Gaussian Sum Filter Research Articles

Rational-quadratic kernel-based maximum correntropy Kalman filter for the non-Gaussian noises

In this paper, a rational-quadratic kernel-based maximum correntropy Kalman filter (RKMCKF) algorithm is proposed to improve the estimation accuracy for non-Gaussian noise interference. Firstly, the RKMCKF algorithm is derived to eliminate the singular matrix produced by multi-dimensional non-Gaussian noise disturbance. Secondly, the upper limit is analyzed to provide a theoretical range for kernel bandwidth, which is beneficial for the selection of proper kernel bandwidths and boosting the precision of state estimation. Furthermore, the boundness of the state estimation error is verified to manifest the RKMCKF algorithm stability. Finally, under different types of non-Gaussian noise, the proposed RKMCKF algorithm is demonstrated to promote the accuracy of state estimation compared with the conventional Kalman filter, Gaussian sum filter, Huber filter, and maximum correntropy Kalman filter through the simulations.

VB-Based Gaussian Sum Cubature Kalman Filter for Adaptive Estimation of Unknown Delay and Loss Probability

The traditional Kalman filter assumes that all measurements can be obtained in real time, which is invalid in practical engineering. Therefore, a variational Bayesian- (VB-) based Gaussian sum cubature Kalman filter is proposed to solve the nonlinear tracking problem of multistep random measurement delay and loss (MRMDL) with unknown probability. First, the measurement model with MRMDL is modified by Bernoulli random variables. Then, the expression of the likelihood function is reformulated as a mixture of multiple Gaussian distributions, and the cubature rule is used to improve the estimation accuracy under the framework of Gaussian sum filter in the process of time update. Finally, by constructing a hierarchical Gaussian model, the unknown and time-varying measurement delay and loss probability are estimated in real time with the state jointly using the VB method in the measurement update stage. The algorithm does not need to calculate the equivalent noise covariance matrix so as to avoid the possible division by zero operation, which improves the stability of the algorithm. Simulation results for a target tracking problem show that the proposed algorithm has a better performance in the presence of MRMDL and can estimate the unknown measurement delay and loss probability accurately.

Attitude estimation for remote sensing satellite CBERS-4 using unscented Gaussian sum filter

Attitude estimation for remote sensing satellite CBERS-4 using unscented Gaussian sum filter

State estimation for a class of nonlinear non‐Gaussian cyber–physical systems under false data injection attacks

AbstractIn this article, we consider the state estimation problem for nonlinear cyber–physical systems with non‐Gaussian process noises under actuator false data injection attacks from the perspective of defenders. The process noises and actuator false data injection attacks herein are regarded as non‐Gaussian noises. Then, the prior density of the state is considered as a sum of Gaussians with unknown covariance matrixes. The partial variational Bayesian method is applied to approximate the unknown covariance matrixes, and the unscented Gaussian sum filter is used for state estimation as well as decreasing the computing complexity. Finally, some simulation results are presented to show the effectiveness of the proposed state estimation method.

Aircraft Parameter Estimation Using Gaussian Sum Filter With Lyapunov Stability Analysis

The use of Gaussian Sum Filter (GSF) as a state/parameter estimation tool for aerodynamic modelling is relatively unexplored in the literature. In this paper GSF has been employed for aircraft parameter estimation. The stability analysis of GSF has also been discussed using the normalized Lyapunov energy functional and the conditions for the convergence of the GSF are derived by using the proposed nonlinear observer which is a novel approach. An approach to prune the weights of GSF based on Pearson coefficient is considered. The performances of the three recursive parameter estimation algorithms: Extended Kalman Filter (EKF), the simplified version of the GSF and the improved version of the GSF are evaluated for aircraft parameter estimation using simulated flight data in the MATLAB implementations. The numerical results show that pruning the weights reduces the computational load by nearly 75%, and still provides good performance compared to the conventional GSF.

Parallel Kalman filter group integrated particle filter method for the train nonlinear operational status high-precision estimation under non-Gaussian environment

For the problem of multi-mode state estimation in actual train operation, this paper proposes a nonlinear non-gaussian high-precision parallel Kalman filter group (NN-HEKFG) integrated Particle Filter. A multi-model Gaussian decomposition of the probability density function for state equations and measurement equations is performed, and each local state model is represented by a multi-dimensional high-order polynomial to establish the expanded dimensional state model. Then, by updating the mean and variance of the local state expanded dimensional model and in turn solving the particle filtering posterior probability density distribution function, the global estimation results are obtained. In reducing the number of Gaussian terms, a new parameter reduction criterion is established, which can effectively carry out the re-identification of parameters such as weights and means, so as to avoid the problem of parameter explosion. The superiority of NN-HEKFG over particle filters and Gaussian sum filters and its effectiveness for train running state estimation are verified by simulating the multi-model running state of trains.

복수 탐지정보의 공간분포 사전지식을 활용한 고해상도 차량용 레이더 표적추적 필터 설계

This paper deals with the problem of target tracking using high-resolution automotive radars. Due to the enhanced angular resolution, a high-resolution radar allows a single target to produce multiple detections originated from the main scatters with complicated spatial distribution Thus, the target tracking problem is formulated as a nonlinear state estimation associated with the multi-modal measurement noise distribution. As a practical resolution to this problem, the statistical properties of the multi-detection is modeled by approximating the reflected target signal strength as Gaussian mixture, then a Gaussian sum filter is designed using the prior knowledge about the model. To prevent the divergence of the number constituents of the posterior distribution, the weights of the pairing hypotheses between the multiple detections and the modes of their distribution are evaluated and the mode management technique is applied in implementing the tracking filter. Accordingly, the proposed filter is able to improve computational efficiency while maintaining the estimation performance. Through simulations for typical automotive target tracking scenarios, it is demonstrated that our approach provides reliable target tracking performance compared to the existing filters.

Gaussian Mixture Filtering with Nonlinear Measurements Minimizing Forward Kullback-Leibler Divergence

A Gaussian mixture filter is proposed for the state estimation of dynamical systems with nonlinear measurements. The filter is derived by solving an assumed density filtering problem where Kullback-Leibler (KL) divergence from the assumed posterior, which is a Gaussian mixture, to the true posterior (which we call forward KL divergence) is minimized. The approximate solution to this problem gives an iterative measurement update by which the weights, means and covariances of the assumed posterior are optimized. The resulting Gaussian mixture filter is shown to be a generalization of the (damped) posterior linearization filter to Gaussian mixture posteriors. The performance of the proposed filter is illustrated and compared to alternatives on a challenging example of target tracking in a sensor network and on an example of tracking with a radar. The results show that the proposed filter can outperform Gaussian filters as well as the Gaussian sum filter and the Gaussian sum particle filter yielding results very close to a bootstrap particle filter when the number of components in the assumed posterior is sufficiently large.

A Bayesian Filtering Method for Wiener State-Space Systems Utilizing a Piece-wise Linear Approximation

In this paper, we develop a filtering algorithm for Wiener systems written in state-space form which considers correlated noise sources. The output non-linearity is approximated by using a piece-wise linear function. The probability function of the output signal conditioned to the system state is written as a Gaussian mixture distribution. A Gaussian sum filter algorithm to obtain the a posteriori probability density function of the state given the current and past output is developed. The associated statistics of the system state are obtained. The benefits of our proposal are illustrated via numerical simulations.

Gaussian Mixture Midway-Merge for Object SLAM With Pose Ambiguity

In this letter, we propose a novel method to merge a Gaussian mixture on matrix Lie groups and present its application for a simultaneous localization and mapping problem with symmetric objects. The key idea is to predetermine the weighted mean called a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">midway point</i> and merge Gaussian mixture components at the associated tangent space. Through this rule, the covariance matrix captures the original density more accurately, and the need for the back-projection is spared when compared to the conventional merge. We highlight the midway-merge by numerically evaluating dissimilarity metrics of density functions before and after the merge on the rotational group. Furthermore, we experimentally discover that the rotational error of symmetric objects follows heavy-tailed behavior and formulate the Gaussian sum filter to model it by a Gaussian mixture noise. The effectiveness of our approach is validated through virtual and real-world datasets.

Kernel-based ensemble gaussian mixture filtering for orbit determination with sparse data

In this paper, a modified kernel-based ensemble Gaussian mixture filtering (EnGMF) is introduced to produce fast and consistent orbit determination capabilities in a sparse measurement environment. The EnGMF is based on kernel density estimation (KDE) to combine particle filters and Gaussian sum filters. This work proposes using Silverman’s rule of thumb to reduce the computational burden of KDE. Equinoctial orbital elements are used to improve the accuracy of the KDE bandwidth parameter in the modified EnGMF. A bi-fidelity approach to propagation and an adaptation algorithm for selecting the appropriate number of particles are also applied to the EnGMF to reduce the computational burden with an acceptable loss in accuracy for long time propagation. Through numerical simulation, the proposed implementation is compared to state-of-the-art approaches in terms of accuracy, consistency, and computational speed.

Gaussian sum state estimators for three dimensional angles-only underwater target tracking problems

Gaussian sum filters are considered to be more accurate in terms of estimation accuracy when compared to the conventional algorithms. In this work, Gaussian sum state estimation algorithms are implemented for three dimensional angles-only target tracking problem. Shifted Rayleigh filter (SRF) has been considered as the most accurate estimation algorithms for bearings-only tracking, with moderate computational load. Therefore, SRF formulated in the Gaussian sum framework is developed for solving three dimensional angles-only target tracking problems. The estimation accuracy of the developed algorithm, and other Gaussian as well as Gaussian sum algorithms is validated in terms of percentage track-loss and root mean square error (RMSE).

Combined Macroscopic and Microscopic Multi-Agent Control For Multi-Target Tracking

We propose a method to address a multi-target tracking problem in which a multi-agent network tries to track the predicted density path of a team of multiple mobile targets (multi-target system). Our modeling and control approach consists of a macroscopic level and a microscopic level. At the macroscopic level, the multi-agent network tries to predict the density path of the multi-target system whereas at the microscopic level, the members of the multi-agent network try to compute the individual actions that will allow the density (macroscopic state) of the network to track the density of the multi-target system while avoiding collisions among themselves. In our proposed solution, the probability density of the multi-target system is characterized as a (time-varying) GM density which is estimated by an adaptive Gaussian sum filter. This GM density serves as a reference tracking density in a decentralized motion coordination control scheme which is based on a variation of Lloyd's algorithm to compute the individual (microscopic) control inputs of the members of the multi-agent network. Because the density prediction problem (at the macroscopic level) and the motion coordination problem (at the microscopic level) are coupled with each other, they are solved iteratively and their solutions are combined accordingly. Finally, numerical simulation results are provided to illustrate the effectiveness of our proposed algorithm.

Gaussian Mixture Based Motion Prediction for Cluster Groups of Mobile Agents

This paper presents a Gaussian mixture model based method for estimation of multiple mobile agent trajectories over time following a social force motion model. From an initial observation, groups of agents are formed using a bottom-up agglomerate clustering approach. Each cluster constitutes a component of the Gaussian mixture model, with mean position of the cluster members. An Adaptive Gaussian Sum filter estimates the future trajectories of the agent clusters using the Gaussian mixture model input. The proposed algorithm is capable of reducing the computation time compared to the standard Adaptive Gaussian Sum filter and provides solutions with lower variance due to utilization of agent groups. The efficacy of the proposed method is demonstrated via extensive numerical simulations.

A Two-Filter Approach for State Estimation Utilizing Quantized Output Data.

Filtering and smoothing algorithms are key tools to develop decision-making strategies and parameter identification techniques in different areas of research, such as economics, financial data analysis, communications, and control systems. These algorithms are used to obtain an estimation of the system state based on the sequentially available noisy measurements of the system output. In a real-world system, the noisy measurements can suffer a significant loss of information due to (among others): (i) a reduced resolution of cost-effective sensors typically used in practice or (ii) a digitalization process for storing or transmitting the measurements through a communication channel using a minimum amount of resources. Thus, obtaining suitable state estimates in this context is essential. In this paper, Gaussian sum filtering and smoothing algorithms are developed in order to deal with noisy measurements that are also subject to quantization. In this approach, the probability mass function of the quantized output given the state is characterized by an integral equation. This integral was approximated by using a Gauss–Legendre quadrature; hence, a model with a Gaussian mixture structure was obtained. This model was used to develop filtering and smoothing algorithms. The benefits of this proposal, in terms of accuracy of the estimation and computational cost, are illustrated via numerical simulations.

Abridged Gaussian sum extended Kalman filter for nonlinear state estimation under non-Gaussian process uncertainties

Studies on Extended Kalman Filter (EKF) that explicitly consider non-Gaussian process uncertainties are scarce in the literature. In the current study, a novel approach referred to as Abridged Gaussian Sum-Extended Kalman Filter (AGS-EKF) is developed to improve EKF performance for such systems. AGS-EKF performs a modified version of the EKF formulation that accounts for non-zero mean process uncertainties. Contrary to the conventional Gaussian Sum Filter (GSF), the process uncertainties involved in the prior estimation step of AGS-EKF follow an overall Gaussian mixture model that approximates the actual non-Gaussian process uncertainty distribution. Hence, AGS-EKF computes the EKF calculation only once thus avoiding the need to perform multiple EKFs to capture all the Gaussian components in the mixture as in GSF approach. Since AGS-EKF considers the overall Gaussian mixture of the process uncertainties, the method avoids the biased estimations that may appear in GSF. In addition to AGS-EKF, an adaptation in the GSF framework is proposed to improve the estimation accuracy of GSF. Case studies have shown that AGS-EKF offers higher accuracy of the estimation in comparison with GSF in shorter computational times.

Cauchy kernel-based maximum correntropy Kalman filter

Non-Gaussian noise processing is a difficult and hot spot in the study of filters. A currently effective method to deal with non-Gaussian noise is replacing the minimum mean square error criterion with the maximum correntropy criterion. Based on the maximum correntropy criterion, maximum correntropy Kalman filter, which usually uses the Gaussian kernel function to define the distance between vectors, is developed. However, when the non-Gaussian noise is multi-dimensional, maximum correntropy Kalman filter tends to break down due to the appearance of singular matrices. To overcome the drawback, a novel filter named Cauchy kernel-based maximum correntropy Kalman filter is proposed, which utilises the Cauchy kernel function to define the distance between vectors. Due to the insensitive feature to the kernel bandwidth and thick-tailed characteristic of the Cauchy kernel function, Cauchy kernel-based maximum correntropy Kalman filter can effectively avoid filter faults and has a better stability. Simulation results demonstrate the excellent performance of the proposed algorithm by comparing it with other conventional methods, such as Kalman filter, ideal Kalman filter, Huber-based filter, Gaussian sum filter and maximum correntropy Kalman filter.

Extended, Unscented Kalman, and Sigma Point Multiple Distribution Estimation Filters for Nonlinear Discrete State-Space Models

The extended and unscented Kalman multiple distribution estimation filters (EKMDEF/UKMDEF) were recently proposed for nonlinear continuous-discrete state-space models and their superior filtering accuracy was shown in the simulation of satellite reentry. The EKMDEF/UKMDEF is based on the multiple distribution estimation (MDE) for the filtered state probability density function (PDF) and this letter provides its alternate derivation with a comparison of the Gaussian sum filter (GSF). This result sheds light on the relationship between the EKMDEF/UKMDEF and the GSF and a new filter that is more stable than the EKMDEF/UKMDEF can be designed. The performance of the proposed filter is examined for nonlinear discrete state-space models using benchmark simulation problems and is compared with those of the representative filters including particle filters (PFs).

Analytic Minimum Mean-Square Error Bounds in Linear Dynamic Systems With Gaussian Mixture Noise Statistics

For linear dynamic systems with Gaussian noise, the Kalman filter provides the Minimum Mean-Square Error (MMSE) state estimation by tracking the posterior. Similarly, for systems with Gaussian Mixture (GM) noise distributions, a bank of Kalman filters or the Gaussian Sum Filter (GSF), can provide the MMSE state estimation. However, the MMSE itself is not analytically tractable. Moreover, the general analytic bounds proposed in the literature are not tractable for GM noise statistics. Hence, in this work, we evaluate the MMSE of linear dynamic systems with GM noise statistics and propose its analytic lower and upper bounds. We provide two analytic upper bounds which are the Mean-Square Errors (MSE) of implementable filters, and we show that based on the shape of the GM noise distributions, the tighter upper bound can be selected. We also show that for highly multimodal GM noise distributions, the bounds and the MMSE converge. Simulation results support the validity of the proposed bounds and their behavior in limits.

Thermal Sensor Placement and Thermal Reconstruction Under Gaussian and Non-Gaussian Sensor Noises for 3-D NoC

On-chip thermal sensors are essential for temperature management in 3-D network-on-chip (NoC) systems. However, due to the physical (area and power) or economical constraints, the number of sensors is limited. Therefore, the two critical issues we face are: 1) how to figure out an efficient thermal sensor placement with the limited number of sensors and 2) how to reconstruct the entire thermal profile based on sensor observations. Another major issue for the thermal reconstruction is the sensor measurement accuracy. Thus, online accurate full-chip thermal reconstruction under Gaussian and non-Gaussian noises is another great challenge. In this paper, a greedy thermal sensor placement algorithm maximizing the rank of the observability Gramian is proposed. A good placement algorithm always relies on a specific reconstruction method. The proposed placement algorithm is designed for the state-space-based thermal model, thus the combination of the proposed placement algorithm and the Kalman filter-based reconstruction method provides a high reconstruction accuracy under Gaussian noise. For accurate temperature reconstruction under non-Gaussian noise, the Gaussian-Sum filter is applied to 3-D NoC. Compared with the Kalman filter, the Gaussian-Sum filter can reduce the root-mean-squared-error and the max error by 29.27%–35% and 33.26%–40.6%, respectively. A reusable architecture for the Kalman filter and the Gaussian-Sum filter has been proposed. Its hardware implementation details are presented in this paper. Besides, the performance and the area are evaluated as well.