Non-regular processes and singular Kalman filtering

Contrary to the continuous-time case, a discrete-time process y can be represented by minimal linear models (see (1.1) 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 has been studied for several decades in connection with the so-called “invariant directions” of the Riccati equation. For a singular D, a reduction in the order of the Riccati equation is in general possible. In this paper, we provide an explanation of this phenomenon from the classical point of view of “zero flippin” 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, zeroflipping 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.