Censored regression distributed functional link adaptive filtering algorithm over nonlinear networks
Introduction
The wireless sensor network (WSN) is a multi-hop self-organizing network system formed by a large number of sensor nodes deployed in the monitoring area to communicate with each other [1], [2], [3]. It is an important technology in the underlying network of the Internet of Things (IoT). As the perception of external information, WSN has become a development trend of the IoT technology. Distributed estimation over WSNs extracts data collected by nodes distributed in geographic areas, thereby estimating system parameters of interest from noise measurements [4], [5]. Centralized estimation is performed by a central processor that fuses all nodes together for estimation [6], [7], while distributed estimation, after obtaining the estimated value on each node, is used for data exchange with direct neighbors in the network. Distributed estimation performs estimation tasks on each node and communicates with nodes from the same neighborhood to exchange information, which reduces the demand for communication resources and enhances the reliability and robustness of the network [8], [9]. For different application scenarios, a number of distributed algorithms have been developed. For example, the distributed multi-error filtered-x least mean square (DMEFxLMS) algorithm was developed for distributed active noise control (ANC) networks [10], and the diffusion augmented complex LMS (DACLMS) algorithm was proposed for distributed complex networks [11], etc. The -stable distribution is considered to be an effective modeling method for impulsive noise signals that are often encountered in real-life [12]. By minimizing the real-time fractional powers of an error measure, the authors in [13] developed a class of distributed adaptive filtering solutions for -stable signals. More distributed processing methods of -stable noise can be found in [14], [15].
The above algorithms assume that the input-output of each node is a linear regression model, and do not consider the nonlinear behavior of the system. However, when the system exhibits nonlinearity, the reduced performance of these algorithms cannot meet the requirements of distributed estimation. To the best of our knowledge, the literature is limited on distributed nonlinear estimation, including kernel space-based [16], [17], Volterra filter (VF)-based [18], [19], and spline filter (SF)-based [20] distributed estimation algorithms. The principle of the kernel method is to map the input samples into the high-dimensional reproducing kernel Hilbert space of linear modeling, and then use the corresponding kernel function for nonlinear approximation [16]. However, conventional kernel adaptive methods suffer from the high computational load due to the increase of dictionary dimension [21]. To alleviate this problem, several methods have been proposed [22], [23]. Based on the principles of set-membership filtering, two computationally efficient kernel adaptive filters were developed in [23]. The VF is derived from the Taylor expansion with memory, which belongs to the linear-in-parameters (LIP) nonlinear filter, i.e., its output depends linearly on coefficients [24]. The main bottleneck of VF is that the computational complexity increases exponentially as the order of polynomials increases [25]. The SF characterizes a nonlinear system by using a series of local interpolation polynomials, which adapts a small subset of the parameters at each time step [20].
In many engineering distributed applications, the measurement data of a nonlinear system may be partially censored due to the saturation of sensors. For example, distributed acoustically coupled sensor networks are usually used to deal with nonlinear active noise problems, where the primary or secondary path may exhibit nonlinear distortions [26]. When the measurement of the primary or secondary disturbance exceeds the range of acoustic sensors, part of the statistical information of the recorded signal will be lost or censored. Algorithms based on the censored regression (CR) process were proposed to eliminate the estimation bias caused by censored observations [27], [28]. Note that due to incomplete observations, the CR problem is an intentional nonlinear process independent of system nonlinearity. Since current CR-based distributed algorithms only focus on solving the censored measurement of distributed linear estimation, the bias compensation of CR models in distributed nonlinear networks is still a promising research gap.
On the other hand, to model the nonlinearity of the system, the functional link network (FLN) has been considered as a good alternative to VF in recent years due to its fewer parameters, simple structure, and efficient calculation. The FLN resorts to point-wise orthogonal polynomial expansions of the current input signal, which are then linearly combined by an adaptive filter to obtain the output. Therefore, an FLN is also a LIP filter with the same flexibility as the VF. Inspired by the attractive properties of the FLN, it is natural to consider developing a new distributed nonlinear algorithm. However, expansion functions based on different orthogonal polynomials lead to different diffusion FLNs (DFLNs), and the modeling capabilities of various DFLNs for nonlinear distributed networks need to be further compared.
In this work, we focus on solving distributed nonlinear estimation problems. By developing different DFLNs, a class of distributed function link LMS (DFLMS) algorithm is proposed, in which the input signals expanded by DFLNs are represented in a unified fashion. Moreover, to compensate the bias caused by CR measurement, the CR-distributed functional link adaptive filtering (CR-DFLAF) algorithm is further proposed. Specifically, the main contributions are outlined as follows.
1) A class of DFLMS algorithm is derived for distributed nonlinear networks, in which four types of DFLNs including Chebyshev DFLN (CDFLN), Hermite DFLN (HDFLN), Legendre DFLN (LDFLN), and trigonometric DFLN (TDFLN), are respectively used as controllers to model the nonlinearity of the system.
2) Consider censored measurements at distributed nodes that are corrupted by additional noise with unknown standard deviation. Based on the probit regression, the proposed CR-DFLAF algorithm corrects the bias of censored measurements by constructing an adaptive estimator and adopts the least square criterion and distributed collaboration strategy to obtain the estimate of the parameters of interest.
3) For the proposed approaches, the stability condition of the learning rate is derived, and the corresponding computational complexity is provided.
4) Simulation results of two types of distributed nonlinear networks verify the effectiveness of the proposed algorithms.
The paper is organized as follows. Section 2 reviews the distributed nonlinear network model and censored regression. In Section 3, the DFLMS and CR-DFLAF algorithms are derived. In Section 4, the performance of the proposed algorithms is analyzed. In Section 5, simulations are conducted to verify the performance of the proposed algorithms. Conclusions are provided in Section 6.
: Normal font stands for scalars. Uppercase and Lowercase boldface letters denote matrices and vectors, respectively. Operators and are transpose and expectation. The symbol is the set of real numbers.
Section snippets
Distributed nonlinear network model
The considered model is a spatial network, composed of nodes. At each time instant , each node receives the random data , where is the node input, is the order of the input, and is the scalar measurement. The nonlinear relationship between and is as follows [16], [29]:where denotes the nonlinear behavior of nodes, is the unknown vector of the system parameters, and represents
DFLN model
By developing the DFLN, the general form of the expanded input signal is given bywhere , is the expansion function of DFLN, and is the nonlinear order. A generic set of functional link for the -order trigonometric DFLN (TDFLN) can be expressed by
Performance analysis
In this section, we only focus on the analysis of the CR-DFLAF algorithm based on the -order TDFLN, since the analysis of the DFLMS algorithm is similar to that of the DLMS algorithm. The weight vector and intermediate estimate vector are vectors with dimension .
Numerical simulations
In this section, we perform experimental studies to evaluate the effectiveness of the proposed algorithms, where the distributed network is considered to be composed of nodes, as shown in Fig. 1. The variance of the measurement noise and the SNRs of each node are shown in Fig. 2. The original input signal is a uniform noise over the interval .
Conclusions and discussions
In this work, we focused on the distributed nonlinear network and used DFLN to model the nonlinear characteristics of the system. Based on the diffusion strategy, the DFLMS algorithm was proposed for distributed nonlinear estimation. In particular, four DFLNs were developed to model the nonlinearity of the system. In the case of censored distributed nonlinear networks, the CR-DFLAF algorithm was further proposed by constructing an adaptive estimator and combining with the least square
CRediT authorship contribution statement
Kai-Li Yin: Data curation, Formal analysis, Writing – original draft, Methodology. Yi-Fei Pu: Writing – original draft, Funding acquisition, Writing – review & editing. Lu Lu: Investigation, Writing – original draft, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the National Key Research and Development Program Foundation of PR China under Grant 2018YFC0830300, China South Industries Group Corporation (Chengdu) Fire Control Technology Center Project (non-secret) under Grant HK20-03, National Natural Science Foundation of PR China under Grant 62171303 and Grant 61901285, China Postdoctoral Science Foundation under Grant 2020T130453, the Fundamental Research Funds for the Central Universities under Grant 2021SCU12063, and
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