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Publication# Normalization and correlation of cross-nested logit models

Abstract

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, Papolas 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 Papolas conjecture.

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Ontological neighbourhood

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The course introduces the theoretical foundations to choice modeling and describes the steps of operational modeling.

Discrete choice

In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods (e.g. first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis.

Mode choice

Mode choice analysis is the third step in the conventional four-step transportation forecasting model of transportation planning, following trip distribution and preceding route assignment. From origin-destination table inputs provided by trip distribution, mode choice analysis allows the modeler to determine probabilities that travelers will use a certain mode of transport. These probabilities are called the modal share, and can be used to produce an estimate of the amount of trips taken using each feasible mode.

Correlation

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

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