Euclidean Quantum Mechanics and Universal Nonlinear Filtering

An important problem in applied science is the continuous nonlinear filtering problem, i.e., the estimation of a Langevin state that is observed indirectly. In this paper, it is shown that Euclidean quantum mechanics is closely related to the continuous nonlinear filtering problem. The key is the configuration space Feynman path integral representation of the fundamental solution of a Fokker-Planck type of equation termed the Yau Equation of continuous-continuous filtering. A corollary is the equivalence between nonlinear filtering problem and a time-varying Schr\"odinger equation.Comment: 19 pages, LaTeX, interdisciplinar

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces

Development of Nonlinear Filtering Algorithms of Digital Half-Tone Images

This chapter is devoted to solving the problem of algorithms and structures investigations for Radio Receiver Devices (RRD) with the aim of the nonlinear filtering of Digital Half-Tone Images (DHTI) representing the discrete-time and discrete-value random Markovian process with a number of states greater than two. At that, it is assumed that each value of the DHTI element is represented by the binary g-bit number, whose bits are transmitted via digital communication links in the presence of Additive White Gaussian Noise (AWGN). The authors present the qualitative analysis of the optimal DHTI filtering algorithm. The noise immunity of the optimal radio receiver device for the DHTI filtering with varying quantization and dimension levels is investigated

Dynamic Models and Nonlinear Filtering of Wave Propagation in Random Fields

In this paper, a general model of wireless channels is established based on the physics of wave propagation. Then the problems of inverse scattering and channel prediction are formulated as nonlinear filtering problems. The solutions to the nonlinear filtering problems are given in the form of dynamic evolution equations of the estimated quantities. Finally, examples are provided to illustrate the practical applications of the proposed theory.Comment: 12 pages, 1 figur

Phase Transitions in Nonlinear Filtering

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur. We also discuss closely related problems in the setting of conditional Markov random fields.Comment: 51 page

The Hitchhiker's Guide to Nonlinear Filtering

Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest `filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete- and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.Comment: 64 page

White noise theory of robust nonlinear filtering with correlated state and observation noises

In the direct white noise theory of nonlinear filtering, the state process is still modeled as a Markov process satisfying an Ito stochastic differential equation, while a finitely additive white noise is used to model the observation noise. In the present work, this asymmetry is removed by modeling the state process as the solution of a (stochastic) differential equation with a finitely additive white noise as the input. This makes it possible to introduce correlation between the state and observation noise, and to obtain robust nonlinear filtering equations in the correlated noise cas