A maximum likelihood model for estimating origin-destination matrices

We describe a model for estimating an origin-destination matrix from an observed sample matrix, when the volumes on a subset of the links of the network and/or the total productions and attractions of the zones are known. The elements of the observed sample matrix are assumed to be integers that are obtained from independent Poisson distributions with unknown means. A maximum likelihood model is formulated to estimate these means, yielding an estimation of the "true" origin- destination matrix which is consistent with the observed link volumes. Conditions for existence and uniqueness of a solution are discussed. A solution algorithm based on the cyclic coordinate descent method is developed and its convergence properties are analyzed. The special case of the matrix estimation problem, in which marginal totals are given instead of link volumes, is considered separately; a numerical example is used to illustrate the problem. Using results about the asymptotic behavior of the distribution of the likelihood function, tests may be derived that allow statistical inferences on the consistency of the available data. Finally, an extension of the model is studied in which the observed volumes are Poisson-distributed as well.