 Research
 Open Access
Adaptive link selection algorithms for distributed estimation
 Songcen Xu^{1}Email author,
 Rodrigo C. de Lamare^{1, 2} and
 H. Vincent Poor^{3}
https://doi.org/10.1186/s1363401502724
© Xu et al. 2015
 Received: 26 May 2015
 Accepted: 23 September 2015
 Published: 6 October 2015
Abstract
This paper presents adaptive link selection algorithms for distributed estimation and considers their application to wireless sensor networks and smart grids. In particular, exhaustive searchbased least mean squares (LMS) / recursive least squares (RLS) link selection algorithms and sparsityinspired LMS / RLS link selection algorithms that can exploit the topology of networks with poorquality links are considered. The proposed link selection algorithms are then analyzed in terms of their stability, steadystate, and tracking performance and computational complexity. In comparison with the existing centralized or distributed estimation strategies, the key features of the proposed algorithms are as follows: (1) more accurate estimates and faster convergence speed can be obtained and (2) the network is equipped with the ability of link selection that can circumvent link failures and improve the estimation performance. The performance of the proposed algorithms for distributed estimation is illustrated via simulations in applications of wireless sensor networks and smart grids.
Keywords
 Adaptive link selection
 Distributed estimation
 Wireless sensor networks
 Smart grids
1 Introduction
Distributed signal processing algorithms have become a key approach for statistical inference in wireless networks and applications such as wireless sensor networks and smart grids [1–5]. It is well known that distributed processing techniques deal with the extraction of information from data collected at nodes that are distributed over a geographic area [1]. In this context, for each specific node, a set of neighbor nodes collect their local information and transmit the estimates to a specific node. Then, each specific node combines the collected information together with its local estimate to generate an improved estimate.
1.1 Prior and related work
Several works in the literature have proposed strategies for distributed processing which include incremental [1, 6–8], diffusion [2, 9], sparsityaware [3, 10], and consensusbased strategies [11]. With the incremental strategy, the processing follows a Hamiltonian cycle, i.e., the information flows through these nodes in one direction, which means each node passes the information to its adjacent node in a uniform direction. However, in order to determine an optimum cyclic path that covers all nodes (considering the noise, interference, path loss, and channels between neighbor nodes), this method needs to solve an NPhard problem. In addition, when any of the nodes fails, data communication through the cycle is interrupted and the distributed processing breaks down [1].
In distributed diffusion strategies [2, 10], the neighbors for each node are fixed and the combining coefficients are calculated after the network topology is deployed and starts its operation. One potential risk of this approach is that the estimation procedure may be affected by poorly performing links. More specifically, the fixed neighbors and the precalculated combining coefficients may not provide an optimized estimation performance for each specified node because there are links that are more severely affected by noise or fading. Moreover, when the number of neighbor nodes is large, each node requires a large bandwidth and transmit power. In [12, 13], the idea of partial diffusion was introduced for reducing communications between neighbor nodes. Prior work on topology design and adjustment techniques includes the studies in [14–16] and [17], which are not dynamic in the sense that they cannot track changes in the network and mitigate the effects of poor links.
1.2 Contributions
The adaptive link selection algorithms for distributed estimation problems are proposed and studied in this chapter. Specifically, we develop adaptive link selection algorithms that can exploit the knowledge of poor links by selecting a subset of data from neighbor nodes. The first approach consists of exhaustive searchbased least mean squares (LMS)/ recursive least squares (RLS) link selection (ESLMS/ESRLS) algorithms, whereas the second technique is based on sparsityinspired LMS/RLS link selection (SILMS/SIRLS) algorithms. With both approaches, distributed processing can be divided into two steps. The first step is called the adaptation step, in which each node employs LMS or RLS to perform the adaptation through its local information. Following the adaptation step, each node will combine its collected estimates from its neighbors and local estimate, through the proposed adaptive link selection algorithms. The proposed algorithms result in improved estimation performance in terms of the mean square error (MSE) associated with the estimates. In contrast to previously reported techniques, a key feature of the proposed algorithms is that the combination step involves only a subset of the data associated with the best performing links.
In the ESLMS and ESRLS algorithms, we consider all possible combinations for each node with its neighbors and choose the combination associated with the smallest MSE value. In the SILMS and SIRLS algorithms, we incorporate a reweighted zero attraction (RZA) strategy into the adaptive link selection algorithms. The RZA approach is often employed in applications dealing with sparse systems in such a way that it shrinks the small values in the parameter vector to zero, which results in better convergence and steadystate performance. Unlike prior work with sparsityaware algorithms [3, 18–20], the proposed SILMS and SIRLS algorithms exploit the possible sparsity of the MSE values associated with each of the links in a different way. In contrast to existing methods that shrink the signal samples to zero, SILMS and SIRLS shrink to zero the links that have poor performance or high MSE values. By using the SILMS and SIRLS algorithms, the data associated with unsatisfactory performance will be discarded, which means the effective network topology used in the estimation procedure will change as well. Although the physical topology is not changed by the proposed algorithms, the choice of the data coming from the neighbor nodes for each node is dynamic, leads to the change of combination weights, and results in improved performance. We also remark that the topology could be altered with the aid of the proposed algorithms and a feedback channel which could inform the nodes whether they should be switched off or not. The proposed algorithms are considered for wireless sensor networks and also as a tool for distributed state estimation that could be used in smart grids.

We present adaptive link selection algorithms for distributed estimation that are able to achieve significantly better performance than existing algorithms.

We devise distributed LMS and RLS algorithms with link selection capabilities to perform distributed estimation.

We analyze the MSE convergence and tracking performance of the proposed algorithms and their computational complexities, and we derive analytical formulas to predict their MSE performance.

A simulation study of the proposed and existing distributed estimation algorithms is conducted along with applications in wireless sensor networks and smart grids.
This paper is organized as follows. Section 2 describes the system model and the problem statement. In Section 3, the proposed link selection algorithms are introduced. We analyze the proposed algorithms in terms of their stability, steadystate, and tracking performance and computational complexity in Section 4. The numerical simulation results are provided in Section 5. Finally, we conclude the paper in Section 6.
Notation: We use boldface upper case letters to denote matrices and boldface lower case letters to denote vectors. We use (·)^{ T } and (·)^{−1} to denote the transpose and inverse operators, respectively, (·)^{ H } for conjugate transposition and (·)^{∗} for complex conjugate.
2 System model and problem statement
where \(\mathcal {N}_{k}\) denotes the set of neighbors of node k including node k itself, c _{ kl }≥0 is the combining coefficient, and ψ _{ l }(i) is the local estimate generated by node l through its local information.

Some nodes may face high levels of noise or interference, which may lead to inaccurate estimates.

When the number of neighbors for each node is high, large communication bandwidth and high transmit power are required.

Some nodes may shut down or collapse due to network problems. As a result, local estimates to their neighbors may be affected.
Under such circumstances, a performance degradation is likely to occur when the network cannot discard the contribution of poorly performing links and their associated data in the estimation procedure. In the next section, the proposed adaptive link selection algorithms are presented, which equip a network with the ability to improve the estimation procedure. In the proposed scheme, each node is able to dynamically select the data coming from its neighbors in order to optimize the performance of distributed estimation techniques.
3 Proposed adaptive link selection algorithms
In this section, we present the proposed adaptive link selection algorithms. The goal of the proposed algorithms is to optimize the distributed estimation and improve the performance of the network by dynamically changing the topology. These algorithmic strategies give the nodes the ability to choose their neighbors based on their MSE performance. We develop two categories of adaptive link selection algorithms; the first one is based on an exhaustive search, while the second is based on a sparsityinspired relaxation. The details will be illustrated in the following subsections.
3.1 Exhaustive search–based LMS/RLS link selection
is the local estimator and ψ _{ l }(i) is calculated through (7) or (11), depending on the algorithm, i.e., ESLMS or ESRLS. With different choices of the set Ω _{ k }, the combining coefficients c _{ kl } will be recalculated through (5), to ensure condition (6) is satisfied.
At this stage, the main steps of the ESLMS and ESRLS algorithms have been completed. The proposed ESLMS and ESRLS algorithms find the set \(\widehat {\Omega }_{k}(i)\) that minimizes the error pattern in (16) and (17) and then use this set of nodes to obtain ω _{ k }(i) through (18).
The ESLMS/ESRLS algorithms are briefly summarized as follows: Step 1 Each node performs the adaptation through its local information based on the LMS or RLS algorithm. Step 2 Each node finds the best set Ω _{ k }(i), which satisfies (17). Step 3 Each node combines the information obtained from its best set of neighbors through (18).
The details of the proposed ESLMS and ESRLS algorithms are shown in Algorithms 1 and 2. When the ESLMS and ESRLS algorithms are implemented in networks with a large number of small and low–power sensors, the computational complexity cost may become high, as the algorithm in (17) requires an exhaustive search and needs more computations to examine all the possible sets Ω _{ k }(i) at each time instant.
3.2 Sparsity–inspired LMS/RLS link selection
Figure 2 a shows a standard type of sparsityaware processing. We can see that, after being processed by a sparsityaware algorithm, the nodes with small MSE values will be shrunk to zero. In contrast, the proposed SILMS and SIRLS algorithms will keep the nodes with lower MSE values and reduce the combining weight of the nodes with large MSE values as illustrated in Fig. 2 b. When compared with the EStype algorithms, the SILMS/RLS algorithms do not need to consider all possible combinations of nodes, which means that the SILMS/RLS algorithms have lower complexity. In the following, we will show how the proposed SILMS/SI–RLS algorithms are employed to realize the link selection strategy automatically.
 1.
The element with largest absolute value e _{ kl }(i) in \(\hat {\boldsymbol e}_{k}\) will be kept as e _{ kl }(i).
 2.
The element with smallest absolute value will be set to −e _{ kl }(i). This process will ensure the node with smallest error pattern has a reward on its combining coefficient.
 3.
The remaining entries will be set to zero.
Note that the condition \(c_{k,j}\rho \varepsilon \frac {{\text {sign}}({\hat {e}_{k,j}}) }{1+\varepsilon \xi _{\text {min}}}\geq 0\) is enforced in (28). When \(c_{k,j}\rho \varepsilon \frac {{\text {sign}}({\hat {e}_{k,j}}) }{1+\varepsilon \xi _{\text {min}}}= 0\), it means that the corresponding node has been discarded from the combination step. In the following time instant, if this node still has the largest error, there will be no changes in the combining coefficients for this set of nodes.

The process in (28) satisfies condition (6), as the penalty and reward amounts of the combining coefficients are the same for the nodes with maximum and minimum error, respectively, and there are no changes for the rest nodes in the set.

When computing (28), there are no matrix–vector multiplications. Therefore, no additional complexity is introduced. As described in (24), only the jth element of \(\boldsymbol c_{k}, \hat {\boldsymbol e}_{k}\) and jth column of Ψ _{ k } are used for calculation.
For the neighbor node with the largest MSE value, after the modifications of \(\hat {\boldsymbol e}_{k}\), its e _{ kl }(i) value in \(\hat {\boldsymbol e}_{k}\) will be a positive number which will lead to the term \(\rho \varepsilon \frac {{\text {sign}}({\hat {e}_{k,j}})}{1+\varepsilon \xi _{\text {min}}}\) in (28) being positive too. This means that the combining coefficient for this node will be shrunk and the weight for this node to build ω _{ k }(i) will be shrunk too. In other words, when a node encounters high noise or interference levels, the corresponding MSE value might be large. As a result, we need to reduce the contribution of that node.
In contrast, for the neighbor node with the smallest MSE, as its e _{ kl }(i) value in \(\hat {\boldsymbol e}_{k}\) will be a negative number, the term \(\rho \varepsilon \frac {{\text {sign}}({\hat {e}_{k,j}})}{1+\varepsilon \xi _{\text {min}}}\) in (28) will be negative too. As a result, the weight for this node associated with the smallest MSE to build ω _{ k }(i) will be increased. For the remaining neighbor nodes, the entry e _{ kl }(i) in \(\hat {\boldsymbol e}_{k}\) is zero, which means the term \(\rho \varepsilon \frac {{\text {sign}}({\hat {e}_{k,j}})}{1+\varepsilon \xi _{\text {min}}}\) in (28) is zero and there is no change for the weights to build ω _{ k }(i). The main steps for the proposed SILMS and SIRLS algorithms are listed as follows: Step 1 Each node carries out the adaptation through its local information based on the LMS or RLS algorithm. Step 2 Each node calculates the error pattern through (20). Step 3 Each node modifies the error vector \(\hat {\boldsymbol e}_{k}\). Step 4 Each node combines the information obtained from its selected neighbors through (28).
The SILMS and SIRLS algorithms are detailed in Algorithm 3. For the ESLMS/ESRLS and SILMS/SIRLS algorithms, we design different combination steps and employ the same adaptation procedure, which means the proposed algorithms have the ability to equip any diffusiontype wireless networks operating with other than the LMS and RLS algorithms. This includes, for example, the diffusion conjugate gradient strategy [30]. Apart from using weights related to the node degree, other signal dependent approaches may also be considered, e.g., the parameter vectors could be weighted according to the signaltonoise ratio (SNR) (or the noise variance) at each node within the neighborhood.
4 Analysis of the proposed algorithms
where ω _{0} denotes the optimum Wiener solution of the actual parameter vector to be estimated, and ω _{ k }(i) is the estimate produced by a proposed algorithm at time instant i.
Assumption II: The input signal vector x _{ l }(i) is drawn from a stochastic process, which is ergodic in the autocorrelation function [22].
Assumption III: The M×1 vector q(i) represents a stationary sequence of independent zero mean vectors and positive definite autocorrelation matrix \(\boldsymbol Q\,=\,\mathbb {E}\left [\boldsymbol q(i)\boldsymbol q^{H}(i)\right ]\), which is independent of x _{ k }(i), n _{ k }(i) and ε _{ l }(i).
Assumption IV (Independence): All regressor input signals x _{ k }(i) are spatially and temporally independent. This assumption allows us to consider the input signal x _{ k }(i) independent of \(\boldsymbol \omega _{l}(i), l\in \mathcal {N}_{k}\).
4.1 Stability analysis
In general, to ensure that a partially connected network performance can converge to the global network performance, the estimates should be propagated across the network [31]. The work in [14] shows that it is central to the performance that each node should be able to reach the other nodes through one or multiple hops [31].
where ⊗ denotes the Kronecker product.
Before we proceed, let us define \(\boldsymbol X=\boldsymbol I\mathcal {\boldsymbol M}\mathcal {D}\). We say that a square matrix X is stable if it satisfies X ^{ i }→0 as i→∞. A known result in linear algebra states that a matrix is stable if, and only if, all its eigenvalues lie inside the unit circle. We need the following lemma to proceed [9].
Lemma 1.
Let C _{ G } and X denote arbitrary N M×N M matrices, where C _{ G } has real, nonnegative entries, with columns adding up to one. Then, the matrix \(\boldsymbol Y=\boldsymbol {C_{G}^{T}}\boldsymbol X\) is stable for any choice of C _{ G } if, and only if, X is stable.
Proof.
Assume that X is stable, it is true that for every square matrix X and every α>0, there exists a submultiplicative matrix norm ·_{ τ } that satisfies X_{ τ }≤τ(X)+α, where the submultiplicative matrix norm A B≤A·B holds and τ(X) is the spectral radius of X [32, 33]. Since X is stable, τ(X)<1, and we can choose α>0 such that τ(X)+α=v<1 and X_{ τ }≤v<1. Then we obtain [9]
so Y ^{ i } converges to the zero matrix for large i. Therefore, Y is stable.
where λ _{ max }(R _{ k }) is the largest eigenvalue of the correlation matrix R _{ k }. The difference between the ESLMS and SILMS algorithms is the strategy to calculate the matrix C. Lemma 1 indicates that for any choice of C, only X needs to be stable. As a result, SILMS has the same convergence condition as in (43). Given the convergence conditions, the proposed ESLMS/ESRLS and SILMS/SIRLS algorithms will adapt according to the network connectivity by choosing the group of nodes with the best available performance to construct their estimates.
4.2 MSE steadystate analysis
For the proposed adaptive link selection algorithms, we will derive the EMSE formulas separately based on (47) and we assume that the input signal is modeled as a stationary process.
4.2.1 ES–LMS
At each time instant, each node will generate data associated with network covariance matrices A _{ k } with size N×N which reflect the network topology, according to the exhaustive search strategy. In the network covariance matrices A _{ k }, a value equal to 1 means nodes k and l are linked and a value 0 means nodes k and l are not linked.
Coefficients α _{ kl } for different sets of Ω _{3}
\(\{2,3\}\left \{ \begin {array}{l} \alpha _{31}=0\\ \alpha _{32}=1\\ \alpha _{33}=1\\ \alpha _{34}=0\\ \alpha _{35}=0 \end {array} \right. \)  \(\{3,5\}\left \{\begin {array}{l} \alpha _{31}=0\\ \alpha _{32}=0\\ \alpha _{33}=1\\ \alpha _{34}=0\\ \alpha _{35}=1 \end {array} \right. \)  \(\{2,3,5\}\left \{\begin {array}{l} \alpha _{31}=0\\ \alpha _{32}=1\\ \alpha _{33}=1\\ \alpha _{34}=0\\ \alpha _{35}=1 \end {array} \right. \) 
The parameters c _{ kl } will then be calculated through Eq. (5) for different choices of matrices A _{ k } and coefficients α _{ kl }. After α _{ kl } and c _{ kl } are calculated, the error pattern for each possible Ω _{ k } will be calculated through (16) and the set with the smallest error will be selected according to (17).
4.2.2 SILMS
For each node k, at time instant i, after it receives the estimates from all its neighbors, it calculates the error pattern e _{ kl }(i) for every estimate received through Eq. (20) and finds the nodes with the largest and smallest errors. An error vector \( \hat {\boldsymbol e}_{k}\) is then defined through (23), which contains all error patterns e _{ kl }(i) for node k.
and the term ‘error pattern’ e _{ kl } in (60) is from the modified error vector \( \hat {\boldsymbol e}_{k}\).
Under this situation, after the time instant i goes to infinity, the parameters h _{ kl } for each neighbor node of node k can be obtained through (61) and the quantity h _{ kl } will be deterministic and can be taken out from the expectation.
4.2.3 ESRLS
On the basis of (72), we have that when i tends to infinity, the MSE approaches the MMSE in theory [22].
4.2.4 SIRLS
In conclusion, according to (62) and (74), with the help of modified combining coefficients, for the proposed SItype algorithms, the neighbor node with lowest MSE contributes the most to the combination, while the neighbor node with the highest MSE contributes the least. Therefore, the proposed SItype algorithms perform better than the standard diffusion algorithms with fixed combining coefficients.
4.3 Tracking analysis
In this subsection, we assess the proposed ESLMS/RLS and SILMS/RLS algorithms in a nonstationary environment, in which the algorithms have to track the minimum point of the error performance surface [34, 35]. In the timevarying scenarios of interest, the optimum estimate is assumed to vary according to the model ω _{0}(i+1)=β ω _{0}(i)+q(i), where q(i) denotes a random perturbation [32] and β=1 in order to facilitate the analysis. This is typical in the context of tracking analysis of adaptive algorithms [22, 32, 36, 37].
4.3.1 ESLMS
4.3.2 SILMS
4.3.3 ESRLS
4.3.4 SIRLS
In conclusion, for timevarying scenarios, there is only one additional term \(M\sigma _{x,k}^{2}\text {tr}\{\boldsymbol Q\}\) in the MSE expression for all algorithms, and this additional term has the same value for all algorithms. As a result, the proposed SItype algorithms still perform better than the standard diffusion algorithms with fixed combining coefficients, according to the conclusion obtained in the previous subsection.
4.4 Computational complexity
Computational complexity for the adaptation step per node per time instant
Adaptation method  Multiplications  Additions 

LMS  2M+1  2M 
RLS  4M ^{2}+16M+2  4M ^{2}+12M−1 
Computational complexity for the combination step per node per time instant
Algorithms  Multiplications  Additions 

ES–LMS/RLS  \(M(t+1){\frac {\mathcal {N}_{k}!}{t!(\mathcal {N}_{k}t)!}}\)  \(Mt{\frac {\mathcal {N}_{k}!}{t!(\mathcal {N}_{k}t)!}}\) 
SI–LMS/RLS  \((2M+4)\mathcal {N}_{k}\)  \((M+2)\mathcal {N}_{k}\) 
Computational complexity per node per time instant
Algorithm  Multiplications  Additions 

ESLMS  \(\left [{\frac {(t+1)\mathcal {N}_{k}!}{t!(\mathcal {N}_{k}t)!}}+8\right ]M+2\)  \(\left [{\frac {\mathcal {N}_{k}!}{(t1)!(\mathcal {N}_{k}t)!}}+8\right ]M\) 
ESRLS  \(4M^{2}+\left [{\frac {(t+1)\mathcal {N}_{k}!}{t!(\mathcal {N}_{k}t)!}}+16\right ]M+2\)  \(4M^{2}+\left [{\frac {\mathcal {N}_{k}!}{(t1)!(\mathcal {N}_{k}t)!}}+12\right ]M1\) 
SILMS  \((8+2\mathcal {N}_{k})M+4\mathcal {N}_{k}+2\)  \((8+\mathcal {N}_{k})M+2\mathcal {N}_{k}\) 
SIRLS  \(4M^{2}+(16+2\mathcal {N}_{k})M+4\mathcal {N}_{k}+2\)  \(4M^{2}+(12+\mathcal {N}_{k})M+2\mathcal {N}_{k}1\) 
5 Simulations
In this section, we investigate the performance of the proposed link selection strategies for distributed estimation in two scenarios: wireless sensor networks and smart grids. In these applications, we simulate the proposed link selection strategies in both static and timevarying scenarios. We also show the analytical results for the MSE steadystate and tracking performances that we obtained in Section 4.
5.1 Diffusion wireless sensor networks
In this subsection, we compare the proposed ESLMS/ESRLS and SILMS/SIRLS algorithms with the diffusion LMS algorithm [2], the diffusion RLS algorithm [38], and the singlelink strategy [39] in terms of their MSE performance. A reducedcommunication diffusion LMS algorithm with a performance comparable or worse to the standard diffusion LMS algorithm, which has been reported in [40], may also be considered if a designer needs to reduce the required bandwidth.
where q(i) is an independent zero mean Gaussian vector process with variance \({\sigma ^{2}_{q}}= 0.01\) and β=0.9998.
5.2 MSE analytical results
5.3 Smart Grids
where ω _{0}(i) is the state vector of the entire interconnected system and X _{ k }(ω _{0}(i)) is a nonlinear measurement function of bus k. The quantity n _{ k }(i) is the measurement error with mean equal to zero and which corresponds to bus k.
All busses are corrupted by additive white Gaussian noise with variance \(\sigma ^{2}_{n,k}\in \,[\!0.001,0.01]\). The step size for the standard diffusion LMS [2], the proposed ESLMS, and SILMS algorithms is 0.15. The parameter vector ω _{0} is set to an allone vector. For the diffusion RLS, ESRLS, and SIRLS algorithms, the forgetting factor λ is set to 0.945 and δ is equal to 0.001. The sparsity parameters of the SILMS/RLS algorithms are set to ρ=0.07 and ε=10. The results are averaged over 100 independent runs. We simulate the proposed algorithms for smart grids under a static scenario.
6 Conclusions
In this paper, we have proposed the ESLMS/RLS and SILMS/RLS algorithms for distributed estimation in applications such as wireless sensor networks and smart grids. We have compared the proposed algorithms with existing methods. We have also devised analytical expressions to predict their MSE steadystate performance and tracking behavior. Simulation experiments have been conducted to verify the analytical results and illustrate that the proposed algorithms significantly outperform the existing strategies, in both static and timevarying scenarios, in examples of wireless sensor networks and smart grids.
Declarations
Acknowledgements
This research was supported in part by the US National Science Foundation under Grants CCF1420575, CNS1456793, and DMS1118605.
Part of this work has been presented at the 2013 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vancouver, Canada and 2013 IEEE International Workshop on Computational Advances in MultiSensor Adaptive Processing, Saint Martin.
The authors wish to thank the anonymous reviewers, whose comments and suggestions have greatly improved the presentation of these results.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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