Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

An approximation for the nonlinear filtering problem,with error bound †

For the standard continuous-time nonlinear filtering problem an approximation approach is derived. The approximate filter is given by the solution to an appropriate discrete-time approximating filtering problem that can be explicitly solved by a finite-dimensional procedure. Furthermore an explicit upper bound for the approximation error is derived. The approximating problem is obtained by first approximating the signal and then using measure transformation to express the original observation process in terms of the approximating signal     

基于噪声先验知识的LMS自适应噪声对消法的水声微弱信号提取

讲述了LMS自适应噪声对消法的数学原理,设计了一种基于噪声参考信号的噪声对消原理结构图,并对初步提取的信号做Fourier变换,设计合理的系数阈值经行滤波,Fourier逆变换的信号与理想信号做性能对比.仿真实验表明:基于噪声参考信号的噪声对消算法呈现出滤波阶数少,收敛速度快,精度良好,提取信号效果良好等优点.汾河二库湖试测试实验结果也验证了该算法具有良好的高效性和实用性.     

Particle Approximations for a Class of Stochastic Partial Differential Equations

The paper presents a particle approximation for a class of nonlinear stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The new results permit the treatment of filtering problems where the signal noise is no longer independent of the observation noise.     

Risk Sensitive Filtering with Poisson Process Observations

In this paper we consider risk sensitive filtering for Poisson process observations. Risk sensitive filtering is a type of robust filtering which offers performance benefits in the presence of uncertainties. We derive a risk sensitive filter for a stochastic system where the signal variable has dynamics described by a diffusion equation and determines the rate function for an observation process. The filtering equations are stochastic integral equations. Computer simulations are presented to demonstrate the performance gain for the risk sensitive filter compared with the risk neutral filter. Accepted 23 July 1999     

An Examination of the Use of Adaptive Filtering in Forecasting

Adaptive filtering is a technique for preparing short- to medium-term forecasts based on the weighting of historical observations, in a similar way to moving average and exponential smoothing. However, adaptive filtering, as it has been developed in electrical engineering, attempts to distinguish a signal pattern from random noise, rather than simply smoothing the noise of past data. This paper reviews the technique of adaptive filtering and investigates its applications and limitations for the forecasting practitioner. This is done by looking at the performance of adaptive filtering in forecasting a number of time series and by comparing it with other forecasting techniques.     

Filtering of discrete-time systems hidden in discrete-time random measures

This paper is concerned with the filtering problem of a discrete-time signal hidden in discrete-time random measures. Using measure change techniques as discussed in [1], recursive estimates of the signal are obtained. Also, a finite-state signal is discussed and filters of functionals of the signals are derived.     

A nonsmooth optimization problem in envelope constrained filtering

In envelope-constrained filtering, the filter is optimized subject to the constraint that the filter response to a given signal lies within a specified envelop or mask. In this note, we develop an efficient method for solving a class of nonsmooth optimization problems which covers the envelope- constrained filtering problem as a special case.     

Imprecise expectations for imprecise linear filtering

In most sensor measure based applications, the raw sensor signal has to be processed by an appropriate filter to increase the signal-to-noise ratio or simply to recover the signal to be measured. In both cases, the filter output is obtained by convoluting the sensor signal with a supposedly known appropriate impulse response. However, in many real life situations, this impulse response cannot be precisely specified. The filtered value can thus be considered as biased by this arbitrary choice of one impulse response among all possible impulse responses considered in this specific context. In this paper, we propose a new approach to perform filtering that aims at computing an interval valued signal containing all outputs of filtering processes involving a coherent family of conventional linear filters. This approach is based on a very straightforward extension of the expectation operator involving appropriate concave capacities.     

Linear Filtering for Time-Delay Systems

A linear filtering problem is studied in which the signal process(x(t):t 0) is described by a stochastic differential equationwhere time delays are present in both the noise input and thex variable. By means of a transformation new to the filteringliterature, we reduce the signal equation to a delay-free stochasticevolution equation; this permits us to solve the problem byapplication of the theory of infinite-dimensional linear filtering.     

Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter

In this work we study connections between various asymptotic properties of the nonlinear filter. It is assumed that the signal has a unique invariant probability measure. The key property of interest is expressed in terms of a relationship between the observation σ field and the tail σ field of the signal, in the stationary filtering problem. This property can be viewed as the permissibility of the interchange of the order of the operations of maximum and countable intersection for certain σ-fields. Under suitable conditions, it is shown that the above property is equivalent to various desirable properties of the filter such as

(a) uniqueness of invariant measure for the signal,

(b) uniqueness of invariant measure for the pair (signal, filter),

(c) a finite memory property of the filter,

(d) a property of finite time dependence between the signal and observation σ fields and

(e) asymptotic stability of the filter.

Previous works on the asymptotic stability of the filter for a variety of filtering models then identify a rich class of filtering problems for which the above equivalent properties hold.     

Finite-dimensional filter for a class of nonlinear systems with correlated noises

The purpose of this article is to compute an explicit formula for the unnormalized conditional density for the filter associated with a nonlinear filtering problem with correlated noises and a signal process with nonlinear terms in the drift. This article extends the result of Daum to nonlinear filtering systems with correlated noises and incorporates both the Kalman–Bucy and Bene? filters as particular cases.     

Équations du filtrage non linéaire de la prédiction et du lissage

We establish equations of non linear filtering, prediction (extrapolation) and smoothing (interpolation) in the case where the signal is a non degenerate diffusion process, and the observation is a noisy functional of the signal. We consider both the case of observation noise correlated with the signal, and the opposite case where we establish “robust” form of the equations. We study finally the case of unbounded coefficients, and the case where there is a feedback from the observation to the signal.     

Approximate filter for a two-dimensional nonlinear diffusion observed in a correlated low noise channel

The purpose of this article is to study a nonlinear filtering problem when the signal is a two-dimensional process from which only the second component is noisy and when only its first (and unnoisy) component is observed in a correlated low noise channel. We propose an approximate finite-dimensional filter and we prove that the filtering error converges to zero. The order of magnitude of the error between the approximate filter and the optimal filter, as the observation noise vanishes, is computed.     

Filltering of a partially observed process in the case of a high signal –to–noise ratio for correlated systems

This paper concerns a nonlinear filtering problem with correlated noises in the case of a high signal–to–noise ratio, when only one component of the signal is observed. We compute an approximate filter for the unnormalized filter associated to the system and derive both a Zakai and a Kushner-Stratonovitch type equation for the approximate filter     

Least-squares linear estimation of signals from observations with Markovian delays

The least-squares linear estimation of signals from randomly delayed measurements is addressed when the delay is modeled by a homogeneous Markov chain. To estimate the signal, recursive filtering and fixed-point smoothing algorithms are derived, using an innovation approach, assuming that the covariance functions of the processes involved in the observation equation are known. Recursive formulas for filtering and fixed-point smoothing error covariance matrices are obtained to measure the goodness of the proposed estimators.     

Least-squares linear estimation of signals from observations with Markovian delays

The least-squares linear estimation of signals from randomly delayed measurements is addressed when the delay is modeled by a homogeneous Markov chain. To estimate the signal, recursive filtering and fixed-point smoothing algorithms are derived, using an innovation approach, assuming that the covariance functions of the processes involved in the observation equation are known. Recursive formulas for filtering and fixed-point smoothing error covariance matrices are obtained to measure the goodness of the proposed estimators.     

On Hölder solutions of the integro-differential Zakai equation

The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.     

Nonlinear filtering for jump-diffusions

The paper treats the nonlinear filtering problem for jump-diffusion processes. The optimal filter is derived for a stochastic system where the dynamics of the signal variable is described by a jump-diffusion equation. The optimal filter is described by stochastic integral equations.     

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