Optimal Nonlinear Filtering Research Articles

Non-Linear Adapted Spatio-Temporal Filter for Single-Trial Identification of Movement-Related Cortical Potential

The execution or imagination of a movement is reflected by a cortical potential that can be recorded by electroencephalography (EEG) as Movement-Related Cortical Potentials (MRCPs). The identification of MRCP from a single trial is a challenging possibility to get a natural control of a Brain–Computer Interface (BCI). We propose a novel method for MRCP detection based on optimal non-linear filters, processing different channels of EEG including delayed samples (getting a spatio-temporal filter). Different outputs can be obtained by changing the order of the temporal filter and of the non-linear processing of the input data. The classification performances of these filters are assessed by cross-validation on a training set, selecting the best ones (adapted to the user) and performing a majority voting from the best three to get an output using test data. The method is compared to another state-of-the-art filter recently introduced by our group when applied to EEG data recorded from 16 healthy subjects either executing or imagining 50 self-paced upper-limb palmar grasps. The new approach has a median accuracy on the overall dataset of 80%, which is significantly better than that of the previous filter (i.e., 63%). It is feasible for online BCI system design with asynchronous, self-paced applications.

Optimal nonlinear filtering of M-PSK signals against a background of harmonic interference with a random initial phase

Objectives. The widespread use of radio data transmission systems using signals with multiposition phase shift keying (MPSK) is due to their high noise immunity and the simplicity of constructing the transmitting and receiving parts of the equipment. The conducted studies have shown that the presence of non-fluctuation interference, in particular, harmonic interference, in the radio channel significantly reduces the noise immunity of receiving discrete information. The energy loss in this case, depending on the interference intensity, can range from fractions of dB to 10 db or more. Therefore, interference suppression is an important task for such radio systems. The aim of the work is to synthesize and analyze an algorithm for optimal nonlinear filtering of MPSK signals against a background of harmonic interference with a random initial phase.Methods. The provisions of the theory of optimal nonlinear signal filtering and methods of statistical radio engineering are used.Results. The synthesis and analysis of the algorithm of optimal nonlinear filtering of MPSK signals against the background of harmonic interference with a random initial phase are carried out. The synthesized receiver contains a discrete symbol evaluation unit, two phase-locked frequency circuits of reference generators that form evaluation copies of the signal and interference, and cross-links between them. Analytical expressions are obtained that allow calculating the dependences of the bit error probability on the signal-to-noise ratio and the interference intensity µ. It is established that uncompensated fluctuations of the initial phase of the useful signal have a greater effect on the receiver noise immunity than similar fluctuations of the phase of harmonic interference, especially with low positional signals.Conclusions. Comparison of the obtained results with the results obtained in the case when there are no harmonic interference compensation circuits shows that the use of the obtained phase filtering algorithms allows for almost complete suppression of harmonic interference. Thus, if µ = 0.5 and the probability of error is 10−2, the energy gain at M = 2 is about 2.5 dB, at M = 4 – about 6 dB, at M = 8 and M = 16 – at least 10 dB.

Optimal nonlinear fi ltering of stochastic processes in rescue radar

Subject and Purpose. Smoke, fog, avalanches, debris of collapsed structures and other optically opaque obstacles in both natural and man-made disasters make optical sensors useless for detecting victims. Electromagnetic waves of the decimeter range penetrate well almost all obstacles, reflect from the trapped people and return to the radar receiver. Due to the breathing and heartbeat, the human-reflected sounding signals get the Doppler phase modulation, which is an information signal. These information signals and their properties provide the subject matter for the present research with the aim to create optimal methods and algorithms of random event processing for the prompt location of survivors by rescuers. Method and Methodology. The method of stochastic analysis of the fluctuation Doppler spectra of reflected sounding signals shows that the information signals have properties of conditional Markov processes. Results. The problem of optimal nonlinear filtering of conditional Markov processes entering the radar signal processing unit has been examined closely. An optimal adaptive filter has been proposed to reduce the masking effect of interferences caused by non-stationary noises and sounding signal reflections from stationary objects. The optimality criterion is the minimum mean square error function whose current value is evaluated in real time during the filtering process as the statistics is accumulated. The filter coefficients are calculated by the recurrent, steepest descent algorithm. The real-time work is carried out through the use of fast Fourier transform algorithms. Conclusion. The structure of the optimal adaptive filter to be built into the radar signal processing unit has been developed. Real radar signals have shown that the optimal filtering during the signal processing in systems designed for detecting live people by their breathing and heartbeat facilitates the interpretation of the observed signals. Some spectra of real signals generated by human breathing and heartbeat are presented.

Optimal linear mean square filter for the operation mode of continuous‐time Markovian jump linear systems

This paper makes a further foray on the study of the filtering problem for the class of Markov jump linear systems (MJLSs). The authors shall be particularly interested in the filtering problem for the Markov jump parameter (the operation mode). Previous result in the literature on this problem has been obtained by Wonham, which has derived an optimal non-linear filter for this problem. The main hindrance with Wonham's result, in the context of the optimal control problem for MJLS with partial observation of the operation mode, is that it introduces a great deal of non-linearity in the Hamilton-Jacobi-Belman equation, which makes it difficult to get an explicit closed solution for the control problem. Motivated, in part, by this, the main contribution of this paper is to devise an optimal linear filter for the mode operation, which they believe could be more favourable in the solution of the control problem with partial observations. In addition, relying on Murayama's stochastic numerical method and the results by Chenggui-Yuan, they carry out simulation of Wonham's filter, and the one devised in this paper, in order to compare their performances.

RBFNN-Based Minimum Entropy Filtering for a Class of Stochastic Nonlinear Systems

This paper presents a novel minimum entropy filter design for a class of stochastic nonlinear systems, which are subjected to non-Gaussian noises. Motivated by stochastic distribution control, an output entropy model is developed using a radial basis function neural network, while the parameters of the model can be identified by the collected data. Based upon the presented model, the filtering problem has been investigated, while the system dynamics have been represented. As the model output is the entropy of the estimation error, the optimal nonlinear filter is obtained based on the Lyapunov design, which makes the model output minimum. Moreover, the entropy assignment problem has been discussed as an extension of the presented approach. To verify the presented design procedure, a numerical example is given, which illustrates the effectiveness of the presented algorithm. The contributions of this paper can be summarized as follows: 1) an output entropy model is presented using a neural network; 2) a nonlinear filter design algorithm is developed as the main result; and 3) a solution of the entropy assignment problem is obtained, which is an extension of the presented framework.

Integrated quantum flow and hidden Markov chain approach for resisting DDoS attack and C-Worm

A denial of service attack becomes a major security issue in the network service provisioning. New worm detection methods are not capable to analyze and detect exponentially rising abnormal traffic patterns. In this paper, a new method called an integrated quantum flow and hidden Markov chain approach (IQF-HMC) is introduced in the internet service provisioning. The Quantum flow measured network traffic with features of origination source, nature of the data traffic, and time duration. The standard classes of traffic pattern are evaluated with training sample and the entropy of test traffic data flow patterns are analyzed and compared to detect and resist the abnormal traffic flooding attack. In addition, the quantum phase shift is done on incoming data traffic pattern enables the server to identify the abnormal cross traffic. After that, hidden Markov chain (HMC) is integrated with quantum flow model to restrict the propagation of uncontrolled malicious traffic by camouflage-Worm. Hidden Markov chain adapted a dynamic Bayesian network to evaluate the camouflaging worm propagation with optimal nonlinear filtering. This integrated method improves the performance of the secured network communication in the internet scenario. The performance parameters are DDoS flood attack resistance rate, execution time, true positive rate and memory utilization.

Practical optimal nonlinear filter with estimated noise probability density function

Practical optimal nonlinear filter with estimated noise probability density function

Optimal continuous-discrete nonlinear finite memory filter with a discrete predictions

We consider the following problem: using the discrete time measurements of the state variables of a continuous stochastic object and obtain the most precise estimate of these variables. To accelerate the estimating process, we synthesize the optimal structure of a discrete final-dimensional nonlinear filter with a piecewise-constant prediction, remembering the last few measurements in its state vector. The dimension of this vector (i.e., the filter memory volume) can be selected arbitrarily to balance the desired measurement precision and available processing speed for the measurements. We obtain the mean-square optimal structure functions of the filter expressed via the corresponding probability distributions and a chain of Fokker—Planck—Kolmogorov equations to find those distributions. We describe the procedure to obtain the structural functions of the filter numerically by means of the Monte-Carlo method. Also, we provide simple numeric-analytic approximations to the proposed filters: they are compared with approximations to the known filters. An example of the construction of such approximations is considered.

On robustness of discrete time optimal filters

A new result on stability of an optimal nonlinear filter with respect to small perturbations on every step is established.

Engineering Synthesis of Nonlinear Spatial Selection with Artificial Intelligence Elements to Suppress Critical Interference of Background in Aviation and Space-Based Opto-Electronic Devices

The previous authors’ works have shown that the system of quasi-optimal linear spatial filtering, due to the restriction of this class of filters, related to the superposition principle, has very limited capacity to suppress the most critical interference spatially inhomogeneous background. Such partial suppression does not meet extreme approach requirements for providing high probability characteristics to detect small targets in the most difficult background conditions. In this regard, there is a conclusion that it is necessary to find a different approach, in which the result of the system operation in complex background does not depend on the level of the background noise at the input. This article performs an engineering synthesis of the system with the artificial visual intelligence elements, which recognizes a class of the small-sized radiating objects with the suppression of the most critical interference through nonlinear topological selection. Consideration of this problem begins with the formation of the filter-discriminator aperture, which is a basis for this theory, «echoing» with the theory of optimal nonlinear filtering spatial Poisson processes. Thus, formation of the optimized nonlinear filter structure is based on the optimal linear filter (Wiener filter) structure. As a result, there are three versions of filter apertures (4-, 8- and 16-connected ones), with one of which later providing operations of the object shape discrimination. The focus of the article is, mainly, on the 8-connected aperture, as the average in balance of efficiency and complexity option. The article pays considerable attention to development of signs and algorithms to select the objects by size and shape. It shows that selection on a uniform background is possible by the maximum value of the first derivative and to separate the most critical form of Markov’s field inhomogeneities and background brightness, as the fragments of component boundaries of macrostructure background – edges («steps») and targets, you must use the anisotropy coefficients of the first and second order. The result of using the system, most effectively separating of signs, was the formation of the so-called «areas of selection». Mathematical models of the target and interference of parametrically-stochastic approach, as well as simulation on a semi-natural test frame with a fairly complex background «scene», proved the fundamental capability of the synthesized circuits to suppress radically the sources of interference signals, which are tens and even hundreds times greater than the signals from targets at the input. Thus, unlike the method based on optimal linear filtering, this approach allows the nonlinear «blockers» to suppress completely interference previously recognized by topological discriminators. Control sampling of points of the semi-natural frame enabled us to estimate conditional probabilities of a true solution of the target detection problem. The estimate has shown that the obtained probability of missing the target and false alarm are quite acceptable the values that make up a few percent. Obtained results show the incomparable advantages in complex background situation of presented non-linear method as compared to its traditional linear analogues.

Asymptotically Optimal Discrete-Time Nonlinear Filters From Stochastically Convergent State Process Approximations

We consider the problem of approximating optimal in the Minimum Mean Squared Error (MMSE) sense nonlinear filters in a discrete time setting, exploiting properties of stochastically convergent state process approximations. More specifically, we consider a class of nonlinear, partially observable stochastic systems, comprised by a (possibly nonstationary) hidden stochastic process (the state), observed through another conditionally Gaussian stochastic process (the observations). Under general assumptions, we show that, given an approximating process which, for each time step, is stochastically convergent to the state process, an approximate filtering operator can be defined, which converges to the true optimal nonlinear filter of the state in a strong and well defined sense. In particular, the convergence is compact in time and uniform in a completely characterized measurable set of probability measure almost unity, also providing a purely quantitative justification of Egoroff's Theorem for the problem at hand. The results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating optimal nonlinear filters, such as approximate grid based techniques, known to perform well in a variety of applications.

Friction estimation in wind turbine blade bearings

Summary The influence of friction on wind turbines (WTs) is often seen as undesirable as it has been proven to be the root cause of different severe damages, where the most vulnerable WT's components are bearings. Accurate modelling and identification of friction affects are thus required in order to characterise frictional behaviour and prevent severe damages. This paper addresses the problem on the real-time estimation of friction levels that vary in time with different rates in the WT's blade bearings. Because of ageing or damage progression in bearings, friction parameters change in time. Knowing the material properties and a damage model, monitoring of the friction can be used for condition monitoring of WT's blade bearings. The considered nonlinear identification methods suitable for estimation of fast varying parameters are the extended Kalman filter (EKF), the unscented Kalman filter, the particle filter (PF) and the PF combined with the EKF. This paper also proposes a novel modification of a differential evolution algorithm for the identification of parameters that slowly vary in time. The main contribution of this paper is the algorithm, which combines the modified differential evolution algorithm with optimal nonlinear filters for the estimation of parameters which vary at different rates. A LuGre friction model and a realistic WT sensor setup are selected. The techniques are benchmarked against simulated WT data, and their performances are compared and discussed. Copyright © 2015 John Wiley & Sons, Ltd.

The Testing Filtering and Majorized Algorithm of a Kind of Nonstationary Stochastic Transmission System in Intelligence Control

In intelligence control, applying the method of optimal non-linear filtering and majorized algorithm, this paper discusses the optimal control of a kind of incomplete data and continuous nonstationary stochastic process; yields two optimal control mathematical models in these two situations; illustrates how to establish the optimal coding and decoding of the nonstationary stochastic process; and provides an effective and reliable approach for the optimal control of such a process.

Performance assessment, diagnosis, and optimal selection of non-linear state filters

Non-linear state filters of different approximations and capabilities allow for real-time estimation of unmeasured states in non-linear stochastic processes. It is well known that the performance of non-linear filters depends on the underlying numerical and statistical approximations used in their design. Despite the theoretical and practical interest in evaluating the performance of non-linear filtering methods, it remains one of the most complex problems in the area of state estimation. We propose the use of posterior Cramér–Rao lower bound (PCRLB) or mean square error (MSE) inequality as a filtering performance benchmark. Using the PCRLB inequality, we develop assessment and diagnosis tools for monitoring and evaluating the performance of non-linear filters. Using the PCRLB inequality-based performance assessment tool, an optimal non-linear filter switching strategy is proposed for state estimation in general non-linear systems. The non-linear filter switching strategy is an optimal performance strategy, which maintains high filtering performance under all operating conditions. The complex, high dimensional integrals involved in the computation of the PCRLB inequality-based non-linear filter assessment and diagnosis tools are approximated using sequential Monte-Carlo (SMC) methods. The utility and efficacy of the developed tools are illustrated through a numerical example.

Particle Kalman Filtering: A Nonlinear Bayesian Framework for Ensemble Kalman Filters

This paper investigates an approximation scheme of the optimal nonlinear Bayesian filter based on the Gaussian mixture representation of the state probability distribution function. The resulting filter is similar to the particle filter, but is different from it in that the standard weight-type correction in the particle filter is complemented by the Kalman-type correction with the associated covariance matrices in the Gaussian mixture. The authors show that this filter is an algorithm in between the Kalman filter and the particle filter, and therefore is referred to as the particle Kalman filter (PKF). In the PKF, the solution of a nonlinear filtering problem is expressed as the weighted average of an “ensemble of Kalman filters” operating in parallel. Running an ensemble of Kalman filters is, however, computationally prohibitive for realistic atmospheric and oceanic data assimilation problems. For this reason, the authors consider the construction of the PKF through an “ensemble” of ensemble Kalman filters (EnKFs) instead, and call the implementation the particle EnKF (PEnKF). It is shown that different types of the EnKFs can be considered as special cases of the PEnKF. Similar to the situation in the particle filter, the authors also introduce a resampling step to the PEnKF in order to reduce the risk of weights collapse and improve the performance of the filter. Numerical experiments with the strongly nonlinear Lorenz-96 model are presented and discussed.

A New Convergent Approximation of the Optimal Recursive Parameter Estimator: Averaged Asymptotic Dynamic Analysis

AbstractA key feature of a new recursive parameter estimator is its fast convergence. Here averaging theory supports this claim of fast convergence for system parameters, which heretofore has been supported only by example and by analogy with optimal nonlinear filtering. Averaging theory is also used to investigate possible computational shortcuts. Finally, it is proven that the likelihood function has a unique maximum for stable, observable, single output systems in state space form when the parameters are asymptotically identifiable and true values are in the model set. Then the new parameter estimator must converge globally with probabiltity one to true parameter values when the trajectories of the parameter estimates remain in a certain bounded set.

On the Filtering Problem for Continuous-Time Markov Jump Linear Systems with no Observation of the Markov Chain

We consider in this paper the optimal stationary dynamic linear filtering problem for continuous-time linear systems subject to Markovian jumps in the parameters (LSMJP) and additive noise (Wiener process). It is assumed that only an output of the system is available and therefore the values of the jump parameter are not accessible. It is a well known fact that in this setting the optimal nonlinear filter is infinite dimensional, which makes the linear filtering a natural numerically treatable choice. The goal is to design a dynamic linear filter such that the closed loop system is mean square stable and minimizes the stationary expected value of the mean square estimation error. It is shown that an explicit analytical solution to this optimal filtering problem is obtained from the stationary solution associated to a certain Riccati equation. It is also shown that the problem can be formulated using a linear matrix inequalities (LMI) approach, which can be extended to consider convex polytopic uncertainties on the parameters of the possible modes of operation of the system and on the transition rate matrix of the Markov process. As far as the authors are aware of, this is the first time that this stationary filtering problem (exact and robust versions) for LSMJP with no knowledge of the Markov jump parameters is considered in the literature. Finally, we illustrate the results with an example.

Optimizing the signal processing of detector arrays, using the window method

Based on earlier results of optimizing spatiotemporal signal processing by optimal nonlinear filtering methods, this article discusses ways to implement spatial processing in optoelectronic systems with detector arrays. It is pointed out that it is expedient to use the window method, and examples are presented of schemes for processing and selecting the window parameters in problems involving estimating the coordinates of the radiation source and its intensity in the detection problem. The analysis is carried out in terms of the Poisson signal model, which can be applied to most cases of the practical use of the detectors.

Designing pareto optimal nonlinear filters for image processing

We consider a method of designing multidimensional polynomial filters characterized by an interval of a discrete functional Volterra series. This method allows designing Pareto optimal nonlinear filters by several criteria. We prove that finding a set of Pareto optimal alternatives is equivalent to minimizing a weighted target function. The designed nonlinear filter is characterized by the plane tangent to a convex optimal solving function, whose curvature is determined by how contradictory the given criteria are. We show examples of polynomial filter design for image processing and compare them with known filters of the same class.

The Bispectrum and Bicoherence for Quadratically Nonlinear Systems Subject to Non-Gaussian Inputs

In the analysis of data from nonlinear systems both the bispectrum and the bicoherence have emerged as useful tools. Both are frequently used to detect the influence of a nonlinear system on the joint probability distribution of the system input. Previous work has provided an analytical expression for the bispectrum of a quadratically nonlinear system output if the input is stationary, jointly Gaussian distributed. This work significantly generalizes the previous analysis by providing an analytical expression for the bispectrum of the response of quadratically nonlinear systems subject to stationary, jointly non-Gaussian inputs possessing arbitrary auto-correlation function. The expression is then used to determine the optimal input probability density function for detecting a quadratic nonlinearity in a second-order system. It is also shown how the expression can be used to design an optimal nonlinear filter for detecting deviations from normality in the probability density of a signal.