Normalization and Correlation of Cross-nested Logit Models

Normalization and Correlation of Cross-nested Logit Models


M Bierlaire, EPFL, CH; E Abbé, T Toledo, Massachusetts Institute of Technology, US


We provide a detailed analysis about the normalization and the correlation structure of Cross-Nested Logit models. We propose a method to define a CNL with a given correlation structure.


Demand analysis is more and more critical in the transportation context. Discrete choice models methodology provide an appropriate framework to capture the behaviour of the actors of transportation systems and, consequently, to forecast the demand. Recently, the cross-nested logit (CNL) model has received significant attention in the literature to capture decisions such as mode choice (Vovsha, 1997), departure time choice (Small, 1987) and route choice (Vovsha and Bekhor, 1998). Its general structure is appealing. Indeed, this model has a closed form probability formula, and allows for a wide range of correlation structures to be modeled. As shown by Bierlaire (forthcoming), various instances of the Cross-Nested Logit (CNL) model have been proposed in the literature. They are more or less the same, some being more specific as they constrain some parameters to fixed values.
The issue of normalization of these models is not easy in practice. We provide a detailed analysis of this problem, and show that the normalization proposed by Wen and Koppelman (2001) is indeed correct. We emphasize the relation between the parameters of the CNL and the Alternative Specific Constants.
In the second part of the paper, we analyze the correlation structure of the CNL. We show that the conjecture by Papola (2004) is erroneous. In fact, Papola?s result is based on the assumption that the relation between the underlying NL correlations and the overall CNL correlations is linear. We show in the paper that this relation is actually based on a maximum.
We propose a method to define a CNL with a given correlation structure. This is particularly important in applications such as route choice analysis, where the correlation of the alternatives can be somehow obtained from the network topology. We illustrate our method on a simple example, where we compare the correct results with Papola?s conjecture.

Bierlaire, M. (forthcoming). A theoretical analysis of the cross-nested logit model, Annals of operations research . Accepted for publication.
McFadden, D. (1978). Modelling the choice of residential location, in A. Karlquist et al. (ed.), Spatial interaction theory and residential location, North-Holland,Amsterdam, pp. 75?96.
Papola, A. (2004). Some developements on the cross-nested logit model, Transportation Research B 38(9): 833?851.
Small, K. (1987). A discrete choice model for ordered alternatives, Econometrica 55(2): 409?424.
Vovsha, P. (1997). Cross-nested logit model: an application to mode choice in the Tel-Aviv metropolitan area, Transportation Research Record 1607: 6?15.
Vovsha, P. and Bekhor, S. (1998). The link-nested overcoming the route overlapping problem, Transportation 1645: 133?142.
Wen, C.-H. and Koppelman, F. S. (2001). The generalized Transportation Research B 35(7): 627?641.


Association for European Transport