Robust Multikernel Maximum Correntropy Filters

In wireless sensor networks (WSN), measurements are always corrupted by outliers or impulsive noise. Cubature information filtering (CIF) is founded based on minimum mean square error (MMSE) criterion, which is not applicable to non-Gaussian noise. Hence, a novel robust CIF (RCIF) is derived based on maximum correntropy criterion (MCC) to enhance the robustness of state estimation in the local node. For the information fusion, weighted average consensus (WAC) based distributed RCIF (DRCIF) is founded to improve the stability of sensor networks and the accuracy of state estimation. The estimation error of DRCIF is proved to be bounded in mean square. Numerical simulations are provided to evaluate the effectiveness of proposed algorithms.

To date, most linear and nonlinear Kalman filters (KFs) have been developed under the Gaussian assumption and the well-known minimum mean square error (MMSE) criterion. In order to improve the robustness with respect to impulsive (or heavy-tailed) non-Gaussian noises, the maximum correntropy criterion (MCC) has recently been used to replace the MMSE criterion in developing several robust Kalman-type filters. To deal with more complicated non-Gaussian noises such as noises from multimodal distributions, in this article, we develop a new Kalman-type filter, called minimum error entropy KF (MEE-KF), by using the minimum error entropy (MEE) criterion instead of the MMSE or MCC. Similar to the MCC-based KFs, the proposed filter is also an online algorithm with the recursive process, in which the propagation equations are used to give prior estimates of the state and covariance matrix, and a fixed-point algorithm is used to update the posterior estimates. In addition, the MEE extended KF (MEE-EKF) is also developed for performance improvement in the nonlinear situations. The high accuracy and strong robustness of MEE-KF and MEE-EKF are confirmed by experimental results.

This paper presents novel kernel adaptive filters with feedback, namely, kernel recursive maximum correntropy with multiple feedback (KRMC-MF) and its simplified version, a linear recurrent kernel online learning algorithm based on maximum correntropy criterion (LRKOL-MCC). In LRKOL-MCC and KRMC-MF, single output and multiple outputs based on single delay are utilized to construct their feedback structure, respectively. Compared with the minimum mean square error criterion, the maximum correntropy criterion (MCC) adopted by LRKOL-MCC and KRMC-MF captures higher order statistics of errors. The proposed filters are, therefore, robust against outliers. Therefore, the past information can be reused to improve filtering performance in terms of the steady-state mean square error. The convergence characteristics of the filter parameters in LRKOL-MCC and KRMC-MF are also derived. Simulations on chaotic time-series prediction and nonlinear regression illustrate the desirable accuracy and robustness of the proposed filters.

Adaptive channel equalization is a signal processing technique to mitigate inter-symbol interference in a time dispersive channel. For adaptive equalization, minimum mean square error (MMSE) criterion-based reproducing kernel Hilbert spaces (RKHS) approaches such as the kernel least mean squares (KLMS) algorithm and its variants have been suggested in the literature for nonlinear channels. Another optimality criterion, based on minimum bit/symbol error rate (MBER/MSER), is a better choice for adapting an equalizer as compared to MMSE criterion. A kernel-based minimum symbol error rate (KMSER) equalization algorithm combines minimum symbol error rate (MSER)-based approaches with RKHS techniques. However, most algorithms in RKHS such as KMSER/KLMS require infinite storage requirement and hence cannot be practically implemented. To curtail the infinite memory requirement, and make adaptive algorithm suitable for implementation with finite memory and processing power, we propose quantized KMSER (QKMSER) and fixed-budget quantized KMSER (FBQKMSER)-based equalizers in this paper. In this paper, we derive the dynamical equation for MSE evolution of the QKMSER and FBQKMSER and find their performance to be asymptotically close to the MSE behavior of the KMSER. Also, it is found via simulations that the tracking performance of FBQKMSER is better than all the compared algorithms in this paper which is particularly useful for non-stationary channels.

This paper investigates the kernel entropy based extended Kalman filter (EKF) as the navigation processor for the Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS). The algorithm is effective for dealing with non-Gaussian errors or heavy-tailed (or impulsive) interference errors, such as the multipath. The kernel minimum error entropy (MEE) and maximum correntropy criterion (MCC) based filtering for satellite navigation system is involved for dealing with non-Gaussian errors or heavy-tailed interference errors or outliers of the GPS. The standard EKF method is derived based on minimization of mean square error (MSE) and is optimal only under Gaussian assumption in case the system models are precisely established. The GPS navigation algorithm based on kernel entropy related principles, including the MEE criterion and the MCC will be performed, which is utilized not only for the time-varying adaptation but the outlier type of interference errors. The kernel entropy based design is a new approach using information from higher-order signal statistics. In information theoretic learning (ITL), the entropy principle based measure uses information from higher-order signal statistics and captures more statistical information as compared to MSE. To improve the performance under non-Gaussian environments, the proposed filter which adopts the MEE/MCC as the optimization criterion instead of using the minimum mean square error (MMSE) is utilized for mitigation of the heavy-tailed type of multipath errors. Performance assessment will be carried out to show the effectiveness of the proposed approach for positioning improvement in GPS navigation processing.

The kernel adaptive filters (KAFs) based on the minimum mean square error (MMSE) criterion in reproducing kernel Hilbert space (RKHS) improve the performance of linear adaptive filters but result in instability issues and large burdens of computation and memory in impulsive noises. To this end, a novel Nystrom kernel recursive generalized maximum correntropy (NKRGMC) with probability density rank-based quantization (PRQ) sampling (NKRGMC-PRQ) algorithm is proposed to improve filtering performance, robustness, and computational efficiency of the traditional KAFs in this letter. In a fixed dimensional network structure, the proposed NKRGMC-PRQ algorithm can achieve a comparable performance to KAFs with low computational complexity. Monte Carlo simulations are conducted to validate the superiorities of NKRGMC-PRQ in terms of filtering accuracy, computational complexity, and robustness.

As a new measure of similarity, the correntropy can be used as an objective function for many applications. In this letter, we study Bayesian estimation under maximum correntropy (MC) criterion. We show that the MC estimation is, in essence, a smoothed maximum a posteriori (MAP) estimation, including the MAP and the minimum mean square error (MMSE) estimation as the extreme cases. We also prove that under a certain condition, when the kernel size in correntropy is larger than some value, the MC estimation will have a unique optimal solution lying in a strictly concave region of the smoothed posterior distribution.

Autonomous Underwater Vehicles (AUVs) need accurate navigation and positioning data to carry out their operations in challenging marine environments. However, non-Gaussian environmental noise and sensor outliers in underwater environments degrade the positioning accuracy and stability of traditional algorithms. In such circumstances, the typical Unscented Kalman Filter (UKF) performs worse according to the Minimum Mean Square Error (MMSE) criteria. To reduce non-Gaussian environmental disturbances, a Maximum Correntropy Criterion (MCC) is created. This study extends the MCUKF by utilizing a mixture correntropy framework and an effective iterative optimization technique, and an improved Sage-Husa filtering mechanism is integrated into the iterative algorithm to address time-varying noise dynamics in complex scenarios. These changes increase the algorithm's resilience and flexibility in demanding underwater conditions. The algorithm's capability is confirmed using simulation and sea experiments, which show notable gains in Root Mean Square Error and navigation accuracy. Experimental results show that the proposed method outperforms other existing algorithms in terms of navigation accuracy performance in our simulations and experiments. To guarantee algorithm performance and resilience for the non-Gaussian time-varying measurement noise issue that AUVs encounter in intricate maritime settings, and to additionally enhance the precision and dependability of localization, this study uses the MCC-UKF to combat the environment's non-Gaussian noise, and the Levenberg-Marguardt iterative method to improve reliability of numbers and estimation precision, while the Maximum Mixture Correntropy Criterion (MMCC) serves the purpose of improve the filter flexibility and convergence speed, and finally, the improved Sage-Husa algorithm is combined to improve the robustness to time-varying noise. The following are the study's primary contributions. (1) In our study, we suggest an enhanced UKF algorithm that combines the MCC and iterated cost function, and adopt a Levenberg-Marquardt optimization strategy to accelerate the convergence. The mixture correntropy criterion was added in the construction of the LMIMC-UKF framework in order to achieve the synergistic enhancement of convergence performance and calculate accuracy. (2) Targeting the issue of time-varying environment noise interference, a fusion approach of iterative tuning and improved Sage-Husa filtering is suggested to achieve the systematic improvement of algorithm robustness. (3) By combining the Monte Carlo simulation with maritime measurement data, the algorithm's technical viability is confirmed, and the parameter sensitivity is quantitatively examined by methodical simulation tests. According to the results, the suggested algorithm performs noticeably better than the current maximum correntropy filtering techniques in terms of anti-interference capability, iteration speed, and navigation accuracy.

Adaptive channel equalisation is a signal processing technique to mitigate inter-symbol interference (ISI) in a time dispersive channel. To this end, the use of least mean squares (LMS) algorithm and its variants is widespread since they minimise the minimum mean squared error (MMSE) criteria by online stochastic gradient algorithms and they asymptotically tend to the optimal Weiner solution for linearly separable channels. The kernel least mean squares (KLMS) algorithm and its variants are based on the MMSE based algorithms for non-linear channels. However, as has been pointed out in the literature, the minimum bit/symbol error rate (MBER/MSER) criterion is a better choice for adapting an equaliser as compared to the traditional approaches based on MMSE criterion. In this paper, we propose a novel equaliser that is inspired from the recently proposed MSER adaptation by Gong et al. using the kernel trick for non-linear channel equalisation.

A robust adaptive filtering algorithm based on the convex combination of two adaptive filters under the maximum correntropy criterion (MCC) is proposed. Compared with conventional minimum mean square error (MSE) criterion-based adaptive filtering algorithm, the MCC-based algorithm shows a better robustness against impulsive interference. However, its major drawback is the conflicting requirements between convergence speed and steady-state mean square error. In this letter, we use the convex combination method to overcome the tradeoff problem. Instead of minimizing the squared error to update the mixing parameter in conventional convex combination scheme, the method of maximizing the correntropy is introduced to make the proposed algorithm more robust against impulsive interference. Additionally, we report a novel weight transfer method to further improve the tracking performance. The good performance in terms of convergence rate and steady-state mean square error is demonstrated in plant identification scenarios that include impulsive interference and abrupt changes.

Multiple-input multiple-output (MIMO) systems can achieve high data-rate and high capacity transmission. In MIMO systems, the scheme that weight substreams based on minimum mean square error (MMSE) criterion at the transmitter using feedback channel state information (CSI) achieves good performance (denoted hereafter as MMSE precoder). However, if the optimum number of substreams is not selected based on CSI and methods for reducing the performance degradation due to feedback delay is not used, MMSE precoder cannot achieve the full performance possible. In this paper, we propose an MMSE precoder with mode selection for MIMO systems, which selects both the number of substreams and the modulation scheme for each substream to minimize the average BER at a fixed rate. We evaluate the BER of the proposed MMSE precoder with two methods for reducing performance degradation due to feedback delay, channel prediction and receive weight robust to feedback delay. We also evaluate the BER of the proposed MMSE precoder with the method that combines channel prediction with receive weight robust to feedback delay. Simulation results show that the BER of the proposed MMSE precoder is improved compared to that of the conventional MMSE precoder using the fixed number of substreams. We also show that the method that combines channel prediction with receive weight robust to feedback delay can achieve good BER even when the large feedback delay exists.

Global Navigation Satellite Systems (GNSSs) observables, such as those of the Global Positioning System (GPS), are frequently affected by multipath effects that cause unpredictable signal interference at the receiver, posing significant challenges for accurate state estimation in complex environments with non-Gaussian noise or outliers. The traditional extended Kalman filter (EKF), based on the minimum mean square error (MMSE) criterion, assumes Gaussian noise distributions and exhibits degraded performance under non-Gaussian conditions. To overcome this limitation, the minimum error entropy (MEE) criterion was proposed to reduce random uncertainty in estimation error distributions; however, due to its translation invariance property, MEE may inadvertently increase bias when errors contain systematic offsets, leading to poor convergence. In contrast, the maximum correntropy criterion (MCC) concentrates the error probability density function (PDF) around zero, enabling effective entropy adjustment even in the presence of bias and achieving superior error convergence. This paper presents the centered error entropy (CEE) extended Kalman filter (CEE-EKF) that integrates the complementary merits of both MEE and MCC approaches to overcome their individual limitations. Experimental validation in complex nonlinear GPS environments with non-Gaussian noise demonstrates that the CEE-EKF significantly outperforms individual algorithms in noise suppression, particularly exhibiting enhanced robustness and accuracy when handling outliers. These results offer an effective approach to enhancing the reliability of GPS navigation in challenging real-world environments, and the algorithm can be readily extended to other GNSS applications.

The kernel least mean square (KLMS) algorithm is the simplest algorithm in kernel adaptive filters. However, the network growth of KLMS is still an issue for preventing its online applications, especially when the length of training data is large. The Nystrom method is an efficient method for curbing the growth of the network size. In this paper, we apply the Nystrom method to the KLMS algorithm, generating a novel algorithm named kernel least mean square based on the Nystrom method (NysKLMS). In comparison with the KLMS algorithm, the proposed NysKLMS algorithm can reduce the computational complexity, significantly. The NysKLMS algorithm is proved to be convergent in the mean square sense when its step size satisfies some conditions. In addition, the theoretical steady-state excess mean square error of NysKLMS supported by simulations is calculated to evaluate the filtering accuracy. Simulations on system identification and nonlinear channel equalization show that the NysKLMS algorithm can approach the filtering performance of the KLMS algorithm by using much lower computational complexity, and outperform the KLMS with the novelty criterion, the KLMS with the surprise criterion, the quantized KLMS, the fixed-budget QKLMS, and the random Fourier features KLMS.

We generalize the multiuser (CDMA spread spectrum) communication systems to the fading environments. We first extend Verdu's (1986) conventional optimum receiver to Rayleigh-fading environments and then evaluate its performance. Having no knowledge of the received power at the receiving end, we therefore need an estimator to efficiently estimate the received signal strength of each user in fading environments. A linear minimum mean square error (LMMSE) unbiased estimator is proposed to attain this goal. By using the minimum mean square error (MMSE) Bayesian estimation, we further propose the LMMSE bit estimator for efficient demodulation. Its performance is close to the optimum multiuser receiver but with a much simpler polynomial complexity. To further reduce the complexity, we extend the LMMSE estimator to the sequential LMMSE estimator. In sequential estimation, we do not need to implement the matched filter banks and to perform the matrix inversion when estimating. In addition, it converges after approximately 2k iterations, where k is the number of users. With this fast convergence property and the simple structure, the sequential LMMSE estimator provides an attractive alternative to the implementation of a multiuser system.

In present days, there has been a race among the communication systems in different schemes and generations. Gradually the transmission rate of data is increasing tremendously and breaking the previous generation's record. Now people are thinking of transmitting data beyond imagination. Using multiple antennas both at transmitter and receiver side, MIMO wireless technology efficiently provides very high spectral efficiency, boost the data transmission rate, enhance link reliability and improve system coverage. While transmitting data over mobile radio channels at high bit rates, the impulse response of the channel can be extended over a lot of symbol periods, leading fading and Inter-symbol interference (lSI). Equalizers are used to minimize Fading and interference effects. The analysis of the paper is studying the performance of Minimum mean square error method in terms of BER for MIMO (2×2) and (3×3) wireless channels. Binary Phase Shift Keying (BPSK) modulation technique was used in Rayleigh flat fading channel. Simulating in MATLAB results show that Minimum mean square error (MMSE) receivers is good for noise free channel and is successful in removing ISI. Successive interference method with optimal ordering outperforms the MMSE however a few problems arise because of iterative nature of the algorithm.