Silverman Algorithm and the structure of Discrete-Time Stochastic Systems

It is a well-known, yet poorly understood fact that, contrary to the continuous-time case, the same discrete-time process y can be represented by minimal linear models (see (1.1a), (1.1b) below) which may either have a non-singular or a singular D matrix. In fact, models with D=0 have been commonly used in the statistical literature. On the other hand, for models with a singular D matrix the Riccati difference equation of Kalman filtering involves in general the pseudo-inversion of a singular matrix. This “cheap filtering” problem, dual to the better known “cheap control” problem, has been studied for several decades in connection with the so-called “invariant directions” of the Riccati equation. For a singular D, a reduction of the order of the Riccati equation is in general possible. The reasons for such a reduction do not seem to be completely clear either. In this paper we provide an explanation of this phenomenon from the classical point of view of “zero flipping” among minimal spectral factors. Changing D's occurs whenever zeros are “flipped” from z=∞ to their reciprocals at z=0. It is well known that for finite zeros, the zero-flipping process takes place by multiplication of the underlying spectral factor by a suitable rational all-pass matrix function. For infinite zeros, zero flipping is implemented by a dual version of the Silverman structure algorithm. Using this interpretation, we derive a new algorithm for filtering of non-regular processes, based on a reduced-order Riccati equation. We also obtain a precise characterization of the reduction of the order of the Riccati equation which is afforded by zeros either at z=∞ or at the origin. This order reduction has traditionally been associated with the study of invariant directions, a point of view which, as we show, does not capture the essence of the phenomenon.