A Novel Robust Student’s t-Based Cubature Information Filter with Heavy-Tailed Noises

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.


Introduction
To obtain the reliable and precise information, multisensor systems have become more and more popular in a wide range of applications such as cooperative tracking, autonomous navigation, signal processing, and guidance [1,2]. Therefore, how to best fuse the observations from different sensors has gained much attention in several decades. Kalman filter (KF), one of the most popular state estimation methods, has been used extensively in many multisensor estimation problems [3,4]. However, the KF is a linear state estimation method, and it cannot get an optimal estimation result with non-Gaussian process and measurement noises. But in many engineering applications, such as tracking agile target in clutter with bearing sensors, the system is nonlinear and the heavy-tailed noises are presented [5,6]. In this case, the performance of the classical KF degrades dramatically.
To apply the KF in nonlinear system and improve the accuracy of nonlinear approximation, the first order linearization-based extended Kalman filter (EKF) [7], the unscented transformation-based unscented Kalman filtering (UKF) [8,9], quadrature rule-based Gauss-Hermite filtering (GHF) [10], polynomial interpolation-based divided difference filtering (DDF) [11], and the spherical-radical rulebased cubature Kalman filter (CKF) [12,13] have been proposed successively. Although the information filter (IF) is an algebraically equivalent form of Kalman filter, it is more suitable for multisensor estimation, since the IF can reduce the computational burden significantly and be decentralized easily [14]. Soon, the abovementioned filters were formulated in the information type and applied to the multisensor estimation, such as extended information filter (EIF) [15], unscented information filter (UIF) [16], and cubature information filter (CIF) [17]. Unfortunately, the performance of these information filters will decrease in the presence of heavy-tailed noises, since they all derived based on the Gaussian assumption.
To tackle the heavy-tailed non-Gaussian noise, some Huber-based information filters have been proposed by minimizing a cost function which is a combined l 1 and l 2 norm [18], such as Huber-based UIF [19] and Huber-based CIF [20]. Nevertheless, the Huber's weight function cannot converge to zero even as the error approaches infinity. It indicates the measurements with large errors still will be employed, and the estimation performance will be deteriorated [5,21]. Besides Huber's technique, some maximum correntropy (MC) information filters such as MC-UIF [22] and MC-CIF [5] have been proposed by employing the maximum correntropy criterion as the optimality criterion. However, the estimation covariance cannot be developed due to the lack of theoretical basis, which will degrade the performance of MC Kalman filters.
Since the Student's t distribution is a reasonable model of the heavy-tailed noise, some Student's t-based Kalman filters (STKF) have been proposed by modeling the heavy-tailed noises as the Student's t distribution [23][24][25][26][27]. To obtain the approximate solution of posterior estimations, the posterior probability density function (PDF) has also been approximated by the Student's t distribution [21,23], in which the growth of the dof parameter in STKF degrades the estimation accuracy obviously. To tackle with the difficulty in choosing scale matrix and dof parameter, variational Bayesian (VB) approach is employed to jointly estimate the state vector, scale matrix, and dof parameter [24,25]. In [28], the inaccurate process noise covariance noise and measurement noise covariance matrix is estimated by choosing inverse Wishart priors. In [29], both skewed noise and heavy-tailed noise are considered, whereas Gaussian scale mixture distributions are used to model one-step prediction and likelihood PDFs. Despite the excellent work mentioned above, there have been relatively few results concerning the nonlinear multisensor estimation problem with heavy-tailed process, and measurement noises have not been considered, which serves as the main motivation of this paper.
To cope with the nonlinear multisensor estimation problem with heavy-tailed process and measurement noises, a novel robust Student's t-based cubature information filter (NRSTCIF) is proposed in this paper. At first, the one-step predicted state vector and nominal covariance is propagated based on the cubature rule. Then, the predictive PDF and the likelihood PDF are transformed into a hierarchical Gaussian form to obtain the approximate solution of posterior PDF. Next, the posterior densities are approximated iteratively based on the variational Bayesian (VB) approximation method, where the information matrix and information state are updated by summing the local information contributions. Finally, the state, scale matrix, and posterior densities are estimated after fixed point iterations.
The rest of the paper is organized as follows. In Sec. II, the problem is formulated. Sec. III gives the derivation of the proposed filter. The simulations are conducted in Sec. IV, and the conclusions are drawn in Sec. V.

Problem Formulation
Consider a discrete-time stochastic nonlinear system described by where x k ∈ R n is the n-dimensional state, z k ∈ R m is the m -dimensional measurement, and w k ∈ R n and v k ∈ R m are zero mean heavily tailed process noise and measurement noise with nominal covariance Q k and R k , respectively.
Since v k is a heavy-tailed measurement noise, its distribution is approximated as where Stðv k ; 0, R k , υÞ denotes the Student's t probability distribution function (PDF) of v k , with mean 0 and scale matrix R k , υ is the dof parameter. Based on the measurement equation and the distribution of measurement noise, the likelihood PDF pðz k jx k Þ can be formulated as where hðx k Þ is the mean vector of z k . Due to the heavy-tailed process noise, the predictive PDF pðx k jZ k−1 Þ is also approximated as a Student's t distribution.
wherex k|k−1 is the mean vector, μ is the dof parameter, and T k is the scale matrix which will be approximated as the inverse Wishart distribution in the next section.

Brief Review of Extended Information Filter.
Based on the state model given by (1), the predicted information matrix Y k|k−1 and information stateŷ k|k−1 can be formulated as [16,17].
where Φ is the state-transition matrix which can be computed based on (1). Then, the updated information matrix Y k|k and information stateŷ k|k can be formulated aŝ where i k and I k are information state contribution and information matrix contribution, respectively. They are given by where z k is the measurement at time k and H k is the measurement matrix.
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The Transformation of the Predictive PDF and the Likelihood PDF.
To get the approximate solution of posterior PDF pðx k jZ k Þ, the predictive PDF pðx k jZ k−1 Þ and the likelihood PDF pðz k jx k Þ given by (5) and (4) are written as an infinite mixture of Gaussian distribution [24], i.e., where ς k and η k are auxiliary parameters, Gðω ; a, bÞ is the Gamma PDF of auxiliary parameter ω. Gðω ; a, bÞ is defined by where Γð⋅Þ denotes the Gamma function. Then, ln Gðω ; a, bÞ can be formulated as Since the process noise is a zero mean vector with nominal covariance Q k , the state vector and its associated nominal covariance can be propagated bŷ where χ i,k−1 ði = 1, 2,⋯,2nÞ is the i-th cubature point generated by where S k−1 is the Cholesky decomposition of P k−1 satisfying To accurately describe the predictive PDF, the VB approximation method is applied to dynamically estimate the scale matrix T k rather than set nominal covariance P k|k−1 as T k . Then, the PDF of T k can be appropriated as where IWðT k ; t k , Τ k Þ denotes the inverse Wishart distribution, whose functional form will not change between prior and posterior, t k is the dof parameter, and Τ k is inverse scale matrix. Γ n ð⋅Þ and trð⋅Þ denote the n-variate gamma function and the trace operation, respectively. Then, ln ½IWðT k ; t k , Τ k Þ can be formulated as According to the property of inverse Wishart distribution, we have To capture the statistics of T k , the nominal covariance P k|k−1 is chosen as the mean value of pðT k Þ. Exploiting (19), the inverse scale matrix Τ k is given by where the nonnegative tuning parameter τ = t k − n − 1.
Based on the (10), (11), and (17), the predictive PDF pðx k jZ k−1 Þ and the likelihood PDFpðz k jx k Þ can be transformed as follows: Note that we consider the case that the heavy-tailed process noise statistics, i.e., the scale matrix T k , is unknown and time varying, which is usually the case in practical applications especially for target tracking [29]. Therefore, the scale matrix T k of process noise is modeled as inverse Wishart distribution and estimated dynamically. For measurement noise, we assume that the noise statistics R k is known based on the prior information of sensors; therefore, it can be chosen wisely rather than estimated.

Derivation of Measurement Update Based on VB Approximation
Method. To estimate the state x k , the joint 3 International Journal of Aerospace Engineering posterior PDF pðx k , ς k , T k , η k jZ k Þ is required to be approximated. By making use of the VB approximation method, we have where qð⋅Þ is the approximate posterior PDF, which could be obtained by minimizing the following function.
where KL denotes the Kullback-Leibler divergence function.
where Θ = ½x k , ς k , T k , η k is a set consists of x k , ς k , T k and η k , φ is an arbitrary element of Θ and Θð−φÞ denotes the set Θ without φ, and c φ is the constant related to φ. Since the PDF given by (21) and (22) are conditional independent, pðΘ, Z k Þ could be factored as Substituting (21) and (22) into (26), we have Then, ln pðΘ, Z k Þ can be expressed as [24].
By updating one element of Θ and keeping the other fixed at their last estimated values, the VB-marginals given by (25) can be solved iteratively.

into (25) results in
According to the Bayesian theory, the posterior PDF can be approximated by Exploiting (38), p ði+1Þ ðz k jx k Þ and p ði+1Þ ðx k jZ k−1 Þ can be written as where R Since Gaussian distribution can approximate the posterior PDF more accurate than the Student's t distribution [24], the approximate posterior PDF q ði+1Þ ðx k Þ can be formulated by According to the basic equations of information filter, Based on the linear error propagation, the cross covariance P  (8) and (9) yields Using (43) to simplify (44) and (45) results in The predicted measurementẑ ði+1Þ k|k−1 and cross covariance P ði+1Þ xz,k|k−1 are computed as follows: and S ði+1Þ k|k−1 is the Cholesky decomposition of P ði+1Þ k|k−1 . Based on the information contributions given by (46) and(47), the information state and matrix are updated aŝ Then, the estimated state and covariance can be recovered as The main advantage of the IF is its ability to fuse multisource measurements simply by adding the information contributions to the information matrix and state. Suppose N sensors is available, the measurement equations can be given by where the subscript κ denotes the κ-th sensor, v κ,k~S tðv κ,k ; 0, R κ,k , υÞ is the measurement noise.
Then, the information matrix and information state can be updated byŷ where the information contributions of κ-th sensor are computed by To better illustrate the proposed filtering algorithm, the computational procedures are summarized in Figure 1.

Simulation
To demonstrate the feasibility of the proposed algorithm, a target tracking problem is investigated in this section. The target moves in a plane with an unknown speed and unknown constant turn rate, it is observed by multiple radars in clutter. Due to the rapid motion and clutter, the heavytailed noises are introduced.
The state equation is given by where the state vector x k is defined as x k = ½x k , _ x k , y k , _ y k , ω k T , ω k is the turn rate, T = 1s is the sampling interval, and w k−1 is the process noise generated by where p = 0:05 is probability of outlier, Q k−1 is given by The measurement equation of j-th radar is given by where ðx j,s , y j,s Þ is the position of j-th radar, v k is the measurement noise generated by and R k is given by The locations of the four sensors are given as   International Journal of Aerospace Engineering The initial state estimationx 0|0 is generated randomly based on the true initial value x 0 and initial covariance P 0 given by x 0 = 1000m, 300m/s, 1000m, 0,−3 ∘ /s ½ T P 0 = diag 100m 2 , 10m 2 /s 2 , 100m 2 , 10m 2 /s 2 , 100mrad 2 /s 2 Â Ã T In the simulation, the dof parameter, tuning parameter, and NOI of the proposed filter are set as μ = υ = 5, τ = 5, and d = 10, respectively. The root mean square error (RMSE) is chosen as the performance metric. Based on 1000 indepen-dent Monte Carlo simulation runs, Figure 2 compares the performance of cubature information filter (CIF) [17], Huber-based cubature information filter (HCIF) [19], Student's t-based cubature information filter (STCIF) [23], and the proposed filter. It can be seen that the CIF has the biggest RMSEs among four filters. This is not surprise, since the CIF is essentially a minimizer of a weighted least-squares criterion, which is sensitive to heavy-tailed noises. In addition, one can also observe that the RMSEs of the proposed filter are smaller than the existing HCIF and STCIF, which indicates the superiority of the proposed filter.
The average computational costs of the various filters for each Monte-Carlo run are given in Table 1. It can be seen that   International Journal of Aerospace Engineering the computational time of the NRSTCIF is the longest, and that of the CIF is the shortest. It takes the NRSTCIF about 5 times as long as the CIF. Therefore, the algorithm proposed in this paper does not increase the computation time dramatically.
To clarify the relation between estimation accuracy and NOI, another 1000 independent Monte Carlo simulation runs with different NOIs are conducted based on the proposed filter, and the simulation results are summarized in Figure 3. It can be seen that the averaged RMSEs decrease as the NOI increase when NOI = 1, 2, and 3, but the averaged RMSEs are almost same when NOI ≥4. Therefore, it is unnecessary to choose the NOI as big as possible, and the result is an important reference for NOI choosing. Figure 4 shows the RMSEs of the CIF, HCIF, and the proposed filter with different probabilities of outlier p = 0:01, 0:02, ⋯, 0:1. It can be seen the proposed filter is superior to the CIF and HCIF for all values of p. And the averaged RMSEs of CIF and HCIF increase more rapidly than the proposed NRSTCIF as the increase of p, which indicate the superiority of the NRSTCIF is more and more significant. In other words, the proposed NRSTCIF is more promising when the probability of outlier is larger.

Conclusions
In this paper, a novel robust Student's t-based cubature information filter is proposed to handle the heavy-tailed process and measurement noises in nonlinear multisensor estimation. Based on the cubature rule, the one-step predicted state and the nominal covariance are propagated. To get the approximate solution of the posterior PDF, the predictive PDF and the likelihood PDF described by Student's t distribution are transformed into a hierarchical Gaussian form. To approximate the joint posterior PDF, the VB approximation method is utilized. After fixed point iterations, the state, scale matrix, and posterior densities are estimated. And the simulation results demonstrate the superiority of the proposed filter.

Data Availability
There is no underlying data related to the submission.