Univariate Nonstationary Growth Model Research Articles

Robust Cubature Kalman Filter for Moving-Target Tracking with Missing Measurements.

Handling the challenge of missing measurements in nonlinear systems is a difficult problem in various scientific and engineering fields. Missing measurements, which can arise from technical faults during observation, diffusion channel shrinking, or the loss of specific metrics, can bring many challenges when estimating the state of nonlinear systems. To tackle this issue, this paper proposes a technique that utilizes a robust cubature Kalman filter (RCKF) by integrating Huber's M-estimation theory with the standard conventional cubature Kalman filter (CKF). Although a CKF is often used for solving nonlinear filtering problems, its effectiveness might be limited due to a lack of knowledge regarding the nonlinear model of the state and noise-related statistical information. In contrast, the RCKF demonstrates an ability to mitigate performance degradation and discretization issues related to track curves by leveraging covariance matrix predictions for state estimation and output control amidst dynamic disruption errors-even when noise statistics deviate from prior assumptions. The performance of extended Kalman filters (EKFs), unscented Kalman filters (UKFs), CKFs, and RCKFs was compared and evaluated using two numerical examples involving the Univariate Non-stationary Growth Model (UNGM) and bearing-only tracking (BOT). The numerical experiments demonstrated that the RCKF outperformed the EKF, EnKF, and CKF in effectively handling anomaly errors. Specifically, in the UNGM example, the RCKF achieved a significantly lower ARMSE (4.83) and ANCI (3.27)-similar outcomes were observed in the BOT example.

Sequential Fusion Filter for State Estimation of Nonlinear Multi-Sensor Systems with Cross-Correlated Noise and Packet Dropout Compensation

This paper is concerned with the problem of state estimation for nonlinear multi-sensor systems with cross-correlated noise and packet loss compensation. In this case, the cross-correlated noise is modeled by the synchronous correlation of the observation noise of each sensor, and the observation noise of each sensor is correlated with the process noise at the previous moment. Meanwhile, in the process of state estimation, since the measurement data may be transmitted in an unreliable network, data packet dropout will inevitably occur, leading to a reduction in estimation accuracy. To address this undesirable situation, this paper proposes a state estimation method for nonlinear multi-sensor systems with cross-correlated noise and packet dropout compensation based on a sequential fusion framework. Firstly, a prediction compensation mechanism and a strategy based on observation noise estimation are used to update the measurement data while avoiding the noise decorrelation step. Secondly, a design step for a sequential fusion state estimation filter is derived based on an innovation analysis method. Then, a numerical implementation of the sequential fusion state estimator is given based on the third-degree spherical-radial cubature rule. Finally, the univariate nonstationary growth model (UNGM) is combined with simulation to verify the effectiveness and feasibility of the proposed algorithm.

Characteristic analysis and filtering algorithm design for UNGM model

The univariate non-stationary growth model (UNGM) is widely used in the verification of nonlinear filters, and the unscented Kalman filter (UKF) is often used as the reference filter for comparative analysis when using this model to evaluate the filter performance. However, due to the strong nonlinearity of UNGM and the change of model properties with different parameter settings, the estimation misalignment problem due to different reasons will occur when UKF is used for filtering. To solve these problems, this paper analyzes the complex characteristics of UNGM in filtering process, and proposes an UKF with sliding sampling module(SSUKF). The algorithm is optimized on the basis of UKF, and can effectively deal with the complex characteristics of UNGM by sampling and analyzing the filtering information in the filtering process and correcting the distribution of Sigma points in real time. SSUKF is applied to UNGM under different parameters and compared with UKF and bootstrap particle filter(BPF). The simulation results show that SSUKF can effectively solve the misalignment problem when UKF is applied to UNGM, and the calculation speed is better than BPF. Compared with UKF, SSUKF is suitable as a benchmark filter for evaluating the performance of nonlinear filters using UNGM.

Design of robust Gaussian approximate filter and smoother with latency probability identification

The Kalman filter is a linear optimal estimator when it can accurately receive measurements and the noise obeys a Gaussian distribution. However, the measurement noise may have heavy-tailed characteristics because of outlier interference. Actually, real-world measurement information may have one-step randomly delayed measurements (ORDMs) with unknown latency probability (LP) due to exogenous disturbances. Moreover, systems are usually not linear. To overcome the influence of the state estimator of nonlinear systems from heavy-tailed measurement noise (HMN) and the unknown LP, this paper focuses on the robust Gaussian approximate (GA) filter and smoother for the above issues. First, the one-step predicted probability density function (PDF) is modeled as a Gaussian distribution, and the likelihood PDF is modeled as a Normal–Gamma–Beta (NGBM) distribution. Furthermore, novel robust Gaussian approximate filter and smoother are proposed based on the variational Bayesian technique. They provide general estimation frameworks for estimating state of nonlinear systems with HMN and ORDMs, and different nonlinear estimators can be implemented using different numerical techniques. In addition, the computational complexity of the proposed algorithms is calculated. Finally, taking a univariate nonstationary growth model, a passive ranging problem, and target tracking as examples, the effectiveness and superiority of the proposed robust algorithms contrasted with the existing algorithms are shown. The outcomes show that the proposed methods have higher estimation precision; however, the computational burden is marginally increased.

Grid-Based Bayesian Filters With Functional Decomposition of Transient Density

The paper deals with the state estimation of nonlinear stochastic dynamic systems with special attention to grid-based Bayesian filters such as the point-mass filter (PMF) and the marginal particle filter (mPF). In the paper, a novel functional decomposition of the transient density describing the system dynamics is proposed. The decomposition approximates the transient density in a closed region. It is based on a non-negative matrix/tensor factorization and separates the density into functions of the future and current states. Such decomposition facilitates a thrifty calculation of the convolution involving the density, which is a performance bottleneck of the standard PMF/mPF implementations. The estimate quality and computational costs can be efficiently controlled by choosing an appropriate decomposition rank. The performance of the PMF with the transient density decomposition is illustrated in a terrain-aided navigation scenario and a problem involving a univariate non-stationary growth model.

An improved unscented Kalman filter for nonlinear systems with one-step randomly delayed measurement and unknown latency probability

In this paper, an improved unscented Kalman filter is presented to achieve state estimation for nonlinear systems with one-step randomly delayed measurement and unknown latency probability. Firstly, a Bernoulli random variable is introduced to describe the situation of randomly delayed measurement. Then, a joint prior probability density function is obtained. Finally, in the improved unscented Kalman filter algorithm, the state vector and the latency probability are estimated by utilizing a variational Bayesian technique, and the covariance matrices are computed by using the unscented transform. A univariate nonstationary growth model and a Lorenz system are included to verify that the proposed algorithm can not only implement enhanced estimation accuracy compared with the existing methods but also provide the latency probability accurately.

Maximum Correntropy Square-Root Cubature Kalman Filter for Non-Gaussian Measurement Noise

Cubature Kalman filter (CKF) is widely used for non-linear state estimation under Gaussian noise. However, the estimation performance may degrade greatly in presence of heavy-tailed measurement noise. Recently, maximum correntropy square-root cubature Kalman filter (MCSCKF) has been proposed to enhance the robustness against measurement outliers. As is generally known, the square-root algorithms have the benefit of low computational complexity and guaranteed positive semi-definiteness of the state covariances. Therefore, MCSCKF not only possesses the advantages of square-root cubature Kalman filter (SCKF), but also is robust against the heavy-tailed measurement noise. Nevertheless, MCSCKF is prone to the numerical problems. In this paper, we propose a new maximum correntropy square-root cubature Kalman filter (NMCSCKF) based on a cost function which is obtained by a combination of weighted least squares (WLS) to handle the Gaussian process noise and maximum correntropy criterion (MCC) to handle the heavy-tailed measurement noise. Compared to MCSCKF, the proposed method is more time-efficient and most importantly, it avoids the numerical problem. A univariate non-stationary growth model and a multi-rate vision/IMU integrated attitude measurement model are used to demonstrate the superior performance of the proposed method.

Hummingbirds optimization algorithm-based particle filter for maneuvering target tracking

As a commonly used filtering method for nonlinear non-Gaussian systems, particle filters (PFs) have been successfully applied in the field of maneuvering target tracking. However, particle impoverishment is a major obstacle to the PF performance. To overcome this defect, this paper combines the hummingbirds optimization algorithm (HOA) with a standard PF and proposes an HOA-based PF (HOA-PF) for maneuvering target tracking. The proposed filter treats the particles as individual hummingbirds, simulates the honey-collecting process of hummingbirds in nature and moves the particles as a whole to the high-likelihood region by performing self-searching and guided-searching phases. Moreover, to enhance the particle diversity, the mutation method of the following birds in the HOA is improved. Thus, the distribution of particles in the HOA-PF is reasonable. The results of experiments on the univariate nonstationary growth model and the maneuvering target tracking problem demonstrate the effectiveness of the proposed method.

Modification of unscented Kalman filter using a set of scaling parameters

This work, based on the standard unscented Kalman filter (UKF), proposes a modified UKF for highly non-linear stochastic systems, assuming that the associated probability distributions are normal. In the standard UKF with 2n + 1 sample points, the estimated mean and covariance match the true mean and covariance up to the third order, besides, there exists a scaling parameter that plays a crucial role in minimising the fourth-order errors. The proposed method consists of a computationally efficient formulation of the unscented transform that incorporates N − 1 , N ≥ 2 , constant parameters to scale 2 n ( N − 1 ) + 1 sample points such that the 2 N th order errors are minimised. The scaling parameters are obtained by solving a set of algebraic equations. Through rigorous analytical processes and numerical simulations, it is demonstrated that the new filter provides consistent estimates and the estimation error of the modified UKF is smaller than that of the standard UKF. With the help of a well-studied case, univariate non-stationary growth model, the authors evaluate the estimation performance of the new technique using 4n + 1 sample points over 100 independent runs.

Estimating Parameters of Nonlinear Systems Using the Elitist Particle Filter Based on Evolutionary Strategies

In this paper, we present the elitist particle filter based on evolutionary strategies EPFES as an efficient approach to estimate the statistics of a latent state vector capturing the relevant information of a nonlinear system. Similar to classical particle filtering, the EPFES consists of a set of particles and respective weights which represent different realizations of the latent state vector and their likelihood of being the solution of the optimization problem. As main innovation, the EPFES includes an evolutionary elitist-particle selection scheme which combines long-term information with instantaneous sampling from an approximated continuous posterior distribution. In this paper, we propose two advancements of the previously published elitist-particle selection process. Further, the EPFES is shown to be a generalization of the widely-used Gaussian particle filter and thus evaluated with respect to the latter: First, we consider the univariate nonstationary growth model with time-variant latent state variable to evaluate the tracking capabilities of the EPFES for instantaneously calculated particle weights. This is followed by addressing the problem of single-channel nonlinear acoustic echo cancellation as a challenging benchmark task for identifying an unknown system of large search space: the nonlinear acoustic echo path is modeled by a cascade of a parameterized preprocessor to model the loudspeaker signal distortions and a linear FIR filter to model the sound wave propagation and the microphone. By using long-term information, we highlight the efficacy of the well-generalizing EPFES in estimating the preprocessor parameters for a simulated scenario and a real smartphone recording. Finally, we illustrate similarities between the EPFES and evolutionary algorithms to outline future improvements by fusing the achievements of both fields of research.

Huber‐based Unscented Kalman Filters with theq‐gradient

This study presents the Huber-based unscented Kalman filters with the q-gradient (HUKF-Q). As an extension of the classical gradient vector based on the concept of Jackson's derivative, the q-gradient can be utilised to improve the optimisation performance of the Huber method, significantly. Combining the Huber method based on the q-gradient into state estimation based on the unscented transformation, generates the novel HUKF-Q. The Cramér–Rao lower bound is introduced as a performance measure metric. Compared with the conventional HUKF, the proposed HUKF-Q can achieve better filtering accuracy and robustness. In addition, the impact of the tuning parameter q on the filtering performance is discussed by simulations. Simulations on the two examples of univariate non-stationary growth model and bearings only tracking model, confirm the superior performance of the proposed HUKF-Q.

Auxiliary Truncated Unscented Kalman Filtering for Bearings-Only Maneuvering Target Tracking.

Novel auxiliary truncated unscented Kalman filtering (ATUKF) is proposed for bearings-only maneuvering target tracking in this paper. In the proposed algorithm, to deal with arbitrary changes in motion models, a modified prior probability density function (PDF) is derived based on some auxiliary target characteristics and current measurements. Then, the modified prior PDF is approximated as a Gaussian density by using the statistical linear regression (SLR) to estimate the mean and covariance. In order to track bearings-only maneuvering target, the posterior PDF is jointly estimated based on the prior probability density function and the modified prior probability density function, and a practical algorithm is developed. Finally, compared with other nonlinear filtering approaches, the experimental results of the proposed algorithm show a significant improvement for both the univariate nonstationary growth model (UNGM) case and bearings-only target tracking case.

Comparison of Estimation Accuracy of EKF, UKF and PF Filters

Abstract Several types of nonlinear filters (EKF – extended Kalman filter, UKF – unscented Kalman filter, PF – particle filter) are widely used for location estimation and their algorithms are described in this paper. In the article filtering accuracy for non-linear form of measurement equation is presented. The results of complex simulations that compare the quality of estimation of analyzed non-linear filters for complex non-linearities of state vector are presented. The moves of maneuvering object are described in two-dimensional Cartesian coordinates and the measurements are described in the polar coordinate system. The object dynamics is characterized by acceleration described by the univariate non-stationary growth model (UNGM) function. The filtering accuracy was evaluated not only by the root-mean-square errors (RMSE) but also by statistical testing of innovations through the expected value test, the whiteness test and the WSSR (weighted sum squared residual) test as well. The comparison of filtering quality was done in the MATLAB environment. The presented results provide a basis for designing more accurate algorithms for object location estimation.

Improved Dynamic Neighborhood Adaptive Particle Swarm Optimized Particle Filter for Integrated Navigation System

Particle filter based on particle swarm optimization (PSO-PF) is not precise enough and trapping in local optimum easily, it is not able to meet the requirement of modern navigation system. To solve the problems, a new particle filter based on dynamic clone particle swarm optimization (DPSO-PF) is presented in this paper. This improved filter enables the particles to fit the condition better and then reach the goal of global optimization through orthogonal initialization, clonal selection and local searching of self-learning, accordingly a best balance is achieved between optimal exploring and convergence rate. Finally univariate nonstationary growth model and integrated navigation model are used for simulation experiment and the results indicate that this new filter improves the precision of GPS/INS integrated navigation system.

An improved particle filtering algorithm for aircraft engine gas-path fault diagnosis

In this article, an improved particle filter with electromagnetism-like mechanism algorithm is proposed for aircraft engine gas-path component abrupt fault diagnosis. In order to avoid the particle degeneracy and sample impoverishment of normal particle filter, the electromagnetism-like mechanism optimization algorithm is introduced into resampling procedure, which adjusts the position of the particles through simulating attraction–repulsion mechanism between charged particles of the electromagnetism theory. The improved particle filter can solve the particle degradation problem and ensure the diversity of the particle set. Meanwhile, it enhances the ability of tracking abrupt fault due to considering the latest measurement information. Comparison of the proposed method with three different filter algorithms is carried out on a univariate nonstationary growth model. Simulations on a turbofan engine model indicate that compared to the normal particle filter, the improved particle filter can ensure the completion of the fault diagnosis within less sampling period and the root mean square error of parameters estimation is reduced.

Latency probability estimation of non‐linear systems with one‐step randomly delayed measurements

In this study, the authors focus on estimating the unknown constant latency probability of non-linear systems with one-step randomly delayed measurements using maximum likelihood (ML) criterion. A new latency probability estimation algorithm is proposed based on an expectation maximisation approach to obtain an approximate ML estimation of latency probability. The proposed algorithm consists of expectation step (E-step) and the maximisation step (M-step). In the E-step, the expectation of the complete data log-likelihood function is approximately computed based on the currently estimated latency probability, and in the M-step, the approximate expectation is maximised using the Newton approach. The efficacy of the proposed algorithm is illustrated in a numerical example concerning univariate non-stationary growth model.

Design of Gaussian Approximate Filter and Smoother for Nonlinear Systems with Correlated Noises at One Epoch Apart

In this study, the authors investigate the filtering and smoothing problems of nonlinear systems with correlated noises at one epoch apart. A pseudomeasurement equation is firstly reconstructed with a corresponding pseudomeasurement noise, which is no longer correlated with the process noise. Based on the reconstructed measurement model, new Gaussian approximate (GA) filter and smoother are derived, from which Kalman filter and smoother can be obtained for linear systems. For nonlinear systems, different GA filters and smoothers can be developed through utilizing different numerical methods for computing Gaussian-weighted integrals involved in the proposed solution. Numerical examples concerning univariate nonstationary growth model, passive ranging problem, and target tracking show the efficiency of the proposed filtering and smoothing methods for nonlinear systems with correlated noises at one epoch apart.

Particle filter with one-step randomly delayed measurements and unknown latency probability

In this paper, a new particle filter is proposed to solve the nonlinear and non-Gaussian filtering problem when measurements are randomly delayed by one sampling time and the latency probability of the delay is unknown. In the proposed method, particles and their weights are updated in Bayesian filtering framework by considering the randomly delayed measurement model, and the latency probability is identified by maximum likelihood criterion. The superior performance of the proposed particle filter as compared with existing methods and the effectiveness of the proposed identification method of latency probability are both illustrated in two numerical examples concerning univariate non-stationary growth model and bearing only tracking.

A New Adaptive Square-Root Unscented Kalman Filter for Nonlinear Systems with Additive Noise

The Kalman filter (KF), extended KF, and unscented KF all lack a self-adaptive capacity to deal with system noise. This paper describes a new adaptive filtering approach for nonlinear systems with additive noise. Based on the square-root unscented KF (SRUKF), traditional Maybeck’s estimator is modified and extended to nonlinear systems. The square root of the process noise covariance matrixQor that of the measurement noise covariance matrixRis estimated straightforwardly. Because positive semidefiniteness ofQorRis guaranteed, several shortcomings of traditional Maybeck’s algorithm are overcome. Thus, the stability and accuracy of the filter are greatly improved. In addition, based on three different nonlinear systems, a new adaptive filtering technique is described in detail. Specifically, simulation results are presented, where the new filter was applied to a highly nonlinear model (i.e., the univariate nonstationary growth model (UNGM)). The UNGM is compared with the standard SRUKF to demonstrate its superior filtering performance. The adaptive SRUKF (ASRUKF) algorithm can complete direct recursion and calculate the square roots of the variance matrixes of the system state and noise, which ensures the symmetry and nonnegative definiteness of the matrixes and greatly improves the accuracy, stability, and self-adaptability of the filter.

Robust derivative-free Kalman filter based on Huber's M-estimation methodology

In this study, a discrete-time robust nonlinear filtering algorithm is proposed to deal with the contaminated Gaussian noise in the measurement, which is based on a robust modification of the derivative-free Kalman filter. By interpreting the Kalman type filter (KTF) as the recursive Bayesian approximation, the innovation is reformulated capitalizing on the Huber's M-estimation methodology. The proposed algorithm achieves not only the robustness of the M-estimation but also the accuracy and flexibility of the derivative-free Kalman filter for the nonlinear problems. The reliability and accuracy of the proposed algorithm are tested in the Univariate Nonstationary Growth Model.