoured distributions Asymptotic dependence random variables X, Y are asymptotically dependent if λ(x, Y ) = lim u 1 pr(f X (X ) u F Y (Y ) u) > 0 spatial : extremogram λ ( extreme value correlogram ) λ(s 1, s 2 ) = λ(x (s 1 ), X (s 2 )) 0
3/25 Plan Multivariate regular variation and ellipticity Results from the literature Density of the MRV limit measure The max-stable limit process Construction Elliptical domain of attraction Limits for threshold exceedances Conditional multivariate constructions Spatial extensions Inference Likelihood-based In perspective : Quantile-based Application to daily maxima of wind speeds Conclusion
4/25 Multivariate regular variation and ellipticity Elliptically contoured distributions stochastic polar representation X = RAU + M random radius R 0 covariance matrix Σ = AA T U uniform on the Euclidean unit sphere, independent of R median vector M spatial : elliptical random field
5/25 Multivariate regular variation and ellipticity Gaussian-based examples Gaussian W : asymptotic independence However, triangular arrays with non-degenerate limit lim log(n) ( 1 1 T ) Σ n n lead to asymptotic dependence : Huesler-Reiss (multivariate), Brown-Resnick (spatial) Y W with power-tailed Y : asymptotic dependence, e.g. student s t with (df/y ) 2 Gamma(df/2, 2) in the spatial domain : t process
6/25 Multivariate regular variation and ellipticity Multivariate regular variation on R d Definition with pseudo-polar coordinates pr( X tr, X/ X ) pr( X t) = r α Q( ) for t angular measure Q on the unit sphere index of regular variation α > 0 Equivalent definition by vague convergence on R d \ {0} Radon measure η with η(r d \ {0}) > 0 and η(tb) = t α η(b) vague convergence : for some a(t), t pr(x/a(t) ) = η( ) for t
7/25 Multivariate regular variation and ellipticity [Hult and Lindskog, 2002] ; [Hashorva, 2006] For X = RAU + M, asymptotic dependence multivariate regular variation R is univariate regularly varying X j is univariate regularly varying for all j X j is univariate regularly varying for some j Indeed, for M = 0, with the Mahalanobis norm X = X T Σ 1 X : X = R, X/ X = AU and Q( ) = P(AU ) η is elliptically contoured, characterized by α and Σ
8/25 Multivariate regular variation and ellipticity Exhaustiveness of the t extreme value dependence For t random elements, we observe α = df all asymptotically dependent limits are limits for a t random element extremal elliptical = extremal-t
9/25 Multivariate regular variation and ellipticity Density of the MRV limit measure Density representation of η (cf. [Opitz, 2013b]) if t pr(r/a(t) x) x α, then the density of η is η(dx) = c α (x T Σ 1 x) 0.5(α+d) with c α = απ 0.5(d 1) Σ 0.5 Γ(0.5(α + 1)) 1 Γ(0.5(α + d)) Sketch of the proof : pseudo-polar coordinates density r α change to Euclidean coordinates the Jacobian is the same as in the Gaussian case conditional density is of the elliptical t type (see Appendix of [Dombry et al., 2013] for calculation)
10/25 Multivariate regular variation and ellipticity Density of the MRV limit measure Standardized marginal scale Often we use a standardized version η, η (B) = η(cb α ) with cb α = {c sign(x) abs(x) α x B} such that η ({x x j x 0 }) = η ({x x j x 0 }) = x 1 0 for x 0 > 0. original, corr. = 0.5 standardized (α = 0.5) standardized (α = 2)
The max-stable limit process Construction Max-stability Definition Z is max-stable if a n > 0, b n such that for iid copies Z 1, Z 2,... of Z. max i=1,...,n a n 1 (Z i b n ) d = Z 1 max. domain of attraction limit for componentwise maxima model blockwise maxima or threshold exceedances spatial : max-stable process Spectral constructions {Z (s)} = { max i=1,2,... V ix i (s)} with V i PRM(v 2 dv) and EX 1 (s) + = 1 pr(z (s) z) = exp( 1/z) for z > 0 (unit Fréchet) construction and interpretation of parametric models 11/25 simulation and conditional simulation
12/25 The max-stable limit process Construction The extremal elliptical max-stable process arises as the max-stable limit e.g. of random t fields characterized by correlation function and index α > 0 finite-dimensional distributions G(z) = exp( η([, z] c )) with α-fréchet marginal distributions η ( (, x] C ) is given in terms of mv. t probabilities ([Nikoloulopoulos et al., 2009]) Spectral construction ([Opitz, 2013a]) {V i } a Poisson process PRM(αv (α+1) ) on (0, ) W i iid standard Gaussian with the targeted correlation function ( ) α } {Z (s)} = { c α max V iw i (s) i=1,2,...
13/25 The max-stable limit process Construction Sketch of the proof W.l.o.g. V 1 = max i V i. V 1 W 1 MDA(Z ) (cf. Lemma 3.1 in [Segers, 2012]) finite-dimensional distributions of V 1 W 1 : elliptical with regularly varying R when Cov(s 1, s 2 ) = 0, the extremogram takes values in ]0, 0.5[ (depending on α) limit 0.5 for α 0 limit 0 for α extremal Gaussian process ([Schlather, 2002]) is special case (α = 1) conditional sampling via the framework of [Dombry et al., 2013]
14/25 The max-stable limit process Elliptical domain of attraction Elliptical domain of attraction ([Opitz, 2013a]) X = {X (s), s S} a random process with finite-dimensional elliptical distributions correlation < 1 when s j1 s j2 Two equivalent conditions : At least one of the finite-dimensional distributions for d 2 is in a multivariate MDA with asymptotic dependence. At least one of the univariate marginal distributions of X is regularly varying. Then a max-stable limit process Z exists for X with the extremal elliptical dependence structure. Sketch of the proof. Apply iteratively bv./mv. results.
Limits for threshold exceedances Conditional multivariate constructions Distributions for multivariate simple exceedances threshold vector u > 0 at least one component x j exceeds u j, i.e. max j x j /u j 1 Negative values allowed 0-truncated (mv. Pareto) Y := (X [max X j /u j 1]) H H(dx) = η([, u] c ) 1 η(dx) Y 0 := (X + [max X j /u j 1]) H 0 marginal transformations for mv. generalized Pareto, similar [Rootzén and Tajvidi, 2006] 5/25
16/25 Limits for threshold exceedances Conditional multivariate constructions Sampling from H and H 0 from H ( with negative values) : For u = 1 and a correlation matrix A (A ) T = Σ, R Pareto(α) and X = RA U, we obtain Y = [ ( )] X max X j 1 H. j Conditional distributions H(x 2 x 1 ) are of the elliptical t type (when max(x 1 ) 1). from H 0 (0-truncated) : replace X by X + in Y.
17/25 Limits for threshold exceedances Spatial extensions Spatial extensions for simple exceedance modeling with negative values : use limit measure η from regular variation of processes (see e.g. [Hult and Lindskog, 2005]) 0-truncated : Pareto processes [Ferreira and de Haan, 2012] : for simple exceedances [Dombry and Ribatet, 2013] : for exceedances of a homogeneous cost functional l, applied on the standard scale Cannot construct the processes, but e.g. the finite-dimensional simple exceedance distributions.
18/25 Inference Inference Objective : model extremes of multivariate financial data see also [Klüppelberg et al., 2008], [Li and Peng, 2009] and [Krajina, 2012] who estimate the dependence structure from all data spatial extremes in environmental data use the standardized scale to estimate the dependence structure Marginal standardization of data Need unit Fréchet margins for standard max-stable process unit Pareto tails for standard exceedance process, e.g. π t1 or 4 Unif( 1, 1) Par(1) (0-truncated for Pareto models)
19/25 Inference Likelihood-based inference for the simple exceedance approach with negative values pairwise likelihood full likelihood (if dimension not too large) composite likelihood based on conditional t density if X i (s j ) = x > u, add density term for X i (s j ) (X i (s j ) = x) for other approaches (Pareto, max-stable) pairwise likelihood Consistency and asymptotic normality clear for iid replications but : extremal elliptical process is not mixing if spatio-temporal modeling, need temporal mixing
20/25 Inference In perspective : Quantile-based Perspective : Quantile-based inference Conditional distributions in the simple exceedance framework On the standard scale with η, Ys def 1 s 2 = (y ) 1 [Y (s 1 ) Y (s 2 ) = y ] tα+1 α independent of y, with median Corr(s 1, s 2 ) in t α+1. Idea : if we knew α, could estimate the empirical correlogram difficulty : a priori estimation of α use location-scale invariant quantile expression for estimation
Inference In perspective : Quantile-based Example use C s1 s 2 (α) = q 0.875 q 0.625 + q 0.375 q 0.125 q 0.75 q 0.25 with q p = q s1 s 2 (p) 1/α and q from the standard t α+1 distribution minimize to obtain ˆα. log C s 1 s 2;emp(α) min s 1,s 2 C s1 s 2;theor(α) Consistency : Is the minimum (asymptotically) unique? yes! β 0 2 4 6 8 10 12 5 2 0.4 0.8 0.8 2 5 21/25 0 2 4 6 8 10 12 α
22/25 Application to daily maxima of wind speeds Application : Daily maxima of mean hourly wind speeds (cf. [Engelke et al., 2012] for a similar analysis) 24 inland stations in the Netherlands n = 2640 days marginal standardization to π t 1 negative standard values values below the median use simple exceedances in the regular variation limit process threshold u = n/40 composite likelihood with conditional t-densities here : stable covariance function, geometric anisotropy
23/25 Application to daily maxima of wind speeds Goodness-of-fit Conditional t-distributions are elliptical extract fitted radii R from RAU + M check their distribution QQ-plot : obs. 5 0 5 10 15 0 5 10 15 20 theor.
24/25 Application to daily maxima of wind speeds Bivariate fit A realisation (km/h) extremogram 0.0 0.2 0.4 0.6 0.8 1.0 50 40 30 20 10 0 100 200 300 400 distance
25/25 Conclusion Conclusion flexibility choice of covariance structure choice of estimation approach Huesler-Reiss / Brown-Resnick are limiting cases for α interpretability ellipticity on the α-pareto scale conditional t distributions random set approaches in spatial modeling
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