Estimation of multivariate critical layers: Applications to rainfall data

Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 Estimation of multivariate critical layers: Applications to rainfall data Elena Di Bernardino, CNAM, Paris, Département IMATH International Conference on Risk Analysis ICRA 6/RISK 2015 Barcelona, May 26-29, 2015

Contents 1 2 Nonparametric estimation Parametric estimation 3 Geyser data: illustration 4 Estimation results 5 Estimation results

(Multivariate) Return Period The notion of Return Period (RP) is frequently used in environmental sciences for the identication of dangerous events, and provides a means for rational decision making and risk assessment. Roughly speaking, the RP can be considered as an analogue of the Value-at-Risk in Economics and Finance, since it is used to quantify and assess the risk. During the last years, researchers in environmental elds joined eorts to properly answer the following crucial question: How is it possible to calculate the critical design event(s) in the multivariate case?. In this sense, a possible consistent theoretical framework for the calculation of the design event(s) and the associated return period(s) in a multi-dimensional environment, is proposed, e.g., by Salvadori et al. (2011), Salvadori et al. (2012), Gräler et al. (2013) Multivariate return period using the notion of upper and lower level sets of multivariate probability distribution F and of the associated Kendall's measure.

Critical layers Consider the (nonnegative) real-valued random vector X = (x 1,..., X d ) such that X FX = C(F X1,..., F Xd ), with FX : R d + [0, 1]. Denition (Critical layer) The critical layer L(α) associated to the multivariate distribution function FX of level α (0, 1) is dened as L(α) = {x R d + : FX(x) = α}. Then L(α) is the iso-hyper-surface (with dimension d 1) where F equals the constant value α. The critical layer L(α) partitions R d into three non-overlapping and exhaustive regions: L < (α) = {x R d : FX(x) < α}, L(α) = the critical layer itself, L > (α) = {x R d : FX(x) > α}.

Multivariate RP and Critical layers Event of interest is of the type {X A}, where A is a non-empty Borel set in R d collecting all the values judged to be dangerous according to some suitable criterion. A natural choice for A is the set L > (α) Then RP > (α) = t, where P[X L > t > 0 is the (α)] (deterministic) average time elapsing between X k and X k+1, k IN. Then, the considered Return Period can be expressed using Kendall's function RP > (α) = t 1 1 K C (α), where K C (α) = P [ X L < (α) ] = P [C(U 1,..., U d ) α], for α (0, 1).

Goals and ideas This talk aims at: Giving a parametric representation of the multivariate distribution F of a random vector X, here representing rain measurements Giving direct estimation procedure for this representation Giving closed parametric expressions, both for critical layers L(α) and Return Periods RP > (α) adapting this methodology to some asymmetric dependencies (as, for instance, non-exchangeable random vectors (nested copula structure).

1 It can represent some kind of a priori belief on dependence structure of the data. Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 Proposed distributional model Nonparametric estimation Parametric estimation We consider the following model, F (x 1,..., x d ) = T C 0(T 1 F 1 1(x 1),..., T 1 d F d (x d )), or equivalently F (x 1,..., x d ) = C( F 1(x 1),..., F d (x d )), with C(u 1,..., u d ) = T C 0(T 1 (u 1),..., T 1 (u 1)) F i (x) = T T 1 i F i (x), for i I, where F 1,..., F d are given parametric initial marginal cumulative distribution functions, and where C 0 is a given initial Archimedean copula 1

2 Conditions on transformations such that C is a copula are discussed for example in Durante et al. (2010), Di Bernardino and Rullière (2013a), Di Bernardino and Rullière (2013b). Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 Archimedean copulas Nonparametric estimation Parametric estimation C φ (u 1,..., u d ) = φ(φ 1 (u 1) +... + φ 1 (u d )), where the function φ is called the generator of the Archimedean copula C φ. The function T : [0, 1] [0, 1] is a continuous and increasing function on the interval [0, 1], with T (0) = 0, T (1) = 1, with supplementary assumptions that will be chosen to guarantee that C is also a copula 2. Internal transformations T i : [0, 1] [0, 1] are continuous non-decreasing functions, such that T i (0) = 0, T i (1) = 1, for i I.

Diagonal section Nonparametric estimation Parametric estimation Firstly, we propose some (classical) estimators of the diagonal section δ 1 of a copula, δ 1(u) = C(u,..., u), u [0, 1]. For any u [0, 1], the empirical estimation for the diagonal of C and its inverse, i.e., { δemp (u) = Ĉ(u,..., u), 1 { δ emp (u) = arginf x [0, 1]; 1 } emp δ (x) u. 1 At a relative integer order k Z, the self-nested diagonals estimators are dened as δ k (u) = δ 1... δ 1(u), (k times), k IN δ k (u) = δ 1... δ 1(u), (k times), k IN δ 0(u) = u.

Nonparametric estimation Parametric estimation Denition (Non-parametric estimators of T and T i ) For a given arbitrary couple (x 0, y 0) (0, 1) 2, a non-parametric estimator of T is given by T (x) = δ r(x) (y 0), for all x (0, 1) with r(x) such that δ 0 r(x)(x 0) = x, where δ 0 r(x) refers to the self nested diagonal of the initial Archimedean copula C 0, i.e., ( ) δr(x)(x 0 0) = φ 0 d r(x) φ 1 (x 0 0) and r(x) = 1 ( φ 1 ln ln (x) ) 0. d φ 1 (x 0 0) In particular, if C 0 is the independence copula:, r(x) = 1 ln d ln ( i I, non-parametric estimators T i are T i (x) = F i F 1 i T (x). ln x ln x0 ). For any

A class of parametric transformation Nonparametric estimation Parametric estimation In practice, we will propose some parametric estimators for the transformations T and T i, by requiring that these transformations are passing through a nite set of points coming from the non-parametric estimators proposed above. Univariate distribution F F(x) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 x

A class of parametric transformation Nonparametric estimation Parametric estimation We take back from Bienvenüe and Rullière (2012): Denition (Conversion and transformation functions) Let f any bijective increasing function from R to R. It is said to be a conversion function. The transformation T f : [0, 1] [0, 1] is dened as 0 if u = 0, T f (u) = logit 1 (f (logit(u))) if 0 < u < 1, 1 if u = 1. Remark: transformations function are chosen in a way to be easily invertible (T f T g = T f g, T 1 f = T f 1). We will use (composited) hyperbolic conversion functions: ( x m h f (x) = H m,h,p1,p2,η(x) = m h + (ep 1 + e p 2 ) x m h 2 (e p 1 e p 2 ) with m, h, p1, p2 R, and one smoothing parameter η R. H 1 m,h,p1,p2,η (x) = H m, h, p1, p2,η(x). 2 ) 2 + e η p 1 +p 2 2

Nonparametric estimation Parametric estimation Denition (Smooth estimation of T and T i, i I ) Let Q and Q i, i I be given sets of quantile levels. Let η R and η i R, i I be given smoothing parameters. One denes { T = H θ,η, T i = H θi,η i, i I, One can dene the complete vector parameter Θ = ( θ1,..., θ d, θ, η 1,..., η d, η), with θ i = (m i, h i, p i 1, p i 2), for i 1,..., d

Geyser data: illustration The previous parametric model allows to get various analytical results for both the transformed multivariate distribution function; its associated critical layers; Kendall's function and multivariate return periods. Firstly we obtain the parametric expression for the transformed copula C: C(u 1,..., u d ) = φ( φ 1 (u 1) +... + φ 1 (u d )), where φ(t) = T (φ0(t)), where φ 0 is the generator associated to the initial copula C 0 and T is the smooth estimator of the external transformation. The corresponding estimated transformed multivariate distribution is given by: F Θ (x 1,..., x d ) = H θ,η C 0(H 1 θ1,η1 F 1(x 1),..., H 1 θ d,η d F d (x d )), where Θ is the complete estimated vector parameter.

Geyser data: illustration The associated transformed parametric α critical-layers are given by L Θ (α) = {(F 1 1 H θ1,η1 (u 1),..., F 1 d H θd,η (u d )), d (u 1,..., u d ) (0, 1) d, C 0(u 1,..., u d ) = H 1 θ,η (α)}. The estimated transformed Kendall distribution K C can be easily written as: K C (α) = α + d 1 i=1 1 ( ) i ( ) φ 1 (T 1 (i) ( ) (α)) 0 T φ0 φ 1 (T 1 (α)), 0 i! for α (0, 1), where φ(t) = T (φ 0(t)), and the notation f (i) corresponds to the i th derivatives of a function f. Then, the associated multivariate Return Period RP > (α) = t 1. 1 K C (α)

Geyser data: illustration Geyser data: transformed (parametric) bivariate density Non-parametric Parametric estimation (without optimization). Data : 272 eruptions of the Old Faithful geyser in Yellowstone National Park. Each observation consists of two measurements: the duration (in min) of the eruption (X ), and the waiting time (in min) before the next eruption (Y ). Figure : Level curves of transformed density f (x1, x 2 ) and Old Faithful geyser data (red points). Left: parameter setting η = 0.9, η1 = 9, η 2 = 7.5; (right) parameter setting η = 0.9, η 1 = 4, η 2 = 4.

Geyser data: illustration Geyser data: transformed (parametric) c.d.f. F and L(α) Figure : (Left) Transformed distribution F (x1, x 2 ) whit associated transformed level curves (red curves). (Right) Black points are plotted at empirical tail probabilities on the diagonal, calculated from the empirical bivariate distribution (in log scale). Red line is 1 F (x, x) (in log scale). The vertical dotted lines show estimate of 95% VaR (i.e., the univariate quantile in log scale). Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015

Geyser data: illustration Geyser data: transformed (parametric) marginal F 1 Univariate parametric fit F_1 Tail probabilities for F_1 Fn(x) 0.0 0.2 0.4 0.6 0.8 1.0 log(1 F_1) 10 8 6 4 2 0 2 3 4 5 x 0.0 0.5 1.0 1.5 2.0 2.5 log(x) Figure : Duration (in min) of the eruption data. (Left) F1 (red) and the empirical distribution function of eruption data (black). (Right) Black points are plotted at empirical tail probabilities calculated from empirical distribution function (in log scale). Red line is 1 F1 (x) (in log scale). The vertical dotted lines show estimate of 95% VaR (univariate quantile) for the eruption data.

Geyser data: illustration Geyser data: transformed (parametric) marginal F 2 Univariate parametric fit F_2 Tail probabilities for F_2 Fn(x) 0.0 0.2 0.4 0.6 0.8 1.0 log(1 F_2) 10 8 6 4 2 0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 x 0.0 0.5 1.0 1.5 2.0 2.5 log(x) Figure : Black points are plotted at empirical tail probabilities calculated from the empirical marginal distributions (in log scale). Red line is 1 F1 (x) (in log scale) (left) and 1 F2 (x) (in log scale) (right). The vertical dotted lines show estimate of 95% VaR (i.e., the univariate quantiles in log scale).

Estimation results 5 dimensional rainfall data-set (India and Sri-Lanka) Data-set has 797 lines giving monthly precipitation in decimetres, from 1893-01 to 2013-06. Excluded dates in this period are those for which at least one eld Precip (mm) was missing, zero or non-numerical.

Estimated vector of parameters Θ Estimation results We take as initial copula C 0 the independent one, and the initial margins F i (x) = 1 e x, i I. We obtain the complete estimated vector of parameters Θ. Parameters Θ m h ρ 1 ρ 2 η θ for external T -0.576 0.576-0.0566-0.185-1 θ for T 1 1.509-0.089-0.211 0.0624-3 θ for T 2 0.532 0.888 0.216 0.244-3 θ for T 3 0.921 0.499-0.0057-0.083-3 θ for T 4 1.097 0.323 0.067-0.001-3 θ for T 5 1.147 0.274-0.102 0.116-3 Then we get the transformed multivariate copula C and distribution function F Θ.

Goodness-of-t test Estimation results We perform a goodness-of-t test based on the empirical process in order to test the quality of the adjustment of copula C on these multivariate data. An approximate p-value for S n can be obtained by means of a parametric bootstrap-based procedure. For the transformed copula C, we obtain a p value= 0.37129. Copula under H 0 S n S B n S C n A n Gumbel-Hougaard 0.00331 0.00495 0.00454 0.03465 Clayton 0.00381 0.00980 0.00704 0.00981 Frank 0.00617 0.00941 0.00819 0.08416 t-student 0.00495 0.00592 0.00498 0.00963 Normal 0.00980 0.00719 0.00454 0.00205 Joe 0.00819 0.00495 0.00454 0.00916 Table : The bootstrapped p values for dierent goodness-of-t tests for competitor copula families on the considered 5-dimensional rainfall data. In all cases, the number of Monte Carlo experiments is xed at N = 1000. Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015

y y Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 y Classical parametric models Estimation results Errors F(X1, X4) Clayton model Errors F(X1, X4) Frank model Errors F(X1, X4), 5 dim model 1 2 3 4 0.08 0.06 0.04 0.02 1 2 3 4 0.08 0.06 0.04 0.02 1 2 3 4 0.08 0.06 0.04 0.02 0.00 0.00 0.00 1 2 3 4 5 x 1 2 3 4 5 x 1 2 3 4 5 x Figure : Errors F (X1,X4)(x, y) F n(x, y), for (x, y) in a lattice of 100 100 points, where F n is the empirical distribution function and F (X1,X4) is parametric model with Gamma marginals and Clayton copula (left), Frank copula (centre panel), and our transformed model F (right panel). Black cross represents the maximum error in the considered lattice. Black dots represent the associated rainfall data (X 1, X 4).

y y Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 y Classical parametric models Estimation results Errors F(X2, X5) Clayton model Errors F(X2, X5) Frank model Errors F(X2, X5), 5 dim model 1 2 3 4 5 0.08 0.06 0.04 0.02 1 2 3 4 5 0.08 0.06 0.04 0.02 1 2 3 4 5 0.08 0.06 0.04 0.02 0.00 0.00 0.00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x Figure : Errors F (X2,X5)(x, y) F n(x, y), for (x, y) in a lattice of 100 100 points, where F n is the empirical distribution function and F (X2,X5) is parametric model with Gamma marginals and Clayton copula (left), Frank copula (centre panel) and our transformed model F (right panel). Black cross represents the maximum error in the considered lattice. Black dots represent the associated rainfall data (X 2, X 5).

Estimation results Diagonal fit ln(1 F(x,x,x,x,x)) 4 3 2 1 0 Empirical Transformed copula Gumbel Frank Clayton 0 2 4 6 8 x Figure : Estimation of the 5-dimensional survival diagonal in logarithmic scale.

Choice of clusters Estimation results Aim: exibility of the proposed model and associated estimation procedure. We adapt our methodology in the case of some asymmetric dependencies (as, for instance, non-exchangeable random vectors). Figure : Left: Correlation matrix of the considered rainfall data. Right: Dendrogram resulting to the hierarchical cluster analysis on the set of dissimilarities produced by the Euclidian distance on the rainfall data. Red boxes show the two considered clusters.

Two child/cluster models Estimation results First (tri-variate) cluster composed by stations (X 2, X 3, X 5); second (bivariate) cluster is (X 1, X 4). The multivariate distribution for the cluster A = {2, 3, 5} is assumed to be written: F A (x 2, x 3, x 5) = T A C 0(T 1 A F 2(x 2), T 1 A with Fi = T A T 1 A i F i, for i A. F 3(x 3), T 1 A F 5(x 5)), Parameters F A m h ρ 1 ρ 2 η θ A -2.418-0.168-0.831-0.517-2 θ A2-0.262 0.747-0.136-0.1 35-4 θ A3 1.711 0.576-0.389-0.315-4 θ A5 2.061 0.300-0.401-0.104-4

Two child/cluster models Estimation results The multivariate distribution for the cluster B = {1, 4} is assumed to be written: F B (x 1, x 4) = T B C 0(T 1 B F 1(x 1), T 1 B F 4(x 4)), with Fi = T B T 1 B i F i, for i B. Parameters F B m h ρ 1 ρ 2 η θ B 0.168 0.776-0.168-0.271-2 θ B1 1.823 0.094-0.277-0.003-4 θ B4-0.449 0.491 0.032-0.029-4 The whole 5 dimensional distribution is assumed to be written: ( 1 F (x 1, x 2, x 3, x 4, x 5) = T C 0 T F A (x 2, x 3, x 5), T 1 ) F B (x 1, x 4), where C(u, v) = T C0 ( T 1(u), T 1 (v) ) is referred as the root copula at point (u, v).

Root model Estimation results To estimate the external root transformation T we rstly construct a bivariate pseudo data-set: Z 1 = F A (X 1, X 4), Z 2 = F B (X 2, X 3, X 5). Then we t on this bivariate data-set a model F (Z1,Z2)(z 1, z 2) = T C 0(T 1 F1(z 1), T 1 F2(z 2)), (1) with Fi = T T 1 F 1 i, for i = 1, 2. Parameters F (Z1,Z2) in (1) m h ρ 1 ρ 2 η θ -1.225 0.126-0.243-0.150-2 θ 1 0.401 0.682-0.055-0.384-4 θ 2 0.478 0.864 0.081-0.250-4

y y Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 y Estimation results Errors F(X1, X4) Transformed Nested model Errors F(X2, X5) Transformed Nested model Errors F(X3, X4) Transformed Nested model 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.08 0.06 0.04 0.02 1 2 3 4 5 0.08 0.06 0.04 0.02 1 2 3 4 0.08 0.06 0.04 0.02 0.00 0.00 0.00 1 2 3 4 5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.5 1.0 1.5 2.0 2.5 3.0 x x x Figure : Errors F (Xi,X j )(x, y) F n(x, y), for (x, y) in a lattice of 100 100 points, where F n is the empirical distribution function and F (Xi,X j ) is the parametric nested model for (i, j)= (1, 4) (left), (i, j)= (2, 5) (centre panel), (i, j)= (3, 4) (right). Black cross represents the maximum error in the considered lattice. Black dots represent the associated rainfall data (X i, X j ).

Critical Layers for nested model Estimation results Figure : Left: 2-dimensional critical-layers L B (α) with α = 0.2, 0.5, 0.9. Associated non-parametric empirical critical-layers are drawn in blue dashed lines. Right: 3-dimensional critical layers L A (α) with α = 0.3, 0.9. Black dots represent rainfall data (X 1, X 4) (left) and (X 2, X 3, X 5) (right).

Bibliography Estimation results Di Bernardino, E. and Rullière, D. (2015). Estimation of multivariate critical layers: Applications to rainfall data. Journal de la Société Française de Statistique, Vol. 156 (1), 11-50. Di Bernardino, E. and Rullière, D. (2013). Distortions of multivariate distribution functions and associated level curves: Applications in multivariate risk theory. Insurance: Mathematics and Economics, 53(1):190-205. Di Bernardino, E. and Rullière, D. (2013). On certain transformations of Archimedean copulas : Application to the non-parametric estimation of their generators. Dependence Modeling 1(1): 1-36. Di Bernardino, E. and Rullière, D. (2015). On tail dependence coecients of transformed multivariate Archimedean copulas. Preprint available on HAL. Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015

Estimation results Thank you for your attention