This paper considers the scenario of noisy inputs and compressive diffusion (for reducing communication load) with noisy links over sensor networks. We first study the implementation of diffusion bias-compensated Bayesian adaptation (DBCBA) for noisy inputs, which outperforms the existing solution. Next, an average-estimate step is applied to lessen the impact of link noise in the full diffusion case, yielding a diffusion average-estimate bias-compensated Bayesian adaptation (DABCBA) algorithm. A Bayes-based adaptation construction step is then presented to reconstruct the compressed diffusion information in the presence of link noise, resulting in a compressive DBCBA (CDBCBA) algorithm whose mean and mean-square behaviors are analyzed and the closed-form expression of the steady-state mean-square deviation is derived. In addition, estimators are devised for the input and output noise variances. The excellent performance of our algorithms is demonstrated via numerical examples while the theoretical calculation aligns closely with the simulation results.

To improve the learning performance of the conventional diffusion least mean square (DLMS) algorithms, this article proposes Bayesian-learning-based DLMS (BL-DLMS) algorithms. First, the proposed BL-DLMS algorithms are inferred from a Gaussian state-space model-based Bayesian learning perspective. By performing Bayesian inference in the given Gaussian state-space model, a variable step-size and an estimation of the uncertainty of information of interest at each node are obtained for the proposed BL-DLMS algorithms. Next, a control method at each node is designed to improve the tracking performance of the proposed BL-DLMS algorithms in the sudden change scenario. Then, a lower bound on the variable step-size of each node of the proposed BL-DLMS algorithms is derived to maintain the optimal steady-state performance in the nonstationary scenario (unknown parameter vector of interest is time-varying). Afterward, the mean stability and the transient and steady-state mean square performance of the proposed BL-DLMS algorithms are analyzed in the nonstationary scenario. In addition, two Bayesian-learning-based diffusion bias-compensated LMS algorithms are proposed to handle the noisy inputs. Finally, the superior learning performance of the proposed learning algorithms is verified by numerical simulations, and the simulated results are in good agreement with the theoretical results.

Affine projection (AP) algorithm has been widely used in many applications because of the fast convergence ability for colored inputs. To reduce high steady-state errors resulting from weight-noise correlations, this letter presents an interval-extraction mechanism for AP algorithm, named IEAP. In the procedure, we recommend using an interval between each data point of the output vector in the AP update. Finally, by means of theoretical mean square deviations analysis and computer simulations, the interval-extraction method is demonstrated to be effective.

Sensor networks are an indispensable part of the Internet of Things (IoT), where sensors perform data acquisition and information processing tasks to obtain the parameters of interest so that IoT-based monitoring, diagnosis and other systems respond quickly to the changing conditions, instantaneous faults, etc. Distributed estimation algorithms are usually employed to estimate the parameters of interest in these IoT-based applications. However, when sensor networks have highly correlated input signals and nonstationary behavior in which the parameters of interest are time-varying, conventional distributed estimation algorithms suffer from severely degraded learning performance due to the large eigenvalue spread in the covariance matrix of the input signals and the random perturbation of the parameters of interest. To address these problems, this paper proposes two diffusion Bayesian subband adaptive filter (DBSAF) algorithms from a Bayesian learning perspective. As the highly-correlated input signal is whitened in a multiband structure and an estimate of the uncertainty in the parameters of interest is obtained by performing Bayesian inference, the proposed DBSAF algorithms are able to achieve better learning performance in comparison with the competing diffusion algorithms. The transient and steady-state mean square error performance of the proposed DBSAF algorithms are analyzed, and are verified by numerical simulations. A lower bound on the time-varying step-size is derived to maintain the optimal steady-state performance in nonstationary scenarios. A new method for the estimation of the noise variance is also proposed. Numerical simulations demonstrate the excellent learning performance of the proposed algorithms in comparison with benchmark algorithms.