Optimal linear estimator of origin-destination flows with redundant data

Suppose given a network endowed with a multiflow. We want to estimate some quantities connected with this multiflow, for instance the value of an s-t flow for one of the sources-sinks pairs s-t, but only measures on some arcs are available, at least on one s-tcocycle (set of arcs having exactly one endpoint in a subset X of vertices with s∈X and t∉X). These measures, supposed to be unbiased, are random variables whose variances are known. How can we combine them optimally in order to get the best estimator of the value of the s-t flow? This question arises in practical situations when the OD matrix of a transportation network must be estimated. We will give a complete answer for the case when we deal with linear combinations, not only for the value of an s-t flow but also for any quantity depending linearly from the multiflow. Interestingly, we will see that the Laplacian matrix of the network plays a central role.