Stochastic Filtering Problem Research Articles

Grid-Based Bayesian Beam Tracking With Multiple Observations for Millimeter Wave Channels

In this paper, we study a beam tracking problem for millimeter wave (mmWave) channels in massive multiple-input and multiple-output communication systems. We formulate the beam tracking problem as a stochastic filtering problem by modeling the dynamic state equation and the observation equation. Then, given prior information of the channel, we propose an efficient grid-based Bayesian beam tracking algorithm by leveraging the sparsity of the mmWave channel. Our proposed algorithm examines neighboring transmit and receive beam pairs and exploits multiple observations on the angular domain representation in the sense of the Bayesian statistics. Numerical results show that the proposed Bayesian method with multiple observations outperforms conventional schemes in terms of both the beam tracking accuracy and computational complexity.

Nonlinear stochastic position and attitude filter on the Special Euclidean Group 3

This paper formulates the pose (attitude and position) estimation problem as nonlinear stochastic filter kinematics evolved directly on the Special Euclidean Group 3 (SE(3)). This work proposes an alternate way of potential function selection and handles the problem as a stochastic filtering problem. The problem is mapped from SE(3) to vector form, using the Rodriguez vector and the position vector, and then followed by the definition of the pose problem in the sense of Stratonovich. The proposed filter guarantees that the errors present in position and Rodriguez vector estimates are semi-globally uniformly ultimately bounded (SGUUB) in mean square, and that they converge to small neighborhood of the origin in probability. Simulation results show the robustness and effectiveness of the proposed filter in presence of high levels of noise and bias associated with the velocity vector as well as body-frame measurements.

Credit Scoring with Ego-Network Data

This article investigates a stochastic filtering problem whereby the bor-rower’s hidden credit quality is estimated using ego-network signals. The hidden credit quality process is modeled as a mean reverting Ornstein-Ulehnbeck process. The lender observes the borrower’s behavior modeled as a continuous time diffusion process. The drift of the diffusion process is driven by the hidden credit quality. At discrete fixed times, the lender gets ego-network signals from the borrower and the borrower’s direct friends. The observation filtration thus contains continuous time borrower data augmented with discrete time ego-network signals. Combining the continuous time observation data and ego-network information, we derive filter equations for the hidden process and the properties of the conditional variance. Further, we study the asymptotic properties of the conditional variance when the frequency of arrival of ego-network signals is increased.

Bobrovsky–Zakai-Type Bound for Periodic Stochastic Filtering

Mean-squared-error (MSE) lower bounds are commonly used for performance analysis and system design. Recursive algorithms have been derived for computation of Bayesian bounds in stochastic filtering problems. In this letter, we consider stochastic filtering with a mixture of periodic and nonperiodic states. For periodic states, the modulo- T estimation error is of interest and the MSE lower bounds are inappropriate. Therefore, in this case, the mean-cyclic error and the MSE risks are used for estimation of the periodic and nonperiodic states, respectively. We derive a Bobrovsky-Zakai-type bound for mixed periodic and nonperiodic stochastic filtering. Then, we derive a recursive computation method for this bound in order to allow its computation in dynamic settings. The proposed recursively-computed mixed Bobrovsky-Zakai bound is useful for the design and performance analysis of filters in stochastic filtering problems with both periodic and nonperiodic states. This bound is evaluated for a target tracking example and is shown to be a valid and informative bound for particle filtering performance.

Conditional reachability of uncertain Max Plus Linear systems

The reachability analysis problem of Max Plus Linear (MPL) systems has been properly solved using the Difference-Bound Matrices approach. In this work, the same approach is considered in order to solve the reachability analysis problem of MPL systems subjected to bounded noise, disturbances and/or modeling errors, called uncertain MPL (uMPL) systems. Moreover, using the results on the reachability analysis of uMPL systems, we solve the conditional reachability problem, herein defined as the support calculation of the probability density function involved in the stochastic filtering problem.

A hidden-Gamma model-based filtering and prediction approach for monotonic health factors in manufacturing

In the context of Smart Monitoring and Fault Detection and Isolation in industrial systems, the aim of Predictive Maintenance technologies is to predict the happening of process or equipment faults. In order for a Predictive Maintenance technology to be effective, its predictions have to be both accurate and timely for taking strategic decisions on maintenance scheduling, in a cost-minimization perspective. A number of Predictive Maintenance technologies are based on the use of “health factors”, quantitative indicators associated with the equipment wear that exhibit a monotone evolution. In real industrial environment, such indicators are usually affected by measurement noise and non-uniform sampling time. In this work we present a methodology, formulated as a stochastic filtering problem, to optimally predict the evolution of the aforementioned health factors based on noisy and irregularly sampled observations. In particular, a hidden Gamma process model is proposed to capture the nonnegativity and nonnegativity of the derivative of the health factor. As such filtering problem is not amenable to a closed form solution, a numerical Monte Carlo approach based on particle filtering is here employed. An adaptive parameter identification procedure is proposed to achieve the best trade-off between promptness and low noise sensitivity. Furthermore, a methodology to identify the risk function associated to the observed equipment based on previous maintenance data is proposed. The present study is motivated and tested on a real industrial Predictive Maintenance problem in semiconductor manufacturing, with reference to a dry etching equipment.

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

The stochastic filtering problem: a brief historical account

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

Combining syndromic surveillance and ILI data using particle filter for epidemic state estimation

Designing effective mitigation strategies against influenza outbreak requires an accurate prediction of a disease’s future course of spreading. Real time information such as syndromic surveillance data and influenza-like-illness (ILI) reports by clinicians can be used to generate estimates of the current state of spreading of a disease. Syndromic surveillance data are immediately available, in contrast to ILI reports that require data collection and processing. On the other hand, they are less credible than ILI data because they are essentially behavioral responses from a community. In this paper, we present a method to combine immediately-available-but-less-reliable syndromic surveillance data with reliable-but-time-delayed ILI data. This problem is formulated as a non-linear stochastic filtering problem, and solved by a particle filtering method. Our experimental results from hypothetical pandemic scenarios show that state estimation is improved by utilizing both sets of data compared to when using only one set. However, the amount of improvement depends on the relative credibility and length of delay in ILI data. An analysis for a linear, Gaussian case is presented to support the results observed in the experiments.

Stochastic filtering of a random Fibonacci sequence: Theory and applications

This paper conceives a stochastic filtering problem to estimate, from noisy measurements, the numbers of the random Fibonacci sequence. The dynamical system is amenable to an exact solution, being the convolution of Bernoullian and Gaussian variables, yielding a closed form for the equations of the filter. The derived optimal filter has exponential computational complexity, thus a suboptimal filter with affordable computational load is conceived. The stochastic filter performance is then evaluated with two applications: one of theoretical value and one for a more practical application. More precisely the first case study estimates the Viswanath constant from noisy measurements of the random Fibonacci sequence. It is shown how the filter performs well in estimating the Viswanath constant even if the noise is significantly increased. The second case study refers to a model of malware propagation in a computer network. In this case, it is assumed that there is a random rate for the infection, assuming that a finite time is needed before a computer is infected. A random generalized Fibonacci sequence fits well in this case. Additional applications are possible in view of the fact that several systems both in biology and economy are well represented by Fibonacci binary random trees.

Circumventing the Feature Association Problem in SLAM

In autonomous applications, a vehicle requires reliable estimates of its location and information about the world around it. To capture prior knowledge of the uncertainties in a vehicle's motion response to input commands and sensor measurements, this fundamental task has been cast as probabilistic Simultaneous Localization and Map building (SLAM). SLAM has been investigated as a stochastic filtering problem in which sensor data is compressed into features, which are consequently stacked in a vector, referred to as the map. Inspired by developments in the tracking literature, recent research in SLAM has recast the map as a Random Finite Set (RFS) instead of a random vector, with huge mathematical consequences. With the application of recently formulated Finite Set Statistics (FISST), such a representation circumvents the need for fragile feature management and association routines, which are often the weakest component in vector based SLAM algorithms. This tutorial demonstrates that true sensing uncertainty lies not only in the spatial estimates of a feature, but also in its existence. This gives rise to sensor probabilities of detection and false alarm, as well as spatial uncertainty values. By re-addressing the fundamentals of SLAM under an RFS framework, it will be shown that it is possible to estimate the map in terms of true feature number, as well as location. The concepts are demonstrated with short range radar, which detects multiple features, but yields many false measurements. Comparison of vector, and RFS SLAM algorithms shows the superior robustness of RFS based SLAM to such realistic sensing defects.

A measurement policy in stochastic linear filtering problems

With reference to a (discrete time) linear system filtering problem, we consider the problem of online deciding at which instant we actually measure and process the output values. We propose a convenient measurement policy, able to sensibly reduce the measurement cost, while keeping the estimate accuracy at satisfactory levels. By some probabilistic arguments we analyse the upper bounds for the measurement and non-measurement time intervals.

Filtering for a class of nonlinear MIMO uncertain time-delay stochastic systems

In this article, a high-gain nonlinear observer-based approach is proposed for filtering of nonlinear stochastic systems with time delays. The system under consideration contains parameter uncertainties, stochastic disturbances, time delays, as well as Lipschitz-like nonlinearities. The objective is to design a robust filter such that the dynamics of the estimation error is guaranteed to be stochastically exponentially ultimately bounded in a sense of mean square. A new high-gain filter idea is introduced to solve the stochastic filtering problem by using the Lyapunov method and stochastic analysis techniques. The design of the proposed filter does not necessitate the resolution of any dynamics systems and its expression is explicitly given, i.e. its calibration is achieved through the choice of a single parameter, and does not rely on solving any matrix inequalities by numerical computation. A simulation example is given to illustrate the performance of the proposed filter.

Robust filtering of stochastic uncertain systems on an infinite time horizon

In this paper, we consider a robust filtering problem for continuous time stochastic uncertain systems.The uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input. This uncertainty is assumed to satisfy a certain relative entropy constraint. The solution to a specially parameterized risk-sensitive stochastic filtering problem is used to construct a filter for the original uncertain system which guarantees an optimal worst-case filtering error. The corresponding minimax optimal filter is obtained by solving a pair of algebraic Riccati equations.

Patterns and structure in systems governed by linear second-order differential equations

This article represents a survey of transmutation ideas and their interaction with typical physical problems. For linear second-order differential operatorsP andQ one deals with canonical connectionsB:P→Q (transmutations) satisfyingQB=BP and the related transport of ‘structure’ between the theories ofP andQ. One can study in an intrinsic manner, e.g., Parseval formulas, eigenfunction expansions, integral transform, special functions, inverse problems, integral equations, and related stochastic filtering and estimation problems, etc. There are applications in virtually any area where such operators arise.

A new performance measure for stochastic optimization in Hilbert space

A stochastic optimization theory predicated on the minimization of the memoryless part of an error covariance operator is formulated. The existence of an appropriate memoryless part transformator is verified and its properties delineated. The resultant memoryless part of the error covariance operator is then used as a performance measure in a stochastic filtering problem which is minimized in the partial ordering of the positive hermitian operators. This results in an explicit solution to the filtering problem which formally replicates the classical solution obtained via a mean squared error criterion but which bypasses the restrictive hypotheses required to guarantee the existence of a mean squared error.

A Stability Theory for the Linear-Quadratic-Gaussian Problem for Systems with Delays in the State, Control, and Observations

The estimation and control of linear stochastic systems with delays in the state, control, and observations are studied. First, the deterministic optimal control problem with quadratic cost over an infinite time interval is examined. Using an extended notion of stabilizability, the existence and characterization of the optimal control law is obtained. Using the additional assumption of detectability the optimal closed-loop system is shown to be $L^2 $-stable. Next, the stochastic filtering problem is studied. A new version of the duality relations between optimal control and filtering is developed. This combined with a suitable notion of detectability, is exploited to show convergence of the filter gains. Under the additional assumption of stabilizability, the optimal stationary filter is shown to be $L^2 $-stable. Finally, by putting together the optimal control and filtering results, a stable constant stochastic control law is obtained for the linear-quadratic-Gaussian problem.

Some filtering results in Riemann manifolds

Some stochastic filtering problems are formulated and solved where the observations are described by stochastic differential equations in the tangent bundle of a compact Riemann manifold. The stochastic differential equations in the tangent bundle are formulated and solved by using the notion of parallelism of vectors along a curve. A stochastic differential equation is obtained for the evolution of the estimate of a suitable process given the observations. An example is solved that exploits the linear filtering results.