Duncan-Mortensen-Zakai Equation Research Articles

Sampling and filtering with Markov chains

A new continuous-time Markov chain rate change formula is proven. This theorem is used to derive existence and uniqueness of novel filtering equations akin to the Duncan–Mortensen–Zakai equation and the Fujisaki–Kallianpur–Kunita equation but for Markov signals with general continuous-time Markov chain observations. The equations in this second theorem have the unique feature of being driven by both the observations and the process counting the observation transitions. A direct method of solving these filtering equations is also derived. Most results apply as special cases to the continuous-time Hidden Markov Models (CTHMM), which have become important in applications like disease progression tracking, The corresponding CTHMM results are stated as corollaries. Finally, application of our general theorems to Markov chain importance sampling, rejection sampling and branching particle filtering algorithms is also explained and these are illustrated by way of disease tracking simulations.

Solving nonlinear filtering problems with correlated noise based on Hermite–Galerkin spectral method

Nonlinear filtering problem has important applications in various fields. One of the core issues in nonlinear filtering is to numerically solve the Duncan–Mortensen–Zakai (DMZ) equation, which is an evolution equation satisfied by the unnormalized conditional density of state process under noisy observations, in a real-time and memoryless manner. When the noise in observations is correlated to the state process, the DMZ equation we need to deal with is a second-order stochastic partial differential equation. In this paper, we will propose an algorithm to solve the DMZ equation in this case, based on Hermite–Galerkin spectral method. According to this method, the DMZ equation is converted into a system of linear stochastic differential equations generated by the observation process. The effects of different discretization schemes on this stochastic differential system will also be discussed. Moreover, rigorous convergence analysis of the algorithm is given under mild conditions. Numerical results show that the method proposed in this paper can provide an instantaneous and accurate estimation to the state process of the system.

Numerical Solution of the Problem of Filtering Estimates Information Impact on the Electorate

The formulation and numerical scheme for solving the problem of filtering estimates of the informational impact of mass media on the electorate, allowing with a high degree of accuracy at a given observation interval to estimate the number of individuals in society who prefer a certain political subject (opinion), are proposed in the article. A mathematical model for assessing the information impact on the electorate during election campaigns, which boils down to solving a stochastic differential equation – the equation of state, forms the basis of the formulation of the problem. When compiling a model for filtering information impact estimates, it is proposed to reduce the study of the equation of state to a numerical solution of the Duncan–Mortensen–Zakai equation by introducing an additional observation equation, which is obtained from the equation of state when evaluating its stochastic components (observed agitation intensities) by methods of polyspectral analysis. In the projection formulation of the Galerkin method, when reducing to a system of linear differential equations and obtaining its solution in a recursive estimation scheme when sampling the analysis interval into subintervals and using the matrix exponential method, the Duncan–Mortensen–Zakai equation is solved. For a visual comparison of the effectiveness of the generated numerical solution to the problem of filtering information impact assessments, calculations were carried out on test examples.

Solving Nonlinear Filtering Problems in Real Time by Legendre Galerkin Spectral Method

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan–Mortensen–Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy.

Direct Method for Time-Varying Nonlinear Filtering Problems

This paper discusses how to solve a filtering problem for a class of continuous nonlinear time-varying systems via the Duncan–Mortensen–Zakai (DMZ) equation. In this paper, the original DMZ equation is changed into the Kolmogorov forward equation (KFE) by exponential transformations in each time interval, and then, under some assumptions, the KFE can be transformed into a time-varying Schrodinger equation, which can be solved explicitly. The novelty of this paper lies in how to transform the KFE into the Schrodinger equation. As a direct application, the results of the paper “Nonlinear filtering and time varying Schrodinger equation” are extended for time-varying Yau systems.

Real Time Solution of the Nonlinear Filtering Problem without Memory II

It is well known that the nonlinear filtering problem has important applications in both military and commercial industries. The central problem of nonlinear filtering is to solve the Duncan–Mortensen–Zakai (DMZ) equation in real time and in a memoryless manner. The purpose of this paper is to show that, under very mild conditions (which essentially say that the growth of the observation $|h|$ is greater than the growth of the drift $|f|$), the DMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution $u_R$ of the DMZ equation on the ball $B_R$ with $u_R|_{\pat B_R}=0$. The error of this approximation is bounded by a function of R which tends to zero as R goes to infinity. The solution $u_R$ can in turn be approximated efficiently by an algorithm depending only on solving the observation-independent Kolmogorov equation on $B_R$. In theory, our algorithm can solve basically all engineering problems in real time. Specifically, we show that the solution obtained from our algorithms converges to the solution of the DMZ equation in the $L^1$ sense. Equally important, we have a precise error estimate of this convergence, which is important in numerical computation.

New Algorithms in Real Time Solution of the Nonlinear Filtering Problem

It is well known that the filtering theory has important applications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation algebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner in theory. 1. Introduction. It is well known that the filtering theory has important ap- plications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The nonlinear filtering problem involves the estimation of a stochastic process x = {xt} (called the signal or state process) that cannot be observed directly. Information con- taining x is obtained from observations of a related process y = {yt} (the observation process). The goal of nonlinear filtering is to determine the conditional density �(t,x) of xt given the observation history of {ys: 0 6 s 6 t}. In the late 1960s, Duncan (Du), Mortensen (Mo) and Zakai (Za) independently derived the Duncan-Mortensen-Zakai (DMZ) equation for the nonlinear filtering theory which the conditional probability density �(t,x) must satisfy. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation al- gebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner theoretically.

Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise

The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index ( H 1 , H 2 ) with 0.5 < H 1 , H 2 < 1 , is studied and a suitable version of the Bayes’ formula for the optimal filter is obtained. Two types of spatial “fractional” analogues of the Duncan–Mortensen–Zakai equation are also derived: one tracks the evolution of the unnormalized optimal filter along an arbitrary “monotone increasing” (in the sense of partial ordering in R 2 ) one-dimensional curve in the plane, while the other describes the dynamics of the filter along paths that are truly two-dimensional. Although the paper deals with the two-dimensional parameter space, the presented approach and results extend to d -parameter random fields with arbitrary d ≥ 3 .

Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs

In this note, we develop a real-time and accurate solution for nonlinear filtering problems based on the Gaussian distribution. Specifically, we present an explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation of the Yau filtering system, which includes the linear filtering system and exact filtering system. The solution is given in terms of a solution of a system of ordinary differential equations. In particular, our method can be implemented in hardware. The complexity of our algorithms is the same as those of Kalman-Bucy filters in the case of linear filtering systems.

Finite-dimensional filters with nonlinear drift. XV. New direct method for construction of universal finite-dimensional filter

We consider the explicit solution of Duncan-Mortensen-Zakai (DMZ) equation for the finite-dimensional filtering system. We show that under certain conditions, the nonlinear filtering system can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation. Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.

Finite-dimensional filters with nonlinear drift X: explicit solution of DMZ equation

In this note, we consider the explicit solution of Duncan-Mortensen-Zakai (DMZ) equation for the finite-dimensional filtering system. We show that Yau's (1990, 1994) filtering system (/spl delta/f/sub j///spl delta/x/sub i/)-(/spl delta/f/sub i///spl delta/x/sub i/)=c/sub ij/=constant for all (i,j) can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation. Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.

Hamilton–Jacobi–Bellman Equations for the Optimal Control of the Duncan–Mortensen–Zakai Equation

We study a class of Hamilton–Jacobi–Bellman (HJB) equations associated to stochastic optimal control of the Duncan–Mortensen–Zakai equation. The equations are investigated in weighted L2 spaces. We introduce an appropriate notion of weak (viscosity) solution of such equations and prove that the value function is the unique solution of the HJB equation. We apply the results to stochastic optimal control problems with partial observation and correlated noise.

Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation

Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation

Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system

We consider the Duncan-Mortensen-Zakai (DMZ) equation for the Kalman-Bucy filtering system and Benes filtering system. We show that this equation can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation, Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation.