A metric for ARMA processes

Autoregressive-moving-average (ARMA) models seek to express a system function of a discretely sampled profess as a rational function in the z-domain. Treating an ARMA model as a complex rational function, we discuss a metric defined on the set of complex rational functions, We give a natural measure of the "distance" between two ARMA processes. Our paper concentrates on the mathematics behind the problem and shows that the various algebraic structures endow our choice of metric with some interesting and remarkable properties, which we discuss. We suggest that the metric can be used in at least two circumstances: i) in which we have signals arising from various models that are unknown (so we construct the distance matrix and perform cluster analysis) and ii) where there are several possible models M-i, all of which are known, and we wish to find which of these is closest to an observed data sequence modeled as M.