Nonlinear Gaussian Filter Research Articles

Nonlinear Gaussian Filtering With Network-Induced Delay in Measurements

This article designs an advanced Gaussian filtering algorithm for improving accuracy in the presence of time-delay in measurements. The proposed method uses a Bernoulli random variable and a geometric random variable to reformulate the delay modeling strategy. Subsequently, the traditional Gaussian filtering method for the modified measurement model is rederived. The proposed method precludes two major drawbacks of the existing delay filtering methods, including <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> knowledge of many delay probabilities and an ambiguous selection of an upper bound of delay. Thus, the proposed method outperforms the existing delay filtering methods and the same is validated from the simulation results. The proposed method is a general modification of the traditional Gaussian filtering and applies to all conventionally popular Gaussian filters.

Compressed Positive Quadrature Filter

A compressed positive quadrature filter (CPQF) is proposed in this technical note. Accuracy, efficiency, and numerical stability are the main concerns for quadrature-based nonlinear Gaussian filtering, but desirable performance in all these three aspects can hardly be simultaneously achieved in previously proposed nonlinear Gaussian filters, such as unscented Kalman filter, Gauss–Hermite quadrature filter (GHQF), and sparse grid quadrature filter. To improve the overall capacity of nonlinear Gaussian filtering techniques, the CPQF is proposed by numerically solving the constrained moment matching equations. By enforcing the positivity of the quadrature weights, the CPQF is numerically as stable as the GHQF. Through exploiting the underlying sparsity while solving the moment matching equations, the CPQF can also achieve the same level of accuracy as the GHQF using far fewer quadrature points. Simulations are conducted to evaluate the effectiveness of the CPQF, and the numerical results demonstrate its superior performance over conventional Gaussian filters.

Bayesian Approximation Filtering With False Data Attack on Network

Very often, a measurement is transmitted through network systems before it is available for filtering. The network systems, designed with several communication channels, are prone to cyber-attacks. The cyber-attack often injects false data to alter the original measurement. This article develops a modified Bayesian approximation filtering method for nonlinear filtering with measurements altered due to cyber-attack. The proposed development is within the scope of nonlinear Gaussian filtering. It considers the false data to have either additive or multiplicative effect over the original measurement. Subsequently, two modified measurement models are introduced to model the possibility of false data stochastically. Then, the traditional nonlinear Gaussian filtering method is redesigned for the modified measurement models to deal with the false data attack. The proposed modification is applicable to any of the existing nonlinear Gaussian filters, such as extended Kalman filter, unscented Kalman filter, cubature Kalman filter, and Gauss–Hermite filter. The simulation results show an enhanced estimation accuracy for the proposed modification over the traditional nonlinear Gaussian filtering in the presence of false data.

Nonlinear Gaussian Filter with Multi-Step Colored Noise

Color noise is a special kind of noise often occurring in localization systems, and it is more suitable than the general Gaussian white noise to model time dependence due to time delay or high-frequency sampling. This paper derives a nonlinear Gaussian filtering framework for multi-step colored noise systems using noise whitening techniques and Bayes rule. Meanwhile, the cubature rule is used to solve the Gaussian-weighted integral in the proposed Gaussian filtering framework, resulting in an analytic form of posterior state estimate. Compared with the existing nonlinear filtering algorithms, the proposed method has obvious advantages in colored noise systems because it fully takes into account the time dependence of colored noise. Finally, the effectiveness and advantages of the proposed algorithm are verified with a classical target tracking system.

Real-time parameter identification of ship maneuvering response model based on nonlinear Gaussian Filter

In order to solve the problem of parameter identification of nonlinear ship motion model in ship autonomous navigation control, a real-time parameter identification method based on nonlinear Gaussian filtering algorithm and nonlinear ship response model is proposed. It is proved theoretically that the influence of parameter drift on parameter identification can be reduced by increasing the number of observers and filters, and the system identification accuracy can be improved. The validity of the proposed method is verified by parameter identification experiments based on Zig-zag motion simulation data of Mariner standard ship model. Simulation results show that compared with EKF, the nonlinear Gaussian filter algorithm can effectively improve the parameter identification accuracy and reduce the computational complexity. The application of parallel structure is helpful to improve the identification accuracy and convergence rate of nonlinear Gaussian filter algorithm.

Robust Linearly Constrained Square-Root Cubature Kalman Filter for Mismatched Nonlinear Dynamic Systems

This letter addresses the robust state estimation problem for mismatched nonlinear dynamic systems. In this context, we are interested in nonlinear filters able to provide consistent estimates while mitigating the impact of possible modeling errors. We focus on the cubature Kalman filter (CKF) and its square-root form (SCKF), being the most promising nonlinear Gaussian filters, the latter providing improved numerical stability. We leverage on the recently introduced linearly constraint Kalman filter, and derive its sigma-point-based counterparts, the linearly constrained CKF (LC-CKF) and SCKF (LC-SCKF). Such formulation can be exploited, using the correct constraints, to mitigate a large number of model misspecifications. To assess the validity of this approach, we consider an illustrative vehicle navigation problem. Results are provided to support the discussion and show the good performance of the new filters with respect to the state-of-the-art, which are robust to different types of mismatches between the assumed dynamic system model and the true one.

Gaussian Recursive Filter for Nonlinear Systems with Finite-step Correlated Noises and Packet Dropout Compensations

Abstract This paper is focused on the nonlinear state estimation problem with finite-step correlated noises and packet loss. Firstly, by using the projection theorem repeatedly, the mean and covariance of process noise and measurement noise in the condition of measurements before the current epoch are calculated. Then, based on the Gaussian approximation recursive filter (GASF) and the prediction compensation mechanism, one-step predictor and filter with packet dropouts are derived, respectively. Based on these, a nonlinear Gaussian recursive filter is proposed. Subsequently, the numerical implementation is derived based on the cubature Kalman filter (CKF), which is suitable for general nonlinear system and with higher accuracy compared to the algorithm expanded from linear system to nonlinear system through Taylor series expansion. Finally, the strong nonlinearity model is used to show the superiority of the proposed algorithm.

Gaussian Filter for Nonlinear Networked Systems With Synchronously Correlated Noises and One- Step Randomly Delayed Measurements and Multiple Packet Dropouts

In this paper, a Gaussian recursive filter is proposed for nonlinear networked systems with synchronously correlated noises and one-step randomly delayed measurements and multiple packet dropouts. Since each packet at the sensor side is sent only once and the system has one-step delay and multiple packet dropouts, the data processing center may receive zero/one/two measurements. By augmenting the system state at last epoch to the current one, employing two Bernoulli distributed random variables to describe the phenomena of delay and packet dropouts and using the prediction compensation mechanism, a novel model is derived. At the same time, to deal with the problem of correlated noise, Gaussian approximation recursive filter (GASF) is used. Based on the above processing, a nonlinear Gaussian recursive filter and corresponding numerical implementation based on cubature Kalman filter (CKF) are given. Compared with the existing results in linear systems, the proposed algorithm is more general and has higher estimation accuracy in nonlinear systems. The simulation results show the effectiveness of the proposed algorithm.

Data-Driven Enhanced Nonlinear Gaussian Filter

Nonlinear Gaussian filters are usually developed from numerical quadrature rules to approximate the mean and covariance. However, the Gaussian assumption may not be viable after propagation of the estimate through nonlinear dynamics, which may lead to degraded performance. In this brief, we propose an enhanced nonlinear Gaussian quadrature filter in which the quadrature points/weights are generated by a data-driven arbitrary polynomial chaos (aPC) method without relying on the Gaussian assumption. A set of Monte Carlo samples are first propagated through nonlinear dynamics. The statistic moments can be calculated directly from these Monte Carlo samples. The enhanced quadrature points/weights are generated from these moments using the aPC method. Since such quadrature points contain higher order statistic information of the propagated distribution, they can better represent the state distribution and provide a more accurate estimate. Numerical examples show the effectiveness of the proposed filter.

A Method of Stable Extended Kalman filter

The extend kalman filter provides an stability solution to the approximate nonlinear-Gaussian filter problem. Due to the limitation of the word length of the processor, the rounding errors is easy to occur in the calculation, which causes the covariance matrix to lose its positive definite and convergence. In order to solve this problem, The proposed stable extended kalman filter (SEKF) method is applied to the problem of non-positive and divergence in recursive filter. This method introduces the matrix decomposition and constructs a covariance square root adaptive filter which is used to solve the numerical stability problem of the extend kalman filter. Experimental results show that SEKF can effectively guarantee symmetric positive definite in recursive calculation, and perform remarkably better than EKF algorithm.

Gaussian Filter for Nonlinear Stochastic Uncertain Systems With Correlated Noises

In this paper, a nonlinear Gaussian filter is designed for nonlinear stochastic uncertain system with correlated noises. Because the networked systems with random delays and packet losses can be transformed into those with correlated multiplicative noises in the state and measurement matrices, we consider the stochastic uncertain system with synchronously correlated multiplicative noises. The process and observation additive noises are one-step autocorrelated, respectively. Process and observation noises are two-step forward cross-correlated. Based on the abovementioned conditions, we proposed a nonlinear Gaussian recursive filter by using an alternative formulation and a new cubature Kalman filter is given on the basis of the third-degree spherical-radial rule. In order to compare with the algorithm proposed in this paper, a new version based on the extended Kalman filter is developed in Appendix B. In the simulation part, we give two simulation examples to show the effectiveness of the proposed algorithm.

Maximum Correntropy Derivative-Free Robust Kalman Filter and Smoother

We consider the problem of robust estimation involving filtering and smoothing for nonlinear state space models which are disturbed by heavy-tailed impulsive noises. To deal with heavy-tailed noises and improve the robustness of the traditional nonlinear Gaussian Kalman filter and smoother, we propose in this work a general framework of robust filtering and smoothing, which adopts a new maximum correntropy criterion to replace the minimum mean square error for state estimation. To facilitate understanding, we present our robust framework in conjunction with the cubature Kalman filter and smoother. A half-quadratic optimization method is utilized to solve the formulated robust estimation problems, which leads to a new maximum correntropy derivative-free robust Kalman filter and smoother. Simulation results show that the proposed methods achieve a substantial performance improvement over the conventional and existing robust ones with slight computational time increase.

A Variational Bayesian Based Strong Tracking Interpolatory Cubature Kalman Filter for Maneuvering Target Tracking

In tracking a maneuverable target, a proper estimation method with better filtering accuracy, stronger robustness, and faster convergence speed is crucial to the tracking system. The performance of conventional nonlinear Gaussian approximate filters may decline when the target engages in abrupt state changes or the noise covariance matrix is unknown and time-varying. In order to overcome these problems, a new variational Bayesian-based strong tracking interpolatory cubature Kalman filter (VB-STICKF) is deduced in this paper. Gaussian weighted integrals in the nonlinear filter are performed using the interpolatory cubature rule, which has better numerical characteristics for maneuvering target tracking. Moreover, by introducing the strong tracking filter into ICKF, the fading factor is used to modify the predicted error covariance, and the residual sequences are forced to be orthogonal, thus the decreasing performance resulted by the states change and the uncertain process noise could be effectively prevented. Furthermore, the measurement noise can be estimated online by variational Bayesian approach based on the inverse wishart distribution, the robustness of dealing with the uncertain measurement noise is improved. The detailed derivation of VB-STICKF for the general nonlinear models is presented in the paper. A target tracking problem with model uncertainty and time-varying process and measurement noise is utilized to test the performance of the proposed filter, the experimental results of three different scenarios demonstrate the improved filtering performance of the deduced VB-STICKF algorithm.

Hermite Polynomial Uncorrelated Conversion Filter for Bearings-Only Tracking

Passive bearings-only target tracking is a challenging estimation problem due to high nonlinearity and low observability. In this paper, a novel Hermite polynomial uncorrelated conversion filter is proposed within the linear minimum mean square error framework to improve the target tracking performance. The uncorrelated conversion is a nonlinear transformation of the original measurement that is uncorrelated with it. It can be regarded as a pseudomeasurement to augment the measurement space so that additional information can be exploited to update the state estimate in a nonlinear way. Due to the orthogonality property of the Hermite polynomials in the probability domain, each uncorrelated conversion is constructed as a Hermite polynomial, which is not only uncorrelated with the original measurement but also uncorrelated with other Hermite polynomial uncorrelated conversions automatically. This method enables a systematic way to generate multiple uncorrelated conversions. Moreover, it is shown that, under the linear measurement model, the Hermite polynomial uncorrelated conversion filter is equivalent to the linear minimum mean square error estimator. The simulation results illustrate the superiority of the Hermite polynomial uncorrelated conversion filter over the conventional nonlinear Gaussian filters in the sense of the root mean square error and consistency.

Complete offline tuning of the unscented Kalman filter

The unscented Kalman filter (UKF) is a widely used nonlinear Gaussian filter. It has the potential to deal with highly nonlinear dynamic systems, while displaying computational cost of the same order of magnitude as that of the extended Kalman filter (EKF). The quality of the estimates produced by the UKF is dependent on the tuning of both the parameters that govern the unscented transform (UT) and the two noise covariance matrices of the system model. In this paper, the tuning of the UKF is framed as an optimization problem. The tuning problem is solved by a new stochastic search algorithm and by a standard model-based optimizer. The filters tuned with the proposed algorithm and with the standard model-based optimizer are numerically tested against other nonlinear Gaussian filters, including two UKF tuned with state-of-the-art tuning strategies. One of these strategies relies on online tuning and the other on offline tuning.

Anti-interference Tracking Methods of Maneuvering Target for Structure Random Jump Systems

In this article, a study of anti-interference tracking method of maneuvering target for nonlinear structure random jump systems (SRJSs) with random interference was investigated. Inview of the random interference problem of the tracking system, the nonlinear Gaussian approximation filtering (NGAF) was applied to achieve anti-interference tracking of maneuvering target in observation noises environment with the pip interference signal. Inview of the defects of the NGAF algorithm, bootstrap filtering (BSF) algorithm of SRJSs was applied to avoid the loss of information caused by neglecting the higher-order terms. A meaningful example of radar/IR dual-mode compound seeker is presented to illustrate the effectiveness of the authors’methods, the performances of extended Kalman filtering (EKF), NGAF and BSF in terms of stability, accuracy and computational complexity were compared. The purpose of this paper was to demonstrate the effectiveness of applying the NGAF and BSF on anti-interference target tracking problems of SRJSs, which in the past the factors of randomness and structure uncertainties characteristics had been rarely considered for the studies of maneuvering target tracking, mostly the structures and the parameters of the tracking system were invariant and certainty, and had typically been solved by Kalman or extended Kalman filters.

Posterior inference on parameters of stochastic differential equations via non-linear Gaussian filtering and adaptive MCMC

This article is concerned with Bayesian estimation of parameters in non-linear multivariate stochastic differential equation (SDE) models occurring, for example, in physics, engineering, and financial applications. In particular, we study the use of adaptive Markov chain Monte Carlo (AMCMC) based numerical integration methods with non-linear Kalman-type approximate Gaussian filters for parameter estimation in non-linear SDEs. We study the accuracy and computational efficiency of gradient-free sigma-point approximations (Gaussian quadratures) in the context of parameter estimation, and compare them with Taylor series and particle MCMC approximations. The results indicate that the sigma-point based Gaussian approximations lead to better approximations of the parameter posterior distribution than the Taylor series, and the accuracy of the approximations is comparable to that of the computationally significantly heavier particle MCMC approximations.

The combined effect of spatial compounding and nonlinear filtering on the speckle reduction in ultrasound images

Recently, a spatial compounding ultrasound imaging method was presented that utilizes a conventional 64-element phased array transducer with two unfocused pistons, each placed at one of the sides of the phased array transducer. This method is augmented here by inclusion of nonlinear filtering of the compounded images. The combined effects of the specific spatial compounding and nonlinear filtering on speckle reduction in the generated ultrasound images are studied and evaluated in two stages: First, the image quality is studied when nonlinear filtering is used as part of the spatial compounding. The study is performed by simulations using the Field II program, by processing several B-mode images of a kidney. The second stage compares the results obtained by the simulations to those obtained by in vitro laboratory experiments. Five different compounding strategies and two nonlinear filters, Gaussian and anisotropic diffusion, are investigated and evaluated in terms of image quality parameters—contrast and signal-to-noise ratio. It is shown that the combination of “averaging + nonlinear Gaussian filtering” produces the greatest improvement of image quality. When compared to a conventional phased array imaging system, the spatial compounding method that includes the conventional 64-element phased array transducer with two unfocused pistons, and employs the “averaging + nonlinear Gaussian filtering” strategy, obtains improvement in SNR that has reached 334%. Thus, though this method necessitates a somewhat wider probe, it produces significantly improved images.

Usefulness of noise adaptive non-linear Gaussian filter in FDG-PET study

In positron emission tomography (PET) studies, shortening transmission (TR) scan time can improve patient comfort and increase scanner throughput. However, PET images from short TR scans may be degraded due to the statistical noise included in the TR image. The purpose of this study was to apply non-linear Gaussian (NLG) and noise adaptive NLG (ANLG) filters to TR images, and to evaluate the extent of noise reduction by the ANLG filter in comparison with that by the NLG filter using phantom and clinical studies. In phantom studies, pool phantoms of various diameters and injected doses of 2-deoxy-2-[18F]fluoro-D-glucose (FDG) were used and the coefficients of variation (CVs) of the counts in the TR images processed with the NLG and ANLG filters were compared. In clinical studies, two normal volunteers and 13 patients with tumors were studied. In volunteer studies, the CV values in the liver were compared. In patient studies, the standardized uptake values (SUVs) of tumors in the emission images were obtained after processing the TR images using the NLG and ANLG filters. In phantom studies, the CV values in the TR images processed with the ANLG filter were smaller than those in the images processed with the NLG filter. When using the ANLG filter, their dependency on the phantom size, injected dose of FDG and TR scan time was smaller than when using the NLG filter. In volunteer studies, the CV values in the images processed with the ANLG filter were smaller than those in the images processed with the NLG filter, and were almost constant regardless of the TR scan time. In patient studies, there was an excellent correlation between the SUVs obtained from the images with a TR scan time of 7 min processed with the NLG filter (x) and those obtained from the images with a TR scan time of 4 min processed with the ANLG filter (y) (r = 0.995, y = 1.034x - 0.075). Our results suggest that the ANLG filter is effective and useful for noise reduction in TR images and shortening TR scan time while maintaining the quantitative accuracy of FDG-PET studies.

Clinical application of noise adaptive non-linear Gaussian filter in FDG-PET study

Clinical application of noise adaptive non-linear Gaussian filter in FDG-PET study