Diffusion adaptive filtering algorithm based on the Fair cost function

To better perform distributed estimation, this paper, by combining the Fair cost function and adapt-then-combine scheme at all distributed network nodes, a novel diffusion adaptive estimation algorithm is proposed from an M-estimator perspective, which is called the diffusion Fair (DFair) adaptive filtering algorithm. The stability of the mean estimation error and the computational complexity of the DFair are theoretically analyzed. Compared with the robust diffusion LMS (RDLMS), diffusion Normalized Least Mean M-estimate (DNLMM), diffusion generalized correntropy logarithmic difference (DGCLD), and diffusion probabilistic least mean square (DPLMS) algorithms, the simulation experiment results show that the DFair algorithm is more robust to input signals and impulsive interference. In conclusion, Theoretical analysis and simulation results show that the DFair algorithm performs better when estimating an unknown linear system in the changeable impulsive interference environments.


Proposed the DFair algorithm
In this section, the DFair algorithm is developed. Firstly, an adaptive filtering algorithm based on the Fair cost function is proposed. Then we modify this adaptive filtering algorithm by extending at all network agents to develop the DFair algorithm.
The adaptive filter algorithm based on the fair cost function. Let W(i) be the system estimated weight vector with length M, X(i) the input signal vector of the adaptive filter at iteration i, and the prediction error e(i) between the desired signal d(i) and the actual output y(i) can be expressed by where W o (M × 1 ) is the parameter of interest system, which needs to be estimated and v(i) is the measurement interference. Fair adaptive filter aims to minimize the Fair cost function defined as where δ > 0 is the cut-off value. According to the steepest descent method, the system estimated weight vector update of the Fair adaptive filter is where sgn() is the symbolic function, and µ is the step size.
The adaptive diffusion filter based on the fair cost function. Our previous paper's research considers a network of N sensor nodes distributed over a geographic area (as Fig. 1) 18,28,41 . We assume an undirected graph so that if agent n-1 is a neighbor of agent n, then agent n-1 is also a neighbor of agent n. We assign a pair of nonnegative scaling weights to the edge connecting n and n-1. A network is connected if paths with nonzero scaling weights can be found linking any two distinct agents in both directions, either directly when they are neighbors or by passing through intermediate agents when they are not neighbors. In this way, information can flow in both directions between any two agents in the network. X n (i) and d n (i) are the input signals and observation output signals at agent n, respectively.
An adaptive network equips the network's nodes with local learning rules or local adaptive filters. The available communication topology is then employed to efficiently implement a cooperation protocol among the nodes to exploit spatial and temporal information efficiently. Different learning rules allied with different cooperation protocols give rise to different adaptive networks. Based on Fig. 1, using the local cost function J local n (W(i)) = δ 2 |e n( i)| δ − log 1 + |e n( i)| δ , we seek the optimal linear estimator that minimizes the global cost function: where at each time instant i, each sensor node n ∈ {1, 2, · · · , N} has access to some zero-mean random process {d n (i), X n (i)} , d n (i) is a scalar and X n (i) is a regression vector ( M × 1 ). Suppose these measurements output follows a standard model given by:  www.nature.com/scientificreports/ where W o (M × 1 ) is the unknown system parameter vector with length M, and v n (i) is the measurement interference with variance σ 2 v,n and each node has a different value of v n (i). In 9 , the DLMS algorithm is obtained by minimizing a linear combination of the local MSE: where the set of nodes that are connected to n (including n itself) is denoted by N n and is called the neighborhood of nodes n. The weighting coefficients a l,n are real and satisfy l∈N n a l,n = 1 . a l,n forms a nonnegative combination matrix A.
The DLMS algorithm obtains the estimation via two steps, adaptation and combination. According to the order of these two steps, the updating equation of the DLMS algorithm can be expressed as where µ n is the step size (learning rate), and ϕ n (i) is the local estimates at node n.
Based on Eq. (7), the derivative of the local instantaneous approximations for W n (i) can be formulated as: As shown in Fig. 1 for the framework of the ATC diffusion strategy, the diffusion algorithms first update the local intermediate estimate by the steepest descent method. Then, each node combines the local intermediate estimates from its neighbors. The steps of the diffusion strategy for distributed estimation are as follows: (1) adaptation step, utilizing the stochastic gradient descent method, the intermediate estimate of node k for the parameter update is derived as: where µ n is a learning step-size and W n (i) is the estimate of W o for node n at the time instant i.
(2) combination step, in this step, the node receives all intermediate estimates from its neighbors as follows: For simplicity, a summary of the DFair algorithm procedure based on the analysis presented above is given in Table 1.

Performance analysis
The DFair adaptive filtering algorithms, including mean behavior and computational complexity, will be discussed in this subsection. Firstly, to facilitate analysis and expression, we define some equations at agent n and time i, Ŵ n (i) = W o − W n (i) , φ n (i) = W o − ϕ n (i) , which are then collected to form the system weight error vector and intermediate system weight error vector, i.e., W(i) = col{W 1 , and e(i) = col{e 1 (i), e 2 (i), · · · , e N (i)}.

Mean weight error vector behavior.
To facilitate performance analysis, we make the following assumptions: Assumption 1 All measurement interferences are independent of any other signals.
Assumption 2 X(i) is zero-mean Gaussian, temporally white, and spatially independent with R xx,n = E X n (i)X T n (i) .  www.nature.com/scientificreports/ Assumption 3 The regression vector X n (i) is independent of Ŵ k j for all k and j < i.
The DFair algorithm will be obtained. Equations (9) and (10) From Eq. (14), one can see that the asymptotic unbiasedness of the DFair algorithm can be guaranteed if the matrix is a block-diagonal matrix and it can be easily verified that it is stable if its block-diagonal entries I − µ n R xx,n is stable. So, the condition for stability of the mean weight error vector (as Eq. (15)) is given by where ρ max denotes the maximal eigenvalue of R xx,n . So, based on Eqs. (13) and (15), we obtain E Ŵ (∞) = 0.
Parameters δ for the proposed algorithm. The choice of δ in Eq. (8) plays a vital role in the performance of the DFair algorithm. When |e n (i)|/δ → 0 , μ n (i) = 0 for each time i ≥ 0 and each agent n. When |e n (i)|/δ → ∞ , μ n (i) = µ n for each time i ≥ 0 and each agent n. So, when large μ n (i) = µ n lead to a large MSD and even cause loss of convergence, while a small μ n (i) = 0 degrade the tracking speed of the DFair algorithm, which means that a large step-size responds quickly to plant changes during the initial convergence, and then a tiny step-size is used as the algorithm approaches its steady state. In other words, the DFair algorithm has been presented to obtain a fast convergence rate and a small steady-state error. Therefore, it is necessary to discuss the value of δ under the different intensities of impulsive interference. We set six experiment groups in a system identification application to choose the optimum cut-off value δ under different input signals, impulsive interference, and different network structures. Another method that can get the optimum cut-off value δ based on the theory derivation method for different input signals, various impulsive interferences, and various network structures. For the theory derivation method, although the optimal parameters δ of the proposed two algorithms are obtained based on minimizing the mean-square deviation (MSD) at the current time. The specific derivation methods are similar so that readers can refer to our previous published paper for details 43 . However, the optimal parameters δ must be time-varying with MSD, which increases the complexity of the algorithm. Therefore, for the sake of simplicity and this paper mainly discusses the design of a novel cost function structure, iterative formulas will increase the computational complexity. So, in this paper, find the approximate optimal parameter value δ of the proposed diffusion adaptive filtering algorithm by designing multiple sets of simulation experiments in different situations. Several experiments were performed in a system identification application in the presence of impulsive interference and Gaussian noise. Gaussian noise is a white Gaussian random process with zero mean and variance equal to 0.01. Impulsive interference is a Bernoulli-Gaussian distribution 18 that was added to the unknown system output also. The Bernoulli-Gaussian impulsive interference, v(i) = f (i)g(i) is a product of a Bernoulli process g(i) and a Gaussian process f (i) , where f (i) is a white Gaussian random process with zero mean and variance σ 2 f , and g(i) = {0, 1} is a Bernoulli process with the probabilities p(1) = Pr and p(0) = 1 − Pr.
In this part, nodes of network topology are set as N = 20, and the input regresses X n (i) of this distributed network are assumed to be spatiotemporally independent zero-mean white Gaussian distributed with different covariance matrixes R xx,n . The impulsive interference is also assumed to be spatially and temporally independent distributed with power σ 2 f . For the adaptation and combination weights, we apply the uniform rule (i.e., a l,n = 1/N n , where the set of nodes that are connected to n is denoted by N n ). We evaluate the relative efficiency of different δ estimators based on their MSD to evaluate the performance of DFair, where 28,41 . Also, the independent Monte Carlo number is 10, and each run has 800 iteration numbers. The different probability density of impulsive interference is considered 0%, 20%, 40%, 60%. www.nature.com/scientificreports/  Table 1). The DFair algorithm has only two more multiplication operations than the DLMS algorithm. But for DPLMS, there are need more multiplication operations when µ n (i) = µ n α n (i) in Eq. (16) Table 1 and Eq. (15) 42 will add more multiplication operations. So, the computational complexity of the DFair algorithm smaller than RDLMS 29 , DNLMM 30 , DGCLD 42 , and DPLMS 41 algorithms.

Simulation results
In this paper, we focus on the distributed adaptive filtering algorithm and compare the DFair algorithm with the RDLMS 29 , DNLMM 30 , DGCLD 42 , and DPLMS 41 algorithms in linear system identification under different types of input signal and impulsive interference. In this part, to demonstrate the robustness performance of the proposed DFair algorithm in the presence of different intensity levels of impulsive interference and input signal, we set several group simulation experiments with different impulsive interference and different input signal types. For this unknown linear system, we set M = 8, and the parameters vector was selected randomly. Each distributed network topology consists of N = 20 nodes. An impulsive interference with a Bernoulli-Gaussian distribution 18 was added to the system output (as described in "Computational complexity" section). Besides, we set the impulsive interference as spatiotemporally independent. For the adaptation weights in the adaptation step and combination weights in the combination step, we apply the uniform rule i.e. a l,n = 1/N n . We use the network MSD to evaluate the performance of diffusion algorithms 28,41 . Simulation experiment 1. Illustrating our proposed DFair algorithm is more robust to the input signal than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. In this experiment, there have the same network topology and the same impulsive interference. If any two network topology nodes are declared neighbors, connect probability greater than or equal to 0.2, the network topology is shown in Fig. 5. The MSD iteration curves for RDLMS ( µ equal to 0.4), DNLMM ( µ equal to 0.4), DGCLD ( µ equal to 0.4), DPLMS, and DFair ( µ equal to 0.4) algorithms in Figs. 6, 7, and 8 are different types of the input signal when the measurement interference in an unknown linear system is impulsive interference with Pr = 0.4, σ 2 f = 0.09 and the cut-value δ = 0.1 for the DFair algorithm. Besides, the independent Monte Carlo number is 10, and each run has 800 iteration numbers. Figures 6, 7, and 8 show that although different input signals are used, the DFair algorithm still has a faster convergence rate and lowest steady-state estimated error than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. Besides, the DFair algorithm is more robust to the input signal. In a word, from Simulation experiment 1, we can get the DFair algorithm superior to the RDLMS, DNLMM, DGCLD, and DPLMS algorithms with different input signals, impulsive interference when using the same distributed network topology.

Simulation experiment 2.
Illustrating the DFair algorithm is more robust to σ 2 f for impulsive interference with a constant Pr and faster convergence rate and lower steady-state estimated error than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. This experiment has the same network topology: the same Pr of impulsive interference and the same input signal. If any two nodes in network topology are declared neighbors, a certain radius for each node is larger than or equal to 0.3, and the network topology is shown in Fig. 9(Left). The MSD iteration curves for RDLMS ( µ equal to 0.3), DNLMM ( µ equal to 0.3), DGCLD ( µ equal to 0.3), DPLMS, and DFair ( µ equal to 0.3) algorithms in Fig. 10 with Pr = 0.4 and the cut-value δ = 0.1 for the DFair algorithm. Also, the independent Monte Carlo number is 10, and each run has 800 iteration numbers.
In this experiment, we want to show that the DFair algorithm is more robust to different the probability density of impulsive interference, so we set four sub-experiments with different density σ 2 f of impulsive interference, the same Pr of impulsive interference, the same input signal, and the same distributed network topology. From   www.nature.com/scientificreports/ Fig. 10, we can find that although different probability density of impulsive interference is considered, the DFair algorithms have a slightly faster rate than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. The DFair algorithm still has a minor steady-state error than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. In a word, from Simulation experiment 2, we can observe that the DFair algorithm is more robust to impulsive interference than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms.

Simulation experiment 3.
To illustrate our algorithm, it is more robust to Pr for impulsive interference with a constant σ 2 f and has a faster convergence rate and lower steady-state error than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. In this experiment, there have the same network topology, same σ 2 f for impulsive interference and the same input signal. If any two nodes in network topology are declared neighbors, a certain radius for each node is larger than or equal to 0.3, and the network topology is shown in Fig. 11(Left). The MSD iteration curves for RDLMS ( µ equal to 0.4), DNLMM ( µ equal to 0.4), DGCLD ( µ equal to 0.4), DPLMS, and DFair ( µ equal to 0.4) algorithms in Fig. 12 with σ 2 f = 0.04 and the cut-value δ = 0.1 for the DFair algorithm. Besides, the independent Monte Carlo number is 10, and each run has 800 iteration numbers.  www.nature.com/scientificreports/ In this experiment, four sub-experiments with different Pr of impulsive interference were set with the same density of impulsive interference, the same input signal, and the same distributed network topology. From Fig. 12, we can find that although different Pr of impulsive interference is considered, the DFair algorithm has a slightly faster rate than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. The DFair algorithm still has a minor steady-state error than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. In a word, from Simulation experiment 3, we can observe that the DFair algorithm is more robust to impulsive interference than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms.

Conclusion
This paper proposed a novel diffusion algorithm by using the Fair cost function, namely the DFair algorithm. The method is developed to combine and modify the DLMS algorithm and the Fair cost function at all distributed network nodes. Compared with some existing distributed adaptive filtering algorithms, the DFair algorithm has low computational complexity. The theoretical analysis demonstrates that the DFair algorithm can effectively estimate from an M-estimation cost function perspective. Besides, theoretical mean behavior interpreted that the DFair algorithm can achieve accurate estimation under the convergence interval. Besides, experimental simulation results showed that the DFair algorithm is more robust to the input signal and impulsive interference than the RDLMS, DNLMM, DGCLD, and DPLMS algorithms. Overall, the DFair algorithm has superior