Multivariate Extreme Value models Michel Bierlaire Transport and - PowerPoint PPT Presentation

Multivariate Extreme Value models Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique F´ ed´ erale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 1 / 68

Outline Outline Introduction 1 Multivariate Extreme Value distribution 2 MEV model 3 Examples of MEV models 4 Cross nested logit model 5 Network MEV model 6 M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 2 / 68

Introduction Logit Assumptions Random utility: U in = V in + ε in ε in is i.i.d. EV (Extreme Value) distributed ε in is the maximum of many r.v. capturing unobservable attributes, measurement and specification errors. i.i.d. independent and identically distributed. M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 3 / 68

Introduction Relax the independence assumption Multivariate distribution       ε 1 n U 1 n V 1 n  .   .   .  . . .  =  +     . . . U Jn V Jn ε Jn that is U n = V n + ε n and ε n is a vector of random variables. M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 4 / 68

Multivariate Extreme Value distribution Outline Introduction 1 Multivariate Extreme Value distribution 2 MEV model 3 Examples of MEV models 4 Cross nested logit model 5 Network MEV model 6 M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 5 / 68

Multivariate Extreme Value distribution Multivariate Extreme Value distribution Definition ε n = ( ε 1 n , . . . , ε Jn ) follows a multivariate extreme value distribution if it has the CDF F ε n ( ε 1 n , . . . , ε Jn ) = e − G ( e − ε 1 n ,..., e − ε Jn ) , where G : R J n + → R + is a positive function with positive arguments. Valid CDF must verify three properties F ε n ( ε 1 n , . . . , −∞ , . . . , ε J n n ) = 0 . F ε n (+ ∞ , . . . , + ∞ ) = 1 . For any set of � J n ≤ J n distinct indices i 1 , . . . , i � J n , ∂ � J n F ε n ( ε 1 n , . . . , ε J n n ) ≥ 0 . ∂ε i 1 n · · · ∂ε i � Jn n M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 6 / 68

Multivariate Extreme Value distribution The limit property Valid CDF F ε n ( ε 1 n , . . . , −∞ , . . . , ε J n n ) = 0 . MEV F ε n ( ε 1 n , . . . , ε J n n ) = e − G ( e − ε 1 n ,..., e − ε Jnn ) . Valid G function G ( y 1 n , . . . , + ∞ , . . . , y J n n ) = + ∞ . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 7 / 68

Multivariate Extreme Value distribution The zero property Valid CDF F ε n (+ ∞ , . . . , + ∞ ) = 1 . MEV F ε n ( ε 1 n , . . . , ε J n n ) = e − G ( e − ε 1 n ,..., e − ε Jnn ) . Valid G function G (0 , . . . , 0) = 0 . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 8 / 68

Multivariate Extreme Value distribution The strong alternating sign property Valid CDF ∂ � J n F ε n ( ε 1 n , . . . , ε J n n ) ≥ 0 . ∂ε i 1 n · · · ∂ε i � Jn n MEV F ε n ( ε 1 n , . . . , ε J n n ) = e − G ( e − ε 1 n ,..., e − ε Jnn ) . Valid G function (notation: G i = ∂ G /∂ y i ) The right-hand side changes sign each time it is differentiated. To obtain ≥ 0, G must also change sign each time it is differentiated. For any set of � J n distinct indices i 1 , . . . , i � J n , � J n − 1 G i 1 ,..., i � ( − 1) Jn ≥ 0 . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 9 / 68

Multivariate Extreme Value distribution Homogeneity We need another property: homogeneity A function G is homogeneous of degree µ , or µ -homogeneous, if G ( α y ) = α µ G ( y ) , ∀ α > 0 and y ∈ R J n + . It will imply two results the marginals are univariate extreme value distributions, the choice model has a closed form. M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 10 / 68

Multivariate Extreme Value distribution Marginal distribution i th marginal distribution F ε n (+ ∞ , . . . , + ∞ , ε in , + ∞ , . . . , + ∞ ) = e − G (0 ,..., 0 , e − ε in , 0 ,..., 0) . If G is µ -homogeneous, we have G (0 , . . . , 0 , e − ε in , 0 , . . . , 0) = e − µε in G (0 , . . . , 0 , 1 , 0 , . . . , 0) , or equivalently, G (0 , . . . , 0 , e − ε in , 0 , . . . , 0) = e − µε in +log G (0 ,..., 0 , 1 , 0 ,..., 0) , Define log G (0 , . . . , 0 , 1 , 0 , . . . , 0) = µη , so that � − e − µ ( ε in − η ) � F ε n (+ ∞ , . . . , + ∞ , ε in , + ∞ , . . . , + ∞ ) = exp . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 11 / 68

Multivariate Extreme Value distribution Multivariate Extreme Value distribution CDF F ε n ( ε 1 n , . . . , ε J n n ) = e − G ( e − ε 1 n ,..., e − ε Jnn ) , i th marginal: univariate extreme value distribution � − e − µ ( ε in − η ) � F ε n (+ ∞ , . . . , + ∞ , ε in , + ∞ , . . . , + ∞ ) = exp . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 12 / 68

Multivariate Extreme Value distribution Multivariate Extreme Value distribution Three conditions on G The limit property G ( y 1 n , . . . , + ∞ , . . . , y J n n ) = + ∞ . The strong alternating sign property � J n − 1 G i 1 ,..., i � ( − 1) Jn ≥ 0 . Homogeneity (which implies the zero property) G ( α y ) = α µ G ( y ) , ∀ α > 0 and y ∈ R J n + . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 13 / 68

MEV model Outline Introduction 1 Multivariate Extreme Value distribution 2 MEV model 3 Examples of MEV models 4 Cross nested logit model 5 Network MEV model 6 M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 14 / 68

MEV model Derivation from first principles Probability model P ( i |C n ) = Pr( U in ≥ U jn , ∀ j ∈ C n ) , Random utility U in = V in + ε in . Random utility model P ( i |C n ) = Pr( V in + ε in ≥ V jn + ε jn , ∀ j ∈ C n ) , or P ( i |C n ) = Pr( ε jn − ε in ≤ V in − V jn , ∀ j ∈ C n ) . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 15 / 68

MEV model General derivation Joint distributions of ε n Assume that ε n = ( ε 1 n , . . . , ε J n n ) is a multivariate random variable with CDF F ε n ( ε 1 , . . . , ε J n ) and pdf ∂ J n F f ε n ( ε 1 , . . . , ε J n ) = ( ε 1 , . . . , ε J n ) . ∂ε 1 · · · ∂ε J n Derive the model for the first alternative (wlog) P n (1 |C n ) = Pr( V 2 n + ε 2 n ≤ V 1 n + ε 1 n , . . . , V Jn + ε Jn ≤ V 1 n + ε 1 n ) , P n (1 |C n ) = Pr( ε 2 n − ε 1 n ≤ V 1 n − V 2 n , . . . , ε Jn − ε 1 n ≤ V 1 n − V Jn ) . M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 16 / 68

MEV model Derivation Model P n (1 |C n ) = Pr( ε 2 n − ε 1 n ≤ V 1 n − V 2 n , . . . , ε Jn − ε 1 n ≤ V 1 n − V Jn ) . Change of variables ξ 1 n = ε 1 n , ξ in = ε in − ε 1 n , i = 2 , . . . , J n , that is       ξ 1 n ε 1 n 1 0 · · · 0 0       ξ 2 n − 1 1 · · · 0 0 ε 2 n             . . . . . . = .       . . .             ξ ( J n − 1) n ε ( J n − 1) n − 1 0 · · · 1 0 ξ J n n − 1 0 · · · 0 1 ε J n n M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 17 / 68

MEV model Derivation Model in ε P n (1 |C n ) = Pr( ε 2 n − ε 1 n ≤ V 1 n − V 2 n , . . . , ε Jn − ε 1 n ≤ V 1 n − V Jn ) . Change of variables ξ 1 n = ε 1 n , ξ in = ε in − ε 1 n , i = 2 , . . . , J n , Model in ξ P n (1 |C n ) = Pr( ξ 2 n ≤ V 1 n − V 2 n , . . . , ξ J n n ≤ V 1 n − V J n n ) . Note The determinant of the change of variable matrix is 1, so that ε and ξ have the same pdf M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 18 / 68

MEV model Derivation P n (1 |C n ) = Pr( ξ 2 n ≤ V 1 n − V 2 n , . . . , ξ J n n ≤ V 1 n − V J n n ) = F ξ 1 n ,ξ 2 n ,...,ξ Jn (+ ∞ , V 1 n − V 2 n , . . . , V 1 n − V J n n ) � + ∞ � V 1 n − V 2 n � V 1 n − V Jnn = · · · f ξ 1 n ,ξ 2 n ,...,ξ Jn ( ξ 1 , ξ 2 , . . . , ξ J n ) d ξ, ξ 1 = −∞ ξ 2 = −∞ ξ Jn = −∞ � + ∞ � V 1 n − V 2 n + ε 1 � V 1 n − V Jnn + ε 1 = · · · f ε 1 n ,ε 2 n ,...,ε Jn ( ε 1 , ε 2 , . . . , ε J n ) d ε, ε 1 = −∞ ε 2 = −∞ ε Jn = −∞ M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 19 / 68

MEV model Derivation � + ∞ � V 1 n − V 2 n + ε 1 � V 1 n − V Jnn + ε 1 P n (1 |C n ) = · · · f ε 1 n ,ε 2 n ,...,ε Jn ( ε 1 , ε 2 , . . . , ε J n ) d ε 1 = −∞ ε 2 = −∞ ε Jn = −∞ � + ∞ ∂ F ε 1 n ,ε 2 n ,...,ε Jn P n (1 |C n ) = ( ε 1 , V 1 n − V 2 n + ε 1 , . . . , V 1 n − V J n n + ε 1 ) d ε 1 . ∂ε 1 ε 1 = −∞ The random utility model: P n ( i |C n ) = � + ∞ ∂ F ε 1 n ,ε 2 n ,...,ε Jn ( . . . , V in − V ( i − 1) n + ε, ε, V in − V ( i +1) n + ε, . . . ) d ε ∂ε i ε = −∞ M. Bierlaire (TRANSP-OR ENAC EPFL) Multivariate Extreme Value models 20 / 68