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This paper focuses on adaptive Kalman filtering problem for linear systems with unknown covariances of both dynamic multiplicative noise (multiplicative measurement noise) and additive noises (additive process and measurement noises). A recursive-noise adaptive Kalman filter is proposed to estimate both states and covariances of noises by using the varaitional Bayesian (VB) inference and an indirect method. First, we characterize inverse Wishart priors for both measurement noise covariance and process noise covariance and employ the Student's t-distribution to represent the likelihood function, which is non-Gaussian and affected by mixing multiplicative noise and additive measurement noise. Then, an adaptive Kalman filtering for recursive both noise covariance matrices and dynamic state, is proposed following VB inference. Performance analysis for VB procedures and the proposed filter is provided to ensure the convergence and stability. A target tracking example is provided to validate the effectiveness of the proposed filtering algorithm.

In this article, a new variational adaptive Kalman filter with Gaussian-inverse-Wishart mixture distribution is proposed for a class of linear systems with both partially unknown state and measurement noise covariance matrices. The state transition and measurement likelihood probability density functions are described by a Gaussian-inverse-Wishart mixture distribution and a Gaussian-inverse-Wishart distribution, respectively. The system state vector together with the state noise covariance matrix and the measurement noise covariance matrix are jointly estimated based on the derived hierarchical Gaussian model. Examples are provided to demonstrate the effectiveness and potential of the developed new filtering design techniques.

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O'Connell and N. J. Simm, arXiv:1805.08760] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in [F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Phys. A 49 (2016)] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories., 19 pages, 1 figure

In the application of precise point positioning (PPP), especially in the dynamic mode, the classical Kalman filter (KF) usually produces a large number of estimation errors or diverges when there are gross errors in the observation data or unexpected turbulences occur in target motion state or both of them. For such problem, a variational Bayesian (VB)-based robust adaptive Kalman filtering (VB-RAKF) is proposed in this paper. This filter introduces a classification robust equivalent weight function to resist observation gross error and the inverse Wishart prior to model inaccurate process noise covariance matrix (PNCM). To improve the instantaneous accuracy of state estimation, the VB approach is used to obtain better estimations of inaccurate PNCM. Several sets of observation data collected by IGS reference stations and vehicles are employed to check the robustness and positioning accuracy of the VB-RAKF model. The results show that the VB-RAKF algorithm is more robust than the KF, and can effectively resist the gross error in observation data and control state disturbance. In the IGS reference station tests, when compared to the KF, the static positioning accuracies of the VB-RAKF in the north, east and up directions are improved by 13%, 8% and 22%, respectively, and the simulated dynamic positioning accuracies of the VB-RAKF in the north, east and up directions are improved by 19%, 9% and 21%, respectively. The in-vehicle dynamic test verifies that the VB-RAKF outperforms the KF, and shows that the VB-RAKF has better performance than the KF when dealing with observation data which has obvious gross errors, and similar performance as the KF when gross errors are small.

Diffusion tensor imaging (DTI) is a popular magnetic resonance imaging technique used to characterize microstructural changes in the brain. DTI studies quantify the diffusion of water molecules in a voxel using an estimated 3x3 symmetric positive definite diffusion tensor matrix. Statistical analysis of DTI data is challenging because the data are positive definite matrices. Matrix-variate information is often summarized by a univariate quantity, such as the fractional anisotropy (FA), leading to a loss of information. Furthermore, DTI analyses often ignore the spatial association of neighboring voxels, which can lead to imprecise estimates. Although the spatial modeling literature is abundant, modeling spatially dependent positive definite matrices is challenging. To mitigate these issues, we propose a matrix-variate Bayesian semiparametric mixture model, where the positive definite matrices are distributed as a mixture of inverse Wishart distributions with the spatial dependence captured by a Markov model for the mixture component labels. Conjugacy and the double Metropolis-Hastings algorithm result in fast and elegant Bayesian computing. Our simulation study shows that the proposed method is more powerful than non-spatial methods. We also apply the proposed method to investigate the effect of cocaine use on brain structure. The contribution of our work is to provide a novel statistical inference tool for DTI analysis by extending spatial statistics to matrix-variate data.

Most existing localization schemes necessitate a priori statistical characteristic of measurement noise, which may be unrealistic in practical applications. This paper investigates the variational Bayesian adaptive unscented Kalman filtering (VBAUKF) for received signal strength indication (RSSI) based indoor localization under inaccurate process and measurement noise covariance matrices. First, an inaccurate and slowly varying measurement noise covariance matrix can be estimated by choosing appropriate conjugate prior distribution for an indoor localization model with inaccurate process and measurement noise covariance matrices. By choosing inverse Wishart priors distribution, the state, predicted error and measurement noise covariance matrices are inferred on each time separately. Second, a parameter optimization algorithm is designed to minimize the localization error of VBAUKF until it less than the threshold set in advance. Finally, experimental validation is presented to demonstrate the accuracy and effectiveness of the proposed filtering method for indoor localization.

When the extension state of the non-ellipsoidal extended target (NET) changes, the performance of traditional multiple target tracking algorithms based on the constant number of sub-objects will decrease. To solve this problem, this paper proposes a gamma Gaussian inverse Wishart probability hypothesis density filter for non-ellipsoidal extended targets with varying number of sub-objects, called VN-NET-GGIW-PHD filter. In the proposed filter, each NET is considered as a combination of multiple spatially close sub-objects, and the label management is introduced to realize the association between the NET and corresponding sub-objects. Then, by target spawning and combination, the number of sub-objects for approximating the extension state of each NET can be adjusted automatically. Furthermore, to obtain the partition of the measurement set, an approach based on the clustering by fast search and find of density peaks (CFSFDP) algorithm and expectation maximization (EM) algorithm is proposed. Simulation results show that the proposed filter can adaptively adjust the number of sub-objects and has better performance when the extension state of the NET changes.

In the present research, the features of applying two models for estimating macroeconomic dynamic in the USA are investigated: Bayesian vector autoregression and Bayesian vector autoregression with Markov switching. The research goal is to identify periods, structure of fluctuations and the main directions of interaction of the variables (real US GDP and employment) using Bayesian vector autoregression models. Models with Markov chains include many equations (structures). The switching mechanisms between these structures are controlled by an unobservable variable that follows a first-order Markov process. The analyzed variables were taken from the first quarter of 1953 to the third quarter of 2015. The model parameters were estimated on the basis of a prior for the multivariate normal distribution — the inverse Wishart distribution (a generalization of the Minnesota a priori distribution). Basing on the results of the estimation of the two-dimensional model with Markov Switching the average GDP growth rate and expected duration of phases was calculated. The estimated model is acceptable for describing the US economy and with high accuracy describes the probability of being in a particular phase in different periods of time. On the basis of medium-term forecasts, root mean squared errors of the forecast are calculated and a conclusion is made about the most appropriate model. Within the framework of this paper, impulse response functions are built allowing to evaluate how variables in the model react on fluctuations, shocks.

A slide window variational adaptive Kalman filter is presented in this brief based on adaptive learning of inaccurate state and measurement noise covariance matrices, which is composed of the forward Kalman filtering, the backward Kalman smoothing, and the online estimates of noise covariance matrices. By imposing an approximation on the smoothing posterior distribution of slide window state vectors, the posterior distributions of noise covariance matrices can be analytically updated as inverse Wishart distributions by exploiting the variational Bayesian method, which avoids the fixed-point iterations and achieves good computational efficiency. Simulation comparisons demonstrate that the proposed method has better filtering accuracy and consistency than the existing cutting-edge method.

Moments of multivariate Wishart Distribution are known up to fourth order. But in many contexts, moments of some functions of Wishart distribution and Inverted Wishart Distribution have been found useful in risk theoretic estimation of covariance matrix and its characteristics. In this paper we review moments of some important functions of Wishart and Inverted Distributions.

We consider that existing nonlinear state estimators cannot always yield a satisfactory performance when the distribution of the measurement noise deviates from the assumed Gaussian distribution. This letter investigates the non-linear robust fixed-interval smoothing problem where the measurement vectors are partially disturbed by outliers. The proposed smoother employs a binary indicator variable modelled with a beta-Bernoulli distribution to determine whether the measurement vector is an outlier. Moreover, to address the inaccurate process noise covariance matrix, an inverse Wishart distribution was chosen as the conjugate prior of the process noise covariance matrix. Then, the variational Bayesian inference method was applied to estimate the state trajectory, inaccurate process noise covariance matrix, and a set of binary indicator variables. Traditional target tracking simulation results indicate the effectiveness and stability of the derived smoother, and the proposed smoother has demonstrated a marked improvement in estimation accuracy and robustness.

This paper proposes a robust Poisson multi-Bernoulli mixture (PMBM) filter with inaccurate process and measurement noise covariances. A derivation of the robust PMBM filter is provided for jointly estimating the kinematic state, the predicted state covariance, and the measurement noise covariance. By modeling the augmented state as a Gaussian inverse Wishart inverse Wishart (GIWIW) distribution, a computationally feasible implementation of the robust PMBM (GIWIW-PMBM) filter is given for linear Gaussian systems. To guarantee the conjugacy of the GIWIW distribution, the variational Bayesian (VB) approach is employed to approximate the posterior density. Finally, simulation results show that the GIWIW-PMBM filter has the best overall performance compared to existing state-of-the-art filters regarding computational cost and filtering performance.

Conventional multi-sensor multi-target multi-Bernoulli (MS-MeMBer) filters are based on the assumption that each target produces at most one measurement per time step. However, this assumption is not always reasonable in practice as an extended target can generate multiple measurements per step due to the recent improvement in the sensor resolution. In this case, a potential estimation bias may occur in the current MS-MeMBer filters. Therefore, a novel extended target MS-MeMBer filter and its Gaussian inverse Wishart mixture implementation are given in this paper. Specifically, we modify the update process of the MS-MeMBer filter by assuming that the generation of extended target measurements follows an approximate Poisson-Body model. Simulation results validate that the proposed filter can effectively estimate the shape and position of the extended target.

A prior knowledge of the background parameters such as clutter rate and detection profile is of critical importance in the tracking algorithms under the theory of random finite sets for extended objects which would lead to restrictions in the application. To accommodate this problem, a multiple extended target tracking algorithm based on the generalized labelled multi-Bernoulli (GLMB) filter under the circumstance of unknown clutter rate and detection profile is proposed in this article. After introducing a clutter generator, this new algorithm establishes augmented state space model for targets and clutter and propagates them in parallel by applying multi-class GLMB theory. We then employ Beta to describe detection probability. Target extension is modelled as an ellipse by using gamma Gaussian inverse Wishart distribution. Simulation results indicate that the proposed algorithm has better performance in estimating trajectories and extended shapes compared with the conventional filter having prior knowledge.

This paper proposes a new class of state transition models that afford closed-form predictions for the tracking of extended targets. A key innovation is to employ a non-central inverse Wishart distribution to model the state transition density of the target extent. Importantly, this results in a simplified prediction update that is computationally efficient and improves target tracking performance when compared to state-of-the-art alternatives on standard simulation scenarios.

Aiming at the problem that the performance of adaptive Kalman filter estimation will be affected when the statistical characteristics of the process and measurement of the noise matrices are inaccurate and time-varying in the linear Gaussian state-space model, an algorithm of multi-fading factor and an updated monitoring strategy adaptive Kalman filter-based variational Bayesian is proposed. Inverse Wishart distribution is selected as the measurement noise model and the system state vector and measurement noise covariance matrix are estimated with the variational Bayesian method. The process noise covariance matrix is estimated by the maximum a posteriori principle, and the updated monitoring strategy with adjustment factors is used to maintain the positive semi-definite of the updated matrix. The above optimal estimation results are introduced as time-varying parameters into the multiple fading factors to improve the estimation accuracy of the one-step state predicted covariance matrix. The application of the proposed algorithm in target tracking is simulated. The results show that compared with the current filters, the proposed filtering algorithm has better accuracy and convergence performance, and realizes the simultaneous estimation of inaccurate time-varying process and measurement noise covariance matrices.

The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to $N$ fermions in a Morse potential, which are non-interacting for $\beta=2$. At finite $N$ the distribution of eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar case. The density of fermions in this potential is studied for large $N$, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large $N$ limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valk\'o, Grabsch and Texier, as well as Ossipov, is discussed., Comment: 44 pages, 5 figures, 6 appendices

Estimating correlations among demographic parameters is critical to understanding population dynamics and life‐history evolution, where correlations among parameters can inform our understanding of life‐history trade‐offs, result in effective applied conservation actions, and shed light on evolutionary ecology. The most common approaches rely on the multivariate normal distribution, and its conjugate inverse Wishart prior distribution. However, the inverse Wishart prior for the covariance matrix of multivariate normal distributions has a strong influence on posterior distributions. As an alternative to the inverse Wishart distribution, we individually parameterize the covariance matrix of a multivariate normal distribution to accurately estimate variances (σ 2) of, and process correlations (ρ) between, demographic parameters. We evaluate this approach using simulated capture–mark–recapture data. We then use this method to examine process correlations between adult and juvenile survival of black brent geese marked on the Yukon–Kuskokwim River Delta, Alaska (1988–2014). Our parameterization consistently outperformed the conjugate inverse Wishart prior for simulated data, where the means of posterior distributions estimated using an inverse Wishart prior were substantially different from the values used to simulate the data. Brent adult and juvenile annual apparent survival rates were strongly positively correlated (ρ = 0.563, 95% CRI 0.181–0.823), suggesting that habitat conditions have significant effects on both adult and juvenile survival. We provide robust simulation tools, and our methods can readily be expanded for use in other capture–recapture or capture‐recovery frameworks. Further, our work reveals limits on the utility of these approaches when study duration or sample sizes are small., Specifying hyperpriors (blue) for components of covariance matrices, rather than inverse Wishart priors (red) allows for accurate estimation of process correlations of demographic components.

This study presents an improved multi-target multi-Bernoulli (IMeMBer) gamma Gaussian inverse Wishart (GGIW) filter for tracking multiple extended targets (ETs). The main contribution of this study consists of three parts, first, a novel method is proposed to obtain the unbiased cardinality estimation of multiple targets using the multi-Bernoulli recursion. As a variation of the existing cardinality-balanced MeMBer (CBMeMBer) filter, the presented filter is called the improved MeMBer filter, which overcomes the high detection probability limitation of the CBMeMBer filter. Second, based on the mathematical derivation, the IMeMBer filter is expanded to accommodate the characteristics of the ETs of which each target generates more than one measurement at each time step, and the GGIW method is used for its implementation. The resulting filter simultaneously provides the kinematic, extended and measurement rate states of ETs with an unknown and time-varying number. Third, the simulation results show that the presented filter achieves a considerable performance at the cost of less time, compared to the labelled multi-Bernoulli GGIW filter.

The authors conduct an extensive simulation study to substantially contribute to the question how HB prior parameter settings (i.e. the prior variance and the prior degrees of freedom) affect the performance of HB-CBC models. The statistical performance of HB is evaluated under experimentally varying conditions based on six experimental factors using criteria for goodness-of-fit, parameter recovery and predictive accuracy. The results indicate that the prior degrees of freedom play a negligible role as there is not any noticeable impact on the performance of HB when varying that factor. For increasing prior variance levels overfitting problems occur that markedly affect parameter recovery and model fit, and a number of related interaction effects with regard to the settings for the prior variance can be observed both at the upper and lower level of the HB model. Perhaps the most striking finding however is that the predictive performance of HB-CBC is hardly affected by an increase of the prior variance. Many of our findings regarding the parameter settings of the inverse Wishart prior contrast those reported in previously proposed empirical studies.

As target splitting is not considered in the initial development of δ-generalized labeled multi-Bernoulli (δ-GLMB) filter, the scenarios where the new targets appearing conditioned on the preexisting one are not readily addressed by this filter. In view of this, we model the group target as gamma Gaussian inverse Wishart (GGIW) distribution and derive a δ-GLMB filter based on the group splitting model, in which the target splitting event is investigated. Two simplifications of the approach are presented to improve the computing efficiency, where with splitting detection, we need not to predict the splitting events of all the GGIW components in every iteration. With component combination applied in adaptive birth, a redundant modeling for a newborn target or preexisting target could be avoided. Moreover, a method for labeling performance evaluation of the algorithm is provided. Simulations demonstrate the effectiveness of the proposed approach.

Extended target probability hypothesis density (ET-PHD) filters have recently become popular owing to their relatively simple recursion processes, which makes them suitable for use in applications requiring real-time results, such as radar multi-target tracking. However, in the classic ET-PHD filters, the measurements of different targets are generally considered to be generated at the end of each scan. With this assumption, the measurement time diversity in radar applications cannot be modeled. To address this shortcoming, a novel time-matching ET-PHD filter in which a multi-prediction filtering framework is applied, and the true measurement times are used in the PHD propagation of each cell, i.e., prediction and correction, is proposed in this paper. In addition, a pre-partitioning strategy is employed to reduce the computational complexity of the proposed filter. The results of simulations conducted using the gamma Gaussian inverse Wishart PHD filter indicate that the proposed pre-partitioning-based time-matching ET-PHD filter is superior to standard filters in terms of both estimation accuracy and real-time performance.

In multi-target tracking (MTT), it is hard to guarantee that the dynamic and measurement model mismatches are described by process and measurement noises with suitable covariance matrices (CMs). Meanwhile, model changes caused by target maneuvers and disturbance-corrupted measurements may coexist in a complex environment. Therefore, we propose the problem of MTT under mismatches in both dynamic and measurement models, where the uncertain state prediction error and measurement noise CMs are considered as random variables. Then, a robust generalized labeled multi-Bernoulli (RGLMB) filter framework is derived to recursively propagate the joint posterior density of the multi-target state and the random CMs. By modeling the random CMs as inverse Wishart distributions, on-line random CMs tuning and multi-target state estimation are unified via minimizing the Kullback–Leibler divergence. Simulation results show that the proposed RGLMB filter is robust to different target motion models and noisy radar measurements corrupted by step, sinusoidal and stochastic forms of bias.

This paper proposes an adaptive extended object tracking algorithm with unknown time-varying sensor error covariance in linear state-space models. The proposed algorithm employs Inverse Wishart distribution to describe the full covariance. To produce an analytical solution, the measurement likelihood function introduces a latent variable to obtain an augmented form. Then, the latent variable is involved into the estimated list of quantities. To hold a recursive estimation framework, the proposed algorithm selects variational Bayesian (VB) inference to approximate the joint posterior distribution of estimated quantities. The VB inference minimizes Kullback-Leibler divergence between the true and approximate posterior density to obtain a convergent solution. Simulation experiments with unknown covariance demonstrate the effectiveness of the proposed algorithm.

In surveillance applications, the extent states and measurements of extended targets received by sensors are time-varying. In this paper, we propose a joint tracking and classification (JTC) method for single extended target under the presence of clutter and detection uncertainty. The extent state is modeled as elliptic shape via random matrix model (RMM), and is used as the feature for target classification. To adapt to the time-varying conditions of an extended target, the RMM proposed by Lan et al. is used. Besides, the RMM is integrated into Bernoulli filter to detect an extended target with clutter and detection uncertainty. The resulting method is called joint tracking and classification Gaussian inverse Wishart Bernoulli (JTC-GIW-Ber) filter, and the closed expressions for JTC-GIW-Ber filter recursions are derived under the necessary assumptions and approximations. Comprehensive simulations are carried out to test the performance, and the results demonstrate that the proposed JTC-GIW-Ber filter not only outperforms the JTC-GIW probability hypothesis density (JTC-GIW-PHD) filter and the GIW-Ber filter in extent state estimation, but also outperforms the JTC-GIW-PHD filter in target classification.

In multi-target tracking, the sequential Monte Carlo probability hypothesis density (SMC-PHD) filter is a practical algorithm. Influenced by outliers under unknown heavy-tailed measurement noise, the SMC-PHD filter suffers severe performance degradation. In this paper, a robust SMC-PHD (RSMC-PHD) filter is proposed. In the proposed filter, Student-t distribution is introduced to describe the unknown heavy-tailed measurement noise where the degrees of freedom (DOF) and the scale matrix of the Student-t distribution are respectively modeled as a Gamma distribution and an inverse Wishart distribution. Furthermore, the variational Bayesian (VB) technique is employed to infer the unknown DOF and scale matrix parameters while the recursion estimation framework of the RSMC-PHD filter is derived. In addition, considering that the introduced Student- t distribution might lead to an overestimation of the target number, a strategy is applied to modify the updated weight of each particle. Simulation results demonstrate that the proposed filter is effective with unknown heavy-tailed measurement noise.

Popular measures of reliability for a single-test administration include coefficient α, coefficient λ2, the greatest lower bound (glb), and coefficient ω. First, we show how these measures can be easily estimated within a Bayesian framework. Specifically, the posterior distribution for these measures can be obtained through Gibbs sampling – for coefficients α, λ2, and the glb one can sample the covariance matrix from an inverse Wishart distribution; for coefficient ω one samples the conditional posterior distributions from a single-factor CFA-model. Simulations show that – under relatively uninformative priors – the 95% Bayesian credible intervals are highly similar to the 95% frequentist bootstrap confidence intervals. In addition, the posterior distribution can be used to address practically relevant questions, such as “what is the probability that the reliability of this test is between .70 and .90?”, or, “how likely is it that the reliability of this test is higher than .80?” In general, the use of a posterior distribution highlights the inherent uncertainty with respect to the estimation of reliability measures.

Assuming that the measurement and process noise covariances are known, the probability hypothesis density (PHD) filter is effective in real-time multi-target tracking, however, noise covariance is often unknown and time-varying for an actual scene. To solve this problem, a strong tracking PHD filter based on Variational Bayes (VB) approximation is proposed in this paper. The measurement noise covariance is described in the linear system by the inverse Wishart (IW) distribution. Then, the fading factor in the strong tracking principle uses the optimal measurement noise covariance at the previous moment to control the state prediction covariance in real-time. The Gaussian IW (GIW) joint distribution adopts the VB approximation to jointly return the measurement noise covariance and the target state covariance. The simulation results show that, compared with the traditional Gaussian mixture PHD (GM-PHD) and the VB-adaptive PHD, the proposed algorithm has higher tracking accuracy and stronger robustness in a more reasonable calculation time.

For inference involving a covariance matrix, inverse Wishart priors are often used in Bayesian analysis. To help researchers better understand the influence of inverse Wishart priors, we provide a concrete example based on the analysis of a two by two covariance matrix. Recommendations are provided on how to specify an inverse Wishart prior.

We introduce the new F-Riesz distribution to model tail-heterogeneity in fat-tailed covariance matrix observations. In contrast to the typical matrix-valued distributions from the econometric literature, the F-Riesz distribution allows for different tail behavior across all variables in the system. We study the consistency properties of the maximum likelihood estimator in both static and dynamic models with F- Riesz innovations using both one-step and two-step (targeting) estimation techniques. Allowing for tail-heterogeneity when modeling covariance matrices appears empirically highly relevant. When applying the new distribution to realized covariance matrices of 30 U.S. stocks over a 14 year period, we find huge likelihood increases both in-sample and out-of-sample compared to all competing distributions, including the Wishart, inverse Wishart, Riesz, inverse Riesz, and matrix-F distribution.

This paper proposes a Poisson multi-Bernoulli mixture (PMBM) filter for coexisting point and extended targets, i.e., for scenarios where there may be simultaneous point and extended targets. The PMBM filter provides a recursion to compute the multi-target filtering posterior based on probabilistic information on data associations, and single-target predictions and updates. In this paper, we first derive the PMBM filter update for a generalised measurement model, which can include measurements originated from point and extended targets. Second, we propose a single-target space that accommodates both point and extended targets and derive the filtering recursion that propagates Gaussian densities for point targets and gamma Gaussian inverse Wishart densities for extended targets. As a computationally efficient approximation of the PMBM filter, we also develop a Poisson multi-Bernoulli (PMB) filter for coexisting point and extended targets. The resulting filters are analysed via numerical simulations., Matlab files can be found at https://github.com/Agarciafernandez/Coexisting-point-extended-target-PMBM-filter and https://github.com/yuhsuansia/Coexisting-point-extended-target-PMBM-filter. A relevant multi-object tracking course can be found at https://www.youtube.com/channel/UCa2-fpj6AV8T6JK1uTRuFpw

This paper considers the problem of state estimation with unknown process and measurement noise covariance for a linear Gaussian system. Under the assumption of conjugate priors, inverse Wishart distribution is chosen for both process and measurement noise covariance by introducing a latent variable. Then, the joint state estimation and unknown parameters identification is derived in variational Bayesian framework, and the system state, latent variable, process and measurement noise covariance are updated iteratively. The performance of the proposed algorithm is demonstrated by comparing with the other VB based adaptive filter in a target tracking simulation system.

Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this paper, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors—both subjective and objective—that do not “force eigenvalues apart,” which is a criticism of inverse Wishart and Jeffreys priors. Extensive comparison of these “shrinkage priors” with inverse Wishart and Jeffreys priors is undertaken, with the new priors seeming to have considerably better performance. A number of curious facts about the new priors are also observed, such as that the posterior distribution will be proper with just three vector observations from the multivariate normal distribution—regardless of the dimension of the covariance matrix—and that useful inference about features of the covariance matrix can be possible. Finally, a new MCMC algorithm is developed for this class of priors and is shown to be computationally effective for matrices of up to 100 dimensions.

A polarimetric distributed MIMO radar detector in heterogeneous compound Gaussian clutter is proposed. Based on the modified signal model of the polarimetric distributed MIMO radar, the inverse Gamma distribution assumption on the clutter texture and the complex inverse Wishart distribution assumption on the speckle clutter covariance matrix, the secondary data is used to obtain the maximum posteriori estimation of the texture so as to avoid the integral operation in test statistic. The Bayesian knowledge-aided polarimetric generalized likelihood ratio detector is then acquired. Simulation results show that the proposed detector has a better detection performance than the existing detector.

We propose a new dynamic Nelson--Siegel yield curve model, in which the slope and curvature factor loadings are governed by two time-varying factor-specific decay parameters, and the factor shock variance-covariance (SV) follows a stochastic inverse Wishart process. The importance of incorporating the ensemble of the time-varying factor loadings and SV are demonstrated by comparing the proposed model with simpler specifications in terms of statistical and economic criteria. For the statistical evaluation, we examine the out-of-sample yield curve density forecasting performance. The economic value of the model is measured by the utility gain from the bond portfolio optimization of a Bayesian risk-averse investor. According to our out-of-sample experiment using U.S. monthly yield curve data, the time-varying factor loadings and stochastic variance-covariance accommodate gradual structural changes in the yield curve dynamics around an unconventional monetary policy period, thus improving the predictive accuracy and utility gain.

In this paper, a novel robust Student’s t-based cubature information filter is proposed for a nonlinear multisensor system with heavy-tailed process and measurement noises. At first, the predictive probability density function (PDF) and the likelihood PDF are approximated as two different Student’s t distributions. To avoid the process uncertainty induced by the heavy-tailed process noise, the scale matrix of the predictive PDF is modeled as an inverse Wishart distribution and estimated dynamically. Then, the predictive PDF and the likelihood PDF are transformed into a hierarchical Gaussian form to obtain the approximate solution of posterior PDF. Based on the variational Bayesian approximation method, the posterior PDF is approximated iteratively by minimizing the Kullback-Leibler divergence function. Based on the posterior PDF of the auxiliary parameters, the predicted covariance and measurement noise covariance are modified. And then the information matrix and information state are updated by summing the local information contributions, which are computed based on the modified covariance. Finally, the state, scale matrix, and posterior densities are estimated after fixed point iterations. And the simulation results for a target tracking example demonstrate the superiority of the proposed filter.

Moments of functions of Wishart distributed matrices appear frequently in multivariate analysis. Although a considerable number of such moments have long been available in the literature, they appear in rather dispersed sources and may sometimes be difficult to locate. This paper presents a collection of moments of the Wishart and inverse Wishart distribution, involving functions such as traces, determinants, Kronecker, and Hadamard products, etc. Moments of factors resulting from decompositions of Wishart matrices are also included.

Covariance matrix estimation is a crucial task in adaptive signal processing applied to several surveillance systems, including radar and sonar. In this paper we propose a dynamic learning strategy to track both the covariance matrix of data and its structure (class). We assume that, given the class, the posterior distribution of the covariance is described through a mixture of inverse Wishart distributions, while the class evolves according to a Markov chain. Hence, we devise a novel and general filtering strategy, called multi-class inverse Wishart mixture filter, able to capitalize on previous observations so as to accurately track and estimate the covariance. Some case studies are provided to highlight the effectiveness of the proposed technique, which is shown to outperform alternative methods in terms of both covariance estimation accuracy and probability of correct model selection. Specifically, the proposed filter is compared with class-clairvoyant covariance estimators, e.g., the maximum likelihood and the knowledge-based recursive least square filter, and with the model order selection method based on the Bayesian information criterion.

This paper proposes a variational Bayesian-based cubature Kalman filter to solve the state estimation problem of nonlinear discrete-time systems under dynamic model mismatch and outliers interference. In the proposed filter, the measurement noise is divided into the normal measurement noise and the outlier noise, and two inverse Wishart distributions with different degree of freedom parameters are used to model the unknown normal measurement noise covariance and the unknown outlier noise covariance, respectively. To overcome outliers interference, the judgment factor which follows the Bernoulli-beta distribution is designed to identify the type of measurement noise. Meanwhile, the input vector which follows the Gaussian distribution is introduced to realize the dynamic model revision. The state, the judgment factor, the normal measurement noise covariance, the outlier noise covariance, and the input vector are jointly estimated under the variational Bayesian framework. The numerical simulation is illustrated to verify the performance of the proposed filter and the corresponding results show that the proposed filter has good state estimation performance and robustness.

Use of the Gaussian inverse Wishart probability hypothesis density (GIW-PHD) filter has demonstrated promise as an approach used to track an unknown number of ellipsoidal extended targets. However, based on the assumption that the shape of a target is not ellipsoidal, the performance of the GIW-PHD filter will degrade. This study presents an improved GIW-PHD filter, called a non-ellipsoidal model GIW-PHD (NEM-GIW-PHD) filter, to solve the problem of tracking multiple non-ellipsoidal extended targets. First, an additional target shape estimating approach using B-spline curve is presented for use in the GIW-PHD filter to provide non-ellipsoidal shape parameters. Second, these shape parameters can be used to improve the likelihood function of the GIW-PHD filter. Thus, the identifiability of closely spaced targets with different shapes can be improved. Third, we present an improved shape selection partitioning approach to partition the measurements sets generated by the non-ellipsoidal targets. The simulation results show that the NEM-GIW-PHD filter can achieve higher accuracy than that with the standard GIW-PHD filter in tracking non-ellipsoidal extended targets. In addition, it can provide target shape estimations using the B-spline curve.

We present a JTC-GGIW implementation method, which simultaneously estimates the kinematic, extension, measurement rate and classification states of an extended target.The JTC-GGIW method is applied in the ET-PHD framework, and the presented filter is called the JTC-GGIW-PHD filter.We analyze how a good classification result can improve the performance of tracking.The sizes and shapes of the aircraft carrier of the Nimitz and the frigate of the Oliver Hazard Perry are used to generate simulated data, and the proposed filter is demonstrated on simulated data from real scenarios, compared to the GGIW-PHD filter. This paper presents a novel filter for jointly tracking and classification (JTC) of maneuvering extended targets using the standard probability hypothesis density (PHD) framework. For an extended target, the extended state that describes the target size, shape and orientation is also estimated, in addition to the kinematic state. Assuming that the target size information is known in advance, the presented filter can classify the extended target based on different sizes, instead of based on different kinematic motion modes in point target tracking. By utilizing the known target size information, the presented filter can contribute to a better extended state estimation while classifying, and how a good classification result can improve the estimation is mathematically analyzed. Simulation results show that the presented filter simultaneously provides a superior tracking performance and the correct classification of multiple extended targets, compared to the gamma Gaussian inverse Wishart PHD (GGIW-PHD) filter.

The paper addresses discrete-time event-driven consensus on exponential-class probability densities (including Gaussian, binomial, Poisson, Rayleigh, Wishart, Inverse Wishart, and many other distributions of interest) completely specified by a finite-dimensional vector of so-called natural parameters. First, it is proved how such exponential classes are closed under Kullback-Leibler fusion (average), and how the latter is equivalent to a weighted arithmetic average over the natural parameters. Then, a novel event-driven transmission strategy is proposed in order to trade off the data-communication rate and, hence, energy consumption, versus consensus speed and accuracy. A theoretical analysis of the convergence properties of the proposed algorithm is provided by exploiting the Fisher metric as a local approximation of the Kullback-Leibler divergence. Some numerical examples are presented in order to demonstrate the effectiveness of the proposed event-driven consensus. It is expected that the latter can be successfully exploited for energy- and/or bandwidth-efficient networked state estimation.

This paper deals with the detection problem of a moving target for the distributed airborne multi-input multi-output radar embedded in the compound-Gaussian and non-homogeneous clutters, without assuming that training data are available. A novel detector combing the Bayesian approach and the generalized likelihood ratio test is proposed, where we model the covariance of clutter as an inverse Wishart distribution with an unknown average clutter covariance matrix (ACCM). More precisely, we regard the unknown ACCM as a structured matrix with the Hadamard product form involving two independent parts, composing of the covariance matrix taper (CMT) and the Doppler spectrum component. Further, the proposed detector exploits a nonlinear processing in an iteration fashion to reconstruct sparse signals with the aim of estimating the unknown spectrum of clutters. As to the CMT component, we resort to generalized CMT model to improve estimation accuracy. Numerical simulations are provided to assess the capability of the proposed detector in different complicated scenarios.

In this paper, a novel variational Bayesian (VB)-based adaptive Kalman filter (VBAKF) for linear Gaussian state-space models with inaccurate process and measurement noise covariance matrices is proposed. By choosing inverse Wishart priors, the state together with the predicted error and measurement noise covariance matrices are inferred based on the VB approach. Simulation results for a target tracking example illustrate that the proposed VBAKF has better robustness to resist the uncertainties of process and measurement noise covariance matrices than existing state-of-the-art filters.

In tracking a maneuverable target, a proper estimation method with better filtering accuracy, stronger robustness, and faster convergence speed is crucial to the tracking system. The performance of conventional nonlinear Gaussian approximate filters may decline when the target engages in abrupt state changes or the noise covariance matrix is unknown and time-varying. In order to overcome these problems, a new variational Bayesian-based strong tracking interpolatory cubature Kalman filter (VB-STICKF) is deduced in this paper. Gaussian weighted integrals in the nonlinear filter are performed using the interpolatory cubature rule, which has better numerical characteristics for maneuvering target tracking. Moreover, by introducing the strong tracking filter into ICKF, the fading factor is used to modify the predicted error covariance, and the residual sequences are forced to be orthogonal, thus the decreasing performance resulted by the states change and the uncertain process noise could be effectively prevented. Furthermore, the measurement noise can be estimated online by variational Bayesian approach based on the inverse wishart distribution, the robustness of dealing with the uncertain measurement noise is improved. The detailed derivation of VB-STICKF for the general nonlinear models is presented in the paper. A target tracking problem with model uncertainty and time-varying process and measurement noise is utilized to test the performance of the proposed filter, the experimental results of three different scenarios demonstrate the improved filtering performance of the deduced VB-STICKF algorithm.

A target that generates multiple measurements at each time step is called the extended target and an ellipse can be used to approximate its extension. When the spatial distributions of measurements can reflect its true shape, in this situation the extended target is called a non-ellipsoidal extended target and its complicated extended state cannot be accurately approximated by single ellipse. In view of this, the non-ellipsoidal extended target tracking (NETT) filter was proposed, which uses multiple ellipses (called sub-objects) to approximate the extended state. However, the existing NETT filters are limited to the framework that the number of sub-objects remains still, which does not match the actual tracking situations. When the attitude of the target changes, the view from the sensor on the target may change, then the shape of the non-ellipsoidal extended target varies as well as the reasonable number of sub-objects needed for approximation. To solve this problem, we propose a varying number of sub-objects for non-ellipsoidal extended target tracking gamma Gaussian inverse Wishart (VN-NETT-GGIW) filter. The proposed filter estimates the kinematic, extension and measurement-rate states of each sub-object as well as the number of sub-objects. The simulation results show that the proposed filter can be used for the target changing attitude situation and is more close to the practice application.