Permutation entropy (PE) has been recently suggested as a novel measure to characterize the complexity of nonlinear time series. In this paper, we propose a simple method to address some of PE's limitations, mainly its inability to differentiate between distinct patterns of a certain motif, and the sensitivity of patterns close to the noise floor. The method relies on the fact that patterns may be too disparate in amplitudes and variances and proceeds by assigning weights for each extracted vector when computing the relative frequencies associated with every motif. Simulations were conducted over synthetic and real data for a weighting scheme inspired by the variance of each pattern. Results show better robustness and stability in the presence of higher levels of noise, in addition to a distinctive ability to extract complexity information from data with spiky features or having abrupt changes in magnitude.

Abstract-As a robust nonlinear similarity measure in kernel space, correntropy has received increasing attention in domains of machine learning and signal processing. In particular, the maximum correntropy criterion (MCC) has recently been successfully applied in robust regression and filtering. The default kernel function in correntropy is the Gaussian kernel, which is, of course, not always the best choice. In this work, we propose a generalized correntropy that adopts the generalized Gaussian density (GGD) function as the kernel (not necessarily a Mercer kernel), and present some important properties. We further propose the generalized maximum correntropy criterion (GMCC), and apply it to adaptive filtering. An adaptive algorithm, called the GMCC algorithm, is derived, and the mean square convergence performance is studied. We show that the proposed algorithm is very stable and can achieve zero probability of divergence (POD). Simulation results confirm the theoretical expectations and demonstrate the desirable performance of the new algorithm.

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The maximum correntropy criterion (MCC) has received increasing attention in signal processing and machine learning due to its robustness against outliers (or impulsive noises). Some gradient based adaptive filtering algorithms under MCC have been developed and available for practical use. The fixed-point algorithms under MCC are, however, seldom studied. In particular, too little attention has been paid to the convergence issue of the fixed-point MCC algorithms. In this letter, we will study this problem and give a sufficient condition to guarantee the convergence of a fixed-point MCC algorithm.

Abstract:Sparse adaptive channel estimation problem is one of the most important topics in broadband wireless communications systems due to its simplicity and robustness. So far many sparsity-aware channel estimation algorithms have been developed based on the well-known minimum mean square error (MMSE) criterion, such as the zero-attracting least mean square (ZALMS),which are robust under Gaussian assumption. In non-Gaussian environments, however, these methods are often no longer robust especially when systems are disturbed by random impulsive noises. To address this problem, we propose in this work a robust sparse adaptive filtering algorithm using correntropy induced metric (CIM) penalized maximum correntropy criterion (MCC) rather than conventional MMSE criterion for robust channel estimation. Specifically, MCC is utilized to mitigate the impulsive noise while CIM is adopted to exploit the channel sparsity efficiently. Both theoretical analysis and computer simulations are provided to corroborate the proposed methods.

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Abstract-Kernel adaptive filters have drawn increasing attention due to their advantages such as universal nonlinear approximation with universal kernels, linearity and convexity in Reproducing Kernel Hilbert Space (RKHS). Among them, the kernel least mean square (KLMS) algorithm deserves particular attention because of its simplicity and sequential learning approach. Similar to most conventional adaptive filtering algorithms, the KLMS adopts the mean square error (MSE) as the adaptation cost. However, the mere second-order statistics is often not suitable for nonlinear and non-Gaussian situations. Therefore, various non-MSE criteria, which involve higherorder statistics, have received an increasing interest. Recently, the correntropy, as an alternative of MSE, has been successfully used in nonlinear and non-Gaussian signal processing and machine learning domains. This fact motivates us in this paper to develop a new kernel adaptive algorithm, called the kernel maximum correntropy (KMC), which combines the advantages of the KLMS and maximum correntropy criterion (MCC). We also study its convergence and self-regularization properties by using the energy conservation relation. The superior performance of the new algorithm has been demonstrated by simulation experiments in the noisy frequency doubling problem.