Multinomial logit modal choice models for aggregated data with Box-Cox transformations

Introduction

This project illustrates the implementation of (conditional) multinomial logit mode choice models for aggregated data with Box-Cox transform of the explanatory variable(s).

Input data

The model can be used when observations are not individual but aggregated. This is for instance the case when one observe transported quantities, such as tons, between origins and destinations. The provided sample dataset (sampleData.Rda) contains, for every origin-destination pair, the following data:

• The ID of the origin ORG and the ID of the destination DST
• The unit transportation cost between ORG and DST for three modes
• The total transit time (duration) between ORG and DST for three modes
• The observed transported quantities between ORG and DST for three modes.

The transportation modes are:

• 1 for Trucks (road network)
• 2 for Barges (inland waterways network)
• 3 for Trains (railroad network)

Note that all transportation modes are not necessarily available between all ORG-DST pairs. This is particularly the case for barges, as the inland waterways network is not available everywhere.

Objective of the models

Given the input data, the proposed multinomial logit models try to estimate the quantities transported between every ORG-DST pair using one or two explanatory variables, which are Box-Cox transformed.

Used tools

The models are written in R and use the ‘mlogit’ and ‘mnlogit’ packages.

Why Box-Cox transformations?

The Box-Cox transformation is a well-known power transform. In the framework of modal choice models, it can be useful in order to:

• Improve the log-likelihood of the model.
• Obtain the expected signs of the estimators when explanatory variables are correlated.

The Box-Cox transform is rather easy to apply once the “lambda” to use is known. Obtaining the optimal value(s) of the lambda(s) is, however, not immediate.

Two models are provided in this project: the first one is an univariate model for which only one lambda must be determined. The second is a bivariate model, for which a combination of two lambdas’ (one for each explanatory variable) must be found.

Model 1: The univariate case

The "Logit1.R" script illustrates a simple univariate model that uses the unit transportation cost (or transit time) as only explanatory variable.

"BoxCoxLogit1.R" uses the same explanatory variable, but applies a Box-Cox transform to it. The aim is here to increase the log-likelihood of the model using an “optimal” lambda. Therefore, all the values in the [-2,2] range are tested, with a step of 0.1.

This model is described and used with real datasets in Jourquin B., Estimating Elasticities for Freight Transport Using a Network Model: An Applied Methodological Framework, Journal of Transportation Technologies, 2019, 9, 1-13.

Model 2: The bivariate case

The "Logit2.R" script illustrates the case of a model with two explanatory variables: the unit transportation cost and the transit time (duration). It shows that, without a propre transform of the variables, the sign of an estimator can be wrong. In this case, the estimator for the "duration" variable is positive. This is not expected as the modal share of a transportation mode should decrease when its transit time increases. This unexpected sign is related to the fact that both explanatory variables are heavily correlated.

"BoxCoxLogit2.R" applies a Box-Cox transform, looking for the "best" lambda's, not only to improve the log-likelihood of the model, but also to obtain the expected signs for the estimators.

This model is thoroughly described and discussed when used with real World datasets in Jourquin B. and Beuthe M., Cost, transit time and speed elasticity calculations for the European continental freight transport, Transport Policy, 83, 1-12, 2019.

Output

Both scripts print the estimated model (values of the estimators, t-values, log-likelihood...), the 100 first rows of the input data along with the related estimated quantities and the correlations between observed and estimated quantities.

Use of the estimators in the Nodus software

Nodus is a transportation network modeling software especially designed for multimodal and intermodal freight transport. It offers the possibility to develop user-defined modal choice plugins. It can, for instance, use the estimated parameters by the two models presented in this project.

To start with, the user has to gather the input data needed to estimate the parameters (a dataset which structure is similar to the one of “sampleData.Rda”, but specific to the case to handle). The unit transportation costs and transit times per mode can be obtained by means of an assignment with “save paths” checked (“detailed paths” is not needed) in the Nodus "assignment" panel. In the “header” table resulting from this assignment, one can find, for each OD pair (and each “group of commodities”) the unit transportation costs and transit times (note that these are expressed in seconds and not hours). The observed quantities per mode are to be gathered from the (modal) OD matrixes that are used.

Once the input data prepared, the R script can be run to obtain the estimators. If "printNodusEstimators" is set to TRUE in the R script, the estimators are printed in a format that can directly be cut&pasted in a Nodus “.costs” file. Example:

lambda.0=0.7
(Intercept).2.0=-4.14188571102275
(Intercept).3.0=-4.34237545509826
cost.1.0=-0.709553978510067
cost.2.0=-0.709553978510067
cost.3.0=-0.709553978510067

The "plugin" directory contains two user-defined modal choice plugins for Nodus that can make use of these parameters.

Further improvements

The strategies used to find the optimal lambda's in the two models needs to be improved as the proposed approaches are simple but time consuming.