A DIRECTIONAL NOTION OF MULTIVARIATE EXTREME VALUE ANALYSIS Raúl A. T ORRES D ÍAZ Department of Statistics and Operation Research Universidad de Valladolid PhD. seminar in Mathematical Engineering, Universidad EAFIT joint work with: Elena Di Bernardino, CNAM Paris, Henry Laniado, EAFIT Medellín, Rosa E. Lillo, UC3M Madrid. apple uc3m May 2018 Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 1 / 46
I NTRODUCTION AND M OTIVATION 1 Extreme Value Theory concerns to the limit behavior of the sample extremes, (max or min in the univariate case) BUT 2 Multivariate analysis is mandatory because Extremes are gen- erated by many variables acting jointly with different relation- ships Asymptotic Independence & Asymptotic Dependence, (Pairs relations). Correlations, (Overall relation). apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 2 / 46
D IRECTIONAL P ERSPECTIVE 1 Look at the data with different perspectives to improve the identification and visualization of extremes BUT There are infinite directions to analyze the data, how to select an interesting one?. apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 3 / 46
S TATISTICAL TOOLS FOR E XTREMES 1 The classical tool for Extremes identification is the α − quantile concept BUT There is a lack of a total order in R d . Conditioned to the α − level, there are 2 approaches of estimation, In − Sample and Out − Sample . apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 4 / 46
I MPORTANCE OF α IN THE ESTIMATION In-Sample Vs. Out-Sample α > 1 α ≤ 1 n n Some Observations Non-Observations Available Available Multivariate Extreme Value Standard Estimation Theory Procedures apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 5 / 46
Directional Notions O UTLINE 1 D IRECTIONAL N OTIONS 2 N ON - PARAMETRIC O UT - SAMPLE E STIMATION 3 R EAL CASE STUDY 4 C ONCLUSIONS apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 6 / 46
Directional Notions QR Oriented Orthant. C u ≡ x (Torres et al. 2015) D EFINITION Given x ∈ R d and u ∈ z ∈ R d : || u || = 1 , u i � = 0 for all i = 1 , ..., d � � , the QR oriented orthant with vertex x and direction u is: C u x = { z ∈ R d | R u ( z − x ) ≥ 0 } , d ( 1 , ..., 1 ) ′ and R u is an unique orthogonal matrix obtained 1 where e = √ by a QR decomposition, such that R u u = e . apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 7 / 46
Directional Notions E XAMPLES OF O RIENTED O RTHANTS Examples of oriented orthants in R 2 apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 8 / 46
Directional Notions Directional Multivariate Quantile Q X ( α, u ) ≡ (Laniado et al. 2012) D EFINITION Given u ∈ R d , || u || = 1 and a random vector X with distribution proba- bility P , the α -quantile curve in direction u is defined as: Q X ( α, u ) := ∂ { x ∈ R d : P ( C − u x ) ≥ 1 − α } , where ∂ means the boundary and 0 ≤ α ≤ 1 apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 9 / 46
Directional Notions Directional Multivariate Upper U X ( α, u ) ≡ Level-Set Directional Multivariate Lower L X ( α, u ) ≡ Level-Set D EFINITION Those sets are defined by: U X ( α, u ) := { x ∈ R d : P C − u � � > 1 − α } , x L X ( α, u ) := { x ∈ R d : P � C − u � < 1 − α } . x apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 10 / 46
Directional Notions D IRECTIONAL M ULTIVARIATE L EVEL -S ETS � 1 � 1 �� − 1 , − 1 � � − 1 , 1 � , − 1 � , 1 �� u ∈ U = , , , √ √ √ √ √ √ √ √ 2 2 2 2 2 2 2 2 (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal CLASSICAL DIRECTIONS apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 11 / 46
Directional Notions D IRECTIONAL M ULTIVARIATE L EVEL -S ETS u ∈ U = { ( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 0 ) , ( 0 , − 1 ) } (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal CANONICAL DIRECTIONS apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 11 / 46
Directional Notions D IRECTIONAL M ULTIVARIATE Q UANTILE P ROPERTIES Quasi-Odd Property : Q − X ( α, u ) = −Q X ( α, − u ) . Positive Homogeneity and Translation Invariance : Q c X + b ( α, u ) = c Q X ( α, u ) + b , for all c ∈ R + and b ∈ R d . Orthogonal Quasi-Invariance : Let w and u be two unit vectors. Then, an orthogonal matrix Q exists, such that, Q u = w and Q X ( α, u ) = Q ′ Q Q X ( α, w ) . apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 12 / 46
Non-parametric Out-sample Estimation O UTLINE 1 D IRECTIONAL N OTIONS 2 N ON - PARAMETRIC O UT - SAMPLE E STIMATION 3 R EAL CASE STUDY 4 C ONCLUSIONS apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 13 / 46
Non-parametric Out-sample Estimation R EVIEW ON MULTIVARIATE OUT - SAMPLE ESTIMATION Optimization processes estimation (e.g., Girard and Stupfler (2015)) Estimation of level curves of joint density functions or depth functions (e.g., Cai et al. (2011), Einmahl et al. (2013), He and Einmahl (2017)) Estimation of level curve of either joint distribution or survival functions (e.g. De Haan and Huang (1995)) apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 14 / 46
Non-parametric Out-sample Estimation N ECESSARY BACKGROUND A SSUMPTION 1, A1. The random vector X must be absolutely continuous. A SSUMPTION 2, A2. Given u , R u X possesses positive upper-end points of the marginal distributions. apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46
Non-parametric Out-sample Estimation N ECESSARY BACKGROUND D EFINITION X has first order multivariate regular variation with tail index γ , denoted by RV 1 /γ , if there exists a real-value function φ ( t ) > 0 that is regularly varying at infinity with exponent 1 /γ and a non-zero measure µ ( · ) on the Borel σ − field ¯ R d \{ 0 } such that for every Borel set B, v t P [( φ ( t )) − 1 X ∈ B ] → µ ( B ) , v where → means vague convergence and t → ∞ . A SSUMPTION 3, A3. X has 1st order multivariate regular variation with index γ . apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46
Non-parametric Out-sample Estimation N ECESSARY BACKGROUND D EFINITION X has second order multivariate regular variation if there exist functions φ ( · ) ∈ RV 1 /γ and Λ( t ) → 0 , such that | Λ | ∈ RV π , π ≤ 0 ; satisfying for all relatively compact rectangles B ∈ ¯ R d \{ 0 } , � ( φ ( t )) − 1 X ∈ B � − µ ( B ) t P → ψ ( B ) , Λ( φ ( t )) where ψ ( B ) is finite and not identically zero. A SSUMPTION 4, A4. X has 2nd order multivariate regular variation with indexes ( γ, π ). apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46
Non-parametric Out-sample Estimation D IRECTIONAL R ESULTS R ESULT If X has 1st(2nd) order multivariate regular variation. Then, Q X has 1st(2nd) order multivariate regular variation, for any orthogo- nal transformation Q . C OROLLARY If X has 1st(2nd) order multivariate regular variation. Then the marginals of Q X has 1st(2nd) order multivariate regular variation. apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 16 / 46
Non-parametric Out-sample Estimation C HARACTERIZATION OF Q X ( α, u ) AT HIGH LEVELS Key tools A1-A3 and Q X ( α, u ) = R ′ u Q R u X ( α, e ) Q R u X ( α, e ) ≈ Q R u X ( α, e , θ ) where θ belongs to the unit d − dimensional ball and 0 ≤ θ j ≤ 1 A SYMPTOTIC CHARACTERIZATION OF Q X ( α, u , θ ) x u ( α, θ ) = ( x u , 1 ( α, θ ) , ..., x u , d ( α, θ )) a u , j ( t )( ρ u ( θ ) θ j / t α ) γ − 1 � � = + b u , j ( t ); j = 1 , . . . , d , , γ Q X ( α, u , θ ) = R ′ u Q R u X ( α, e , θ ) . apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 17 / 46
Non-parametric Out-sample Estimation H OW WAS THE CHARACTERIZATION POSSIBLE ? 1 Pre-rotation of X through the orthogonal matrix R u introduced in the QR orthant definition. All the elements with subindex u refer to expressions related to R u X . For instance, F u denotes the joint distribution and its marginals are F u , j , j = 1 , ..., d . 2 Key asymptotic result from the Multivariate Extreme Value Theory. D ISTRIBUTION OF CONVERGENCE OF THE SAMPLE MAXIMA There exist two sequences a u ( ⌊ t ⌋ ) , b u ( ⌊ t ⌋ ) such that, t →∞ t ( 1 − F u ( a u , j ( ⌊ t ⌋ ) x u , j + b u , j ( ⌊ t ⌋ ); j = 1 , . . . , d )) = − ln ( G u ( x u )) , lim apple uc3m where ⌊·⌋ is the floor function. Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46
Non-parametric Out-sample Estimation H OW WAS THE CHARACTERIZATION POSSIBLE ? 3 Marginal high level estimations through extreme value analysis. H IGH LEVEL QUANTILES OF F u , j , j = 1 , ..., d x u , j ( α ) ≈ a u , j ( t )( 1 / t α ) γ − 1 + b u , j ( t ) . γ 4 Polar representation in R d . P OLAR PARAMETRIZATION In R d , any point x can be written in polar coordinates as x = || x || ( x / || x || ) = ρ ( θ ) θ , where ρ ( θ ) ∈ R + and θ belonging to the unit d − dimensional ball. apple uc3m Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46
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