The idea of using estimation algebras to construct finite-dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite-dimensional nonlinear filters. In his talk at the International Congress of Mathematics in 1983, Brockett proposed a classification of all finite-dimensional estimation algebras. Chiou and Yau classify all finite-dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two. In this paper we succeed in classifying all finite-dimensional estimation algebras of maximal rank with state-space dimension equal to three. Thus from the Lie algebraic point of view, we have now understood generically all finite dimensional filters with state-space dimension less than four.

Finite-Dimensional Filters with Nonlinear Drift VIII: Classification of Finite-Dimensional Estimation Algebras of Maximal Rank with State-Space Dimension 4

The nonlinear filtering problem involves the estimation of a stochastic process x = {xt} (called the signal or state process) that cannot be observed directly. Information containing x is obtained from observations of a related process y = {yt} (the observation process). The goal of nonlinear filtering is to determine the conditional expectations of the form E [ φ(xt) : ys, 0 ≤ s ≤ t ] , or perhaps even the computation of the entire conditional density ρ(t, x) of xt given the observation history {ys, 0 ≤ s ≤ t}. When the observations are received sequentially, as in many practical applications, it is preferable that this computation be performed recursively in terms of a statistic θ = {θt}, which can be updated by using only the latest observations

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Real time solution of nonlinear filtering problem without memory I

Ever since the concept of estimation algebra was first introduced by Brockett and Mitter independently, it has been playing a crucial role in the investigation of finite-dimensional nonlinear filters. Researchers have classified all finite-dimensional estimation algebras of maximal rank with state space less than or equal to three. In this paper we study the structure of quadratic forms in a finite-dimensional estimation algebra. In particular, we prove that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂f

 j
 /∂x

 i
 −∂f

 i
 /∂x

 j
 )matrix, wheref denotes the drift term, is a linear matrix in the sense that all the entries in Ω are degree one polynomials. This theorem plays a fundamental role in the classification of finite-dimensional estimation algebra of maximal rank.

Finite-dimensional filters with nonlinear drift, VI: Linear structure of Ω

It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the Duncan–Mortensen–Zakai (DMZ) equation in real time and in a memoryless manner. The purpose of this paper is to show that, under very mild conditions (which essentially say that the growth of the observation $|h|$ is greater than the growth of the drift $|f|$), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution $u_R$ of the DMZ equation on the ball $B_R$ with $u_R|_{\pat B_R}=0$. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution $u_R$ can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on $B_R$. In theory, our algorithm can solve basically all engineering problems in real time. Specifically, we show that the solution obtained from ou...

Real Time Solution of the Nonlinear Filtering Problem without Memory II

We consider the Duncan-Mortensen-Zakai (DMZ) equation for the Kalman-Bucy filtering system and Benes filtering system. We show that this equation can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation, Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.

Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system

The idea of using estimation algebras to construct finite- dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite-dimensional nonlinear filters. In his talk at the International Congress of Mathematics in 1983, Brockett proposed classifying all finite-dimensional estimation algebras. In this paper, all finite-dimensional algebras with maximal rank are classified if the dimension of the state space is less than or equal to two. Therefore, from the Lie algebraic point of view, all finite-dimensional filters are understood generically in the case where the dimension of state space is less than three.

Wen-Lin Chiou

Papers

Finite-Dimensional Filters with Nonlinear Drift II: Brockett's Problem on Classification of Finite-Dimensional Estimation Algebras