Regularized Maximum Correntropy Criterion Kalman Filter for Uncalibrated Visual Servoing in the Presence of Non-Gaussian Feature Tracking Noise.

Some advantages of using cameras as sensor devices on feedback systems are the flexibility of the data it represents, the possibility to extract real-time information, and the fact that it does not require contact to operate. However, in unstructured scenarios, Image-Based Visual Servoing (IBVS) robot tasks are challenging. Camera calibration and robot kinematics can approximate a jacobian that maps the image features space to the robot actuation space, but they can become error-prone or require online changes. Uncalibrated visual servoing (UVS) aims at executing visual servoing tasks without previous camera calibration or through camera model uncertainties. One way to accomplish that is through jacobian identification using environment information in an estimator, such as the Kalman filter. The Kalman filter is optimal with Gaussian noise, but unstructured environments may present target occlusion, reflection, and other characteristics that confuse feature extraction algorithms, generating outliers. This work proposes RMCKF, a correntropy-induced estimator based on the Kalman Filter and the Maximum Correntropy Criterion that can handle non-Gaussian feature extraction noise. Unlike other approaches, we designed RMCKF for particularities in UVS, to deal with independent features, the IBVS control action, and simulated annealing. We designed Monte Carlo experiments to test RMCKF with non-Gaussian Kalman Filter-based techniques. The results showed that the proposed technique could outperform its relatives, especially in impulsive noise scenarios and various starting configurations.

A Robust GPS Navigation Filter Based on Maximum Correntropy Criterion with Adaptive Kernel Bandwidth.

Multiple forms of interference and noise that impact the receiver's capacity to receive and interpret satellite signals, and consequently the preciseness of positioning and navigation, may be present during the processing of Global Positioning System (GPS) navigation. The non-Gaussian noise predominates in the signal owing to the fluctuating character of both natural and artificial electromagnetic interference, and the algorithm based on the minimum mean-square error (MMSE) criterion performs well when assuming Gaussian noise, but drops when assuming non-Gaussian noise. The maximum correntropy criteria (MCC) adaptive filtering technique efficiently reduces pulse noise and has adequate performance in heavy-tailed noise, which addresses the issue of filter performance caused by the presence of non-Gaussian or heavy-tailed unusual noise values in the localizing measurement noise. The adaptive kernel bandwidth (AKB) technique employed in this paper applies the calculated adaptive variables to generate the kernel function matrix, in which the adaptive factor can modify the size of the kernel width across a reasonably appropriate spectrum, substituting the fixed kernel width for the conventional MCC to enhance the performance. The conventional maximum correntropy criterion-based extended Kalman filter (MCCEKF) algorithm's performance is significantly impacted by the value of the kernel width, and there are certain predetermined conditions in the selection based on experience. The MCCEKF with a fixed adaptive kernel bandwidth (MCCEKF-AKB) has several advantages due to its novel concept and computational simplicity, and gives a qualitative solution for the study of random structures for generalized noise. Additionally, it can effectively achieve the robust state estimation of outliers with anomalous values while guaranteeing the accuracy of the filtering.

Application of Adaptive Robust Kalman Filter Base on MCC for SINS/GPS Integrated Navigation.

In this paper, an adaptive and robust Kalman filter algorithm based on the maximum correntropy criterion (MCC) is proposed to solve the problem of integrated navigation accuracy reduction, which is caused by the non-Gaussian noise and time-varying noise of GPS measurement in complex environment. Firstly, the Grubbs criterion was used to remove outliers, which are contained in the GPS measurement. Then, a fixed-length sliding window was used to estimate the decay factor adaptively. Based on the fixed-length sliding window method, the time-varying noises, which are considered in integrated navigation system, are addressed. Moreover, a MCC method is used to suppress the non-Gaussian noises, which are generated with external corruption. Finally, the method, which is proposed in this paper, is verified by the designed simulation and field tests. The results show that the influence of the non-Gaussian noise and time-varying noise of the GPS measurement is detected and isolated by the proposed algorithm, effectively. The navigation accuracy and stability are improved.

2D and 3D Angles-Only Target Tracking Based on Maximum Correntropy Kalman Filters.

In this paper, angles-only target tracking (AoT) problem is investigated in the non Gaussian environment. Since the conventional minimum mean square error criterion based estimators tend to give poor accuracy in the presence of large outliers or impulsive noises in measurement, a maximum correntropy criterion (MCC) based framework is presented. Accordingly, three new estimation algorithms are developed for AoT problems based on the conventional sigma point filters, termed as MC-UKF-CK, MC-NSKF-GK and MC-NSKF-CK. Here MC-NSKF-GK represents the maximum correntropy new sigma point Kalman filter realized using Gaussian kernel and MC-NSKF-CK represents realization using Cauchy kernel. Similarly, based on the unscented Kalman filter, MC-UKF-CK has been developed. The performance of all these estimators is evaluated in terms of root-mean-square error (RMSE) in position and % track loss. The simulations were carried out for 2D as well as 3D AoT scenarios and it was inferred that, the developed algorithms performed with improved estimation accuracy than the conventional ones, in the presence of non Gaussian measurement noise.

Student's t-Kernel-Based Maximum Correntropy Kalman Filter.

The state estimation problem is ubiquitous in many fields, and the common state estimation method is the Kalman filter. However, the Kalman filter is based on the mean square error criterion, which can only capture the second-order statistics of the noise and is sensitive to large outliers. In many areas of engineering, the noise may be non-Gaussian and outliers may arise naturally. Therefore, the performance of the Kalman filter may deteriorate significantly in non-Gaussian noise environments. To improve the accuracy of the state estimation in this case, a novel filter named Student's t kernel-based maximum correntropy Kalman filter is proposed in this paper. In addition, considering that the fixed-point iteration method is used to solve the optimal estimated state in the filtering algorithm, the convergence of the algorithm is also analyzed. Finally, comparative simulations are conducted and the results demonstrate that with the proper parameters of the kernel function, the proposed filter outperforms the other conventional filters, such as the Kalman filter, Huber-based filter, and maximum correntropy Kalman filter.

Variational Bayesian-Based Improved Maximum Mixture Correntropy Kalman Filter for Non-Gaussian Noise.

The maximum correntropy Kalman filter (MCKF) is an effective algorithm that was proposed to solve the non-Gaussian filtering problem for linear systems. Compared with the original Kalman filter (KF), the MCKF is a sub-optimal filter with Gaussian correntropy objective function, which has been demonstrated to have excellent robustness to non-Gaussian noise. However, the performance of MCKF is affected by its kernel bandwidth parameter, and a constant kernel bandwidth may lead to severe accuracy degradation in non-stationary noises. In order to solve this problem, the mixture correntropy method is further explored in this work, and an improved maximum mixture correntropy KF (IMMCKF) is proposed. By derivation, the random variables that obey Beta-Bernoulli distribution are taken as intermediate parameters, and a new hierarchical Gaussian state-space model was established. Finally, the unknown mixing probability and state estimation vector at each moment are inferred via a variational Bayesian approach, which provides an effective solution to improve the applicability of MCKFs in non-stationary noises. Performance evaluations demonstrate that the proposed filter significantly improves the existing MCKFs in non-stationary noises.

Newtonian-Type Adaptive Filtering Based on the Maximum Correntropy Criterion.

This paper provides a novel Newtonian-type optimization method for robust adaptive filtering inspired by information theory learning. With the traditional minimum mean square error (MMSE) criterion replaced by criteria like the maximum correntropy criterion (MCC) or generalized maximum correntropy criterion (GMCC), adaptive filters assign less emphasis on the outlier data, thus become more robust against impulsive noises. The optimization methods adopted in current MCC-based LMS-type and RLS-type adaptive filters are gradient descent method and fixed point iteration, respectively. However, in this paper, a Newtonian-type method is introduced as a novel method for enhancing the existing body of knowledge of MCC-based adaptive filtering and providing a fast convergence rate. Theoretical analysis of the steady-state performance of the algorithm is carried out and verified by simulations. The experimental results show that, compared to the conventional MCC adaptive filter, the MCC-based Newtonian-type method converges faster and still maintains a good steady-state performance under impulsive noise. The practicability of the algorithm is also verified in the experiment of acoustic echo cancellation.

A Robust DOA Estimator Based on Compressive Sensing for Coprime Array in the Presence of Miscalibrated Sensors.

Coprime array with M + N sensors can achieve an increased degrees-of-freedom (DOF) of O ( M N ) for direction-of-arrival (DOA) estimation. Utilizing the compressive sensing (CS)-based DOA estimation methods, the increased DOF offered by the coprime array can be fully exploited. However, when some sensors in the array are miscalibrated, these DOA estimation methods suffer from degraded performance or even failed operation. Besides, the key to the success of CS-based DOA estimation is that every target falls on the predefined grid. Thus, a coarse grid may cause the mismatch problem, whereas a fine grid requires great computational cost. In this paper, a robust CS-based DOA estimation algorithm is proposed for coprime array with miscalibrated sensors. In the proposed algorithm, signals received by the miscalibrated sensors are viewed as outliers, and correntropy is introduced as the similarity measurement to distinguish these outliers. Incorporated with maximum correntropy criterion (MCC), an iterative sparse reconstruction-based algorithm is then developed to give the DOA estimation while mitigating the influence of the outliers. A multiresolution grid refinement strategy is also incorporated to reconcile the contradiction between computational cost and the mismatch problem. The numerical simulation results verify the effectiveness and robustness of the proposed method.

Online Gradient Descent for Kernel-Based Maximum Correntropy Criterion.

In the framework of statistical learning, we study the online gradient descent algorithm generated by the correntropy-induced losses in Reproducing kernel Hilbert spaces (RKHS). As a generalized correlation measurement, correntropy has been widely applied in practice, owing to its prominent merits on robustness. Although the online gradient descent method is an efficient way to deal with the maximum correntropy criterion (MCC) in non-parameter estimation, there has been no consistency in analysis or rigorous error bounds. We provide a theoretical understanding of the online algorithm for MCC, and show that, with a suitable chosen scaling parameter, its convergence rate can be min-max optimal (up to a logarithmic factor) in the regression analysis. Our results show that the scaling parameter plays an essential role in both robustness and consistency.

Robust Variable Selection and Estimation Based on Kernel Modal Regression.

Model-free variable selection has attracted increasing interest recently due to its flexibility in algorithmic design and outstanding performance in real-world applications. However, most of the existing statistical methods are formulated under the mean square error (MSE) criterion, and susceptible to non-Gaussian noise and outliers. As the MSE criterion requires the data to satisfy Gaussian noise condition, it potentially hampers the effectiveness of model-free methods in complex circumstances. To circumvent this issue, we present a new model-free variable selection algorithm by integrating kernel modal regression and gradient-based variable identification together. The derived modal regression estimator is related closely to information theoretic learning under the maximum correntropy criterion, and assures algorithmic robustness to complex noise by replacing learning of the conditional mean with the conditional mode. The gradient information of estimator offers a model-free metric to screen the key variables. In theory, we investigate the theoretical foundations of our new model on generalization-bound and variable selection consistency. In applications, the effectiveness of the proposed method is verified by data experiments.

A Correntropy-Based Proportionate Affine Projection Algorithm for Estimating Sparse Channels with Impulsive Noise.

A novel robust proportionate affine projection (AP) algorithm is devised for estimating sparse channels, which often occur in network echo and wireless communication channels. The newly proposed algorithm is realized by using the maximum correntropy criterion (MCC) and the data reusing scheme used in AP to overcome the identification performance degradation of the traditional PAP algorithm in impulsive noise environments. The proposed algorithm is referred to as the proportionate affine projection maximum correntropy criterion (PAPMCC) algorithm, which is derived in the context of channel estimation framework. Many simulation results were obtained to verify that the PAPMCC algorithm is superior to early reported AP algorithms with different input signals under impulsive noise environments.

Electricity Consumption Forecasting using Support Vector Regression with the Mixture Maximum Correntropy Criterion.

The electricity consumption forecasting (ECF) technology plays a crucial role in the electricity market. The support vector regression (SVR) is a nonlinear prediction model that can be used for ECF. The electricity consumption (EC) data are usually nonlinear and non-Gaussian and present outliers. The traditional SVR with the mean-square error (MSE), however, is insensitive to outliers and cannot correctly represent the statistical information of errors in non-Gaussian situations. To address this problem, a novel robust forecasting method is developed in this work by using the mixture maximum correntropy criterion (MMCC). The MMCC, as a novel cost function of information theoretic, can be used to solve non-Gaussian signal processing; therefore, in the original SVR, the MSE is replaced by the MMCC to develop a novel robust SVR method (called MMCCSVR) for ECF. Besides, the factors influencing users' EC are investigated by a data statistical analysis method. We find that the historical temperature and historical EC are the main factors affecting future EC, and thus these two factors are used as the input in the proposed model. Finally, real EC data from a shopping mall in Guangzhou, China, are utilized to test the proposed ECF method. The forecasting results show that the proposed ECF method can effectively improve the accuracy of ECF compared with the traditional SVR and other forecasting algorithms.

Adaptive Robust Unscented Kalman Filter via Fading Factor and Maximum Correntropy Criterion.

In most practical applications, the tracking process needs to update the data constantly. However, outliers may occur frequently in the process of sensors' data collection and sending, which affects the performance of the system state estimate. In order to suppress the impact of observation outliers in the process of target tracking, a novel filtering algorithm, namely a robust adaptive unscented Kalman filter, is proposed. The cost function of the proposed filtering algorithm is derived based on fading factor and maximum correntropy criterion. In this paper, the derivations of cost function and fading factor are given in detail, which enables the proposed algorithm to be robust. Finally, the simulation results show that the presented algorithm has good performance, and it improves the robustness of a general unscented Kalman filter and solves the problem of outliers in system.

Adaptive Maximum Correntropy Gaussian Filter Based on Variational Bayes.

In this paper, we investigate the state estimation of systems with unknown covariance non-Gaussian measurement noise. A novel improved Gaussian filter (GF) is proposed, where the maximum correntropy criterion (MCC) is used to suppress the pollution of non-Gaussian measurement noise and its covariance is online estimated through the variational Bayes (VB) approximation. MCC and VB are integrated through the fixed-point iteration to modify the estimated measurement noise covariance. As a general framework, the proposed algorithm is applicable to both linear and nonlinear systems with different rules being used to calculate the Gaussian integrals. Experimental results show that the proposed algorithm has better estimation accuracy than related robust and adaptive algorithms through a target tracking simulation example and the field test of an INS/DVL integrated navigation system.

Maximum Correntropy Based Unscented Particle Filter for Cooperative Navigation with Heavy-Tailed Measurement Noises.

In this paper, a novel robust particle filter is proposed to address the measurement outliers occurring in the multiple autonomous underwater vehicles (AUVs) based cooperative navigation (CN). As compared with the classic unscented particle filter (UPF) based on Gaussian assumption of measurement noise, the proposed robust particle filter based on the maximum correntropy criterion (MCC) exhibits better robustness against heavy-tailed measurement noises that are often induced by measurement outliers in CN systems. Furthermore, the proposed robust particle filter is computationally much more efficient than existing robust UPF due to the use of a Kullback-Leibler distance-resampling to adjust the number of particles online. Experimental results based on actual lake trial show that the proposed maximum correntropy based unscented particle filter (MCUPF) has better estimation accuracy than existing state-of-the-art robust filters for CN systems with heavy-tailed measurement noises, and the proposed MCUPF has lower computational complexity than existing robust particle filters.

Electricity Consumption Forecasting Scheme via Improved LSSVM with Maximum Correntropy Criterion.

In recent years, with the deepening of China's electricity sales side reform and electricity market opening up gradually, the forecasting of electricity consumption (FoEC) becomes an extremely important technique for the electricity market. At present, how to forecast the electricity accurately and make an evaluation of results scientifically are still key research topics. In this paper, we propose a novel prediction scheme based on the least-square support vector machine (LSSVM) model with a maximum correntropy criterion (MCC) to forecast the electricity consumption (EC). Firstly, the electricity characteristics of various industries are analyzed to determine the factors that mainly affect the changes in electricity, such as the gross domestic product (GDP), temperature, and so on. Secondly, according to the statistics of the status quo of the small sample data, the LSSVM model is employed as the prediction model. In order to optimize the parameters of the LSSVM model, we further use the local similarity function MCC as the evaluation criterion. Thirdly, we employ the K-fold cross-validation and grid searching methods to improve the learning ability. In the experiments, we have used the EC data of Shaanxi Province in China to evaluate the proposed prediction scheme, and the results show that the proposed prediction scheme outperforms the method based on the traditional LSSVM model.

Diffusion Maximum Correntropy Criterion Based Robust Spectrum Sensing in Non-Gaussian Noise Environments.

Spectrum sensing is the most important task in cognitive radio (CR). In this paper, a new robust distributed spectrum sensing approach, called diffusion maximum correntropy criterion (DMCC)-based robust spectrum sensing, is proposed for CR in the presence of non-Gaussian noise or impulsive noise. The proposed distributed scheme, which does not need any central processing unit, is characterized by an adaptive diffusion model. The maximum correntropy criterion, which is insensitive to impulsive interference, is introduced to deal with the effect of non-Gaussian noise. Simulation results show that the DMCC-based spectrum sensing algorithm has an excellent robust property with respect to non-Gaussian noise. It is also observed that the new method displays a considerably better detection performance than its predecessor (i.e., diffusion least mean square (DLMS)) in impulsive noise. Moreover, the mean and variance convergence analysis of the proposed algorithm are also carried out.

Sparse-Aware Bias-Compensated Adaptive Filtering Algorithms Using the Maximum Correntropy Criterion for Sparse System Identification with Noisy Input.

To address the sparse system identification problem under noisy input and non-Gaussian output measurement noise, two novel types of sparse bias-compensated normalized maximum correntropy criterion algorithms are developed, which are capable of eliminating the impact of non-Gaussian measurement noise and noisy input. The first is developed by using the correntropy-induced metric as the sparsity penalty constraint, which is a smoothed approximation of the ℓ 0 norm. The second is designed using the proportionate update scheme, which facilitates the close tracking of system parameter change. Simulation results confirm that the proposed algorithms can effectively improve the identification performance compared with other algorithms presented in the literature for the sparse system identification problem.

Maximum Correntropy Unscented Kalman Filter for Spacecraft Relative State Estimation.

A new algorithm called maximum correntropy unscented Kalman filter (MCUKF) is proposed and applied to relative state estimation in space communication networks. As is well known, the unscented Kalman filter (UKF) provides an efficient tool to solve the non-linear state estimate problem. However, the UKF usually plays well in Gaussian noises. Its performance may deteriorate substantially in the presence of non-Gaussian noises, especially when the measurements are disturbed by some heavy-tailed impulsive noises. By making use of the maximum correntropy criterion (MCC), the proposed algorithm can enhance the robustness of UKF against impulsive noises. In the MCUKF, the unscented transformation (UT) is applied to obtain a predicted state estimation and covariance matrix, and a nonlinear regression method with the MCC cost is then used to reformulate the measurement information. Finally, the UT is adopted to the measurement equation to obtain the filter state and covariance matrix. Illustrative examples demonstrate the superior performance of the new algorithm.

Generalization analysis of deep CNNs under maximum correntropy criterion.

Convolutional neural networks (CNNs) have gained immense popularity in recent years, finding their utility in diverse fields such as image recognition, natural language processing, and bio-informatics. Despite the remarkable progress made in deep learning theory, most studies on CNNs, especially in regression tasks, tend to heavily rely on the least squares loss function. However, there are situations where such learning algorithms may not suffice, particularly in the presence of heavy-tailed noises or outliers. This predicament emphasizes the necessity of exploring alternative loss functions that can handle such scenarios more effectively, thereby unleashing the true potential of CNNs. In this paper, we investigate the generalization error of deep CNNs with the rectified linear unit (ReLU) activation function for robust regression problems within an information-theoretic learning framework. Our study demonstrates that when the regression function exhibits an additive ridge structure and the noise possesses a finite pth moment, the empirical risk minimization scheme, generated by the maximum correntropy criterion and deep CNNs, achieves fast convergence rates. Notably, these rates align with the mini-max optimal convergence rates attained by fully connected neural network model with the Huber loss function up to a logarithmic factor. Additionally, we further establish the convergence rates of deep CNNs under the maximum correntropy criterion when the regression function resides in a Sobolev space on the sphere.