An EM Approach to OD Matrix Estimation

Consider a \black box" having I input channels and J output channels. Each arrival on an input channel gets routed through the black box and appears on an output channel. The system is monitored for a xed time period and a record is made of the number of arrivals on each input channel and the number of departures on each output channel. The OD (origination-destination) matrix estimation problem is to estimate, for each i and j, the number of arrivals on channel i that depart on channel j. We introduce a Poisson stochastic model and employ the EM algorithm to produce high likelihood estimates. In the case of estimation based on observations over a single time-period, we analyze in detail the xed points of the EM algorithm showing that every vertex of a certain polytope of feasible matrices is a xed point and identifying a speci c interior xed point which is a saddle point for the likelihood function. Research supported by AFOSR through grant AFOSR-91-0359.