Algebraic properties of copulas defined from matrices C ecile - PowerPoint PPT Presentation

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Algebraic properties of copulas defined from matrices C´ ecile Amblard*, St´ ephane Girard**, Ludovic Menneteau*** Krakow, july 2012 *LIG, University Grenoble 1, France, **Inria Grenoble & LJK, France, ***I3M University Montpellier 2, France. 1 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Introduction Extension of the bivariate family [Amblard Girard 2002] c ( u , v ) = 1 + θφ ( u ) φ ( v ) , c ( u , v ) = t φ ( u ) A φ ( v ) where : t { 1 , φ 2 ( u ) , · · · , φ p ( u ) } , - φ ( u ) = - { φ i } is an orthonormal family of functions, - A ∈ R p × p is a symmetric matrix such that t e 1 = (1 , 0 , · · · , 0) . Ae 1 = e 1 , with For which A and φ , c ( u , v ) is a density of copula ? 2 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Plan Definition of the family of copulas C φ , Algebraic properties of the set of convenient matrices A φ and of the copulas family C φ , Dependence properties of the family C φ , Projection on C φ , Examples. 3 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion A new family of copulas Defnition : c ( u , v ) = t φ ( u ) A φ ( v ) - φ ( u ) = t { 1 , φ 2 ( u ) , · · · , φ p ( u ) } , - { φ i } is an orthonormal family of functions, using L 2 ( R ) scalar product : � 1 < φ i , φ j > = φ i ( t ) φ j ( t ) dt , 0 - A ∈ R p × p is a symmetric matrix such that Ae 1 = e 1 , with e 1 = t (1 , 0 · · · , 0) . 4 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion A new family of copulas � � A ∈ R p × p , t A = A , Ae 1 = e 1 , A φ = ∀ ( u , v ) ∈ [0 , 1] 2 , t φ ( u ) A φ ( v ) ≥ 0 � � c : [0 , 1] 2 → R , C φ = c ( u , v ) = t φ ( u ) A φ ( v ) , A ∈ A φ 5 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion A new family of copulas Properties : � 1 0 φ ( t ) dt = e 1 :   � 1 0 1 dv � 1 � 1    0 φ 2 ( v ) dv  φ ( v ) dv =  ,    · · · 0 � 1 0 φ p ( v ) dv = e 1 because { φ i } is orthonormal. 6 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion A new family of copulas A φ is not empty, A 1 = e t 1 e 1 ∈ A φ C φ is a set of copulas density. Positivity : ∀ ( u , v ) ∈ R 2 , c ( u , v ) ≥ 0, Uniform marginals : � 1 � 1 t φ ( u ) A c ( u , v ) dv = φ ( v ) dv , 0 0 t φ ( u ) e 1 , = = 1 7 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion A new family of copulas Each copula of C φ is defined by one unique matrix : t φ ( u ) A φ ( v ) = t φ ( u ) B φ ( v ) ⇒ φ ( u ) t φ ( u ) A φ ( v ) t φ ( v ) = φ ( u ) t φ ( u ) B φ ( v ) t φ ( v ) �� 1 0 φ ( u ) t φ ( u ) A φ ( v ) t φ ( v ) dudv �� 1 ⇒ 0 φ ( u ) t φ ( u ) B φ ( v ) t φ ( v ) dudv = � 1 � 1 0 φ ( u ) t φ ( u ) duA 0 φ ( v ) t φ ( v ) dv ⇒ � 1 � 1 0 φ ( u ) t φ ( u ) duB 0 φ ( v ) t φ ( v ) dv = ⇒ A = B because { φ i } is an orthonormal family. 8 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Examples The copula associated to A 1 = e 1 t e 1 is the independent copula Π, � � 1 0 If p = 2 , necessarily A = and 0 θ c ( u , v ) = 1 + θφ ( u ) φ ( v ) [ Amblard Girard 2002 ], The cubic family [ Nelsen et al. 1997 ] can be written in our formalism (p=3), If { φ i } is an orthonormal family and ∀ ( u , v ) ∈ [0 , 1] 2 t φ ( u ) φ ( v ) ≥ 0 , t φ ( u ) φ ( v ) ∈ C φ I p ∈ A φ and 9 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Algebraic properties of A φ A φ is a convex set, A 1 = e 1 t e 1 ∈ A φ , ( A φ , × ) is a semi group : � 1 t φ ( u ) AB φ ( v ) t φ ( u ) A φ ( y ) t φ ( y ) dyB φ ( v ) , = 0 � 1 ( t φ ( u ) A φ ( y ))( t φ ( y ) B φ ( v )) dy , = 0 ≥ 0 . - ABe 1 = e 1 , - the product × is an associative operator. If I p ∈ A φ , ( A φ , × ) is a monoid. 10 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Algebraic properties of C φ C φ is a convex set, Π ∈ C φ , ( C φ , ⋆ ) is a semi group : - � 1 � c A ⋆ c B ( u , v ) c A ( u , s ) c B ( s , v ) ds , 0 � 1 t φ ( u ) A φ ( s ) t φ ( s ) B φ ( v ) ds , = 0 � 1 t φ ( u ) A φ ( s ) t φ ( s ) dsB φ ( v ) , = 0 t φ ( u ) AI p B φ ( v ) . = - the operator ⋆ is associative, t φ ( u ) φ ( v ) ∈ C φ , ( C φ , ⋆ ) is a monoid . If 11 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Isomorphism between A φ and C φ Definition : T φ : { copulas } → R p × p �� 1 φ ( x ) c ( x , y ) t φ ( y ) dxdy c �→ 0 T φ ( c ) e 1 = e 1 . T φ is an isomorphism between ( A φ , × ) and ( C φ , ⋆ ) : Each matrix of A φ defines a copula of C φ , T φ associates to a copula of C φ its matrix A , T φ ( c A ⋆ c B ) = A × B . 12 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Isomorphism between A φ and C φ T φ ( c ) e 1 = e 1 : �� φ ( x ) c ( x , y ) t φ ( y ) e 1 dydx T φ ( c ) e 1 = �� = φ ( x ) c ( x , y ) φ 1 ( y ) dydx � � 1 = φ ( x ) c ( x , y )1 dydx 0 � 1 = φ ( x )1 dx 0 = e 1 13 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Dependence coefficients Spearman ’s Rho : �� 1 ρ φ � 12 C ( u , v ) dudv − 3 0 = 12 t µ A µ − 3 , � 1 where µ = x φ ( x ) dx . 0 If A = diag { a i , i } , ρ φ = 12 � p i =2 a i , i µ 2 i Tail dependence coefficient : ¯ C ( u , u ) λ φ = P ( V ≥ u | U ≥ u ) = 1 − u = 0 . 14 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Projection on C φ Definition : P ( c )( u , v ) = t φ ( u ) T φ ( c ) φ ( v ) If I p ∈ A φ , P ( c )( u , v ) is a copula : P ( c )( u , v ) = t φ ( u ) T φ ( c ) φ ( v ) �� 1 = t φ ( u ) t φ ( x ) c ( x , y ) φ ( y ) dxdy φ ( v ) �� 1 0 t φ ( u ) φ ( x ) c ( x , y ) t φ ( y ) φ ( v ) dxdy = 0 = c I p ⋆ c ⋆ c I p ( u , v ) if I p ∈ A φ If I p ∈ A φ ; P ( c )( u , v ) ∈ C φ . 15 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Projection on C φ � 1 0 c 1 ⋆ c 2 ( u , u ) du defines a Scalar product . P is an orthogonal projection on ( C φ , <> ) : P ( P ( c )) = P ( c ) : P ( P ( c )( u , v ) = t φ ( u ) T φ ( P ( c )) φ ( v ) �� 1 t φ ( u ) 0 φ ( x ) P ( c )( x , y ) t φ ( y ) dxdy φ ( v ) , = �� t φ ( u ) φ ( x ) t φ ( x ) T φ ( c ) φ ( y ) t φ ( y ) dxdy φ ( v ) = � � = t φ ( u ) φ ( x ) t φ ( x ) dxT φ ( c ) t φ ( y ) φ ( y ) dy φ ( v ) = t φ ( u ) T φ ( c ) φ ( v ) , = P ( c ) . ∀ s ∈ C φ , < c − P ( c ) , s > = 0. 16 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Projection on C φ ∀ s ∈ C φ , < c − P ( c ) , s > = 0 �� c ( u , t ) t φ ( t ) A φ ( u ) dtdu < c , s > = �� c ( u , t ) tr ( t φ ( t ) A φ ( u )) dtdu = �� c ( u , t ) φ ( u ) t φ ( t ) dtduA ) = tr ( = tr ( T φ ( c ) A ) . �� t φ ( u ) T φ ( c ) φ ( t ) t φ ( t ) A φ ( u ) dtdu < P ( c ) , s > = = tr ( T φ ( c ) A ) . 17 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Example : FGM family √ φ ( x ) = 3(1 − 2 x ) , A = diag { 1 , θ } , | θ | ≤ 1 / 3, Copula : c ( u , v ) = 1 + 3 θ (1 − 2 u )(1 − 2 v ) , I p / ∈ A φ . ”Projection ”on C φ : T φ ( c ) = diag { 1 , � θ } �� � θ = 3 c ( x , y )(1 − 2 x )(1 − 2 y ) dxdy �� = 3[4 xyc ( x , y ) dxdy − 1] = ρ c . If | ρ c | ≤ 1 / 3 , P ( c ) is a FGM copula and ρ P ( c ) = ρ c , If | ρ c | > 1 / 3 , P ( c ) is not a copula. 18 / 24

Introduction A new family of copulas Algebraic properties Dependence properties Projection on C φ Examples Conclusion Example : trigonometric family  φ 0 ( x ) = 1 ,  √ φ 2 j − 1 ( x ) = 2 sin(2 π jx ) , A = diag { 1 , θ, θ, · · · } √  φ 2 j ( x ) = 2 cos(2 π jx ) Copula : c k ( x , y ) = 1 + 2 θ [ H k ( x − y ) − 1] , H k ( t ) = sin((2 k + 1) π t ) the Dirichlet Kernel. sin( π t ) � k Spearman’s rho : ρ k ( θ ) = 6 θ 1 π 2 j =1 j 2 ρ 1 (1 / 2) = 3 π 2 ≃ 0 . 3 , ρ 2 (1 / 2) = 15 4 π 2 ≃ 0 . 38 , 98 ρ 3 (4 / 9) = 27 π 2 ≃ 0 . 37 . 19 / 24