On the Mayne-Fraser smoothing formula and stochastic realization theory for nonstationary linear stochastic systems

Optimal Estimation With Missing Observations via Balanced Time-Symmetric Stochastic Models

“…In [3], [4] the Mayne-Fraser formula was analyzed in the context of stochastic realization theory and was shown that it can be formulated in terms a forward and a backward Kalman filter. In a subsequent series of papers, Pavon [5], [6] addressed in a similar manner the hitherto challenging problem of interpolation [7], [8], [9], [10].…”

“…We study the relationship between these systems and the corresponding processes. In particular, we recover as a special case certain results of stochastic realization theory [11], [5], [6], [4] from the 1970's using a novel procedure. This theory provides a normalized and balanced version of the forward-backward duality which is essential for our new formulation of the two-filter Mayne-Fraser-like formula uniformly applicable to intervals with or without observations.…”

“…The purpose of this paper is to develop such a two-filter formula that is universally applicable to smoothing and interpolation based on general records with missing observations. In [3], [4] the Mayne-Fraser formula was analyzed in the context of stochastic realization theory and was shown that it can be formulated in terms a forward and a backward Kalman filter. In a subsequent series of papers, Pavon [5], [6] addressed in a similar manner the hitherto challenging problem of interpolation [7], [8], [9], [10].…”

On time-reversibility of linear stochastic models

“…Time reversal of stochastic systems is central in stochastic realization theory (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8]), filtering (see [9]), smoothing (see [10], [11], [12]) and system identification. The principal construction is to model a stochastic process as the output of a linear system driven by a noise process which is assumed to be white in discrete time, and orthogonal-increment in continuous time.…”

“…We study the relationship between these systems and the corresponding processes. In particular, we recover as a special case certain results of stochastic realization theory ( [1], [5], [10]) from the 1970's using a novel procedure.…”