Semi-parametric estimation of non-stationary Pickands functions

Semi-parametric estimation of non-stationary Pickands functions Linda Mhalla 1 Joint work with: Valérie Chavez-Demoulin 2 and Philippe Naveau 3 1 Geneva School of Economics and Management, University of Geneva, Switzerland 2 Faculty of Business and Economics, University of Lausanne, Switzerland 3 Laboratoire des Sciences du Climat et l'environnement, France NCAR, Boulder 2016 Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 1 / 31

Motivation Modelling multivariate extremes: univariate marginal distributions dependence in extremes (extremal dependence) One should be able to include covariates effects in the model: climate change, seasonality,... univariate case: regression methods have been proposed to model the parameters of extreme values distributions how can we relate the extremal dependence to a set of covariates? Aim: provide a very flexible estimation of a covariate-dependent extremal dependence. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 2 / 31

Overview 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 3 / 31

Multivariate extremes framework 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 4 / 31

Multivariate extremes framework Multivariate Extreme Value Theory Assume Z = (Z 1,..., Z d ) T has a multivariate max-stable distribution with unit-fréchet margins. Then { ( ) } ωi pr(z z) = exp { V (z)} = exp dh(ω 1,..., ω d ), S d 1 d max i=1 where S d 1 = { (ω 1,..., ω d ) R d + : ω 1 +... + ω d = 1 } and H is a finite measure that satisfies S d 1 ω i dh(ω 1,..., ω d ) = 1 for each i. Exponent function V (z) = z i ( 1 + + 1 ) A(ω), z 1 z d where ω = (z 1 / z,..., z d / z ) T S d 1. A(ω) is called the Pickands dependence function. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 5 / 31

Multivariate extremes framework Pickands Dependence function Necessary conditions on A( ) C1) A(ω) is continuous and convex C2) For any ω S d 1 max (ω 1,..., ω d ) A(ω) 1 C3) A(0,..., 0) = 1 and A(e i ) = 1 with e i = (0,..., 0, 1, 0,..., 0), for i = 1,..., d 1 A(ω) 0.5 0.7 0.9 0.0 0.4 0.8 ω Many regularised non-parametric estimators have been proposed. But what if A(ω) varies with a set of covariates X? Benefit from the well-established framework of Generalized Additive Models. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 6 / 31

Multivariate extremes framework Generalized Additive Models 1 A model for Y conditional on X R q E (Y X = x) = g { u β + } g a link function K h k (t k ) (u 1,..., u p ) and (t 1,..., t K ) subsets of (x 1,..., x q ) k=1 h k C 2 (A k ) with A k R admits β R p a vector of parameters h k are smooth functions a finite (m k -)dim basis parametrized by h k = (h 1,, h mk ) R m k a quadratic penalty representation A k h k (t)2 dt = h T k S kh k 1 Textbooks: [Hastie and Tibshirani, 1990, Green and Silverman, 1993] Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 7 / 31

Estimation of non-stationary Pickands functions 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 8 / 31

Estimation of non-stationary Pickands functions GAM for max-stable dependence structures GAM for max-stable dependence structures Let Z(x) = (Z 1 (x),..., Z d (x)) T denote a d-dimensional max-stable process with margins transformed using a regression model to be unit Fréchet, i.e., F (z) = exp( 1/z) a covariate-varying extremal dependence, i.e., A(ω x) Define the max-projection of Z(x) as [ ] Y (ω) x = max F λ 1 {Z 1 (x)},..., F λ d {Z d (x)} with λ i = ω i ( 1 ω 1 + + 1 ω d ) 1 and ω = (ω1,..., ω d ) T S d 1 with non-zero entries. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 9 / 31

Estimation of non-stationary Pickands functions GAM for max-stable dependence structures PROPOSITION The random variable Y (ω) x is Beta distributed with parameter Aω(x), i.e., pr(y (ω) y x) = y A ω(x), for 0 y 1. In particular, E [Y (ω) x] = g 1 {Aω(x)} where the concave link function equals g(u) = u 1 u with 0 < u < 1. The Pickands function Aω(x) is the transformed mean of Y (ω) x and can be related to a set of predictors using GAMs. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 10 / 31

Estimation of non-stationary Pickands functions GAM for max-stable dependence structures A non-stationary extremal dependence The generalized additive model for Aω(x) A ω (x; θ) = u β + K h k (t k ) k=1 (u 1,..., u p ) and (t 1,..., t K ) subsets of {x 1,..., x q } β R p a vector of parameters h k C 2 parametrized by h k = (h 1,..., h mk ) R m k admits a quadratic penalty representation θ = (β, h 1,..., h K ) Θ R d with d = p + K k=1 m k Estimation procedure: Such models are fitted by penalized likelihood maximization. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 11 / 31

Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Penalized log-likelihood The penalized log-likelihood estimator for Aω(x) We define the penalized log-likelihood as l p (θ, γ; ω) = lω(θ) 1 2 K γ k h k (t k ) 2 dt k = lω(θ) 1 K γ k h k S k h k A k 2 k=1 and the penalized maximum log-likelihood estimator as θ n := argmax θ Θ l p (θ, γ; ω). k=1 γ = (γ 1,..., γ K ) R K + Observations at hand {y i (ω), x i } n i=1, then n lω(θ) = [{Aω(x i ; θ) 1} log y i (ω) + log Aω(x i ; θ)] i=1 Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 12 / 31

Estimation of non-stationary Pickands functions Corrected Pickands dependence function estimator Corrected Pickands function estimator Up to this point, the estimation procedure is valid for any d 2 and results in a covariate-varying Pickands function estimator Ã(ω x) := Aω(x; θ n ) 0.6 0.7 0.8 0.9 1.0 1.1 Ã(ω x) ω = 0.98 ω = 0.8 ω = 0.6 ω = 0.4 ω = 0.2 ω = 0.01 x 0 20 40 60 80 100 Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 13 / 31

Estimation of non-stationary Pickands functions Corrected Pickands dependence function estimator Corrected Pickands function estimator Up to this point, the estimation procedure is valid for any d 2 and results in a covariate-varying Pickands function estimator Ã(ω x) := Aω(x; θ n ) 0.6 0.7 0.8 0.9 1.0 1.1 Ã(ω x) ω = 0.98 ω = 0.8 ω = 0.6 ω = 0.4 ω = 0.2 ω = 0.01 x 0 20 40 60 80 100 But, Ã( x) is not a valid Pickands function! Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 13 / 31

Estimation of non-stationary Pickands functions Corrected Pickands dependence function estimator Corrected Pickands function estimator Regularization procedure when d = 2: 1 compute à {(ω i, 1 ω i ) x} = Ã(ω i x), for ω 1,..., ωñ in [0, 1] 2 apply the constrained median smoothing approach (cobs) 2 ĝ γ,l (ω x) = min g ñ i=1 Ã(ω i x) g(ω i x) + γ max g (ω x), ω with conditions C1-C2-C3 on the smoothing spline g(ω x) 3 define Â(ω x) := ĝ γ,l (ω x) Confidence intervals: Use the resampling cases bootstrap 3 : only assumes independence. Remark: Confidence intervals are not guaranteed to be convex. 2 [Ng and Maechler, 2007] 3 [Davison and Hinkley, 1997, Section 6.2.4] Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 14 / 31

Simulation Study 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 15 / 31

Simulation Study No covariate dependence No covariate dependence The aim is to evaluate the impact of the max-projection step along with the constrained smoothing procedure on the estimation of the extremal dependence. The first stage of the estimation procedure boils down to maximising the log-likelihood l (A ω ) = n (A ω 1) log {y i (ω)} + log A ω. i=1 The constrained median smoothing approach is applied to the resulting MLEs for different values of ω [0, 1]. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 16 / 31

Simulation Study No covariate dependence True dependence model: Dirichlet dependence model with β = 3 and different values of α n = 1000 observations ñ = 800 regularly spaced values of ω in [0, 1] Asymptotic 95% confidence intervals α = 0.02 α = 0.2 α = 10 A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 True MLE 95% CI Pickands CFG A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 True MLE 95% CI Pickands CFG A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 True MLE 95% CI Pickands CFG 0.0 0.2 0.4 0.6 0.8 1.0 ω 0.0 0.2 0.4 0.6 0.8 1.0 ω 0.0 0.2 0.4 0.6 0.8 1.0 ω Our estimator is competitive with the [Pickands, 1981] and the [Capéraà et al., 1997] estimators. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 17 / 31

Simulation Study Covariate-dependent Pickands function Logistic dependence with two covariates True dependence model: bivariate logistic dependence model with a covariate-dependent parameter α(t, x) = 1 {x= cata } α 0 sin(α 1 t) + 1 {x= catb } { β0 sin(β 1 t 2 ) + β 3 }, where t is a continuous covariate discretized into 12 equally spaced values, and x a factor covariate with two levels R = 1000 bootstrap samples n = 400 observations for each distinct pair of covariates ñ = 800 regularly spaced values of ω in [0, 1] Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 18 / 31

Simulation Study Covariate-dependent Pickands function A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 1.57 0.0 0.2 0.4 0.6 0.8 1.0 ω A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 3.16 0.0 0.2 0.4 0.6 0.8 1.0 ω A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 3.48 0.0 0.2 0.4 0.6 0.8 1.0 ω A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 2.53 0.0 0.2 0.4 0.6 0.8 1.0 ω A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 1.25 0.0 0.2 0.4 0.6 0.8 1.0 ω A(ω) 0.5 0.6 0.7 0.8 0.9 1.0 t 0.3 0.0 0.2 0.4 0.6 0.8 1.0 ω Figure: Estimates of the Pickands function for the logistic model with weak, mild and strong dependence for cata (top) and catb (bottom) and their 95% bootstrap confidence intervals. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 19 / 31

Application 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 20 / 31

Application Application Data description The data represents daily minimum and maximum temperature for long term stations from the U.S. COOP station network 4, in degrees Celcius. The temperatures were extracted for stations from 46 states and from 1950-01-01 to 2004-12-31. Aim: We are interested in the extremal dependence between daily maxima and daily minima temperatures, i.e., between hot days and warm nights. More specifically, how did this extremal dependence evolve through time? 4 [Climate and Global Dynamics Division, 2010] Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 21 / 31

Application Data analysis: Colorado We use daily maximum and minimum temperatures over all the 151 stations in Colorado. We model marginally monthly maxima of each time series. Temperature ( C) 15 20 25 30 35 40 45 Temperature ( C) 25 15 5 0 5 1950 1960 1970 1980 1990 2000 Time (years) 1950 1960 1970 1980 1990 2000 Time (years) Figure: Monthly maxima of daily maximum temperatures (top) and daily minimum temperatures (bottom). Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 22 / 31

Application Deal with non-stationarity marginally µ max(t, m) = σ max(t, m) = 12 j=1 12 j=1 ξ max(t, m) = ξ maxt (µ 0,j + µ 1,j t) 1 {m=j} σ 0,j 1 {m=j} µ min (t, m) = σ min (t, m) = 12 j=1 12 j=1 ξ min (t, m) = ξ min (µ 0,j + µ 1,j t) 1 {m=j} σ 0,j 1 {m=j} where t and m are the year and month of the observed extreme temperature, respectively. Transform the data marginally to a stationary common scale (standardized Gumbel, unit Fréchet, etc). Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 23 / 31

Application Modelling time-varying Pickands dependence function 1.00 Pickands function 1.0 0.95 0.9 0.90 0.8 0.7 0.85 0.6 0.5 0.0 0.2 0.4 ω 0.6 2000 1990 1980 ) 1970 ars 0.8 ye 1960 e( tim 1.0 1950 0.80 0.75 Figure: Time-dependent Pickands dependence function. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 24 / 31

Application Modelling time-varying Pickands dependence function 1.00 Pickands function 1.0 0.95 0.9 0.90 0.8 0.7 0.85 0.6 0.5 0.0 0.2 0.4 ω 0.6 2000 1990 1980 ) 1970 ars 0.8 ye 1960 e( tim 1.0 1950 0.80 0.75 What happened in 1970? [Meehl et al., 2009]: Since 1970, less extreme low temperatures than expected (warmer nights). Figure: Time-dependent Pickands dependence function. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 24 / 31

Application Covariate-dependent extremal coefficient estimates Extremal coefficient 1.0 1.2 1.4 1.6 1.8 2.0 Colorado 1950 1970 1990 Time (years) Extremal coefficient 1.0 1.2 1.4 1.6 1.8 2.0 California 1950 1970 1990 Time (years) Extremal coefficient 1.0 1.2 1.4 1.6 1.8 2.0 Alabama 1950 1970 1990 Time (years) Figure: Extremal coefficient estimates (smoothed) with 95% bootstrap confidence intervals (R=300). Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 25 / 31

Application Question: How does the extremal dependence evolve through time and altitude? Extremal coefficient 2.0 2.0 1.9 1.9 1.8 1.7 1.8 1.6 1.5 1.7 1950 1960 1970 4000 1.6 1980 3000 Time (years) 1990 2000 500 1000 2000 Altitude (m) 1.5 200 Figure: Time and altitude dependent extremal coefficient. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 26 / 31

Conclusion 1 Multivariate extremes framework 2 Estimation of non-stationary Pickands functions GAM for max-stable dependence structures Corrected Pickands dependence function estimator 3 Simulation Study No covariate dependence Covariate-dependent Pickands function 4 Application 5 Conclusion 6 References Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 27 / 31

Conclusion Conclusion Very flexible framework for the estimation of a valid non-stationary Pickands dependence function in the bivariate case. The estimation procedure is valid in both the stationary and non-stationary cases. Extensions/limitations: Extension of the regularisation procedure to d > 2. Depending on how fine the discretization of the unit simplex is, computational times can be substantial! Application: What other covariates can we include in our model? mean temperatures? ideas? Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 28 / 31

References References Capéraà, P., Fougères, A.-L., and Genest, C. (1997). Thanks for your attention! A nonparametric estimation procedure for bivariate extreme value copulas. BIOMETRIKA, 84:567 577. Climate and Global Dynamics Division, National Center for Atmospheric Research, U. C. f. A. R. (2010). Daily minimum and maximum temperature and precipitation for long term stations from the u.s. coop data. accessed 7 april 2016. Davison, A. and Hinkley, D. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press. Green, P. and Silverman, B. (1993). Nonparametric Regression and Generalized Linear Models: A roughness penalty approach. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Taylor & Francis. Hastie, T. J. and Tibshirani, R. J. (1990). Generalized additive models. London: Chapman & Hall. Meehl, G. A., Tebaldi, C., Walton, G., Easterling, D., and McDaniel, L. (2009). Relative increase of record high maximum temperatures compared to record low minimum temperatures in the u.s. Geophysical Research Letters, 36. Ng, P. and Maechler, M. (2007). A fast and efficient implementation of qualitatively constrained quantile smoothing splines. Statistical Modelling, 7:315 328. Pickands, J. (1981). Multivariate extreme value distributions. volume 49, pages 859 878, 894 902. Proceedings of the 43rd Session of the International Statistical Institute, Vol.2 (Buenos Aires, 1981). Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 29 / 31

References Marginal QQ-plots Empirical quantiles 2 0 2 4 6 2 0 2 4 6 Theoretical quantiles Empirical quantiles 2 0 2 4 6 2 0 2 4 6 Theoretical quantiles Figure: Gumbel marginal QQ-plots for monthly maxima of daily maximum (left) and daily minimum (right) temperatures. Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 30 / 31

References [Meehl et al., 2009] performed a study on the same dataset where the evolution through time of the ratio of annual numbers of record high maximum temperature and record low minimum temperature was of interest. Up to 1970, almost no warming. Since 1970, less extreme low temperatures than expected (warmer nights). Linda Mhalla (UNIGE) Non-stationary Pickands functions Boulder CO, 2016 31 / 31