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Inferences from optimal filtering equation. (English) Zbl 1327.60097
Summary: The processing of stationary random sequences under nonparametric uncertainty is given by a filtering problem when the signal distribution is unknown. A useful signal \((S_n)_{n\geq1}\) is assumed to be Markovian. This assumption allows us to estimate the unknown \((S_n)\) using only an observable random sequence \((X_n)_{n\geq1}\). The equation of optimal filtering of such a signal has been obtained by A. V. Dobrovidov [Autom. Remote Control 44, No. 6, 757–768 (1983); translation from Avtom. Telemekh. 1983, No. 6, 85–98 (1983; Zbl 0539.93076)]. Our result states that when the unobservable Markov sequence is defined by a linear equation with Gaussian noise, the equation of optimal filtering coincides with both the classical Kalman filter and the conditional expectation defined by the theorem on normal correlation.

60G35 Signal detection and filtering (aspects of stochastic processes)
60J05 Discrete-time Markov processes on general state spaces
62M20 Inference from stochastic processes and prediction
62M05 Markov processes: estimation; hidden Markov models
Full Text: DOI arXiv
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