Multivariate Extreme Value models Michel Bierlaire - PowerPoint PPT Presentation

Multivariate Extreme Value models Michel Bierlaire michel.bierlaire@epfl.ch Transport and Mobility LAboratory Multivariate Extreme Value models – p. 1/44

Logit • 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. • Key assumption: Independence Multivariate Extreme Value models – p. 2/44

Relax the independence assumption       U 1 n V 1 n ε 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. Assumption about the random term: multivariate distribution Multivariate Extreme Value models – p. 3/44

Relax the independence assumption A multivariate random variable ε is represented by a density function f ( ε 1 , . . . , ε J ) and � x 1 � x J P ( ε ≤ x ) = · · · f ( ε ) dε J . . . dε 1 −∞ −∞ where x ∈ R J is a J × 1 vector of constants. Multivariate Extreme Value models – p. 4/44

Probit model • Multivariate normal variable N ( µ, Σ) • µ ∈ R J • Σ ∈ R J × J , definite positive • Density function: 2 ( ε − µ ) T Σ − 1 ( ε − µ ) f ( ε ) = (2 π ) − J 2 | Σ | − 1 2 e − 1 Multivariate Extreme Value models – p. 5/44

Probit model Example: trinomial model U 1 = V 1 + ε 1 U 2 = V 2 + ε 2 U 3 = V 3 + ε 3 and ε ∼ N (0 , Σ) . We have P (2) = P ( U i − U 2 ≤ 0 i = 1 , 2 , 3) U 1 − U 2 = V 1 − V 2 + ε 1 − ε 2 U 3 − U 2 = V 3 − V 2 + ε 3 − ε 2 Multivariate Extreme Value models – p. 6/44

Probit model Matrix notation with � � 1 − 1 0 ∆ 2 = 0 − 1 1 � � U 1 − U 2 ∼ N (∆ 2 V, ∆ 2 Σ∆ T ∆ 2 U = 2 ) U 3 − U 2 Multivariate Extreme Value models – p. 7/44

Probit model In general, we have ∆ i U ∼ N (∆ i V, ∆ i Σ∆ T i ) and P ( i ) = � 0 � 0 P (∆ i U ≤ 0) = · · · f (∆ i ε ) d (∆ i ε ) 1 . . . d (∆ i ε ) J − 1 −∞ −∞ with 2 (∆ i ε − ∆ i V ) T (∆ i Σ∆ T f (∆ i ε ) = (2 π ) − J i | − 1 2 e − 1 i ) − 1 (∆ i ε − ∆ i V ) 2 | ∆ i Σ∆ T Multivariate Extreme Value models – p. 8/44

Probit model • The integral of the density function has no closed form • In high dimensions, numerical integration is computationally infeasible • Therefore, the probit model with more than 5 alternatives is very difficult to use in practice Multivariate Extreme Value models – p. 9/44

Relax the independence assumption Assume that ε n is a multivariate random variable with • CDF: F ε n ( ξ 1 , . . . , ξ J n ) ∂ Jn F • pdf: f ε n ( ξ 1 , . . . , ξ J n ) = ∂ξ 1 ··· ∂ξ Jn ( ξ 1 , . . . , ξ J n ) . • The choice probability is P n (1) = Pr( V 2 n + ε 2 n ≤ V 1 n + ε 1 n , . . . , V Jn + ε Jn ≤ V 1 n + ε 1 n ) , or P n (1) = Pr( ε 2 n − ε 1 n ≤ V 1 n − V 2 n , . . . , ε Jn − ε 1 n ≤ V 1 n − V Jn ) . Multivariate Extreme Value models – p. 10/44

Relax the independence assumption 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 1 0 · · · 0 0 ε 1 n       ξ 2 n − 1 1 · · · 0 0 ε 2 n             . . .       . . . = . . . .                   ξ ( J n − 1) n − 1 0 · · · 1 0 ε ( J n − 1) n ξ J n n − 1 0 · · · 0 1 ε J n n and P n (1) = Pr( ξ 2 n ≤ V 1 n − V 2 n , . . . , ξ J n n ≤ V 1 n − V J n n ) . Multivariate Extreme Value models – p. 11/44

Relax the independence assumption P n (1) = Pr( ξ 2 n ≤ V 1 n − V 2 n , . . . , ξ J n n ≤ V 1 n − V J n n ) . • Only J n − 1 inequalities. • ξ 1 n can take any value. • Choice probability = CDF of ( ξ 2 n , . . . , ξ J n n ) evaluated at ( V 1 n − V 2 n , . . . , V 1 n − V J n n ) . P n (1) = 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 ) dξ, ξ 1 = −∞ ξ 2 = −∞ ξ Jn = −∞ Multivariate Extreme Value models – p. 12/44

Relax the independence assumption � + ∞ � V 1 n − V 2 n � V 1 n − V Jnn P n (1) · · · f ξ 1 n ,ξ 2 n ,...,ξ Jn ( ξ 1 , ξ 2 , . . . , ξ J n ) dξ. ξ 1 = −∞ ξ 2 = −∞ ξ Jn = −∞ • Change of variables: determinant 1. • pdf of ( ξ 1 n , . . . , ξ J n n ) = pdf of ( ε 1 n , . . . , ε J n n ) � + ∞ � V 1 n − V 2 n + ε 1 � V 1 n − V Jnn + ε 1 P n (1) = · · · f ε 1 n ,ε 2 n ,...,ε Jn ( ε 1 , ε 2 , . . . , ε J n ) dε. ε 1 = −∞ ε 2 = −∞ ε Jn = −∞ or � + ∞ ∂F ε 1 n ,ε 2 n ,...,ε Jn P n (1) = ( ε 1 , V 1 n − V 2 n + ε 1 , . . . , V 1 n − V J n n + ε 1 ) dε 1 . ∂ε 1 ε 1 = −∞ Multivariate Extreme Value models – p. 13/44

Multivariate Extreme Value model • 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 − εJn ) , where G : R J n + → R + is a positive function with positive arguments. • To be a valid CDF, it must verify the following properties. • (i) the limit property F ε n ( ε 1 n , . . . , −∞ , . . . , ε Jn ) = 0 , or G ( y 1 n , . . . , + ∞ , . . . , y Jn ) = + ∞ . Multivariate Extreme Value models – p. 14/44

Multivariate Extreme Value model • (ii) the zero property F ε n (+ ∞ , . . . , + ∞ ) = 1 . or G (0 , . . . , 0) = 0 . • (iii) the strong alternating sign property: • Any partial derivative of F ε n defines a density function of a marginal distribution. • To be a valid density function, it has to be non negative. • 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 Multivariate Extreme Value models – p. 15/44

Multivariate Extreme Value model F ε n ( ε 1 n , . . . , ε Jn ) = e − G ( e − ε 1 n ,...,e − εJn ) , • (iii) the strong alternating sign property (ctd). • The right-hand side changes sign each time it is differentiated. • To obtain a non negative sign, G must also change sign each time it is differentiated. • For any set of � J n distinct indices i 1 , . . . , i � J n , ( − 1) � J n − 1 G i 1 ,...,i � Jn ≥ 0 . Multivariate Extreme Value models – p. 16/44

Multivariate Extreme Value model We need another property: homogeneity. • A function G is homogeneous of degree µ , or µ -homogeneous, if G ( αy ) = α µ G ( y ) , ∀ α > 0 and y ∈ R J + . • It will imply two results: • the marginals are univariate extreme value distributions, • the choice model has a closed form. Multivariate Extreme Value models – p. 17/44

Multivariate Extreme Value model • i th marginal distribution: F ε n (+ ∞ , . . . , + ∞ , ε in , + ∞ , . . . , + ∞ ) = e − G (0 ,..., 0 ,e − εin , 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 . Multivariate Extreme Value models – p. 18/44

Multivariate Extreme Value model F ε n ( ε 1 n , . . . , ε Jn ) = e − G ( e − ε 1 n ,...,e − εJn ) , � − e − µ ( ε in − η ) � F ε n (+ ∞ , . . . , + ∞ , ε in , + ∞ , . . . , + ∞ ) = exp . • Four properties (actually, three). • Valid CDF. • Marginals: univariate extreme value distribution. • We have a multivariate extreme value distribution. Multivariate Extreme Value models – p. 19/44

MEV: choice model F ε n ( ε 1 n , . . . , ε Jn ) = e − G ( e − ε 1 n ,...,e − εJn ) , � + ∞ ∂F ε 1 n ,ε 2 n ,...,ε Jn P n ( i ) = ( . . . , V in − V ( i − 1) n + ε, ε, V in − V ( i +1) n + ε, . . . ) dε. ∂ε i ε = −∞ As G is µ -homogeneous, G i = ∂G/∂y i is µ − 1 -homogeneous and ∂F ε 1 n ,ε 2 n ,...,ε Jn ( . . . , V in − V ( i − 1) n + ε, ε, V in − V ( i +1) n + ε, . . . ) ∂ε i = e − ε G i ( . . . , e − V in + V ( i − 1) n − ε , e − ε , e − V in + V ( i +1) n − ε , . . . ) � � − G ( . . . , e − V in + V ( i − 1) n − ε , e − ε , e − V in + V ( i +1) n − ε , . . . ) exp = e − ε e − ( µ − 1) ε e − ( µ − 1) V in G i ( . . . , e V ( i − 1) n , e V i n , e V ( i +1) n , . . . ) � � − e − µε e − µV in G ( . . . , e V ( i − 1) n , e V in , e V ( i +1) n , . . . ) . exp Multivariate Extreme Value models – p. 20/44