Finite Dimensional Filters Research Articles

On SDEs with marginal laws evolving in finite-dimensional exponential families

In the present paper, given a diffusion coefficient and a curve in an exponential family, we define a drift such that the density of the resulting diffusion process evolves in the prescribed exponential family according to the given curve. As an application to mathematical finance, we construct a family of stock price processes that are equivalent in discrete time while implying arbitrary prices for options written on them. As an application to nonlinear filtering, we construct nonlinear filtering problems admitting a finite-dimensional filter.

A New Suboptimal Approach to the Filtering Problem for Bilinear Stochastic Differential Systems

The aim of this paper is to present a new approach to the filtering problem for the class of bilinear stochastic multivariable systems, consisting in searching for suboptimal state-estimates instead of the conditional statistics. As a first result, a finite-dimensional optimal linear filter for the considered class of systems is defined. Then, the more general problem of designing polynomial finite-dimensional filters is considered. The equations of a finite-dimensional filter are given, producing a state-estimate which is optimal in a class of polynomial transformations of the measurements with arbitrarily fixed degree. Numerical simulations show the effectiveness of the proposed filter.

Finite-dimensional filters with nonlinear drift XIII: Classification of finite-dimensional estimation algebras of maximal rank with state space dimension five

Finite-dimensional filters with nonlinear drift XIII: Classification of finite-dimensional estimation algebras of maximal rank with state space dimension five

A stochastic jump inventory model with deteriorating items

In this paper, a single product continuous time stochastic inventory model for deteriorating items, driven by a. conditional Poisson process, is suggested. It is assumed the process Zt modulating the jump intensities of the Poisson process is a Markov chain. By observing the history of the inventory level, a finite dimensional filter for the conditional distribution of Zt is found. Further filters are found when the conditional Poisson process is replaced by an integer–valued random measure with predictable compensator depending on a right–constant sample paths process yt

Analysis of H/sub 2/ performance robustness with respect to disturbance model uncertainty

A method for evaluating the H/sub 2/ performance robustness of an LTI SISO feedback system with respect to parametric uncertainties in the disturbance model is developed. The disturbance is assumed to be colored noise obtained by passing standard white noise through a linear finite-dimensional filter with uncertain coefficients. The method is based on the converse of a spectral factorization problem. The main result obtained is an explicit formula for the output variance in terms of the uncertain parameters of the coloring filter. This result is illustrated on a ship roll stabilization problem.

A Gaussian-generalized inverse Gaussian finite-dimensional filter

We consider the filtering problem for a partially observable stochastic process {X n,Z n,Y n} n∈ N , solution to a nonlinear system of stochastic difference equations, which provides a stochastic modellization for both the mean and the variance of the Gaussian observation distribution. The noises in the equations are given by two sequences of independent Gaussian random variables and a sequence of independent gamma random variables. We are able to prove that there exists a finite-dimensional filter system for this model, since, for each n, the conditional distribution of ( X n , Z n ) given ( Y 0,…, Y n ) is that of a suitable bivariate Gaussian-generalized inverse Gaussian random variable.

State encoding of Hidden Markov linear prediction models

In this paper, we derive finite-dimensional non-linear filters for optimally reconstructing speech signals in Switched Prediction vocoders, Code Excited Linear Prediction (CELP) and Differential Pulse Code Modulation (DPCM). Our filter is an extension of the Hidden Markov filter.

Discrete time filters for doubly stochastic poisson processes and other exponential noise models

The well-known Kalman filter is the optimal filter for a linear Gaussian state-space model. Furthermore, the Kalman filter is one of the few known finite-dimensional filters. In search of other discrete-time finite-dimensional filters, this paper derives filters for general linear exponential state-space models, of which the Kalman filter is a special case. One particularly interesting model for which a finite-dimensional filter is found to exist is a doubly stochastic discrete-time Poisson process whose rate evolves as the square of the state of a linear Gaussian dynamical system. Such a model has wide applications in communications systems and queueing theory. Another filter, also with applications in communications systems, is derived for estimating the arrival times of a Poisson process based on negative exponentially delayed observations. Copyright © 1999 John Wiley & Sons, Ltd.

Finite-dimensional risk-sensitive filters and smoothers for discrete-time nonlinear systems

Finite-dimensional optimal risk-sensitive filters and smoothers are obtained for discrete-time nonlinear systems by adjusting the standard exponential of a quadratic risk-sensitive cost index to one involving the plant nonlinearity. It is seen that these filters and smoothers are the same as those for a fictitious linear plant with the exponential of squared estimation error as the corresponding risk-sensitive cost index. Such finite-dimensional filters do not exist for nonlinear systems in the case of minimum variance filtering and control.

New finite-dimensional filters for parameter estimation of discrete-time linear Gaussian models

The authors derive a new class of finite-dimensional recursive filters for linear dynamical systems. The Kalman filter is a special case of their general filter. Apart from being of mathematical interest, these new finite-dimensional filters can be used with the expectation maximization (EM) algorithm to yield maximum likelihood estimates of the parameters of a linear dynamical system. Important advantages of their filter-based EM algorithm compared with the standard smoother-based EM algorithm include: 1) substantially reduced memory requirements, and 2) ease of parallel implementation on a multiprocessor system. The algorithm has applications in multisensor signal enhancement of speech signals and also econometric modeling.

Optimal filtering of doubly stochastic auto-regressive processes

In this paper exact finite-dimensional filters are derived for a class of doubly stochastic auto-regressive models. The parameters of the doubly stochastic auto-regressive process vary according to a nonlinear function of a Gauss–Markov process. We develop a difference equation for the evolution of an unnormalized conditional density related to the state of the doubly stochastic auto-regressive process. We then give a characterization of the general solution followed by examples for which the state of the filter is determined by a finite number of sufficient statistics. These new finite-dimensional filters build upon the discrete-time Kalman filter.

Filtering for hybrid systems with Markovian and deterministic switching parameters (*)

We consider the filtering problem for mixed time Gaussian models with switching. In the case where the switching parameter is a Markov chain, we give (infinite dimensional) exact filters for the signal and for the switching parameter. In a second case, where the switching parameter is an arbitrary non-random (but unknown) function of time, we describe an EM (Expectation Maximization) algorithm for estimating the switching parameter, and derive finite dimensional exact filters necessary for carrying out our EM algorithm. All filters are derived using change of measure techniques

Optimal Filtering of Discrete-Time Hybrid Systems

This paper is concerned with discrete-time hybrid filtering of linear non-Gaussian systems coupled by a hidden switching process. An optimal control approach is used to derive a finite-dimensional recursive filter which is optimal in the sense of the most probable trajectory estimate. Models with unknown switching distributions are considered. Extensions to nonlinear hybrid systems are given. Numerical examples are considered and computational experiments are reported. These examples demonstrate that our filtering scheme outperforms popular filtering schemes available in the literature.

Exact filters for certain moments and stochastic integrals of the state of systems with Benes nonlinearity

Finite-dimensional filters for integrals and stochastic integrals of moments of the state for continuous-time nonlinear systems with Benes nonlinearity are derived. These filters can be used with the expectation maximization algorithm to compute maximum likelihood estimates of the model parameters.

Finite Dimensional Filters with Nonlinear Drift, XI: Explicit Solution of the Generalized Kolmogorov Equation in Brockett–Mitter Program

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei–Norman approach to solve the nonlinear filtering problem. In 1990, Tam, Wong, and Yau presented a rigorous proof of the Brocket–Mitter program which allows one to construct finite-dimensional recursive filters from finite–dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential equations and generalized Kolmogorov equation to which the robust Duncan–Mortenser– Zakai equation can be reduced. Thus there remains three fundamental problems in Brockett–Mitter program. The first is the problem of finding explicit solution to the generalized Kolmogorov equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite–dimensional estimation algebras. In this paper, we solve the first problem.

Finite dimensional filters for nonlinear stochastic difference equations with multiplicative noises

We consider the filtering problem for partially observable stochastic processes {X n,Y n} n∈ N , solutions to systems of stochastic difference equations. In the first part of the paper we shall present a simple constructive method to obtain finite dimensional filters in discrete time. Then, applying some well-known results, mainly on the product of independent positive random variables, we shall present new finite dimensional filters and interpret some known results in a more general setting.

Finite-Dimensional Filters for Passive Tracking of Markov Jump Linear Systems

We present new finite-dimensional filters for estimating the state of Markov jump-linear systems, given noisy measurements of the Markov chain. Discrete-time as well as continuous-time models are considered. A robust version of the continuous-time filters is used to derive a discretization which links the continuous and discrete-time results. Simulations compare the robust discretization with direct numerical solutions of the filtering equations. The new filters have applications in the passive tracking of maneuvering targets.

A finite-dimensional filter for hybrid observations

The paper considers a system whose dynamics are described in terms of the hybrid of a VAR(1)-type process and discrete state-space Markov chain. The paper provides a recursive expression for the conditional density/distributions of these two processes, given a set of observations which are a noisy linear combination of these two processes.

Comments on "Finite-dimensional filters with nonlinear drift" [and addendum

Over ten years ago Daum [1986-88] introduced nonlinear filters for discrete time measurements which are more general than the filters recently announced by Yau and Yau [1997] for continuous time observations. The authors say that, Recently Yau and Yau introduced a new direct method to solve the estimation problem... . They factored the problem into two parts: 1) the on-line solution of a finite system of ordinary differential equations (ODEs), and 2) the off-line calculation of the Kolmogorov equation. However, this direct method of factorization was introduced over ten years ago as indicated explicitly in the quote from Daum [1987]. Yau and Yau give a response to Daum's comments.

Addendum To "Finite-Dimensional Filters With Nonlinear Drift": A Response To F. Daum's Comments

Addendum To "Finite-Dimensional Filters With Nonlinear Drift": A Response To F. Daum's Comments