Pearson type VII distribution‐based robust Kalman filter under outliers interference

IET Radar, Sonar & NavigationVolume 13, Issue 8 p. 1389-1399 Research ArticleFree Access Pearson type VII distribution-based robust Kalman filter under outliers interference Yun Peng, Yun Peng School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this authorWu Pan-long, Corresponding Author Wu Pan-long plwu@163.com School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this authorHe Shan, He Shan School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this author Yun Peng, Yun Peng School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this authorWu Pan-long, Corresponding Author Wu Pan-long plwu@163.com School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this authorHe Shan, He Shan School of Automation, Nanjing University of Science & Technology, No. 200 Xiaolingwei, Nanjing, People's Republic of ChinaSearch for more papers by this author First published: 03 June 2019 https://doi.org/10.1049/iet-rsn.2018.5534Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the authors focus on state estimation in a linear discrete time system where measurements are uncertain due to possible outliers. A novel robust Kalman filter based on the Pearson type VII (PTV) distribution is proposed by using the variational Bayesian (VB) method and tested in a target tracking example. To judge whether the measurement is an outlier, a judgement factor that follows the beta-Bernoulli distribution is introduced in the measurement modelling. In addition, the outlier noise is approximated by the PTV distribution considering the heavy tail characteristic and it is jointly estimated with the judgement factor and state using the VB method. The simulation results show that the proposed filter has better estimation accuracy than most existing robust filters. 1 Introduction State estimation problems are quite common in the fields of target tracking, inertial navigation, and signal processing [1-5]. Researchers have proposed a large number of filters to deal with these problems. Among these, the Kalman filter (KF) is the most popular choice [6], due to its low computational complexity and easy implementation in engineering. The KF is essentially a minimum mean square error estimator for a discrete time state space linear system where the process noise and measurement noise follow a Gaussian distribution. In some practical projects, the measurement is subject to interference that may occur during the acquisition process of sensors, converters, and radio transmissions [7, 8]. The measurements then no longer follow a Gaussian distribution and have heavy tail characteristics [9-11], which leads to a terrible performance of the KF. Many filters have been designed for linear systems under non-Gaussian noise to meet the practical project's requirements. The Huber–KF (HKF) is an optimal estimation method based on the l1/l2 mixed norm function [12-14]. Its robustness is better than the l2 norm-based estimation method and it can effectively reduce the impact of the outlier interference. However, when the non-Gaussian noise intensity is large, the HKF has poor estimation accuracy. A maximum correntropy criterion KF (MCCKF) has been proposed to improve the estimation accuracy under non-Gaussian noise with a large intensity by maximising the correntropy of the prediction error and residual error [15-17]. However, it is difficult to establish the correntropy function of the MCCKF because the kernel parameters lack sufficient prior information. To make full use of the heavy tail characteristics of the noise, the Student's t filter (STF) is put forward [10], which models the noise as a Student's t distribution. Unfortunately, as the degree of freedom (dof) parameter grows during the updating process, the heavy tail characteristic of the noise is weakened, which degrades the estimation accuracy. In order to overcome the shortcomings of STF, a possible alternative is the VB–KF, which also models noise as a Student's t distribution, and achieves a joint estimation of state and the dof parameters by using the VB method [18-20]. However, due to the excessive number of iterations, the implementation time of filtering is long. Another new robust KF called the outlier detection KF (OD-KF) can identify the measurement type and update the measurement covariance [21]. However, the OD-KF cannot model outlier noise, which leads to poor estimation accuracy when the outlier probability is large. In fact, the statistical characteristics of outlier noise are quite different from those of normal measurement noise, and some robust filters [18-20] often combine the outlier noise with the normal measurement noise to construct the statistical characteristics of measurement noise and take them as a whole description measurement model. The idea is that if we can distinguish the type of measurement and use the corresponding statistical characteristics to describe the measurement model, then the measurement model is more suitable for describing the system itself. In order to judge the type of measurement, we introduce a judgment factor and model it as a beta-Bernoulli distribution. Then, based on the heavy tail characteristics of outlier noise, it is modelled as a PTV distribution [22]. Finally, the state updating is implemented in the framework of VB. Simulation results show that the robust filter only needs two iterations to converge, so it has high real-time performance, and the proposed filter has higher estimation accuracy than the KF, MCCKF, STF, VB-KF, and OD-KF. Therefore, the proposed filter is more suitable for solving the state estimation problem of discrete linear systems under outliers' interference. The structure of this paper is as follows. In Section 2, the problem under consideration is formulated. In Section 3, the principle of the VB method is reviewed. In Section 4, we propose a robust KF and present the entire filtering process. In Section 5, a target tracking example is illustrated to verify the effectiveness of the proposed filter. Lastly, some conclusions are drawn in Section 6. 2 Background Consider the linear discrete-time system described by the following state space model: (1) (2) where k is the discrete time index, is the state vector, is the state transition matrix, is the measurement vector, is the measurement matrix, n is the dimension of the state vector, m is the dimension of the measurement vector, is the process noise, and is the measurement noise. follows the Gaussian distribution (3) where denotes the probability density function (pdf), denotes the Gaussian distribution with mean vector and covariance matrix , and is the process noise covariance matrix. is the non-Gaussian noise with mean vector . Owing to outlier interference with heavy tail characteristics, includes normal measurement noise and outlier noise. If the type of measurement (normal measurement or outlier) can be identified, the corresponding noise is more suitable than to be used in the measurement equation. Therefore the measurement equation can be rewritten as (4) (5) where is the judgement factor, indicates that is a normal measurement, indicates that is an outlier and is the outlier noise. follows the Gaussian distribution (6) where is the normal measurement noise covariance matrix. Based on the judgement factor, the PDF of is rewritten as (7) , where (8) (9) From (7), we can obtain that (10) Since has heavy tail characteristics, it means that (11) From (10) and (11), we can obtain that (12) where denotes the transpose of the matrix. According to (11) and (12), also has heavy tail characteristics, so it can be modelled as a PTV distribution as follows [20]: (13) where and are the dof parameters, denotes the PTV distribution with mean vector , scale matrix . When , the PTV distribution becomes the Student's t distribution. For the normal measurement, the KF [6] can be used to estimate the state vector (14) (15) (16) (17) (18) where is the predicted state vector, is the corresponding covariance matrix, is the Kalman gain, is the state estimation vector, and is the corresponding covariance estimation matrix. For the outlier, the corresponding noise no longer follows the Gaussian distribution, and the KF cannot achieve an optimal estimation. However, it is difficult to calculate the posterior distribution of due to . In addition, the most important problem is that we do not know how to judge the type of measurement. Even if we design a corresponding filter for the outlier, it is still difficult to decide which filter to choose for state estimation if the type of measurement cannot be judged. In order to solve these problems, this study designs a robust KF that can judge the type of measurement and determine the state estimation via the VB method. Before introducing the proposed filter, it is necessary to know what exactly the VB method is. 3 VB method According to the Bayesian rule, the posterior distribution of the parameter set can be calculated by (19) where is the posterior distribution of the parameter set and Y is the observation set. When the pdf of does not follow the Gaussian distribution, the posterior distribution of is difficult to determine. However, VB can obtain an approximate posterior distribution by means of an iterative approximation, which effectively solves the problem of the posterior distribution of the parameter set under non-Gaussian conditions. To measure the approximation of the approximate distribution, we need the Kullback–Leibler divergence function (20) where , if and only if . is the approximate distribution of the parameter set; the smaller is, the higher the degree of approximation of is. In fact, can be written as (21) where is the lower bound function and is the joint pdf of and . Since is a constant, if we want to minimise , needs to be as large as possible. Based on the theory of mean field, can be written as (22) where is the total number of elements in the parameter set and is the element. After the end of the iteration, according to (22), we can obtain that (23) where represents the integral of except for . At the same time, we can use (21) and (22) to obtain the element distribution , which is obtained from the j + 1st iteration and it is formulated as (24) (25) where represents the statistical expectation of except for . In order to compute at the first iteration, we need to set the initial element distribution . In the process of solving (24), the parameter distribution in the previous iteration step needs to be used. At this time, the iteration process can be terminated by setting a fixed value of iterations N. The parameter distribution that is obtained by the last iteration is the approximate distribution. The VB method is summarised in Algorithm 1 (see Fig. 1). Fig. 1Open in figure viewerPowerPoint Algorithm 1. The pseudo-code of the VB method 4 PTV distribution-based robust KF In this section, a PTV distribution-based robust KF is derived for the linear discrete-time system with measurements (4) and (5). When the measurement is a normal measurement, measurement (4) is used. When the measurement is an outlier, measurement (5) is used. In order to judge the type of measurement automatically, the judgement factor is modelled as a beta-Bernoulli distribution [23] (26) (27) where is assumed to be drawn from a Bernoulli distribution parameterised by , follows the beta distribution, and are the dof parameters, is a beta function, and denotes the beta distribution with dof parameters and . When , and . represents the covariance matrix of variables a and b. In order to achieve the measurement likelihood function, the measurement noise needs to be analysed. For a normal measurement, the corresponding noise follows the Gaussian distribution, so the pdf is calculated using (4) and (6) (28) According to [20], the pdf of is formulated as (29) where is an auxiliary random variable, when , . denotes a gamma distribution with dof parameters and . The PDFs and are modelled as (30) (31) According to (5) and (31), the pdf is formulated as (32) According to (28), (32), and measurement equation, the measurement likelihood function is formulated as (33) It can be seen from (33) that regardless of the type of measurement, the measurement likelihood function is modelled as a Gaussian distribution. In the modelling process, three parameters , , and are introduced. In this study, we assume that , , and are not related to . and are not related to , and the state space model is a first-order hidden Markov model. According to the state space model, we can find that the hidden random variables include , , and . So, the pdf is formulated as (34) In order to estimate state by using the VB method, The parameter set can be composed of , , , and . So, can be rewritten as (35) According to the Chapman–Kolmogorov equation, the pdf is formulated as (36) where denotes the state transition pdf and denotes the state prediction pdf. According to (1) and (3), is formulated as (37) The state posterior PDF is assumed to be (38) Since the product of two normal random variables is proportional to the pdf of another normal. is rewritten as (39) According to (26), (27), (33), (34), and (39), the joint pdf is formulated as (40) Based on the VB method, we can achieve , , , and . Firstly, needs to be calculated. According to (24) and (40), is formulated as (41) (42) (43) where is a constant that makes . is the statistical expectation of variable at the jth iteration and is the trace of matrix . According to (41) and Appendix 1, the approximate distribution is formulated as (44) , where is a constant that makes . Similarly, can be rewritten as (45) where the state vector and corresponding covariance at the j + 1st iteration are given by (46) (47) (48) (49) where is the gain matrix at the j + 1st iteration and is the measurement covariance at the j + 1st iteration. It can be seen from (45) that the state estimated by using the VB method still follows the Gaussian distribution. The most significant essential difference between it and the KF is that it can automatically give a matching measurement covariance according to the implicit characteristics of the measurement. When , it means that the proposed filter considers the measurement as a normal measurement, so . When , it means that the proposed filter considers the measurement as an outlier, so the measurement noise is modelled as a PTV distribution and the proposed filter automatically assigns a measurement covariance using the VB method. In order to achieve the measurement covariance ,we need to calculate and , which can be obtained by the approximate distribution and . According to (24) and (40), is formulated as (50) (51) where is a constant that makes . According to (45), can be rewritten as (52) It is assumed that the initial approximate distribution is given by (53) According to (53), is formulated as (54) The approximate distribution is modelled as a Bernoulli distribution by using (50). In this paper, in order to make the measurement dominate in the effect on , we ignore the effect of , and . According to (26), the probability of is formulated as (55) (56) where is a constant that makes . is formulated as (57) According to (55) and (56), we need to calculate and , which can be obtained by the approximate distribution . Similarly, using (24) and (40) is formulated as (58) where is a constant that makes . According to (27) and (58), the approximate distribution is modelled as a beta distribution. is formulated as (59) where the dof parameters and at the j + 1st iteration are given by (60) (61) According to (59), and are formulated as (62) (63) where is the digamma function. Therefore, we can use (55)–(63) to calculate . In order to calculate , we need to obtain the approximate distribution . According to (24) and (40), is formulated as (64) where is a constant that makes . The approximate distribution is also modelled as a gamma distribution by using (64) and is formulated as (65) where the dof parameters and at the j + 1th iteration are given by (66) (67) According to (65), is formulated as (68) From the filtering process, we can see that the statistical expectations , , , and are obtained from the previous iteration. During the first iteration, the initial statistical expectations of , , and , which are expressed as and need to be obtained. We assume that , , and . According to Appendix 1, is formulated as (69) According to (26) and (27), is formulated as (70) When , is formulated as (71) When , is formulated as (72) where is the initial statistical expectation of . According to (71) and (72), is formulated as (73) where is given by (74) In summary, the proposed filter only needs the previous state, the measurement, and the initial DOF parameters c, , e, and f to achieve a state update. Owing to lack of sufficient prior information, the dof parameters c, d, e, and f need to be selected. In order to select the relevant parameters more reasonably, we discuss the selection principle of c, , e, and f, here we design the corresponding parameter selection method based on residual error. When the process equation is well matched with the actual system if the measurement is a normal measurement, the residual error and are usually smaller than these in the outlier. According to the proposed filter framework, the design parameters should also be guaranteed and , because these conditions can satisfy . Based on the above information, we can obtain that (75) (76) Usually, when we select and to make , it needs to satisfy that (77) When the measurement is an outlier, the residual error and are usually bigger than these in the normal measurement. According to the proposed filter framework, the design parameters should be guaranteed and is usually small. Based on the above information, we can obtain that (78) (79) In order to satisfy (78), a smaller value should be assigned to . Similarly, according to (79), we should select and to make has a large value, so a smaller value should be assigned to and a bigger value should be assigned to . In this study, the parameters are chosen to lie within the range , and . The related results are shown in the later simulation. In fact, the prior distribution of each parameter is not arbitrarily given. It can be found from the proposed filter that the selection of these distributions can ensure that the prior distribution and the posterior distribution of parameters are conjugate distribution. It can greatly reduce the computational complexity. Finally, the proposed filter where the total number of steps is set as is summarised in Algorithm 2 (see Fig. 2). Fig. 2Open in figure viewerPowerPoint Algorithm 2. The pseudo-code of the proposed filter 5 Simulation In this section, a target tracking problem is discussed. The target undergoes uniform motion in two-dimensional space [19]. The process equation is modelled as a constant velocity model and the measurement equation can determine the position of the target, so the state matrix and the state space model are formulated as (80) (81) (82) where and are projections of the target position on the two axes, and are projections of the target velocity on two axes, and is the sample time. Owing to the outlier, the measurement noise follows the Gaussian mixture distribution (83) where is a probability factor that indicates the probability of outlier. is a gain factor and is the outlier covariance matrix. The process noise covariance matrix and the measurement noise covariance matrix are formulated as (84) (85) where and are the noise parameters. From (83), we can see that the normal measurement noise and the outlier noise follow the Gaussian distribution. Compared with the normal measurement noise, the outlier noise has heavy tail characteristics. In this simulation, the KF [6], MCCKF [17], STF [10], OD-KF [21], VB-KF [19], and proposed filter are tested. The kernel size of the MCCKF is set as . The dof parameter of the STF is set as . The dof parameters and the number of iterations of the OD-KF are set as , , and . The dof parameters and the number of iterations of the VB-KF are set as and . The dof parameters and the number of iterations of the proposed filter are set as , , , , and . These filters are coded in MATLAB and the computer the program is run on has an Intel Core i7-7700HQ CPU at 2.80 GHz. The parameters are set as , , , , and . The initial state matrix and corresponding covariance matrix are set as (86) (87) The true trajectory of the target is shown in Fig. 3. To compare the performance of these filters, the root mean square error (RMSE) and the time-averaged RMSE (TRMSE) are used as performance metrics. The averaged of parameter , the RMSE and TRMSE of the position are defined as (88) (89) (90) where is the estimated parameter at the sth Monte Carlo experiment. and are the estimated and true positions at the sth Monte Carlo experiment. is the number of Monte–Carlo experiments, and is the simulation time. Fig. 3Open in figure viewerPowerPoint True trajectory of a target The RMSEs and TRMSEs of the position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter are shown in Fig. 4 and Table 1. It is seen that the proposed filter has better estimation accuracy than other filters. In order to improve the computational efficiency of the proposed filter, it is necessary to choose a reasonable number of iterations. So, the estimation performance of the proposed filter with different numbers of iterations needs to be tested. The proposed filter changes the state update value by changing the measurement covariance matrix, so when the estimated measurement covariance matrix converges, the state update value also converges. Figs. 5 and 6 show the averages of and from the proposed filter with different numbers of iterations. It is seen that the proposed filter needs three iterations to converge. This is the reason why we choose three iterations in this simulation. Table 1. TRMSEs of position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter Filters TRMSE, m KF 187.03 MCCKF 184.14 STF 201.06 OD-KF 64.27 VB-KF 56.02 proposed filter 49.08 Fig. 4Open in figure viewerPowerPoint RMSEs of position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter Fig. 5Open in figure viewerPowerPoint Average of gain factor from the proposed filter with different numbers of iterations Fig. 6Open in figure viewerPowerPoint Average of gain factor from the proposed filter with different numbers of iterations Table 2 shows the computational complexity [24] of the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter. We can see that the computational complexity of the proposed filter is lower than that of the VB-KF and the computational complexity of the proposed filter is almost the same as that of OD-KF. Since different parameters often make the proposed filter have large performance differences, so it is necessary to compare the estimation accuracy of the proposed filter under different parameters. Fig. 7 shows the RMSEs of the position from the proposed filters when and . Figs. 8 and 9 show the RMSEs of the position from the proposed filters when and . From Figs. 7-9, it can be seen that when the other parameters are unchanged, the dof parameter , , or , which is changed within a certain range has little effect on the estimation accuracy of the proposed filter and also has better estimation accuracy than the KF, the MCCKF, the STF, the OD-KF, and the VB-KF. Therefore, it is not difficult to select the appropriate dof parameters for the proposed filter. Table 2. Computational complexity of the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter Filters Number of multiplications and division Number of additions and subtractions Number of inverse operations KF 1 MCCKF 3 STF 1 OD-KF VB-KF proposed filter Fig. 7Open in figure viewerPowerPoint RMSEs of position from the proposed filters when and Fig. 8Open in figure viewerPowerPoint RMSEs of position from the proposed filters when Fig. 9Open in figure viewerPowerPoint RMSEs of position from the proposed filters when Finally, in order to compare the robustness of the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter, we tested them under different measurement noises. Fig. 10 shows the TRMSEs of the position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter under different probability factors. Fig. 11 shows the TRMSEs of position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter under different gain factors. It is seen that when the probability factor increases, the TRMSEs of the position from these filters also increase. However, when the gain factors increases, the TRMSEs of the position from the OD-KF, the VB-KF, and the proposed filter do not change significantly. Similarly, we can also see that the proposed filter has a better estimation accuracy than other filters under different conditions. In summary, the proposed filter has better robustness than other filters. Fig. 10Open in figure viewerPowerPoint TRMSEs of position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter under different probability factors Fig. 11Open in figure viewerPowerPoint TRMSEs of position from the KF, the MCCKF, the STF, the OD-KF, the VB-KF, and the proposed filter under different gain factors 6 Conclusion This study proposes a new robust KF based on a PTV distribution for the linear discrete time system. The filter automatically identifies the outlier by introducing the judgement factor and models the outlier noise as a PTV distribution. Then, the state is estimated by using the VB method. Simulations show that the proposed filter has high real-time performance and better estimation accuracy than other existing filters. Furthermore, the proposed filter is designed for a linear system. The dynamic model is known. In future research, we need to consider how to design robust filtering for an unknown system under outliers' interference. 7 Acknowledgments This work was supported by the National Natural Science Foundation of China (61473153) and the Aeronautical Science Foundation of China (2016ZC59006) 9 Appendix 1 9.1 Expressions for Gaussian distribution, gamma distribution, digamma function, and Beta function The Gaussian distribution, gamma distribution, beta function, and digamma function are formulated as (91) (92) (93) (94) (95) where is the number of dimension of x, is the gamma function, and is the Euler constant. 8 References 1Zhou, Y., Wu, P.L., Li, X.X.: 'Adaptive navigation algorithm under abnormal measurements in liberation-point mission', IEEE Trans. Aerosp. Electron. Syst., 2018, 54, (1), pp. 246– 256 2Wu, P.L., Zhou, Y., Li, X.X.: 'Space-based passive tracking of non-cooperative space target using robust filtering algorithm', Proc. Inst. Mech. Eng. I, J. Syst. Control Eng., 2016, vol. 230, no. 6, pp. 551– 561 3Wu, P.L., Li, X.X., Zhang, L.Z. et al: 'Tracking algorithm with radar and IR sensors using a novel adaptive grid interacting multiple model', IET Sci. Meas. Technol., 2014, 8, (5), pp. 270– 276 4Li, W., Jia, Y.: 'State estimation for jump Markov linear systems by variational Bayesian approximation', IET Control Theory Appl., 2012, 6, (2), pp. 319– 326 5Wen, F., Wang, Z.: 'Distributed Kalman filtering for robust state estimation over wireless sensor networks under malicious cyber attacks', Digit. Signal Process., 2018, 78, pp. 92– 97 6Kalman, R.E.: 'A new approach to linear filtering and prediction problems', J. Basic Eng., 1960, 82, (1), pp. 35– 45 7Nurminen, H., Ardeshiri, T., Piche, R. et al: 'Robust inference for state-space models with skewed measurement noise', IEEE Signal Process. Lett., 2015, 22, (11), pp. 1898– 1902 8Mertens, M., Koch, W., Kirubarajan, T.: 'Exploiting Doppler blind zone information for ground moving target tracking with bistatic airborne radar', IEEE Trans. Aerosp. Electron. Syst., 2014, 50, (1), pp. 130– 148 9Agamennoni, G., Nieto, J.I., Nebot, E.M.: 'Approximate inference in state-space models with heavy-tailed noise', IEEE Trans. Signal Process., 2012, 60, (10), pp. 5024– 5037 10Roth, M., Özkan, E., Gustafsson, F.: ' A student's t filter for heavy tailed process and measurement noise'. Int. Conf. on Acoustics, Speech and Signal Processing, Canada, May 2013, pp. 5770– 5774 11Chen, B., Liu, X., Zhao, H. et al: 'Maximum correntropy Kalman filter', Automatica, 2017, 76, pp. 70– 77 12Karlgaard, C.D., Schaub, H.: 'Huber-based divided difference filtering', J. Guid. Control Dyn., 2007, 30, (3), pp. 885– 891 13Wang, X., Cui, N., Guo, J.: 'Huber-based unscented filtering and its application to vision-based relative navigation', IET Radar Sonar Navig., 2010, 4, (1), pp. 134– 141 14Chang, L., Hu, B., Chang, G. et al: 'Huber-based novel robust unscented Kalman filter', IET Sci. Meas. Technol., 2012, 6, (6), pp. 502– 509 15Chen, B., Príncipe, J.C.: 'Maximum correntropy estimation is a smoothed MAP estimation', IEEE Signal Process. Lett., 2012, 19, (8), pp. 491– 494 16Wang, Y., Zheng, W., Sun, S. et al: 'Robust information filter based on maximum correntropy criterion', J. Guid. Control Dyn., 2016, 39, pp. 1126– 1131 17Izanloo, R., Fakoorian, S.A., Yazdi, H.S. et al: ' Kalman filtering based on the maximum correntropy criterion in the presence of non-Gaussian noise'. Annual Conf. on Information Science and Systems, USA, March 2016, pp. 500– 505 18Zhu, H., Leung, H., He, Z.: 'A variational Bayesian approach to robust sensor fusion based on student-t distribution', Inf. Sci., 2013, 221, pp. 201– 214 19Huang, Y.L., Zhang, Y.G., Li, N. et al: 'A novel robust student's t-based Kalman filter', IEEE Trans. Aerosp. Electron. Syst., 2017, 53, (3), pp. 1545– 1554 20Huang, Y.L., Zhang, Y.G., Shi, P. et al: 'Robust Kalman filters based on Gaussian scale mixture distributions with application to target tracking', IEEE Trans. Syst. Man Cybern., Syst., 2017, pp. 1– 15, doi: https://doi.org/10.1109/TSMC.2017.2778269 21Wang, H., Li, H., Fang, J. et al: 'Robust Gaussian Kalman filter with outlier detection', IEEE Signal Process. Lett., 2018, 25, (8), pp. 1236– 1240 22Sun, J., Kabán, A., Garibaldi, J.M.: 'Robust mixture clustering using Pearson type VII distribution', Pattern Recognit. Lett., 2010, 31, (16), pp. 2447– 2454 23Wan, Q., Duan, H., Fang, J. et al: 'Robust Bayesian compressed sensing with outliers', Signal Process., 2017, 140, pp. 104– 109 24Chiou, Y.S., Tsai, F.: 'A reduced-complexity data-fusion algorithm using belief propagation for location tracking in heterogeneous observations', IEEE Trans. Cybern., 2014, 44, (6), pp. 922– 935 Citing Literature Volume13, Issue8August 2019Pages 1389-1399 FiguresReferencesRelatedInformation