Bayesian nonparametrics for multivariate extremes including censored data Anne Sabourin PhD advisors: Anne-Laure Fougères (Lyon 1), Philippe Naveau (LSCE, Saclay). Joint work with Benjamin Renard, IRSTEA, France. EVT 2013, Vimeiro September 10, 2013 1
Censored Multivariate extremes: Why bother? ex: Daily streamow at 4 neighbouring sites (Gardons, Cévennes, France). Return levels for jointly extreme events? Recent, `clean' series very short Censored historical data availble from archives, variable `perception thresholds' for oods. Earliest: 1604. How to use all dierent kinds of data? 2
Censored data: pairwise plots Univariate series: Bivariate: Mialet 0 500 1000 1500 2000 2500 Anduze 0 1000 2000 3000 4000 5000 6000 Ales 0 500 1000 1500 2000 2500 0 1000 2000 3000 St Jean 0 1000 2000 3000 St Jean 0 1000 2000 3000 St Jean 3
Purposes Take into account as many data as possible Censored likelihood, integration problems Estimate marginal GPD parameters + dependence structure jointly additional Gibbs step in a MCMC algo. Probability of failure regions (return periods for jointly extreme events) Model for excesses (POT-Poisson). 4
Approach: use Dirichlet consistency properties Angular measure model on the simplex. Dirichlet distributions: Nice marginalization and conditioning properties. Dirichlet mixture model for angular measures Boldi & Davison, 2007 ; Sabourin & Naveau, 2013 : Flexible, non parametric. Reversible jumps - MCMC algorithm available. Adaptation needed: censored likelihood and variable threshold Data augmentation 5
0 1 5 10 1st component Excess Data: polar decomposition Vectorial obs Y t = (Y 1,t,..., Y d,t ); i.i.d. Margin Y j,t F j : Pareto above threshold v j. Dependence dened among the X j,t = 1/log F j,t (Y j, t): standard Fréchet Polar coordinates: R = X 1 = j X j : `radius' (norm), W = X R : `angle' X 2nd component 0 1 5 10 w B x S d W R B W simplex S d = {(w 1,... w d ) : w j = 1, w j 0}. 6
Angular measure models Joint distribution of excesses: product measure in polar coordinates. P(R > r, W B) 1 H(B) r H: angular measure, distribution of angles. Non parametric: Valid i w S i d dh(w) = 1 (1 i d) d Inference: Empirically, Einmahl et.al., 2001, Einmahl, Segers, 2009, Guillotte et al, 2011. : No explicit expression of asymptotic variance, Bayesian inference with d = 2 only. Restriction to parametric subclass Gumbel, Coles&Tawn, Cooley et.al.... : logistic family, Pairwise-Beta... Preferred here: Dirichlet Mixture model: Compromise Flexibility (weakly dense family) Uncertainty quantication through parameters, Bayesian implementation. 7
0.00 0.35 0.71 1.06 1.41 0.00 0.35 0.71 1.06 1.41 Dirichlet mixture model for angular measure Dirichlet distribution: generalization of Beta to higher dimensions. 1 location parameter (point on the simplex): `center' + 1 concentration parameter. w2 w2 w3 w1 w3 w1 Dirichlet mixture model, k components: h(w) = k m=1 p m diri m (w). Boldi, Davison, 2007 H valid center of mass of the location parameters = center of the simplex. 8
Issues: moments constraints and censored data (i) Sampling the posterior distribution with MCMC methods. Constraints Sampling issues Re-parametrization: No more constraint, sampling is manageable for d = 5: Sabourin, Naveau, 2013 (ii) Inference with censored data: Variable threshold: changing normalizing constants. Censored data = segments or boxes. Integrating the likelihood (density dr r 2 dh(w)) on rectangles? Sabourin, under review ; Sabourin, Renard, in preparation 9
Issues: moments constraints and censored data (i) Sampling the posterior distribution with MCMC methods. Constraints Sampling issues Re-parametrization: No more constraint, sampling is manageable for d = 5: Sabourin, Naveau, 2013 (ii) Inference with censored data: Variable threshold: changing normalizing constants. Censored data = segments or boxes. Integrating the likelihood (density dr r 2 dh(w)) on rectangles? Sabourin, under review ; Sabourin, Renard, in preparation 9
Poisson model for variable threshold {( t, X ) } t, 1 t n Poisson Process (Leb λ) n n on [0, 1] A u,n λ: ` exponent measure', with Dirichlet Mixture angular component dλ dr dw (r, w) = d r 2 h(w). `A i ' data overlapping threshold: included in Poisson likelihood as { } N ( t 2 t 1 ) 1 n n n A i = 0 10
`Censored' likelihood: model density integrated over boxes Ledford & Tawn, 1996: GEV parametric model, partially extreme data censored at threshold. Here: more general censoring framework, Poisson likelihood dλ, non parametric, no explicit expression. dx Two terms without closed form: Censored regions A i overlapping threshold: exp { (t 2 t 1 )λ(a i )} Classical censoring above threshold censored region dλ dx. 11
Data augmentation One more Gibbs step, no more numerical integration. Objective: sample [θ Obs]. (Parameter space: Θ) Additional variables (replace missing data component): Z Full conditionals [Z i Z j j, θ, Obs] explicit (Thanks Dirichlet): Gibbs sampling. Sample [z, θ Obs] + (augmented distribution) on Θ Z. 12
Censored regions above threshold Censored region dλ dx dx j1:j r : Generate missing components under univariate conditional distributions Z j 1:r [X missing X obs, θ] x2 Augmentation data Z j = [X censored X observed, θ] Censored interval u 2 /n Extremal region u 1 /n x1 Dirichlet Explicit univariate conditionals Exact sampling of censored data on censored interval 13
Censored regions overlapping threshold e (t 2,i t1,i )λ(a i ) augmentation Poisson process N i on E i A i. + Functional ϕ(n i ) X2 Augmentation Z i = PP(τ.λ) on E i φ(#{points in Ai}) U' 1 E i A i u 2 /n Censored region u 1 /n U' 2 X1 [z, θ Obs] }{{}... [N i ]ϕ(n i ) density terms, prior, augmented missing components 14
h Results on simulated data Angular measure: Dirichlet, d = 4, k = 3 mixture components Censoring: same pattern as real data. S3 S4 0 1000 2000 3000 4000 0 1 2 3 4 5 discharge : 0 3350 6700 10051 13401 dependent independent true 0 1000 2000 3000 4000 5000 6000 S3 0.0 0.2 0.4 0.6 0.8 1.0 X3/( X3 + X4 ) 0 1 2 3 return period (years, log scale) Pair (S3, S4): data, predictive a posteriori of the angular measure, quantile curve for S 3. 15
Conclusion Accounting for all types of censored data: manageable using Dirichlet consistency properties (marignalization, conditioning) High dimension (GCM grid, spatial elds)? Impose a reasonable structure (sparse) on Dirichlet parameters Dirichlet Process? Challenges : Discrete random measure continuous framework for GPP's. 16
Bibliographie I M.-O. Boldi and A. C. Davison. A mixture model for multivariate extremes. JRSS: Series B (Statistical Methodology), 69(2):217229, 2007. Sabourin, A., Naveau, P. Bayesian Dirichlet mixture model for multivariate extremes: a re-parametrization. CSDA, 2013. Ledford, A. and Tawn, J. (1996). Statistics for near independence in multivariate extreme values. Biometrika, 83(1):169187. Schnedler, W. (2005). Likelihood estimation for censored random vectors. Econometric Reviews, 24(2):195217. Tanner, M. and Wong, W. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82(398):528540. Van Dyk, D. and Meng, X. (2001). The art of data augmentation. Journal of Computational and Graphical Statistics, 10(1):150. Gómez, G., Calle, M. L., and Oller, R. Frequentist and bayesian approaches for interval-censored data. Statistical Papers, 45(2):139173, 2004. 17
Bibliographie II Hosking, J.R.M. and Wallis, J.R.. Regional frequency analysis: an approach based on L-moments Cambridge University Press, 2005. Neppel, L., Renard, B., Lang, M., Ayral, P., Coeur, D., Gaume, E., Jacob, N., Payrastre, O., Pobanz, K., and Vinet, F. (2010). Flood frequency analysis using historical data: accounting for random and systematic errors. Hydrological Sciences JournalJournal des Sciences Hydrologiques, 55(2):192208. 18
Details: Augmentation Poisson process e (t 2,i t1,i )λ(a i )?? Proposal Z i PP( λ i ) ; dene f i s.t. { E e } j f i (Z i j ) = Lapl Z i (f i ) = e 1 e f d λ i = e n i λ(a i ) [Z, θ O] + [θ] j {[Z j1:r O, θ] } i { [Z i θ]e j f i (Z i j ) }, with [Z i θ]e j f i (Z i j ) dz i = e n i λ(a i ). 19