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. 1/62

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. 2/62

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. 3/62

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. 4/62

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. 5/62

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. 6/62

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. 7/62

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. 8/62

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. 9/62

Relax the independence assumption • If the CDF F ( ε 1 , . . . , ε J ) of the distribution is known ∂ J F f ( ε 1 , . . . , ε J ) = ( ε 1 , . . . , ε J ) ∂ε 1 · · · ∂ε J • The choice probability is P ( i ) = P ( V 1 + ε 1 ≤ V i + ε i , . . . , V J + ε J ≤ V i + ε i ) = P ( ε 1 ≤ V i + ε i − V 1 , . . . , ε J ≤ V i + ε i − V J ) � ∞ = F i ( V i + ε i − V 1 , . . . , ε i , . . . , V i + ε i − V J ) dε i ε i = −∞ where F i = ∂F/∂ε i . Multivariate Extreme Value models – p. 10/62

Relax the independence assumption Operational solutions: • Generalize the logit: the Nested Logit model • Consider a multivariate distribution such that F is known: the Multivariate Extreme Value model Multivariate Extreme Value models – p. 11/62

Nested logit model • Alternatives within a nest share a random term • Random utility of alt. i in nest C m U i = V i + ε i = V i + ε m + ε im • Assume that ε m are independent across m • ε im are i.i.d. EV with scale param. µ m for each m Multivariate Extreme Value models – p. 12/62

Nested logit model Assume that the nest m is given. P ( i | m ) = P ( U i ≥ U j , j ∈ C m ) = P ( V i + ε m + ε im ≥ V j + ε m + ε jm , j ∈ C m ) = P ( V i + ε im ≥ V j + ε jm , j ∈ C m ) Then we have a logit model: e µ m V i P ( i | m ) = � j ∈ C m e µ m V j Multivariate Extreme Value models – p. 13/62

Nested logit model What is the probability of choosing nest C m ? P ( m ) = P ( max U i ≥ max U j , ∀ k � = m ) = i ∈ C m j ∈ C k P ( ε m + max ( V i + ε im ) ≥ ε k + max ( V j + ε jk ) , ∀ k � = m ) i ∈ C m j ∈ C k Note that V i + ε im is EV( V i , µ m ). Therefore ( V i + ε im ) ∼ EV ( ˜ max V m , µ m ) i ∈ C m where 1 � ˜ e µ m V i V m = ln µ m i ∈ C m See prop. 7, page 105, chap. 5 Multivariate Extreme Value models – p. 14/62

Nested logit model We write the random variable i ∈ C m ( V i + ε im ) = ˜ V m + ε ′ max m Therefore, P ( ε m + ˜ m ≥ ε k + ˜ V m + ε ′ V k + ε ′ P ( m ) = k , ∀ k � = m ) P ( ˜ ε m ≥ ˜ = V m + ˜ V k + ˜ ε k , ∀ k � = m ) Looks familiar, doesn’t it? Multivariate Extreme Value models – p. 15/62

Nested logit model P ( m ) = P ( ˜ ε m ≥ ˜ V m + ˜ V k + ˜ ε k , ∀ k � = m ) Assume that ˜ ε m ∼ EV (0 , µ ) . Then e µ ˜ V m P ( m ) = � k e µ ˜ V k Multivariate Extreme Value models – p. 16/62

Nested logit model Putting everything together: e µ ˜ e µ m V i V m P ( i ) = P ( i | m ) P ( m ) = � � j ∈ C m e µ m V j k e µ ˜ V k with 1 � ˜ e µ m V im V m = ln µ m i ∈ C m Back Multivariate Extreme Value models – p. 17/62

Nested logit model Advantages • Nest partitioning is an intuitive concept • Direct extension of logit • Closed form of the model Drawbacks • Limited correlation structure • What is the actual density function f ( ε ) ? Multivariate Extreme Value models – p. 18/62

MEV models Family of models proposed by McFadden (1978) (called GEV) Idea: a model is generated by a function G : R J + → R + From G , we can build • The cumulative distribution function (CDF) • The probability model • The expected maximum utility Multivariate Extreme Value models – p. 19/62

MEV models 1. G is homogeneous of degree µ > 0 , that is G ( αy ) = α µ G ( y ) 2. y i → + ∞ G ( y 1 , . . . , y i , . . . , y J ) = + ∞ , for each i = 1 , . . . , J , lim 3. the k th partial derivative with respect to k distinct y i is non negative if k is odd and non positive if k is even, i.e., for all (distinct) indices i 1 , . . . , i k ∈ { 1 , . . . , J } , we have ∂ k G ( − 1) k ( y ) ≤ 0 , ∀ y ∈ R J + . ∂y i 1 . . . ∂y i k Multivariate Extreme Value models – p. 20/62

MEV models • CDF: F ( ε 1 , . . . , ε J ) = e − G ( e − ε 1 ,...,e − εJ ) e Vi +ln Gi ( eV 1 ,...,eVJ ) j ∈ C e Vj +ln Gj ( eV 1 ,...,eVJ ) with G i = ∂G • Probability: P ( i | C ) = ∂y i . This � is a closed form • Expected maximum utility: V C = ln G ( ... )+ γ where γ is Euler’s µ constant. • Note: P ( i | C ) = ∂V C ∂V i . Multivariate Extreme Value models – p. 21/62

MEV models Euler’s constant � n � � + ∞ 1 � e − x ln xdx = lim γ = − k − ln n n →∞ 0 k =1 Multivariate Extreme Value models – p. 22/62

Proofs We show first that F ( ε 1 , . . . , ε J ) = e − G ( e − ε 1 ,...,e − εJ ) defines a multivariate CDF. • F goes to zero when one ε goes to −∞ e − G ( e − ε 1 ,...,e + ∞ ,...,e − εJ ) F ( ε 1 , . . . , −∞ , . . . , ε J ) = e − G ( e − ε 1 ,..., + ∞ ,...,e − εJ ) = e −∞ = = 0 Multivariate Extreme Value models – p. 23/62

Proofs F ( ε 1 , . . . , ε J ) = e − G ( e − ε 1 ,...,e − εJ ) • F goes to one when all ε go to + ∞ e − G ( e −∞ ,...,e −∞ ) F (+ ∞ , . . . , + ∞ ) = e − G (0 ,..., 0) = e 0 = = 1 Multivariate Extreme Value models – p. 24/62

Proofs F ( ε 1 , . . . , ε J ) = e − G ( e − ε 1 ,...,e − εJ ) • The function ∂ J F f ( ε 1 , . . . , ε J ) = ( ε 1 , . . . , ε J ) ≥ 0 ∂ε 1 · · · ε J so that it defines a PDF. Multivariate Extreme Value models – p. 25/62

Proofs Define recursively Q 1 = G 1 = ∂G/∂y 1 ≥ 0 Q k = Q k − 1 G k − ∂Q k − 1 /∂y k We show recursively that all (signed) terms of Q k are ≥ 0 Assume it true for Q k − 1 Q k − 1 = Q k − 2 G k − 1 − ∂Q k − 2 /∂y k − 1 � �� � � �� � ≥ 0 ≥ 0 As G k = ∂G/∂y k ≥ 0 , we have Q k − 1 G k ≥ 0 Multivariate Extreme Value models – p. 26/62

Proofs Q k − 1 = Q k − 2 G k − 1 − ∂Q k − 2 /∂y k − 1 � �� � � �� � ≥ 0 ≥ 0 ∂Q k − 1 /∂y k = ∂Q k − 2 /∂y k G k − 1 + Q k − 2 ∂G k − 1 /∂y k ∂ 2 Q k − 2 /∂y k − 1 ∂y k − By assumption, each increase of the order of derivatives imposes a change of sign, so that ∂Q k − 1 /∂y k ≤ 0 Therefore Q k = Q k − 1 G k − 1 − ∂Q k − 1 /∂y k � �� � � �� � ≥ 0 ≥ 0 Multivariate Extreme Value models – p. 27/62

Proofs F ( ε 1 , . . . , ε J ) = e − G ( e − ε 1 ,...,e − εJ ) We show recursively that ∂ J F = e − ε 1 · · · e − ε J Q J F ≥ 0 ∂ε 1 · · · ε J By direct derivation, we have ∂F = e − ε 1 G 1 F = e − ε 1 Q 1 F ∂ε 1 Multivariate Extreme Value models – p. 28/62