On the Reference Probability Approach to the Equations of Non-Linear Filtering

The filtering of diffusions from their noisy observations is considered in this paper. The introduction of various reference probability measures and the use of a stochastic Feynman-Kac formula is shown to lead to new and already known filtering equations. In some cases, which include extensions of the Benes filtering problem, the new equations we propose possess a nice Gaussian solution, yielding an explicit finite dimensional filter     

Stochastic evolution equations with general white noise disturbance

Existence and uniqueness theorems are proved for a general class of stochastic linear abstract evolution equations, with a general type of stochastic forcing term. The abstract evolution equation is modeled using an evolution operator (or 2-parameter semigroup) approach and this includes linear partial differential equations and linear differential delay equations. The stochastic forcing term is modeled by defining an Itô stochastic integral with respect to a Hilbert space-valued orthogonal increments process, which can be used to model both Gaussian and non-Gaussian white noise processes. The theory is illustrated by examples of stochastic partial differential equations and delay equations, which arise in filtering problems for distributed and delay systems.     

Optimal speed of detection in generalized Wiener disorder problems

We define a general Wiener disorder problem in which a sudden change in a time profile of unknown size has to be detected in white noise of small intensity. Since both the time of the change and its size are unknown, this problem is considerably harder than standard Wiener disorder problems where the size of the change is assumed to be known a priori. We formulate the problem within the Bayesian framework of nonlinear filtering theory, and use Strassen's functional law of the iterated logarithm to bound stochastic measures which arise in the nonlinear filtering equations. This leads to explicit expressions for the detection delay in the optimal statistics for small noise intensities, and we indicate how these can be used to analyse the detection delays of recursive suboptimal detection algorithms for this problem.     

Nonlinear filtering of semi-Dirichlet processes

Herein, we consider the nonlinear filtering problem for general right continuous Markov processes, which are assumed to be associated with semi-Dirichlet forms. First, we derive the filtering equations in the semi-Dirichlet form setting. Then, we study the uniqueness of solutions of the filtering equations via the Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. Furthermore, we investigate the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.     

Partially Observed Time-Inconsistency Recursive Optimization Problem and Application

In this paper, we study a partially observed recursive optimization problem, which is time inconsistent in the sense that it does not admit the Bellman optimality principle. To obtain the desired results, we establish the Kalman–Bucy filtering equations for a family of parameterized forward and backward stochastic differential equations, which is a Hamiltonian system derived from the general maximum principle for the fully observed time-inconsistency recursive optimization problem. By means of the backward separation technique, the equilibrium control for the partially observed time-inconsistency recursive optimization problem is obtained, which is a feedback of the state filtering estimation. To illustrate the applications of theoretical results, an insurance premium policy problem under partial information is presented, and the observable equilibrium policy is derived explicitly.     

Extension of the Kalman–Bucy Filter to Elementary Linear Systems with Fractional Brownian Noises

We investigate the optimal filtering problem in the simplest Gaussian linear system driven by fractional Brownian motions. At first we extend to this setting the Kalman–Bucy filtering equations which are well-known in the specific case of usual Brownian motions. Closed form Volterra type integral equations are derived both for the mean of the optimal filter and the variance of the filtering error. Then the asymptotic stability of the filter is analyzed. It is shown that the variance of the filtering error converges to a finite limit as the observation time tends to infinity. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear filtering with fractional brownian motion

A Kalman type system of integral equations is obtained for the linear filtering problem in which the noise generating the signal is a fractional Brownian motion with long-range dependence. The error in applying the usual Kalman filter to this problem is determined explicitly for a simple example     

Particle filtering in the Dempster–Shafer theory

This paper derives a particle filter algorithm within the Dempster–Shafer framework. Particle filtering is a well-established Bayesian Monte Carlo technique for estimating the current state of a hidden Markov process using a fixed number of samples. When dealing with incomplete information or qualitative assessments of uncertainty, however, Dempster–Shafer models with their explicit representation of ignorance often turn out to be more appropriate than Bayesian models.The contribution of this paper is twofold. First, the Dempster–Shafer formalism is applied to the problem of maintaining a belief distribution over the state space of a hidden Markov process by deriving the corresponding recursive update equations, which turn out to be a strict generalization of Bayesian filtering. Second, it is shown how the solution of these equations can be efficiently approximated via particle filtering based on importance sampling, which makes the Dempster–Shafer approach tractable even for large state spaces. The performance of the resulting algorithm is compared to exact evidential as well as Bayesian inference.     

A finitely additive white noise approach to nonlinear filtering

An approach to nonlinear filtering theory is developed in which finitely additive white noise replaces the Wiener process in the observation process model. The important case when the signal is a Markov process independent of the noise is investigated in detail. The theory turns out to be simpler than the current theory based on the stochastic calculus. Stochastic partial differential equations are replaced by partial differential equations in which the observation (in the finitely additive set up) occurs as a parameter. Theorems on existence and uniqueness of solutions are obtained. The white noise approach has the advantage that it provides a robust solution to the filtering problem. Furthermore, the robust theory based on the Ito calculus can be recovered from the results of this paper.     

General approach to filtering with fractional brownian noises — application to linear systems

At first a general approach is proposed to filtering in systems where the observation noise is a fractional Brownian motion. It is shown that the problem can be handled in terms of some appropriate semimartingale and analogs of the classical innovation process and fundamental filtering theorem are obtained. Then the problem of optimal filtering is completely solved for Gaussian linear systems with fractional Brownian noises. Closed form simple equations are derived both for the mean of the optimal filter and the variance of the filtering error. Finally the results are explicited in various specific cases     

Dynamics for Controlled 2D Generalized MHD Systems with Distributed Controls

We study the dynamics of a piecewise (in time) distributed optimal control problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal distributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of Generalized MHD equations are derived. Next, the existence of a solution of optimal control problemis proved also optimality system is derived. Finally, the long-time decay properties for the optimal solutions is established.     

Filtering Problem for Discrete Volterra Equations with Combined Disturbances

Abstract

A minimax filtering problem for discrete Volterra equations with combined noise models is considered. The combined models are defined as the sums of uncertain bounded deterministic functions and stochastic white noises. However, the corresponding variational problem turns out to be very difficult for direct solution. Therefore, simplified filtering algorithms are developed. The levels of nonoptimality for these simplified algorithms are introduced as the ratios of the filtering performances for the simplified and optimal estimators.

In opposite to the original variational problem, these levels can be easily evaluated numerically. Thus, simple filtering algorithms with guaranteed performance are obtained. Numerical experiments confirm the efficiency of our approach.     

Solving approximately an optimal nonlinear filtering problem for stochastic differential systems by statistical modeling

An algorithm for solving an optimal nonlinear filtering problem by statistical modeling is proposed. It is based on reducing the filtration problem to an analysis of stochastic systems with terminating and branching paths using the fact that the Duncan-Mortensen-Zakai equations and the generalized Fokker-Planck-Kolmogorov equation are similar in structure. This problem of analysis can be solved approximately by numerical methods for solving stochastic differential equations and modeling inhomogeneous Poisson flows.     

Non Parametric Distributed Inference in Sensor Networks Using Box Particles Messages

This paper deals with the problem of inference in distributed systems where the probability model is stored in a distributed fashion. Graphical models provide powerful tools for modeling this kind of problems. Inspired by the box particle filter which combines interval analysis with particle filtering to solve temporal inference problems, this paper introduces a belief propagation-like message-passing algorithm that uses bounded error methods to solve the inference problem defined on an arbitrary graphical model. We show the theoretic derivation of the novel algorithm and we test its performance on the problem of calibration in wireless sensor networks. That is the positioning of a number of randomly deployed sensors, according to some reference defined by a set of anchor nodes for which the positions are known a priori. The new algorithm, while achieving a better or similar performance, offers impressive reduction of the information circulating in the network and the needed computation times.     

Adaptive filtering

Summary. In this paper, the adaptive filtering method is introduced and analysed. This method leads to robust algorithms for the solution of systems of linear equations which arise from the discretisation of partial differential equations with strongly varying coefficients. These iterative algorithms are based on the tangential frequency filtering decompositions (TFFD). During the iteration with a preliminary preconditioner, the adaptive test vector method calculates new test vectors for the TFFD. The adaptive test vector iterative method allows the combination of the tangential frequency decomposition and other iterative methods such as multi-grid. The connection with the TFFD improves the robustness of these iterative methods with respect to varying coefficients. Interface problems as well as problems with stochastically distributed properties are considered. Realistic numerical experiments confirm the efficiency of the presented algorithms. Received June 26, 1996 / Revised version received October 7, 1996     

A measure space approach to optimal source placement

The problem of optimal placement of point sources is formulated as a distributed optimal control problem with sparsity constraints. For practical relevance, partial observations as well as partial and non-negative controls need to be considered. Although well-posedness of this problem requires a non-reflexive Banach space setting, a primal-predual formulation of the optimality system can be approximated well by a family of semi-smooth equations, which can be solved by a superlinearly convergent semi-smooth Newton method. Numerical examples indicate the feasibility for optimal light source placement problems in diffusive photochemotherapy.     

Mean-Square Filtering Problem for Discrete Volterra Equations

Abstract

The mean-square filtering problem for the discrete Volterra equations is a nontrivial task due to an enormous amount of operations required for the implementation of optimal filter. A difference equation of a moderate dimension is chosen as an approximate model for the original system. Then the reduced Kalman filter can be used as an approximate but efficient estimator. Using the duality theory of convex variational problems, a level of nonoptimality for the chosen filter is obtained. This level can be efficiently computed without exactly solving the full filtering problem.     

The Role of Frame Force in Quantum Detection

A general method is given to solve tight frame optimization problems, borrowing notions from classical mechanics. In this article, we focus on a quantum detection problem, where the goal is to construct a tight frame that minimizes an error term, which in quantum physics has the interpretation of the probability of a detection error. The method converts the frame problem into a set of ordinary differential equations using concepts from classical mechanics and orthogonal group techniques. The minimum energy solutions of the differential equations are proven to correspond to the tight frames that minimize the error term. Because of this perspective, several numerical methods become available to compute the tight frames. Beyond the applications of quantum detection in quantum mechanics, solutions to this frame optimization problem can be viewed as a generalization of classical matched filtering solutions. As such, the methods we develop are a generalization of fundamental detection techniques in radar.      

Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.     

Optimal finite-dimensional solution for a class of nonlinear observation problems

The filtering problem in a differential system with linear dynamics and observations described by an implicit equation linear in the state is solved in finite-dimensional recursive form. The original problem is posed as a deterministic fixed-interval optimization problem (FIOP) on a finite time interval. No stochastic concepts are used. Via Pontryagin's principle, the FIOP is converted into a linear, two-point boundary-value problem. The boundary-value problem is separated by using a linear Riccati transformation into two initial-value problems which give the equations for the optimal filter and filter gain. The optimal filter is linear in the state, but nonlinear with respect to the observation. Stability of the filter is considered on the basis of a related properly linear system. Three filtering examples are given.