Inverse Wishart Distribution Research Articles

A Dirichlet Process Model for Directional-linear Data with Application to Bloodstain Pattern Analysis

Directional data require specialized models because of the non-Euclidean nature of their domain. When a directional variable is observed jointly with linear variables, modeling their dependence adds an additional layer of complexity. A Bayesian nonparametric approach is introduced to analyze directional-linear data. Firstly, the projected normal distribution is extended to model the joint distribution of linear variables and a directional variable with arbitrary dimension projected from a higher-dimensional augmented multivariate normal distribution. The new distribution is called the semi-projected normal distribution (SPN) and can be used as the mixture distribution in a Dirichlet process model to obtain a more flexible class of models for directional-linear data. Then, a conditional inverse-Wishart distribution is proposed as part of the prior distribution to address an identifiability issue inherited from the projected normal and preserve conjugacy with the SPN. The SPN mixture model shows superior performance in clustering on synthetic data compared to the semi-wrapped Gaussian model. The experiments show the ability of the SPN mixture model to characterize bloodstain patterns. A hierarchical Dirichlet process model with the SPN distribution is built to estimate the likelihood of bloodstain patterns under a posited causal mechanism for use in a likelihood ratio approach to the analysis of forensic bloodstain pattern evidence.

Bayesian detection for distributed targets in compound Gaussian sea clutter with lognormal texture

This article investigates the Bayesian detection problem for the distributed targets in the compound Gaussian (CG) sea clutter. The CG sea clutter is formulated as a product of lognormal texture and speckle component with an inverse Wishart distribution covariance matrix (CM). A generalized likelihood ratio test (GLRT) based Bayesian detector, which can operate without training data, is proposed by integrating the speckle CM and estimating the texture using the maximum a posteriori (MAP) criterion. Additionally, three other Bayesian detectors are designed for distributed targets by exploiting the two-step GLRT, the complex-valued Rao, and Wald tests. We first derive the test statistics assuming known texture and speckle CM. Then, by incorporating the MAP-estimated texture components and speckle CM into the test statistics, we present three Bayesian detectors for distributed targets. Finally, simulation experiments validate the detection performance of the proposed Bayesian detectors using both simulated and real sea clutter data.

A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors

This paper proposes a trajectory multi-Bernoulli filter applied to the superpositional sensor model for multi-target tracking in the presence of unknown measurement noise. This filter can provide a Multi-Bernoulli approximation of the posterior density on a set of alive trajectories at the current time step. We also provide a Gaussian mixture (GM) implementation of this filter, employing a mixture of Gaussian and inverse Wishart distributions to represent the combined state of measurement noise and target information. Subsequently, the variational Bayesian (VB) method is employed to approximate the posterior distribution, ensuring its form remains consistent with the prior distribution. This method is capable of directly generating trajectory estimates and can jointly estimate both multi-object tracking and measurement noise covariance. The performance of this algorithm is verified through simulation. Finally, a computationally more efficient L-scan approximation is provided. The simulation results indicate that the filter can achieve robust tracking performance, adapting to unknown measurement noise.

Joint Bayesian estimation of process and measurement noise statistics in nonlinear Kalman filtering

Bayesian filtering aims at sequentially estimating the states and parameters in nonlinear dynamical systems, and finds applications in a variety of engineering fields. Extended and unscented Kalman filter are popular choices for this purpose as they typically achieve a good trade-off between computational complexity and accuracy, while still quantifying posterior uncertainty. However, the accuracy of the identified states and parameters, both in terms of their posterior mean and variance, is heavily influenced by the assumed statistics of both the process and measurement noise terms. Therefore, learning the noise characteristics becomes an essential aspect of Bayesian filtering, but is non-trivial especially in cases where process and measurement noise statistics are jointly non-identifiable. This article proposes a methodology that decomposes measurement and process noise learning in a fully Bayesian setting. For measurement noise estimation, we improve upon a Bayesian method that leverages conjugacy of the inverse-Wishart distribution with respect to the Gaussian likelihood model. For process noise estimation, Bayesian model averaging is coupled with a mutation scheme to compare the performance of competing models while exploring high-probability regions in the space of the unknown process noise variance. The algorithm is first tested on simple numerical models for both state estimation and parameter learning, highlighting the superior accuracy of the proposed measurement noise correction and the benefits of decoupling process and measurement noise learning in non-identifiable settings. Then we demonstrate the usefulness of this algorithm in more challenging problems, namely identification of a multi-degree-of-freedom system under unmeasured excitation, and modeling of the highly nonlinear response of a full-scale bridge pier using experimental data.

Passive underwater tracking with unknown measurement noise statistics using variational Bayesian approximation

This paper considers a bearings-only tracking problem with unknown measurement noise statistics. It is assumed that the measurement noise follows a Gaussian probability density function where the mean and the covariance of the noise are unknown. Here, an adaptive nonlinear filtering technique is proposed, where the joint distribution of the measurement noise mean and its covariance are considered to follow a normal inverse Wishart (NIW) distribution. Using the variational Bayesian (VB) approximation, joint distribution of the target state, the measurement noise mean and covariance is factorized as the product of their individual probability density function (pdf). Minimizing the Kullback-Leibler divergence (KLD) between the factorized and true joint pdfs, probability distributions of the noise mean, covariance and the target states are evaluated. The estimation of states with the proposed VB based method is compared with the maximum a posteriori (MAP) and the maximum likelihood estimation (MLE) based adaptive filtering. Deterministic sigma points are used to realize the filtering algorithms. The proposed adaptive filter with VB approximation is found to be more accurate compared to their corresponding MAP-MLE based counterparts.

Bayesian detection for distributed target with limited training data

The issue of subspace-based distributed target detection with limited training data is addressed in this study. We use the Bayesian method to tackle the issue by assuming that the covariance matrix follows an inverse Wishart distribution. According to the generalized likelihood ratio test, Rao test, and Wald test, three Bayesian detectors are designed. Real data and simulations both attest to the usefulness of the proposed detectors.

Generalised covariance intersection‐Gamma Gaussian Inverse Wishart‐Poisson multi‐Bernoulli Mixture: An intelligent multiple extended target tracking scheme for mobile aquaculture sensor networks

AbstractPoisson multi‐Bernoulli Mixture (PMBM) filter has been known as an available or practical point and multiple extended target tracking (METT) method. The authors present an improved PMBM filter with adaptive detection probability and adaptive newborn distributions, accompanying with an associated distributed fusion strategy for the tracking extended multiple targets. First, the augmented state of unknown and changing target detection probability is assumed as Gamma (GAM) distribution. Second, extended states are described by Inverse Wishart (IW) distribution based on this augmented state, accompanying with dynamic states presented by Gaussian distribution. And then, an adaptive newborn distribution is adopted to describe the newborn targets appearing arbitrarily. Consequently, the closed‐form solutions of the proposed filter can be derived by approximating the intensity of newborn and potential targets to the Gamma Gaussian Inverse Wishart (GGIW) form. Moreover, the fused means that Generalised Covariance Intersection (GCI) is performed in such a large‐scale aquaculture sensor network. Experiments are presented to verify the availability of the GCI‐GGIW‐PMBM method, and comparisons with other METT filters also demonstrate that tracking behaviours are improved largely.

A Variational Bayesian Truncated Adaptive Filter for Uncertain Systems with Inequality Constraints

In this paper, a variational Bayesian (VB) truncated adaptive filter for uncertain systems with inequality constraints is proposed. By choosing the skew‐t and inverse Wishart distributions as the prior information of the measurement noise and predicted error covariance matrix, the state vector, the predicted error covariance matrix, and noise parameters are inferred and approximated by using the VB method. To achieve the inequality‐constrained estimation, the constrained state is computed by truncating the probability density function (PDF) of the estimated state after the variational update stage; the mean and covariance of the constrained state are the first and second moments of the truncated PDF. Considering the model uncertainties where the system dynamics are unpredictable, a multiple model VB truncated adaptive filter is proposed in the interacting multiple model framework. The performances of the proposed algorithms are evaluated via the target tracking simulations and the robot positioning experiments. Results show that the proposed algorithms improve estimation accuracy compared with the existing adaptive filters when the states suffer inequality constraints.

An adaptive outlier-robust Kalman filter based on sliding window and Pearson type VII distribution modeling

Due to inaccurate noise modeling and only using current measurement, existing robust filtering algorithms cannot obtain good state estimation results when measurement is interfered by outliers. In this article, a Pearson type VII (PTV) distribution-based adaptive sliding window outlier-robust Kalman filter (PTVSWAKF) is proposed. According to the characteristics of the parameters, the PTV and inverse Wishart distributions are respectively modeled as prior distributions for heavy-tailed measurement noises (HMN) and covariance. Two different Gamma distributions are modeled as priori distributions for the two degree of freedom (DOF) parameters to achieve adaptive updating of them. By the smooth posterior distribution of the sliding window state values, the state, measurement noise covariance, auxiliary random variable and DOF parameters are jointly iterated and updated to solve the outlier interference problem through variational Bayesian. Finally, the simulation experiments verify the proposed PTVSWAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

Variational Bayesian-based robust adaptive filtering for GNSS/INS tightly coupled positioning in urban environments

Kalman filter (KF) is an efficient approach for state estimation in integrated positioning systems of global navigation satellite systems and inertial navigation systems (GNSS/INS). Unfortunately, GNSS/INS integrated positioning systems are susceptible to state model perturbations and measurement outliers, which can severely degrade KF performance, especially in complex urban environments. To solve this problem, a robust adaptive filtering algorithm based on variational Bayesian (VBRAKF) is presented that can be employed in GNSS/INS tightly coupled positioning systems. In VBRAKF, the robust estimation approach is first introduced into the filter, followed by modeling the predicted covariance matrix using the inverse Wishart distribution, and finally the state is estimated using Variational Bayesian. The robustness is achieved by adjusting the noise by an equivalent weight matrix to reduce gross errors in observations, while the adaptiveness is realized by the variational Bayesian estimation method for the uncertain predicted noise covariance matrix. The positioning performance of the proposed algorithm in urban environments is verified through two sets of land vehicle experiments. The experimental results demonstrate that the 3D positioning accuracy of the VBRAKF algorithm is improved by 32.1 % and 23.5 % in experiment 1 and by 37.1 % and 17.3 % in experiment 2 when compared to KF and Robust Kalman Filter (RKF). The proposed VBRAKF algorithm has better positioning performance in all schemes, can simultaneously control the influence of measurement outliers and inaccurate noise statistics and improves the accuracy of filter estimation.

A Gaussian–Pearson type VII adaptive mixture distribution-based outlier-robust Kalman filter

Considering that existing robust filtering algorithms rely on the selection of initial values of degree of freedom (DOF) parameters in outlier interference environments and cannot effectively cope with unknown non-stationary heavy-tailed measurement noises (HMN), a Gaussian–Pearson type VII (PTV) adaptive mixture distribution-based outlier-robust Kalman filter (GPTVMAKF) is proposed. In order to determine whether the current measurement is a normal value or an outlier, a judgment factor subject to the Beta-Bernoulli distribution is introduced. PTV distribution is used to model HMN caused by outliers, and two Gamma distributions are used to model the two different DOF parameters, which can make the PTV distribution have the adaptive adjustment ability. By introducing the inverse Wishart distribution as the prior distribution of the measurement noise covariance, which is adaptively estimated to cope with the unknown time-varying measurement noises. The state and parameters are jointly estimated by variational Bayesian. Finally, the simulation experiments verify that the proposed GPTVMAKF can obtain more accurate state estimation than existing filters in the environments with varying degrees of HMN and unknown non-stationary HMN.

Student’s t-Based Robust Poisson Multi-Bernoulli Mixture Filter under Heavy-Tailed Process and Measurement Noises

A novel Student’s t-based robust Poisson multi-Bernoulli mixture (PMBM) filter is proposed to effectively perform multi-target tracking under heavy-tailed process and measurement noises. To cope with the common scenario where the process and measurement noises possess different heavy-tailed degrees, the proposed filter models this noise as two Student’s t-distributions with different degrees of freedom. Furthermore, this method considers that the scale matrix of the one-step predictive probability density function is unknown and models it as an inverse-Wishart distribution to mitigate the influence of heavy-tailed process noise. A closed-form recursion of the PMBM filter for propagating the approximated Gaussian-based PMBM posterior density is derived by introducing the variational Bayesian approach and a hierarchical Gaussian state-space model. The overall performance improvement is demonstrated through three simulations.

Modified Strong Tracking Slide Window Variational Adaptive Kalman Filter With Unknown Noise Statistics

The filter performance will be degraded in the measurements with time-varying and unknown noise statistics. To combat the above challenges, a modified strong tracking slide window variational adaptive Kalman filter algorithm is proposed in this article. Firstly, the multiple fading factors are integrated into the proposed algorithm to adjust the error covariance. Next, an improved adaptive slide window method is designed for variational Bayesian Kalman filtering by adaptively adjusting the slide window size and correcting the previous state according to the later state, which improves the estimation accuracy and computational efficiency. Finally, the inverse Wishart distribution is considered for modeling process and measurement noise, and the state vector, as well as noise statistics, are inferred via the variational Bayesian technique without prior noise covariance information. Simulation results demonstrate that the proposed filter algorithm is more robust than existing filters in counteracting measurement and process noise uncertainties.

Adaptive distributed multiple-model filter with uncertainty of process model

Target tracking is a critical application in multiple-sensor state estimation. However, it is subject to uncertainty in the process model due to target maneuvering, which includes motion model uncertainty and process noise uncertainty. In this paper, we present an adaptive distributed multiple-model filter for maneuvering target tracking that solves the two types of uncertainties by variational Bayesian. Initially, a centralized algorithm is developed over multiple-sensor. Using Categorical distribution and inverse Wishart distribution as the conjugate prior of motion model and process noise covariance respectively, the target state, model identity and noise parameters are jointly estimated by variational Bayesian. The algorithm is then extended to a distributed version where prior information and likelihood function achieve consensus through the Alternating Direction Method of Multipliers (ADMM). We also analyze the consensus and convergence of the proposed algorithm. The simulation results demonstrate the effectiveness of the proposed algorithm for maneuvering target tracking under process model uncertainty.

Maximum correntropy criterion variational Bayesian adaptive Kalman filter based on strong tracking with unknown noise covariances

The performance of the current state estimation will degrade in the existence of slow-varying noise statistics. To solve the aforementioned issues, an improved strong tracking maximum correntropy criterion variational-Bayesian adaptive Kalman filter is presented in this paper. First of all, the inverse-Wishart distribution, as the conjugate-prior, is adopted to model the unknown and time-varying measurement and process noise covariances, then the noise covariances and system state are estimated via the variational Bayesian method. Secondly, the multiple fading-factors are obtained and evaluated to modify the prediction error covariance matrix to address the problems associated with inaccurate error estimation. Finally, the maximum correntropy criterion is employed to correct the filtering gain, which improves the filtering performance of the proposed algorithm. Simulation results show that the proposed filter exhibits better accuracy and convergence performance compared to other existing algorithms.

A novel robust IMM filter for jump Markov systems with heavy-tailed process and measurement noises

In some tracking application scenarios, the performances of the traditional interacting multiple model (IMM) approach may degrade dramatically since the presence of outliers in process and measurement noises. Aiming at solving the filtering problem for jump Markov systems containing heavy-tailed process and measurement noises, a new robust student's t-based Gaussian approximate filter utilizing the IMM filtering framework is presented in this paper. Firstly, the heavy-tailed process and measurement noises are assumed to obey student's t-distributions. Secondly, the degree of freedom parameters and the scale matrices are modeled as Gamma distributions and inverse-Wishart distributions respectively. Finally, the system state, degree of freedom parameters, scale matrices and the mode probabilities are inferred simultaneously by introducing the variational Bayesian (VB) theory. Maneuvering target tracking simulation verifies that the designed algorithm achieves superior estimation performances among the state-of-the-art filters.

A Novel State Estimation Approach for Suspension System with Time-Varying and Unknown Noise Covariance

In this paper, a novel state estimation approach based on the variational Bayesian adaptive Kalman filter (VBAKF) and road classification is proposed for a suspension system with time-varying and unknown noise covariance. Using the VB approach, the time-varying noise covariance can be inferred from the inverse-Wishart distribution and then optimized state estimation by the finite sampling posterior probability distribution function (PDF) of noise covariance and backward Kalman smoothing. In addition, a new road classification algorithm based on multi-objective optimization and the linear classifier is proposed to identify the unknown noise covariance. Simulation results for a suspension model with time-varying and unknown noise covariance show that the proposed approach has a higher performance in state estimation accuracy than other filters.

Tuning-Free Bayesian Estimation Algorithms for Faulty Sensor Signals in State-Space

Sensors provide insights into the industrial processes, while misleading sensor outputs may result in inappropriate decisions or even catastrophic accidents. In this article, the Bayesian estimation algorithms are developed to estimate unforeseen signals in sensor outputs without tuning. The optimal Bayesian estimation method is first derived by incorporating a Gaussian distribution specifying potential unmodeled dynamics into the measurement equation. Since its performance depends on tuning parameters, an iterative Bayesian estimation algorithm is developed using the variational inference technique. Specifically, an inverse Wishart distribution is introduced to describe the predicted covariance of abnormal signals. We then estimate it together with the other independent Gaussian distributions to conditionally approximate the joint posterior distribution, by which the effects of tuning parameters can be replaced adaptively. Testing the proposed algorithms through a simulated electromechanical brake model and a real experimental system shows that the proposed algorithm can satisfactorily estimate additive sensor faults online and services as a sensor monitor that simultaneously provides the locations and magnitudes of faulty signals without tuning.

Variational Bayesian-Based Moving Horizon Estimation of Toolface for Rotary Steerable Drilling Tool Systems

This article is concerned with the estimation problem of Toolface for dynamic point-the-bit rotary steerable drilling tool systems. First, considering the unknown frequency and amplitude of vibration during the drilling process, the Toolface system is modeled as a time-varying stochastic system with unknown and time-varying noise covariance matrices. Under the assumption that the noises and their covariance matrices, respectively, obey the Gaussian distribution and the inverse Wishart distribution, the variational Bayesian-based moving horizon estimation algorithm is proposed. Then, the state and the noise covariance matrices are inferred by iteratively updating their approximate posterior probability distributions. Finally, simulations and experiments are provided to demonstrate the effectiveness and superiority of the developed estimation scheme.

An Adaptive and Scalable Multi-Object Tracker Based on the Non-Homogeneous Poisson Process

This paper proposes a new adaptive framework for tracking multiple objects in the presence of data association uncertainty and heavy clutter, either with or without knowledge of the measurement rates and/or target shapes. Built upon an online Gibbs sequential Markov chain Monte Carlo sampling scheme, the adaptive tracker is Bayesian optimal and robust, requiring no additional approximations or measurement partition steps. With a non-homogeneous Poisson process measurement model, our tracker can tackle the data association task with linear computational complexity. Meanwhile, we study generalised inverse Gaussian and inverse Wishart distributions for modelling Poisson rates and object shapes, respectively; these prior models ensure closed-form full conditionals in our online Gibbs sampling steps, under which object states and shapes can be jointly estimated with associations and Poisson rates in a parallel fashion. Furthermore, a fast Rao-Blackwellisation scheme for linear Gaussian dynamics is designed and demonstrated to significantly improve both tracking efficiency and accuracy. We validate the efficacy of our method on real and simulated data.