Both the minimum error entropy (MEE) and mixture MEE (MMEE) are extensively employed in distributed adaptive filters, exhibiting their robustness against non-Gaussian noise by capturing high-order statistical information from network data. However, the fixed shape of the Gaussian kernel function existing in MEE and MMEE restricts their flexibility, leading to reduced robustness and deteriorated performance. To address this issue, a novel diffusion generalized minimum total error entropy (DGMTE) algorithm is first proposed in this letter, using a generalized MEE criterion to significantly improve the performance of error-in-variables models-based algorithms under non-Gaussian noise. Moreover, as a special case of DGMTE, a generalized minimum total error entropy (GMTE) algorithm is also proposed, and the local convergence analysis of DGMTE is given. Finally, simulations show the superiorities of DGMTE in comparison with other representative algorithms.

Adaptive filters utilizing the low-order moments hidden in robust loss functions have achieved desirable performance under Gaussian input and impulsive noises. However, when the input cannot be modeled by Gaussian process and is simultaneously contaminated by outliers, these filters may suffer from misalignment. To this end, applying fractional-order calculus in stochastic gradient descent method, this letter proposes a fractional-order generalized Cauchy kernel loss (FoGCKL) algorithm to model complex$\alpha$-stable process input. The mean square deviation (MSD) is calculated to evaluate the steady-state performance of FoGCKL. To further avoid steady-state jitters and improve filtering accuracy, an enhanced batch method is constructed in FoGCKL using optimized weighted term, generating another enhanced batch FoGCKL (EB-FoGCKL) algorithm. Simulations on system identification verify the correctness of theoretical analysis and demonstrate the superiorities of FoGCKL and EB-FoGCKL.

The performance of model-based (MB) robust adaptive filters is highly sensitive to specific noise distributions and significantly degrades under complex multi-peak noise conditions. To this end, utilizing the universal function approximation capability and high training efficiency of radial basis function (RBF), this letter proposes a novel RBF neural network-assisted minimum likelihood error entropy (MLEE) criterion. To accurately estimate the gradient contribution of previous estimation errors to the current one, a weighted method is constructed in MLEE to improve its convergence rate, ultimately generating RBF-WMLEE algorithm. Simulations on system identification validate the superiorities of proposed RBF-WMLEE under complex multimodal noise conditions.

Quaternion adaptive filters (QAFs) are extensively used in processing three- or four-dimensional signals effectively. However, their performance can significantly deteriorate or even diverge when system inputs and outputs are contaminated by complex noises. Therefore, this letter addresses the issue of parameter estimation in the quaternion errors-in-variables (QEIV) in asymmetric noise. First, a novel robust criterion, called improved quaternion minimum error entropy criterion with fiducial points (IQMEEF), is constructed. Then, a minimum total quaternion error entropy algorithm with fiducial points (MTQEEF) is proposed by integrating the IQMEEF criterion with the total least squares (TLS) method, leveraging stochastic gradient and quaternion generalized Hamilton-real (GHR) calculus theory. Finally, simulations validate the superior performance of MTQEEF in the QEIV model under asymmetric noise environments.

The conventional active noise control (ANC) system relies on the error-correction learning to design the adaptive controller. However, as the residual noise, the correction error is discarded after each iteration and is still not a perfect input to the human ear. Although many efforts have been devoted to improve the performance by reusing the reference signals, the reuse of the past errors of ANC has been rarely studied in the literature. In this paper, we first design a novel error reused ANC (ErANC) system by introducing the information of the past errors for compensating the gap between the primary noise and estimated output. Then, applying the filtered-x least mean square (FxLMS) algorithm to ErANC for designing the controller, a novel ErFxLMS algorithm is developed to improve the noise reduction performance of FxLMS. However, as the correction errors accumulate over time, the computational and storage burdens of ErFxLMS will grow linearly. To address this issue, a sliding window strategy is applied to the ErFxLMS to curb the size of the stored errors, generating the sliding window ErFxLMS (SW-ErFxLMS) algorithm accordingly. Finally, the performance analysis of mean square error is carried out to validate the universal approximation ability of ErFxLMS. Simulation and experimental results illustrate the effectiveness of the proposed algorithms.

The online conjugate gradient (CG) algorithm with error-correction learning discards the error after each iteration, which shows low filtering accuracy in prediction tasks. Moreover, the CG based on the mean-square error (MSE) criterion suffers from poor performance when dealing with impulsive noise. To address these issues, we first construct a novel batched augmented least lncosh model (ALLM) based on the least lncosh (Llncosh) criterion by fully using the discarded errors. Then, a new structure of adaptive filters is designed under the framework of ALLM for online application. Using the least lncosh CG (LLCG) algorithm for the design of controller, a novel robust augmented LLCG (ALLCG) adaptive filter is finally developed, which achieves better filtering performance than the traditional CG and LLCG. Simulations demonstrate the advantages of ALLCG on robustness and prediction accuracy.

The filtered-x least mean square (FxLMS) algorithm has been proposed for an active noise control (ANC) system. However, due to the used mean square error (MSE) criterion, FxLMS suffers from performance degeneration for non-Gaussian noises, dramatically. To address this issue, a novel robust generalized hyperbolic tangent (GHT) criterion is first constructed in this paper. Then, the random Fourier features (RFF) method and the conjugate gradient (CG) method are used to address the nonlinearity existing in ANC and solve the quadratic optimization problem induced by the GHT criterion, respectively. Finally, a novel robust random Fourier conjugate gradient filtered-x generalized hyperbolic tangent (RFCGFxGHT) algorithm is proposed for ANC. The theoretical analyses regarding the convergence and computational complexity of RFCGFxGHT are also derived. Simulation experiments on nonlinear ANC systems corrupted by the synthetic logistic chaotic and$\alpha$-stable noises, as well as real-world functional magnetic resonance imaging (fMRI) and server room noises, are conducted to confirm the effectiveness, robustness, and desirable nonlinear learning ability of the proposed algorithm.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

In this letter, a generalized measurement model (GMM) is established under the attack injecting additive and multiplicative false data, simultaneously. The GMM can be applied to any existing nonlinear Kalman filters (NKFs), such as extended Kalman filter (EKF), cubature Kalman filter (CKF), and geometric unscented Kalman filter (GUF). Based on the constructed GMM, a class of nonlinear Kalman filters with GMM (NKFs-GMM) is proposed to deal with additive false data, multiplicative false data, and simultaneous attacks on additive and multiplicative false data with no increase of computational complexity. Numerical simulations show that the filtering accuracy of NKFs-GMM is superior to that of corresponding NKFs.

Generally, the power system (PS) faces various anomalies such as non-Gaussian measurement noises and gross measurement errors, when the communication link is lost or the phasor synchronization is inaccurate. Therefore, the robust forecasting-aided state estimation (FASE) is crucial for PS stability. This letter develops a novel geometric unscented Kalman filter (GUF) with the generalized loss (GL) (GL-GUF) to estimate PS state for forecasting aid. In contrast to the minimum mean square error (MMSE) criterion in the original GUF framework, GL-GUF combines the strength of the GL in robust information learning against non-Gaussian disturbances and the advantages of GUF in handling strong model nonlinearity with high accuracy and good stability. More importantly, by establishing the linear regression model to obtain residual vectors and introducing the negative log-likelihood of GL, simultaneously, GL-GUF optimizes free parameters to avoid manual parameter tuning. Simulations on IEEE 14-bus and 30-bus test systems confirm the high precision and robustness of GL-GUF for non-Gaussian disturbances.

Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR)$q$-derivative based on the concept of$q$-derivative and quaternion GHR derivative. Then, the product rules of GHR$q$-derivative are deduced, and the results of GHR$q$-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR$q$-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the$q$-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed$q$-QLMS algorithm.