## Improved Estimation of Travel Demand from Traffic Counts by a New Linearization of the Network Loading Map

### Authors

G Flötteröd, M Bierlaire, EPFL, TRANSP-OR Laboratory, CH

### Description

In the context of origin/destination matrix estimation from traffic counts, we present a new linearization of the network loading map that is superior to the usual proportional assignment in congested conditions.

### Abstract

This work is motivated by the widely discussed problem of estimating travel demand from traffic counts. Travel demand is typically given in terms of a (time-dependent) origin/destination (OD) matrix, and the vast majority of OD matrix estimators identifies an OD matrix that minimizes a weighted sum of the traffic count reproduction error and the deviation from a prior OD matrix. The computational feasibility of these approaches depends on the possibility to linearly predict the effect of OD flow variations on the scalar-valued objective function. A recently developed advancement of the established OD matrix estimation methodology even allows for the mathematically consistent calibration of microsimulation-based demand models from traffic counts. The new method also contains a linearization step, which approximates the effect of individual-level choice variations on the traffic counts' log-likelihood function.

The linearization of a scalar-valued function of network conditions with respect to demand levels is usually decomposed in two steps. First, the mapping of demand levels on link flows is approximated by a proportional assignment. Second, the scalar function is analytically linearized with respect to the link flows. The concatenation of both approximations yields the desired linear mapping. This article presents a novel linearization approach that overcomes the inadequacy of a proportional assignment in congested conditions. The basic idea is to run a recursive regression algorithm in parallel to the iterative simulation that fits a linear model of the scalar function given the demand levels. The main challenge in this approach is to deal with the large number of explanatory variables. This problem is solved in the following way.

Given a route choice model and the previous iteration's network conditions, the demand is converted into "desired" path flows. These flows are treated as independent random variables with variances proportional to their flow levels. Implications of this assumption are discussed below. Given the previous iteration's travel times, the desired path flows are then linearly superposed in the network in order to obtain "desired" link flows. The variance of a desired link flow equals the sum of the variances of all desired path flows across that link, and the covariance of two desired link flows equals the sum of the variances of all desired path flows that cross both links. Assuming a strong causal relation between desired link flows and simulated link flows that result when the demand (desired path flows) is loaded on the network, the principal components (PCs) of the desired link flows are identified every few iterations of the simulation. The explanatory variables used in the continuously running regression-based linearization are then generated by (i) turning demand levels into desired link flows and (ii) projecting the desired link flows on the previously identified PCs. That is, the regression observes the true path flows correlation structure, and the assumption of independent desired path flows only affects the generation of the PCs.

Preliminary results are obtained for a set of 379972 synthetically generated paths on a 2459 link network of the Greater Berlin area. PCs are extracted by a numerical procedure the complexity of which is independent of the network size, scales linearly with the number of paths in the network, and does not require an explicit representation of the covariance matrix. To quantify the results, we express the explanatory power of a set of PCs by the sum of their eigenvalues divided by the trace of the covariance matrix of the desired link flows. This value is zero for an empty set of PCs and becomes one when the PCs perfectly represent the covariance structure of the desired link flows. The results indicate that the 10 largest eigenvectors already explain 10% of the covariance structure, 100 eigenvectors explain 50%, and less than 600 eigenvectors are needed to explain 90% of the covariance structure.

Motivated by these observations, our research proceeds as follows. First, we evaluate the explanatory power of a PC-based recursive regression for the approximation of a log-likelihood function of traffic-counts. We expect further dimensional reductions to be possible in this problem domain because of the locality of flow interactions around sensors. Second, we incorporate the new linearization in an existing calibration tool for dynamic traffic microsimulators and evaluate the improvement in calibration precision for different data sets and in conjunction with different microsimulators. Finally, observing that our PC analysis constitutes a very simple ?demand compression? technique, we investigate its use as an analysis tool for the identification of redundant demand parameters and components.

#### Publisher

Association for European Transport