Graph structure and recursive estimation of noisy linear relations
Résumé
This paper examines estimation problems specified by noisy linear relations describing either dynamical models or measurements. Each such problem has a graph structure, which can be exploited to derive recursive estimation algorithms only when the graph is acyclic, i.e. when it is obtained by combining disjoint trees. Aggregation techniques appropriate for reducing an arbitrary graph to an acyclic one are presented. The recursive maximum likelihood estimation procedures that we present are based on two elementary operations, called reduction and extraction, which are used to compress successive observations and discard unneeded variables. These elementary operations are used to derive filtering and smoothing formulas applicable to both linear and arbitrary trees, which are in turn applicable to estimation problems in settings ranging from 1-D descriptor systems to 2-D difference equations to multiscal models of random fields. These algorithms can beviewed as direct generalizations to a far richer setting of Kalman filtering and both two-filter and Rauch-Tung-Striebel smoothing for standard causal state space models.