Gaussian Mixture Filter Research Articles

An improved Bayesian filter for nonlinear systems under multistep randomly delayed and lost measurements

AbstractThis article addresses the Bayesian filtering problem for a class of nonlinear systems under multistep randomly delayed and lost measurements. A new measurement model is established that can characterize the random delay and loss of measurement data. First, an augmented Gaussian mixture filter framework is developed in the case of random delay of measurement data; the posterior probability density function after state augmentation is calculated by marginalizing over delay variables to extract accurate information from delayed measurements. The implementation of the filter is transformed into the computation of nonlinear numerical integrals. Second, under the proposed framework, novel expressions of the mean and covariance are generated by propagating the measurement taken at the previous moment in the event of no new measurement being received. Finally, we present two simulation examples for estimating system states, and the results demonstrate the effectiveness and superiority of our proposed filter.

A Bayesian Approach to Low-Thrust Maneuvering Spacecraft Tracking

Bayesian estimation with an explicit transitional prior is required for a tracking algorithm to be embedded in most multitarget tracking frameworks. This paper describes a novel approach capable of tracking maneuvering spacecraft with an explicit transitional prior and in a Bayesian framework, with fewer than two observations passes per day. The algorithm samples thrust profiles according to a multivariate Laplace distribution. It is shown that multivariate Laplace distributions are particularly suited to track maneuvering spacecraft, leading to a log probability function that is almost linear with the thrust. Principles from rare event simulation theory are used to propagate the tails of the distribution. Fast propagation is enabled by multi-fidelity methods. Because of the diffuse transitional prior, a novel k-nearest-neighbor-based ensemble Gaussian mixture filter is developed and used. The method allows Bayesian tracking of maneuvering spacecraft for several scenarios with fewer than two measurement passes per day and with a mismatch between the true and expected thrust magnitude of up to a factor of 200. The validity domain and statistical significance of the method are shown by simulation through several Monte Carlo trials in different scenarios and with different filter settings.

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.

GSM Filter: A Learning Model of Measurement Noise Distributions for Bayesian Filtering

In this paper, a new universal Bayesian filter with a Gaussian scale mixture (GSM) model, called the Gaussian scale mixture (GSM) filter, for adaptively learning any measurement noise distribution of a state space system is presented. It is demonstrated that the discrete measurement noise distribution of the system takes on a Gaussian sum form when described by a GSM model. It implies that, as the Gaussian sum model does, the GSM model can also be used to approximate any measurement distribution as closely as desired, whatever the probability density function (pdf) of the measurement noise is. If its covariance is known as a priori or can be estimated, it is found that the scale parameter and matrix of the GSM model at each time step can be achieved adaptively to make the following Bayesian inference much simpler. Further analyses show that the GSM model's auxiliary random variable and the system's state posterior at each time step can be inferred by a variational Bayesian (VB) approach adaptively. At each time step, the posterior distribution of system states shown is a Gaussian whatever the measurement noise pdf of the system is. As a result, the Bayesian filtering for nonlinear systems with any measurement noise pdf can be done by our GSM filter, almost as simple as the general Gaussian filters for filtering a nonlinear Gaussian system. Both simulations and experiments are conducted to verify the effectiveness and correctness of the analytical results. The results on the performances of our algorithms fully outperforming those of the existing state of the art Bayesian filters are given.

Particle Filtering and Gaussian Mixtures – On a Localized Mixture Coefficients Particle Filter (LMCPF) for Global NWP

In a global numerical weather prediction (NWP) modeling framework we study the implementation of Gaussian uncertainty of individual particles into the assimilation step of a localized adaptive particle filter (LAPF). We obtain a local representation of the prior distribution as a mixture of basis functions. In the assimilation step, the filter calculates the individual weight coefficients and new particle locations. It can be viewed as a combination of the LAPF and a localized version of a Gaussian mixture filter, i.e., a Localized Mixture Coefficients Particle Filter (LMCPF).

A Gaussian mixture filter with adaptive refinement for nonlinear state estimation

The state estimation of highly nonlinear dynamic systems is difficult because the probability distribution of their states can be highly non-Gaussian. An adaptive Gaussian mixture filter is developed in this work to address this challenge, in which the Gaussian mixture models are refined based on the system's local severity of nonlinearity to attain a high-fidelity estimation of the state distribution. A set of nonlinearity assessment criteria are designed to trigger the splitting of Gaussian components at both the prediction and update stages of Bayesian filtering and the error bound of estimated distribution is established. The new filter has been benchmarked against the existing methods on two challenging problems and it consistently provides among-the-best accuracy with a reasonable computational cost, which proves that it can be used as a reliable state estimator for engineering systems with highly nonlinear dynamics and subject to high magnitudes of uncertainties.

Efficient Gaussian Mixture Modeling of Long-Range 2D Radar Measurements

In some practical radar systems of interest, the standard EKF linearization approach applied to a nonlinear measurement transformation is ineffective. In particular, if measurements are performed in polar coordinates and the crossrange error is much greater than the range error, then the true error region in Cartesian space is no longer well approximated by a Gaussian distribution. This effect is known as the “contact lens” problem, and various approaches have been proposed to alleviate it, including particle and Gaussian mixture (GM) filters. In this work, a method is presented for modeling Cartesian converted measurement distributions which suffer from the contact lens effect using Maximum Likelihood (ML) GM parameters. In order to allow an efficient implementation of this process in a GM Kalman filter, a normalization of the ML parameters is introduced so that parameters can be computed offline and efficiently stored in a lookup table. Simulation results are presented to illustrate the method and confirm the results.

A Gaussian mixture regression model based adaptive filter for non-Gaussian noise without a priori statistic

In many engineering systems, the distribution of measurement noise is unknown and non-Gaussian, such as skewed, multimodal, time-varying distributions and we can not obtain these prior statistics in advance. How to achieve state estimation via this kind of non-Gaussian measurement noises without prior statistic information is a challenging problem. In this paper, a novel Gaussian mixture regression model (GMRM) is proposed to model the unknown non-Gaussian measurement likelihood for Bayesian update to achieve nonlinear state estimation. Without any prior assumption or limitation of measurement noises’ statistics and distributions, the GMRM can still describe the measurement likelihood accurately based on a group of parameters which are adjusted by maximizing the evidence lower bound. Based on the optimized GMRM, a new variational Bayesian Gaussian mixture filter is proposed by using the variational Bayesian approach. To eliminate the influence of the initialization of the introduced parameters, a learning scheme is proposed to adaptively optimize their hyper-parameters based on the historical measurements. Finally, simulation examples are employed to illustrate the effectiveness of the filter.

Gaussian Mixture Filter for Angles-Only Orbit Determination in Modified Equinoctial Elements

A Gaussian mixture orbit determination filter is developed using a state vector that consists of equinoctial-like osculating elements. This new filter seeks to reduce the number of Gaussian mixture elements that are needed in order to accurately model the Bayesian posterior distributions that apply to the angles-only orbit determination problem. The modified version of the equinoctial elements replaces the elements and , whose sum squared is bounded by 1. The two replacement elements are unbounded, and and can be determined from them. This modification allows the orbit determination filter to operate in an unbounded state space. The new state requires the development of a corresponding mixand covariance upper bound due to the particular type of Gaussian mixture filter that is implemented. The upper bound is used in a mixture resampling algorithm in a way which ensures that extended Kalman filter calculations will be sufficiently accurate for each mixand’s computations. The resulting filter is able to reduce the required number of mixands from 5000 to 500 for angles-only orbit determination of a geosynchronous spacecraft.

The New Trend of State Estimation: From Model-Driven to Hybrid-Driven Methods.

State estimation is widely used in various automated systems, including IoT systems, unmanned systems, robots, etc. In traditional state estimation, measurement data are instantaneous and processed in real time. With modern systems’ development, sensors can obtain more and more signals and store them. Therefore, how to use these measurement big data to improve the performance of state estimation has become a hot research issue in this field. This paper reviews the development of state estimation and future development trends. First, we review the model-based state estimation methods, including the Kalman filter, such as the extended Kalman filter (EKF), unscented Kalman filter (UKF), cubature Kalman filter (CKF), etc. Particle filters and Gaussian mixture filters that can handle mixed Gaussian noise are discussed, too. These methods have high requirements for models, while it is not easy to obtain accurate system models in practice. The emergence of robust filters, the interacting multiple model (IMM), and adaptive filters are also mentioned here. Secondly, the current research status of data-driven state estimation methods is introduced based on network learning. Finally, the main research results for hybrid filters obtained in recent years are summarized and discussed, which combine model-based methods and data-driven methods. This paper is based on state estimation research results and provides a more detailed overview of model-driven, data-driven, and hybrid-driven approaches. The main algorithm of each method is provided so that beginners can have a clearer understanding. Additionally, it discusses the future development trends for researchers in state estimation.

Bernoulli filter for tracking maritime targets using point measurements with amplitude

Statistical performance of the Bernoulli Track-Before-Detect (TBD) algorithm for maritime radar is superior to that of the Bernoulli tracking filter which processes, in a conventional manner, the point measurements (PM) produced by a detector. However, the Bernoulli TBD is orders of magnitude computationally more intensive. In this paper we develop a hybrid algorithm that takes advantage of the amplitude information (exploited by the TBD algorithm), but operates in the framework of conventional point-measurement Bayesian tracking. The resulting filter is formulated using the Rao-Blackwell decomposition: a particle filter estimates the target amplitude, while the target position and velocity, given the amplitude, are estimated using a Gaussian mixture filter. Numerical results show that the statistical and computational efficiency of the new hybrid filter is between the two extremes, namely, the Bernoulli TBD and the Bernoulli PM tracker.

Adaptive Weight Update Algorithm for Target Tracking of UUV Based on Improved Gaussian Mixture Cubature Kalman Filter

The Gaussian mixture filter can solve the non-Gaussian problem of target tracking in complex environment by the multimode approximation method, but the weights of the Gaussian component of the conventional Gaussian mixture filter are only updated with the arrival of the measurement value in the measurement update stage. When the nonlinear degree of the system is high or the measurement value is missing, the weight of the Gauss component remains unchanged, and the probability density function of the system state cannot be accurately approximated. To solve this problem, this paper proposes an algorithm to update adaptive weights for the Gaussian components of a Gaussian mixture cubature Kalman filter (CKF) in the time update stage. The proposed method approximates the non-Gaussian noise by splitting the system state, process noise, and observation noise into several Gaussian components and updates the weight of the Gaussian components in the time update stage. The method contributes to obtaining a better approximation of the posterior probability density function, which is constrained by the substantial uncertainty associated with the measurements or ambiguity in the model. The estimation accuracy of the proposed algorithm was analyzed using a Taylor expansion. A series of extensive trials was performed to assess the estimation precision corresponding to various algorithms. The results based on the data pertaining to the lake trial of an unmanned underwater vehicle (UUV) demonstrated the superiority of the proposed algorithm in terms of its better accuracy and stability compared to those of conventional tracking algorithms, along with the associated reasonable computational time that could satisfy real-time tracking requirements.

Adaptive Gaussian Mixture based orbit determination with combined atmospheric density uncertainty consideration

Large initial uncertainties in the semi-major axis or force-model uncertainties, such as atmospheric density uncertainty are key drivers of the along-track uncertainty growth. Long propagation times may result in the need to use filtering algorithms for orbit determination that do not reside to the assumption of Gaussianity for state errors but estimate the entire probability density function. Adaptive Gaussian mixture based filters have shown promising results in the past. Previous research in the field of orbit determination using Gaussian mixture filters has restricted its attention to initial uncertainties in the semi-major axis direction. The present paper focuses on the consequences of including realistic, physics-based descriptions of atmospheric density uncertainty into the covariance propagation of the mixture kernels.It is shown that the neglect of process noise, as has been customary for many years, can lead to undesired characteristics of the probability density function (pdf) estimates and that the inclusion of atmospheric density uncertainty process noise, even in cases where it is not the dominant driver of along-track uncertainty growth, is able to correct these deficiencies. For low orbiting satellites with increased ballistic coefficients or small initial uncertainties in the semi-major axis direction, density uncertainty is the dominant driver of the along-track uncertainty increase. Due to its growth that evolves at least cubic in time, situations may arise which require the usage of Gaussian mixtures also for the process noise when working in Cartesian coordinates. The theoretical foundation for this case is elaborated and an algorithm capable of dynamically switching between a single Gaussian and a Gaussian mixture for the density uncertainty process noise is presented.

Continuous-Discrete Multiple Target Filtering: PMBM, PHD and CPHD Filter Implementations

This article develops models and algorithms for continuous-discrete multiple target filtering, in which the multi-target system is modelled in continuous time and measurements are available at discrete time steps. In order to do so, this paper first proposes a statistical model for multi-target appearance, dynamics and disappearance in continuous time, based on continuous time birth/death processes and stochastic differential equations. The multitarget state is observed at known time instants based on the standard measurement model, and the objective is to compute the distribution of the multi-target state at these time steps. For the Wiener velocity model, we derive a closed-form formula to obtain the best Gaussian Poisson point process fit to the birth density based on Kullback-Leibler minimisation. The resulting discretised model gives rise to the continuous-discrete Gaussian Poisson multi-Bernoulli mixture (PMBM) filter, the continuous-discrete Gaussian mixture probability hypothesis density (PHD) filter and the continuous-discrete Gaussian mixture cardinality PHD (CPHD) filter. The proposed filters are specially useful for multi-target estimation when the time interval between measurements is non-uniform.

Posterior Inference on Parameters in a Nonlinear DSGE Model via Gaussian-Based Filters

This paper studies Gaussian-based filters within the pseudo-marginal Metropolis Hastings (PM-MH) algorithm for posterior inference on parameters in nonlinear DSGE models. We implement two Gaussian-based filters to evaluate the likelihood of a DSGE model solved to second and third order and embed them into the PM-MH: Central Difference Kalman filter (CDKF) and Gaussian mixture filter (GMF). The GMF is adaptively refined by splitting a mixture component into new mixture components based on Binomial Gaussian mixture. The overall results indicate that the estimation accuracy of the CDKF and the GMF is comparable to that of the particle filter (PF), except that the CDKF generates biased estimates in the extremely nonlinear case. The proposed GMF generates the most accurate estimates among them. We argue that the GMF with PM-MH can converge to the true and invariant distribution when the likelihood constructed by infinite Gaussian mixtures weakly converges to the true likelihood. In addition, we show that the Gaussian-based filters are more efficient than the PF in terms of effective computing time. Finally, we apply the method to real data.

Mine weighted centroid positioning algorithm based on improved Gaussian mixture filter

Mine weighted centroid positioning algorithm based on improved Gaussian mixture filter

A New State Estimation Method with Radar Measurement Missing

This paper describes a method that addresses the transient loss of observations in sea surface target state estimations. A six degrees of freedom swing platform fixed with a MiniRadaScan is used to simulate the loss of observations. The state transition model based on the historical observation data fit prediction is designed because the existing state estimation method can only use the system model prediction while the observation is missing. An observation data sliding window width adaptive adjustment strategy is proposed that can improve the fitting accuracy of the state transition model. To solve the problem where the weight value of the Gaussian components of the Gaussian mixture filter is not changed in the time update stage while the observation is missing, an adaptive adjustment strategy for the weight is proposed based on the Chapman-Kolmogorov equation, which can improve the estimation precision under the conditions of the missing observation. The simulation test demonstrates the proposed accuracy and real-time performance of the proposed algorithm.

Particle Gaussian mixture filters-II

In our previous work, we proposed a particle Gaussian mixture (PGM-I) filter for nonlinear estimation. The PGM-I filter uses the transition kernel of the state Markov chain to sample from the propagated prior. It constructs a Gaussian mixture representation of the propagated prior density by clustering the samples. The measurement data are incorporated by updating individual mixture modes using the Kalman measurement update. However, the Kalman measurement update is inexact when the measurement function is nonlinear and leads to the restrictive assumption that the number of modes remains fixed during the measurement update. In this paper, we introduce an alternate PGM-II filter that employs parallelized Markov Chain Monte Carlo (MCMC) sampling to perform the measurement update. The PGM-II filter update is asymptotically exact and does not enforce any assumptions on the number of Gaussian modes. The PGM-II filter is employed in the estimation of two test case systems. The results indicate that the PGM-II filter is suitable for handling nonlinear/non-Gaussian measurement update.

Adaptive Gaussian mixture filter for Markovian jump nonlinear systems with colored measurement noises

This paper considers the state estimation of discrete-time Markovian jump nonlinear systems with colored measurement noises obeying a nonlinear autoregressive process of order n, which is motivated by tracking the maneuvering target under electronic countermeasures with high speed sampling or persistent perturbations. In order to remove the measurement noises correlation, the left zero divisor is explored to reconstruct a new measurement equation via difference approach, with the help of applying statistical linear regression to the colored measurement noise model. Then, a novel hypothesis set constituted of all possible values of multi-step Markov jumping parameters is defined and the posterior probability density of the state is derived recursively. By using Gaussian mixtures to approximate the posterior probability densities, an adaptive Gaussian mixture filter for the considered system is proposed, where the Gaussian components with small weights are pruned adaptively through measuring the Alpha (or Beta) divergence for the original and approximated Gaussian mixtures, to achieve a tradeoff between the estimation accuracy and running time. A maneuvering target tracking accompanied by range gate pull-off with different colored measurement noises cases is simulated to validate the proposed method.

State-of-the-art stochastic data assimilation methods for high-dimensional non-Gaussian problems

This paper compares several commonly used state-of-the-art ensemble-based data assimilation methods in a coherent mathematical notation. The study encompasses different methods that are applicable to high-dimensional geophysical systems, like ocean and atmosphere and provide an uncertainty estimate. Most variants of Ensemble Kalman Filters, Particle Filters and second-order exact methods are discussed, including Gaussian Mixture Filters, while methods that require an adjoint model or a tangent linear formulation of the model are excluded. The detailed description of all the methods in a mathematically coherent way provides both novices and experienced researchers with a unique overview and new insight in the workings and relative advantages of each method, theoretically and algorithmically, even leading to new filters. Furthermore, the practical implementation details of all ensemble and particle filter methods are discussed to show similarities and differences in the filters aiding the users in what to use when. Finally, pseudo-codes are provided for all of the methods presented in this paper.