# 32116

## Автор(ов):

1

Параметры публикации

Тезисы доклада

## Название:

The equation of optimal filtering, Kalman's filter and Theorem on normal correlation

## ISBN/ISSN:

978-609-433-220-3

## Наименование конференции:

• 11th International Vilnius Conference on Probability and Mathematical Statistics (Вильнюс, 2014)

## Наименование источника:

• Proceedings of the 11th International Vilnius Conference on Probability and Mathematical Statistics (Vilnius, 2014)

• Вильнюс

## Издательство:

• TEV Publishers

2014

## Страницы:

182
Аннотация
An important objective of the classical processing of stationary random sequences under nonparametric uncertainty is the problem of filtering in case when the distribution of the underlying signal is unknown (see \cite{Dobrovidov:12}). Let us have partly observable Markov random sequence $(S_n,X_n)_{n\geq1}$, where $S = (S_n)_{n\geq1}$ is unobservable and $X=(X_n)_{n\geq1}$ is observable. The connection between them is given by the following expression $X_n=\varphi(S_n,\eta_n)$, where $(\eta_n\in\mathbb{R})_{n\geq1}$ is an i.i.d random sequence and $\varphi$ is some function. It is assumed that an unknown useful signal $(S_n)_{n\ge1}$ is Markovian. This allows us to construct an estimate of the useful signal, expressed in terms of the distribution density function of an observable random sequence $(X_n)_{n\ge1}$. The equation of the optimal Bayesian estimation \cite{Stratonovich:66} (so called an optimal filtering equation) of such signal has been received by \cite{Dobrovidov:1983}. Our main result is the following. It is proved that when the unobservable Markov sequence is defined by a linear equation with the Gaussian noise $\eta_n$, the equation of optimal filtering coincides both with the classical Kalman's filter and the conditional expectation defined by the theorem on normal correlation (see \cite{ShiryaevLiptser:2001}). As auxiliary result the explicit inversion of the covariance matrix is obtained. The latter matrix is a Toeplitz matrix \citep{Trench:2001} with the shift on the diagonal elements.

## Библиографическая ссылка:

Маркович Л.А. The equation of optimal filtering, Kalman's filter and Theorem on normal correlation / Proceedings of the 11th International Vilnius Conference on Probability and Mathematical Statistics (Vilnius, 2014). Вильнюс: TEV Publishers, 2014. С. 182.