Benes Filters Research Articles

An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations

The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction [Llopis et al. (SIAM J Sci Comput 40(3):A1544–A1565, 2018), Galanis et al. (Geophysicae 24(10): 2451–2460, 2006)], finance [Brigo and Hanzon (Insurance Math Econom 22(1):53–64, 1998), Date and Ponomareva (IMA J Manag Math 22(3): 195–211, 2011), Crisan and Rozovskii (The Oxford handbook of nonlinear filtering, 2011)] and engineering [Myötyri et al. (Reliability Eng Syst Saf 91(2):200–208, 2005)]. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors, see e.g., Gyöngy and Krylov (Stochastic inequalities and applications, Progr. Probab. 56:301–321, 2003), Le Gland(Stochastic partial differential equations and their applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci. 176:177–187, 1992). This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation inspired by [Han et al. (Proc Natl acad Sci 115(34):8505–8510, 2018)]. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.

Expressing stochastic filters via number sequences

We generalize the results presented in [1] regarding the relation between the Kalman filter and the Fibonacci sequence. We consider more general filtering models and relate the finite dimensional Kalman and Benes filters to the Fibonacci sequence and to the Golden Section. We also prove that Fibonacci numbers may be expressed as the convolution of the Fibonacci and Padovan sequence, thus extending the connection between stochastic filtering and Fibonacci sequence to the Padovan sequence.

Parallel computations for Yau filters

In the early 1990s, Yau developed a new class of nonlinear filters, called the Yau Filterswhich contain the Kalman-Bucy filters and the Benes filters as special cases. It has been shown that, from the Lie algebraic point of view, the Yau filters are the most general finite-dimensional filters. Yau and Hu proved that the DMZ equation for a Yau filter can be reduced to a Kolmogorov type partial differential equation and a system of linear differential equations. They noticed that the PDE is independent of the observed data and hence can be solved off-line. An efficient parallel algorithm for the system of ODEs would lead to fast solutions of Yau filters and hence their suitability for real-life applications. In this paper, we have proposed several parallel methods suitable for this system of ODEs.

A comparative study of the Benes filtering problem

This paper studies a non-linear filter due to Benes for which the optimum solution is known. The paper compares the estimation performance of Benes filter to those of well-known approximate filters: the Extended Kalman, the statistical linearisation and the particle filtering. The performance of all these four filters are also compared to the Cramer–Rao lower bound. Thus, the Benes filter is a yardstick to rank the above-mentioned known techniques in terms of performance and computational cost which solve in an approximate manner the problem solved in an optimum way by Benes. This class of non-linear filtering problem has an interest in application problems like the target tracking.

New finite-dimensional risk-sensitive filters: small noise limits

The paper is concerned with continuous-time nonlinear risk-sensitive filters. It is shown that for large classes of nonlinearities entering both the dynamics and measurements, these filters are finite-dimensional generalizations of the Benes filters. Specific examples are discussed. The small noise limiting analog is discussed using change of probability measures.

Finite-dimensional solutions of a modified Zakai equation

In risk-sensitive control a modified Zakai equation arises which includes an extra term related to the exponential running cost. We show that a wide class of finite-dimensional solutions related to the Benes filter is possible as long as nonlinearities in the drift are canceled in an appropriate way by terms in the running cost.

Explicit solution of a Kolmogorov equation

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 seventies, the concept of the estimation algebra of a filtering system was introduced. It has been the major tool in studying the Duncan-Mortensen-Zakai equation. Recently the second author has constructed general finite-dimensional filters which contain both Kalman-Bucy filters and Benes filter as special cases. In this paper we consider a filtering system with arbitrary nonlinear driftf(x) which satisfies some regularity assumption at infinity. This is a natural assumption in view of Theorem 10 of [DTWY] in a special case. Under the assumption on the observation h(x)=constant, we propose writing down the solution of the Duncan-Mortensen-Zakai equation explicitly.

Finite-dimensional filters

The class of optimal nonlinear finite-dimensional recursive filters found by Benes is extended to include cases in which the drift in the state propagation equation is a general linear function plus the gradient of a scalar potential. It is shown that if the state space is one dimensional, then the deterministic systems underlying the Benes filter fall into five classes, depending on the asymptotic behaviour of the state at large times. Only two of these classes can be obtained using the Kalman filter. It is shown that an arbitrary deterministic trajectory can be approximated at small times to an accuracy of O(t$^{5}$) by a trajectory for which the Benes filter is appropriate. The Benes construction is the starting point for the development of new finite-dimensional recursive approximations to the optimal filter. One of the new filters is applied to a simple tracking problem taken from computer vision, and its performance compared with that of the extended Kalman filter.

General finite-dimensional risk-sensitive problems and small noise limits

For a risk-sensitive, partially observed stochastic control problem, the modified Zakai equation includes an extra term related to the exponential running cost. The finite-dimensional solutions of this modified Zakai equation are obtained. These are analogs of the Kalman and Benes filters. The small noise limits of the finite-dimensional risk-sensitive problems are then obtained. These lead to differential games with deterministic disturbances.

Solution of the Zakai equation by separation of variables

An exact expression for the unnormalized conditional density is derived for certain nonlinear estimation problems. The basic method is to solve the Zakai equation using separation of variables. The new filter is a generalization of the Kalman-Bucy filter, the Benes filter, and other nonlinear filters within a unified framework.

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

Exact finite-dimensional nonlinear filters

A new nonlinear filter is derived for continuous-time processes with discrete-time measurements. The filter is exact, and it can be implemented in real time with a computational complexity that is comparable to the Kalman filter. This new filter includes both the Kalman filter and the discrete-time version of the Benes filter as special cases. Moreover, the new theory can handle a large class of nonlinear estimation problems that cannot be solved using the Kalman or discrete-time Benes filters. A simple approximation technique is suggested for practical applications in which the dynamics do not satisfy the required conditions exactly. This approximation is analogous to the so-called "extended Kalman filter" [10], and it represents a generalization of the standard linearization method.

New classes of finite-dimensional nonlinear filters

We derive sufficient conditions for the estimation algebras of two special classes of nonlinear estimation systems to be finite dimensional. By carrying out the Brockett—Wei—Norman program new classes of finite-dimensional filters are obtained. These systems can be regarded respectively as extensions of the Kalman—Bucy filters and the Benes filters.