Normalization and correlation of cross-nested logit models

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1 Normalization and correlation of cross-nested logit models E. Abbe, M. Bierlaire, T. Toledo Lab. for Information and Decision Systems, Massachusetts Institute of Technology Inst. of Mathematics, Ecole Polytechnique Fédérale de Lausanne, Switzerland Transport Research Institute, Technion, Israël European Transport Conference 2005 p./29

2 Introduction GEV models Cross-nested formulations Normalization Variance-covariance structure Simple examples Route choice examples European Transport Conference 2005 p.2/29

3 GEV models U = V + ε, random vector, V R J F ε,...,ε J (y,..., y J ) = e G(e y,...,e y J ). P (i C) = y ig i (y,..., y J ) µg(y,..., y J ) = e Vi+ln Gi(...) j C ev j+ln G j (...), where G i = G y i, y i = e V i G : R J + R +. European Transport Conference 2005 p.3/29

4 GEV models MNL: G(y,..., y J ) = j C y µ j NL: G(y,..., y J ) = M m= ( Jm j= ) µ µm y µ m j CNL: G(y,..., y J ) = M m= ( (α /µ jm y j ) µ m j C ) µ µm European Transport Conference 2005 p.4/29

5 Cross-nested logit models Ordered GEV mode, Small (987) Vovsha (997) G(y,..., y J ) = J+M r= ( w r j y /ρr j j B r ) ρr, G(y,..., y J ) = m ( j C α jm y j ) µ European Transport Conference 2005 p.5/29

6 Cross-nested logit models Ben-Akiva & Bierlaire (999) G(y,..., y J ) = m Wen & Koppelman (200) ( j C ) µ µm α jm y µ m j G(y,..., y J ) = m ( (α n mx n ) µm n N m ) µm European Transport Conference 2005 p.6/29

7 Cross-nested logit models Papola (2004) G(y,..., y J ) = k ( α θ0/θk jk y /θ k j j C k ) θ k θ0 General formulation (as Ben-Akiva & Bierlaire) Appropriate for easy normalization (as Wen & Koppelman) For consistency with GEV, we note µ = /θ 0 and µ k = /θ k European Transport Conference 2005 p.7/29

8 Normalization F ε,...,ε J (y,..., y J ) = e G(e y,...,e y J ). ( M G(y,..., y J ) = (α /µ jm y j ) µ m m= j C Marginal distribution of ε j : F εj (y j ) = exp exp µ y j ln ) µ µm ( M m= α jm µ ). European Transport Conference 2005 p.8/29

9 Normalization Marginal distribution: extreme value with location parameter ( M ) ln m= α jm µ and scale parameter µ. Therefore, ln E[ε j ] = ( M m= α jm µ ) + γ. Not necessarily constant across alternatives M Normalization: α jm = K, (K = makes sense) m= European Transport Conference 2005 p.9/29

10 Normalization Wen & Koppelman (200) the additional condition m α mn =, n provides a useful interpretation with respect to allocation of each alternative to each nest Condition is formally required to obtain an unbiased model The bias can be absorbed by the ASCs, if a full set is in the model Normalization for Ben-Akiva & Bierlaire: M m= µ µm αjm = c, j C European Transport Conference 2005 p.0/29

11 Variance-covariance Multinomial logit model Corr(U i, U j ) = 0 if i j. Nested logit model Corr(U i, U j ) = ( ( µ µ m ) 2 ) δ m (i, j) where δ m (i, j) = { if i and j are both in nest m 0 otherwise European Transport Conference 2005 p./29

12 Variance-covariance Cross-nested logit model (Papola, 2004) Corr(U i, U j ) = Conjecture M m= Exact for limit cases (NL) α im /2 α jm /2 Linear interpolation between limit cases Weights chosen so that, when i = j, ( ( µ ) ) 2. µ m M α im /2 α im /2 = M α im = m= m= European Transport Conference 2005 p.2/29

13 Variance-covariance CNL is equivalent to the model defined by U j = max m=,...,m Û jm where and ε jm are such that Û jm = V j + ln α jm µ + ε jm ε jm is independent of ε ln CDF of ε is F εm,...,ε Jm (y,..., y J ) = e ( P j C e µ my j ) µ µm European Transport Conference 2005 p.3/29

14 Variance-covariance Therefore, The error structure of a CNL is the maximum of error terms of underlying NL models Corr(U i, U j ) = Corr ( max m ε im, max m ε jm ) where Corr( ε im, ε jn ) = ( ( µ ) ) 2 δ m,n. µ m European Transport Conference 2005 p.4/29

15 Variance-covariance Equivalently, Corr(U i, U j ) = Corr ( ( ln αim max m µ + ε im ), max m ( ln αjm µ + ε jm ) where Corr(ε im, ε jn ) = ( ( µ ) ) 2 δ m,n. µ m Assumption of the conjecture: linear Actual relationship: maximum European Transport Conference 2005 p.5/29

16 Variance-covariance Computation of the true variance-covariance from the joint CDF Corr(U i, U j ) = 6µ2 π 2 R2 x i x j 2xixj F εi,εj (x i, x j )dx i dx j 6γ2 π 2, where F εi,ε j (x i, x j ) = e P M m= (α /µ im e x i ) µ m /µ +(α jm e xj ) µ m µ µ m. European Transport Conference 2005 p.6/29

17 Variance-covariance Parameter identification from a given variance-covariance System of equations: Corr(U i, U j ) = c ij J(J )/2 equations m α im = J equations Total: (J 2 + J)/2 equations Assume a full specification: JM α s, M µ s #equations = #unknowns if M = J/2. European Transport Conference 2005 p.7/29

18 Example A B 2 3 Scale parameters equal: µ A = µ B = µ m α A = α B3 = α B2 = α A2 European Transport Conference 2005 p.8/29

19 Example 0.2 Corr(,2) µ/µ m =0.9 Corr(,2) µ/µ m = α A α A2 µ/µ m =0.4 µ/µ m = Corr(,2) Corr(,2) Papola Exact α A2 α A2 European Transport Conference 2005 p.9/29

20 Example Papola s conjecture overestimates the correlation Overestimation increases with µ m Limit cases α A2 = 0 and α A = 0 are exact. European Transport Conference 2005 p.20/29

21 Example 2 A B 2 3 Scale parameters equal: µ A = µ B = µ m α B3 = α B = α A α B2 = α A2 European Transport Conference 2005 p.2/29

22 Example 2 Corr(,2) µ/µ m =0.9 Corr(,2) µ/µ m = α A α A α A α A 0.8 µ/µ m = µ/µ m =0. Corr(,2) Corr(,2) α A α A α A European Transport Conference 2005 p.22/29 α A

23 Example 2 Corr(,2) Corr(,2) µ/µ m =0.4, α A = α A2 µ/µ m =0.4, α A = α A2 Corr(,2) µ/µ m =0., α A = α A2 Corr(,2) µ/µ =0., α =0.5 m A Papola True α A2 European Transport Conference 2005 p.23/29

24 Example 2 Similar conclusions Papola s conjecture exact for α A = α A2, where the model is equivalent to a NL. European Transport Conference 2005 p.24/29

25 Route choice A a D -a O -a-b C D B -b E b Correlation matrix: a 0 b European Transport Conference 2005 p.25/29

26 Route choice A B C D E 2 3 European Transport Conference 2005 p.26/29

27 Route choice Parameters: 7 α s, µ/µ m Equations: 2 from the matrix, 3 from the normalization Missing: 3 equations. We arbitrarily set α A2 = /3 α C2 = /3 µ/µ m = 0.4 Compare the probabilities European Transport Conference 2005 p.27/29

28 Route choice µ/µ m =0.4, a=0.2 µ/µ m =0.4, a= Probability Probability Probability Probability b b Route Probability Route 2 Probability b b Papola Exact b Route b European Transport Conference 2005 p.28/29

29 Conclusions Normalization of CNL: formal motivation Correlation structure: correct formulation comparison with Papola s conjecture Advise: use the correct formulation Other issues: see next paper by Papola... European Transport Conference 2005 p.29/29

Normalization and correlation of cross-nested logit models

Normalization and correlation of cross-nested logit models E. Abbe M. Bierlaire T. Toledo August 5, 25 Abstract We address two fundamental issues associated with the use of cross-nested logit models. On