Gaussian Sum Research Articles

Spacecraft INS/CNS/Pulsar integrated Positioning Navigation and Timing

In this paper, robust algorithms of information fusion between Pulsars timing, ranging and positioning with orbital dynamical model of explorer spacecraft is demonstrated with combination to inertial navigation system. Pulsar/CNS integration is considered in this work and was recently investigated using multiple variant of nonlinear filtering approaches, providing an interesting advances and results. However, due to the colored measurement noise from pulsars time of arrival measure, robust nonlinear filter has been derived based on Gauss Quadrature Kalman filters at different degrees. Moreover, because it has been demonstrated that in the during short period of observation, and minimum photons numbers, the noise affecting the X-Ray signal becomes not a Gaussian distribution. An alternative solution to this problem is proposed using multiple quadrature Kalman filters derived in the Gaussian Sum framework. At last, as an experimental validation, the Crab pulsar data were used to demonstrate the efficiency of the novel robust filtering approach against non linearity and also colored non Gaussian measurement noises.

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.

An Adaptive Gaussian Sum Kalman Filter Based on a Partial Variational Bayesian Method

In this article, we address the online state estimation problem of linear discrete-time systems in the presence of inaccurate and slowly time-varying non-Gaussian measurement noise (NGMN). Recently, the variational Bayesian (VB) method has been successfully used to jointly estimate the system state along with the statistics of the unknown Gaussian measurement noise. However, we prove that the original VB method for the non-Gaussian state-space models, modeled by the Gaussian mixture distributions, is analytically intractable. To overcome this problem, we propose a partial VB-based adaptive Gaussian sum Kalman filter, which uses a feedback-based filtering framework to independently calculate the posterior distribution of the state and posterior distribution of the NGMN. Experimental results demonstrate the effectiveness of the proposed filter.

Gaussian Sum FIR Filtering for 2D Target Tracking

The constant velocity (CV) motion model has been typically used in 2-dimensional (2D) target tracking problems, but it has an uncertain process noise covariance problem. Unlike the Kalman filter (KF), the least square finite impulse response filter (LSFF) does not require noise covariance information and can overcome the uncertain process noise covariance problem. However, the LSFF has a cumbersome problem that is to select a suitable value of design parameter called the horizon size. This paper proposes a Gaussian sum FIR filter (GSFF), where the Gaussian sum method is used to deal with the horizon size in LSFFs. The GSFF overcomes the uncertain process noise covariance problem and can be alternative to existing filters in 2D target tracking. Superior performance of GSFF is demonstrated by comparison with the Gaussian sum KF (GSKF) that is an existing filter to solve the uncertain process noise covariance problem.

Gaussian Sum Shifted Rayleigh Filter for Underwater Bearings-Only Target Tracking Problems

In this paper, a new Gaussian sum (GS) filter with shifted Rayleigh filter (SRF) as proposal has been developed for the bearings-only tracking (BOT) problem. The proposed filter estimates the prior and posterior probability density functions as weighted sum of several Gaussian densities, where the individual Gaussian densities are generated from the SRF. Performance of the proposed filter on the BOT problem is compared with the SRF, the GS sparse-grid Gauss-Hermite filter, and other available GS and quadrature-based filters. Performance of filters is compared in terms of a root-mean-square error (RMSE), track divergence, and computational time. The effect of initial uncertainty, measurement noise covariance, and sampling time on filtering accuracy are also studied. Finally, RMSEs of all the filters are evaluated in comparison with the Cramer-Rao lower bound. From simulation results, it is observed that the performance of the proposed GS-SRF is superior to all other nonlinear filters considered.

An Improved Particle Filtering Algorithm Using Different Correlation Coefficients for Nonlinear System State Estimation.

Particle filtering (PF) algorithm has found an increasingly wide utilization in many fields at present, especially in nonlinear and non-Gaussian situations. Because of the particle degeneracy limitation, various resampling methods have been researched. This article proposed an improved PF algorithm combining with different rank correlation coefficients to overcome the shortcomings of degeneracy. By simulating iteration operation in Matlab, it discovers that the proposed algorithm provides better accuracy than sequential importance resampling, Gaussian sum particle filter, and Gaussian mixture sigma-point particle filters in Gaussian mixture noise.

Ordered space-time structures: Quantum carpets from Gaussian sum theory

The term “quantum carpet” can be observed in many closed quantum systems, where the evolution of a wave function exhibits a carpet-like pattern. Quantum carpet mechanisms are also akin to the classical interference patterns of light. Although the origins of quantum carpets have previously been studied by various researchers, many interesting details are still worth exploring. In this study, we present a unified framework for simultaneously analyzing three different features of quantum carpets: full revival, fractional revival, and diagonal canal. For the fractional revival feature, a complete formula is presented to explain its formation through Gaussian sum theory, in which all essential features, including phases and amplitudes, are captured analytically. We also reveal important relationships between the interference terms of diagonal canals and their geometric interpretations such that a better understanding of the development of diagonal canals can be supported.

A Novel EM Implementation for Initial Alignment of SINS Based on Particle Filter and Particle Swarm Optimization

For nonlinear systems in which the measurement noise parameters vary over time, adaptive nonlinear filters can be applied to precisely estimate the states of systems. The expectation maximization (EM) algorithm, which alternately takes an expectation- (E-) step and a maximization- (M-) step, has been proposed to construct a theoretical framework for the adaptive nonlinear filters. Previous adaptive nonlinear filters based on the EM employ analytical algorithms to develop the two steps, but they cannot achieve high filtering accuracy because the strong nonlinearity of systems may invalidate the Gaussian assumption of the state distribution. In this paper, we propose an EM-based adaptive nonlinear filter APF to solve this problem. In the E-step, an improved particle filter PF_new is proposed based on the Gaussian sum approximation (GSA) and the Monte Carlo Markov chain (MCMC) to achieve the state estimation. In the M-step, the particle swarm optimization (PSO) is applied to estimate the measurement noise parameters. The performances of the proposed algorithm are illustrated in the simulations with Lorenz 63 model and in a semiphysical experiment of the initial alignment of the strapdown inertial navigation system (SINS) in large misalignment angles.

An Efficient Gaussian Sum Filter Based on Prune-Cluster-Merge Scheme

The main problem for the state estimation with Gaussian mixture model is the exponentially growing number of Gaussian components. To solve this problem, an efficient Gaussian sum filter (GSF) based on the prune-cluster-merge (PCM) scheme-based Gaussian mixture reduction method is proposed. First, an adaptive weight-censored pruning strategy named as the j-th order statistical technique is presented to delete components with little contributes to the posterior distribution. Then, the Gauss clustering method is proposed to partition the remainder components into clusters based on the newly defined distribution similarity criterion. Integrating the acquired clusters with the covariance intersection algorithm, the components in the same cluster are merged into a standard Gaussian component by keeping the shape of the original distribution. Meanwhile, an extended integral square error cost function is constructed to optimize the performance of the cluster-merge operation. Finally, an efficient Gaussian sum filter is derived by combining the PCM scheme with extended Kalman filters. Numerical results show that the proposed filter can not only keep a better approximation to the original distribution with fewer Gaussian components comparing with the number-limited GSF and Runnalls's GSF, but achieve a higher cost-effectiveness than the particle filter.

Underwater terrain-aided navigation method based on improved Gaussian sum particle filtering

To solve the nonlinear Bayesian estimation problem in underwater terrain-aided navigation, a terrain-aided navigation method based on improved Gaussian sum particle filter is proposed. This method approximates the Bayesian function using multiple Gaussian components, and the components can be obtained by radial basis function neural network. This method has no resampling process, the particle depletion of particle filtering is eliminated in principle. The simulation shows that the proposed method has good matching performance, which is suitable for autonomous underwater vehicle navigation.

Ensemble Particle Filter Based on KLD and Its Application to Initial Alignment of the SINS in Large Misalignment Angles

Among many auxiliary particle filtering algorithms, the ensemble particle filter (EnPF) has been widely applied in various fields, which can be ascribed to its avoidance of the heavy computational burden resulted by the parallel computation of multifilters. However, the estimation precision of the EnPF, like other auxiliary particle filters, also decreases when the nonlinear extent of a system aggravates. In order to have the advantages of the EnPF promoted into strongly nonlinear systems, this paper first analyzes the reasons why the EnPF deteriorates in strongly nonlinear systems and then proposes an EnPF based on Kullback–Leibler distance (KLD), which is named KLEnPF. KLEnPF is mainly inspired by the Gaussian sum approximation. For the aim of improving the approximation precision, the KLEnPF calculates the optimal bandwidth parameters of Gaussian kernels to minimize the KLD between two distribution functions. Finally, we apply the KLEnPF to the initial alignment of the strapdown inertial navigation system (SINS) in large misalignment angles, which is a strongly nonlinear system. The experimental results of it demonstrate the effectiveness of the KLEnPF.

A novel robust filtering strategy for systems with Non-Gaussian noises

For estimating the states of moving targets in the nonlinear system with non-Gaussian noise, the combination of Gaussian Sum Filter (GSF) and other nonlinear filters has been chosen as the filtering algorithm conventionally. The Smooth Variable Structure Filter (SVSF) is a new predictor-corrector method used for state and parameter estimation, which has good stability and robustness. In this paper we propose a new strategy called the modified GS-EKF-SVSF, which inherits good robustness of Gaussian Sum and Smooth Variable Structure Filter (GS-SVSF) and high accuracy of Gaussian Sum and Extended Kalman Filter (GS-EKF). A nonlinear system with non-Gaussian noise for target tracking is used to test the proposed new strategy. The simulation results demonstrate that our proposed strategy has higher accuracy and better robustness when there are modelling uncertainties existing in the system.

Fixed-point iteration Gaussian sum filtering estimator with unknown time-varying non-Gaussian measurement noise

Abstract This paper addresses the state estimation problem in the presence of unknown time-varying non-Gaussian measurement noise (NGMN) modeled by Gaussian mixture distribution(GMD). Recently, the variational Bayesian (VB)-based estimators have been successfully used in the state estimation problem in the presence of unknown measurement noise. However, we reveal that when the state-space model is non-Gaussian, exploiting the traditional VB method, the joint posterior probability density function of the state and parameters of non-Gaussian measurement noise has no analytical result. To fill this gap, a fixed-point iteration Gaussian sum filtering (FPI-GSF) estimator is proposed to jointly estimate the state and the unknown time-varying NGMN online. The proposed estimator utilizes a fixed-point iteration scheme to iteratively calculate the state posterior distribution through the GSF and noise parameters posterior distribution via the VB method, thus it avoids calculating the joint posterior distribution. Simulation results demonstrate the effectiveness of the proposed estimator.

Nonlinear State and Parameter Estimation for Parabolic Trough Collector

We propose and compare various estimation methods for estimating states and parameters for a Parabolic Trough Collector (PTC) which is a key component of a solar thermal power plant. In particular, we compare popular approaches namely Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF), both of which implicitly assume Gaussian state densities, with a recently proposed approach named Unscented Gaussian Sum Filter (UGSF) which does not make the Gaussianity assumptions. These filters are compared for various scenarios involving estimation of states along with either spatially constant or spatially varying PTC efficiency. The results demonstrate superior performance of EKF and UKF over UGSF. Further, EKF is identified to have the lowest computational burden which is also acceptable for real-time computations. Thus, use of EKF for estimating states in real-time is proposed, which can then be used in advanced control and optimization strategies.

A Close to Optimal Adaptive Filter for Sudden System Changes

Most adaptive filtering algorithms are designed with smooth changes in mind. For instance, the Kalman filter (KF) usually assumes a random walk model for the state, where the state noise is simply a multivalue Gaussian random variable. However, in practice changes can be abrupt. In this letter, we consider the case where the state noise of the KF is impulsive, corresponding to a probability density function (PDF) given by the sum of two Gaussians. This will result in a PDF for the state that is given by a sum of Gaussians, corresponding to an implementation of a Gaussian sum filter, with multiple KFs contributing to the resulting output. Simulation and theoretical considerations show that in steady state only one filter is operating, while several filters are important at transitions. The operation of the algorithm is then similar to optimally detecting sudden changes, and restarting the KF at those moments, resulting in an close optimal step size varying algorithm. Simulations show improved performance of the proposed algorithm when compared to state of the art algorithms.

Design of High-Degree Student’s t-Based Cubature Filters

The Student’s t-based nonlinear filter (STNF) can cope with the filtering problem of nonlinear systems with heavy-tailed process and measurement noises. The key problem in the design of a STNF is how to calculate the Student’s t weighted integral, and the performance of the STNF depends heavily on the used numerical integration technique. In this paper, new high-degree Student’s t spherical-radial cubature rules (STSRCRs) for the Student’s t weighted integral are proposed based on spherical-radial transformation and moment matching methods, from which new high-degree Student’s t-based cubature filters (STCFs) are developed. The proposed high-degree STSRCRs can achieve better approximation to the Student’s t weighted integral as compared with the existing third-degree Student’s t integral rules. As a result, the proposed high-degree STCFs have higher estimation accuracy than the existing STNFs. Simulation results illustrate that the proposed filters can achieve higher estimation accuracy than the existing Gaussian approximate filters, Huber-based nonlinear Kalman filters and STNFs with slightly increased computational complexities, and are computationally much more efficient than the existing Gaussian sum filter and particle filter.

Statistical moments of Gaussian kernel correlation sum and weighted least square estimator of correlation dimension and noise level

In this paper, using non-central chi-squared distribution and polynomial chaos decomposition of a log-normal random variable, we derive the exact expressions for the covariance and variance of the Gaussian kernel correlation sum (Gkcs). The obtained results are combined with U-statistics theory and non-linear models theory to construct the exact confidence intervals for the correlation dimension and for the noise level in the case where deterministic time series are corrupted by additive Gaussian noise. The theoretical results are tested on two continuous chaotic dynamics corrupted by Gaussian noise for different values of signal to noise ratio (SNR). An application to a real data time series has been also conducted.

A Gaussian Sum Method to Analyze Bounded Acceleration Guidance Systems

To analyze a highly nonlinear dynamical system, it is necessary to evaluate the time evolution of the probability density function (pdf) of the states. In this paper, a novel Gaussian sum method is proposed to approximate the non-Gaussian pdf of a dynamical system whose nonlinear elements depend only on a single state variable. Using the Chapman–Kolmogorov equation, a set of recursive equations is derived to propagate the mean, covariance, and weight of the Gaussian sum components. There are several multiple integrals in the recursive equations which are analytically reduced to single integrals. As an application, the proposed approach is applied to statistically analyze the engagement of two pursuers with bounded acceleration against a randomly step evader maneuver. Pursuers are assumed to use the zero-lag version of the recently developed optimal cooperative guidance law (OCGL) or the well-known proportional navigation (PN). The results show that the accuracy of the proposed method is equivalent to Monte Carlo simulation which requires extensive computational effort. Utilizing the proposed analytical Gaussian sum method, OCGL and PN are statistically investigated and compared with each other. The results demonstrate that OCGL is superior to PN, especially when the guidance system is highly in saturation.

Unconstrained Nonlinear State Estimation for Tennessee Eastman Challenge Process

Tennessee Eastman challenge process (TE) has been well known in process control literature as a challenge problem in estimation and control. Complexities in this nonlinear process and corresponding measurement models lead to non-Gaussian densities and estimators which can accommodate the non-Gaussianity can give better state estimates. In this work, we compare and demonstrate the utility of recently developed Sum of Gaussians based Unscented Gaussian Sum Filter (UGSF) for performing nonlinear state estimation of the process. UGSF uses sigma point concept to capture process nonlinearities coupled with a Sum of Gaussians approximation to achieve an accurate representation of non-Gaussian prior densities evolved from the process. The performance of UGSF is compared with the well known Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) approaches, for one of the operating modes of TE process. Results demonstrate the utility of using UGSF as a state estimator for the large-dimensional and highly nonlinear TE process.

Approximate MMSE Estimator for Linear Dynamic Systems With Gaussian Mixture Noise

In this work, we propose an approximate minimum mean-square error filter for linear dynamic systems with Gaussian Mixture (GM) noise. The proposed estimator tracks each component of the GM posterior with an individual filter and minimizes the trace of the covariance matrix of the bank of filters, as opposed to minimizing the MSE of individual filters filters in the commonly used Gaussian sum filter (GSF). Hence, the spread of means in the proposed method is smaller than that of GSF which makes it more robust to removing components. Consequently, reduction schemes with lower computational complexity can be used with the proposed filter without losing estimation accuracy and precision. This is supported through simulations on synthetic data as well as experimental data related to an indoor localization system. Additionally, we show that in two limit cases the state estimation provided by our proposed method converges to that of GSF, and we provide simulation results supporting this in other cases.