 Research
 Open Access
A kind of effective data aggregating method based on compressive sensing for wireless sensor network
 Degan Zhang^{1, 2},
 Ting Zhang^{1, 2},
 Jie Zhang^{3}Email author,
 Yue Dong^{1, 2} and
 Xiaodan Zhang^{4}
https://doi.org/10.1186/s1363801811764
© The Author(s). 2018
 Received: 22 February 2018
 Accepted: 5 June 2018
 Published: 19 June 2018
Abstract
Wireless sensor network (WSN) in the Internet of Things consists of a large number of nodes. The proposal of compressive sensing technology provides a novel way for data aggregation in WSN. Based on the clustering structure of WSN, a kind of effective data aggregating method based on compressive sensing is proposed in this paper. The aggregating process is divided into two parts: in the cluster, the sink node sets the corresponding seed vector based on the distribution of network and then sends it to each cluster head. Cluster head can generate corresponding own random spacing sparse matrix based on its received seed vector and collect data through compressive sensing technology. Among clusters, clusters forward measurement values to the sink node along multihop routing tree. Performance analysis and comparison with the relative methods show that our method is effective and superior to other methods regardless of intracluster or intercluster on the total energy consumption of network.
Keywords
 WSN
 Internet of things
 Compressive sensing
 Data aggregating
 Effective
1 Introduction
Data aggregating is an effective strategy to control energy consumption because the number of transmissions can be reduced after aggregation. Reference [1–4] strives for energy balancing to make the network lifetime maximum. The unbalanced consumption of energy is harmful to network safety and health [5–9]. If the sensor nodes of wireless sensor networks (WSNs) spend their energy in a relatively balanced way, the connectivity among sensor nodes and the sink nodes can be kept for a longer time, making the network segmentation to be postponed. Avalanched quantities of tiny sensor nodes establish WSNs in the Internet of Things. These nodes can monitor all kinds of object information around them in realtime. Since the energy of these sensor nodes is usually very limited, how to ensure complete data aggregating with the minimum energy consumption of nodes has been a very critical issue in WSNs [10–12].
In order to remove redundant portions of the collected data, and control the number of data nodes in WSNs, which can save the energy consumption of nodes, recently, many scholars proposed a compressive sensing (CS) technology, which can collect and reconstruct signal with high probability through sampling points less than the Nyquist sampling theorem [13–18]. According to the sparsity of the signal, compressive sensing technology can decrease the original signal from high dimensional to low dimensional on the nodes. It needn't aggregate the signal and recover it with high probability on the sink node. The proposal of compressive sensing has good performance on image processing and other applications [19–27].
Without using compressive sensing in data aggregation, nodes near the leaves forward a small amount of packets, but those which are close to the sink node need to forward a large number of packets [28–30]. With using compressive sensing in data aggregating, each node simply forwards M packets, so the total transmission number of the network with N nodes is MN. However, transmission quantity is still large. References [6–10] proposed a hybrid protocol. In this protocol, nodes near the leaves forward original data without using compressive sensing, and those which are close to the sink node use compressive sensing technology to transmit data. References [31–35] applied hybrid compressive sensing to the data aggregating and proposed a minimum energy aggregation tree. The previous work directly applies compressive sensing method to the route tree. Since clustering method has many advantages over the routing tree [36–40], compressive sensing method on clustering network is applied. Compared with routing tree data aggregating methods, clustering algorithm generally has a better communication load balance [41, 42]. In addition, previous works ignore the distribution of location information and node distribution, which can contribute that data aggregating consumes less energy in WSNs of the Internet of Things [43–47].
References [13, 14] proposed Toeplitz matrix and proved that it meets the restricted isometry property (RIP). Since the correlation of data collected in a single cluster is relatively large, the sparse matrix to the process of compressive sensing can be used. It can minimize the number of independent random variables, which can reduce the complexity of compressive sensing process, and improve the calculation speed in the meantime.
The literature [14–18] proposed Toeplitz random measurement matrix and proved it. The literature [17–26] proposed quasiToeplitz matrix, semiHadamard matrix, and chaosToeplitz matrix and proved that they met the condition of the RIP. Based on the former researches, some scholars [27–35] proposed random spacing sparse Toeplitz matrix optimized by singular value decomposition (SVD) and apply it in wireless sensor networks.
The Gaussian random matrix requires MNindependent random elements, the general Toeplitz matrix only needs M + N − 1, and the random space sparse Toeplitz matrix needs only ⌈(N + M − 1)/Δ⌉_{Δ = 2, …, 16} independent random elements, so it is possible to further reduce complexity.
The innovation or contribution of this paper is as follows: based on the clustering structure of WSNs, a new data aggregating method based on sparse hybrid compressive sensing is proposed The aggregating process is divided into two parts: in the cluster, the sink node sets the corresponding seed vector based on the distribution of network and then sends it to each cluster head. Cluster head can generate corresponding own random spacing sparse matrix based on its received seed vector and collect data through compressive sensing technology. Among clusters, clusters forward measurement values to the sink node along the multihop routing tree which we built before. Performance analysis and comparison of the experimental results with the relative methods show that our method is effective and superior to other methods regardless of intracluster or intercluster on the total energy consumption of network and the lifetime of network.
2 Modeling based on hybrid compressive sensing for WSN
As shown in Formula (2), the predicted coefficient of measurement matrix is the sum of all the measured coefficients in the cluster. Therefore, in each round, the cluster head generates predicted coefficients; all cluster heads forward the received predicted coefficients to the sink node. When the sink node collected M rounds predicted value, it can recover the original data.
We define the compressive ratio as ρ = M/N, which means that the ratio is between the measurement value M in the process of compressive sensing and the length N of collected signal. It describes the compression efficiency of the entire network.
We define the relative reconstruction error as \( \varepsilon =\frac{{\left\Vert d\overset{\Lambda}{d}\right\Vert}_2^2}{{\left\Vert d\right\Vert}_2^2} \), i.e., the ratio between the absolute error and the true value, where d is the true distance value of a certain node i and its cluster head node and \( \overset{\Lambda}{d} \) is the measurement distance value of a certain node i and its cluster head node.
3 Data aggregating method based on compressive sensing in WSNs
Although compressive sensing technology can effectively reduce the energy consumption of each node in the network, it is directly related to the measurement value M in compressive sensing. When the value of M is large, the energy consumption of nodes remains high. To solve this problem, a novel hybrid compressive sensing data aggregating method is proposed, which mainly consists of four parts: network clustering, building the appropriate intercluster routing tree, compressive sensing data aggregating in clusters, and cluster head transmitting data to the sink node. How to construct the routing tree and evolve the process of compressive sensing in clusters is shown below.
3.1 Network model
 1)
N nodes randomly distribute in a circular perception area (the radius is L); the sink node is at the center of the sensing area (as shown in Fig. 1).
 2)
The sink node has enough data space and the ability of process.
 3)
The initial energy and the transmission rate of each sensor node are the same.
 4)
Nodes can know its own location information using the relative locating technology.
Lemma 1: Suppose that nodes in the wireless sensor network are distributed randomly, data aggregating in the cluster uses sparse matrices. If the cluster head is at the center of this cluster, then nodes consume least energy for each measurement value aggregating process.
where \( {E}_{Tx}^i\left(k,E\left({d}_i\right)\right) \) represents the energy consumption consumed by the i^{th} node when forwarding k bit data to its cluster head. E(d_{ i }) represents the distance expectations from the i^{th} node to its cluster head. As shown in the formula above, the average energy consumption is decided by \( E\left({d}_i^2\right) \). Suppose that the cluster is square and its side length is b and the cluster head’s coordinate (x_{0}, y_{0}). We can use f(x, y) to represent the probability density function of the distance between child nodes to the cluster head:
is true if and only if x_{0} = y_{0} = 0, i.e., the cluster head node is at the center area of the cluster.
Assuming that the network is divided into N_{ c } nonoverlapping clusters, that means N_{ c } nodes are selected as the cluster heads; the other nodes connect to the cluster head near to them.
We also assume that the node can adjust their own energy levels based on real transmission distance. Thus, the energy consumption from node n_{ i } to node n_{ j } is \( {P}_{ij}={d}_{ij}^{\alpha } \). The parameter α depends on the characteristics of the channel, which usually take between 2 and 4 as mentioned by References [13, 14]. Here, we choose α = 2, which is realistic for a typical WSN deployment [13–16]. Eventually, we use the normalized reconstruction error as the CS signal reconstruction error.
3.2 Establishment of intercluster routing tree
Hops are forwarded from current cluster head to other cluster head (NoH), i.e., the node determines the value based on its own communication radius and the distribution of cluster heads in the network.
Lemma 2: Suppose that cluster heads forward measurement values along the intercluster multihop shortest routing tree, so the energy consumption of intercluster will reach to the minimum value.
where d_{ i } represents the transmission distance of the i^{th} data packet. The formula above shows that if h and k are constant, the final result is decided by \( \sum \limits_{i=1}^h{d}_i^2 \).
We propose an iterative algorithm to build distributed intercluster routing. Assuming that all cluster heads have the same transmission radius (R). Within the communication radius, cluster heads can communicate with each other. All cluster heads broadcast the hops from themselves to the sink node to their neighbors. The NoH of cluster head which contains the sink node in their communication radius is set as 1 at the first time of iterating. At the next iteration, these cluster heads broadcast their NoH to their neighbors and set the NoH of those cluster head nodes without NoH to be 2. After a series of iterations, it keeps choosing routing path until no cluster head is left. The algorithm can be abbreviated as the following steps:
3.3 Intracluster data aggregating based on compressive sensing

Step 1: The sink node forwards the seed vector U(u_{ i }),{i = 1,2,…, N} with sparse space △ to every cluster head. Each cluster head determinating its position in the seed vector depends on its position on the backbone tree.

Step 2: Start from its position in the seed vector, the i^{th}cluster head node traverses forward N_{ i } values depends on the number of its intracluster nodes N_{ i }. Then, the cluster head gets its own new sparse seed vector and eventually generates its corresponding submatrix M_{ i } × N_{ i }.

Step 3: NonCH (cluster head, CH) nodes forward their nodes to CH; CHs calculate the received data as M_{ i } measurement values by using the formula y_{ i } = φ_{ i }x_{ i }.

Step 4: CHs forward measurement values to the sink node along the generated forwarding path.

Step 5: The sink node generates the whole measurement matrix based on the whole seed vector U(u_{ i }),{i = 1,2,…, N} and recovers the original data depends on received data y = [y_{1},y_{2},…,y_{ Nc }] by using CS reconstruction algorithm.
4 Analysis of energy consumption in WSN
 1)
Analysis of P_{ intracluster }
 2)
Analysis of P_{toBS}
 3)
Analysis of communication radius of cluster head
Figure 2 shows the comparison of the total hop change in the network when changing the communication range of the cluster head. Figure 3 shows the comparison of the total energy consumption of the network when changing the communication range of the cluster head. The figures above (a) uses Kmeans before data aggregating and (b) uses LEACH before data aggregating.
5 Description of the algorithm
Theorem 1 Assuming that wireless sensor network is clustering uniformly, the intracluster collect data by using compressive sensing technology, sparse matrix is selected as the measurement matrix, cluster head node is at the center of cluster, and intercluster forwards data along the shortest multihop routing tree. Then every time in the data aggregating, the total energy consumption of network is minimum.
where \( {m}_i^{\hbox{'}} \) represents the average number of nodes within i^{th} cluster the first time to participate in a single measurement. In the case of uniform clustering, \( {m}_i^{\hbox{'}}={m}_j^{\hbox{'}} \) and \( {d}_i^2={d}_j^2\left(i,j=1,2,\dots, h,i\ne j\right) \) then \( \sum \limits_{i=1}^h{m}_i^{\hbox{'}} \) and \( \sum \limits_{i=1}^h{d}_i^2 \) reach the minimum. So, \( {\overline{E}}_{\mathrm{total}}(h) \) reaches the minimum.
 1)
The network is clustered by using conventional clustering methods, such as LEACH and Kmeans.
 2)
The aforementioned method is used to construct the intercluster multihop shortest routing tree between cluster heads and the sink node. Each cluster head can get its own NoH. As seen in Formula (13), if M and N_{ c } are certain, the energy consumption of intercluster is only associated with NoH.
 3)
The sink node generates a corresponding sparse seed vector U(u_{ i }),{i = 1,2,…, N} according to the number of nodes in the network and send it to each cluster head.
 4)
Each cluster head (assuming that i^{th} cluster head) using the received seed vector generates its measurement matrix M_{ i } × N_{ i } according to its location and the number of nodes in it.
 5)
In the cluster, data is collected by using compressive sensing technology, then we can get M measurement values of the corresponding cluster head.
 6)
Cluster heads forward M measurement values to the sink node along the intercluster multihop shortest routing tree. Based on Theorem 1, the total energy consumption of network during the data acquisition is minimum, so as to achieve the best performance; otherwise, we use machine learning approach to reconstruct signal and then ensure that the total energy consumption is minimum. Detailed machine learning approach can be found in our relative research works [7–11], because of the length limit of the paper, we ignore the detailed description.
 7)
Since the measurement matrix used in each cluster is generated by the partial seed sparse vector U(u_{ i }),{i = 1,2,…, N}, so the sink node may also generate a total block matrix as the recovery matrix. The sink node recovers the original data by using corresponding reconstruction algorithm.
Because the random space sparse matrix can be dynamically generated by a series of seed vectors, the measurement matrix required for the whole network can be determined by the sink node. On one hand, compared with the Gaussian random matrix, it reduces the number of independent variables; on the other hand, it avoids the problem that nodes cannot save the dynamic measurement matrix while routing path changes in the process of conventional hybrid compressive sensing.
6 Results and discussions
This section provides some simulations and evaluations of this proposed data aggregating method.
6.1 Performance of data aggregating based on random space sparse compressive sensing
We always assess the performance of methods by using the amount of data packet transmission collected by nodes in the network; the space here is △ = 2. We compare six schemes: (a) Kmeans clustering scheme based on random space sparse measurement matrix, (b) LEACH clustering scheme based on random space sparse measurement matrix, (c) Kmeans clustering scheme based on Gaussian measurement matrix, (d) LEACH clustering scheme based on Gaussian measurement matrix, (e) Kmeans clustering scheme without compressive sensing, and (f) LEACH clustering scheme without compressive sensing. The number of nodes is increased from 500 to 1500, the transmission radius nodes is 10, and the compressive ratio is ρ = M/N.
6.2 Simulation and analysis of energy consumption in network
We also deploy 2000 nodes, and L is 100. Firstly, the network is clustered by Kmeans or LEACH, then we get N_{ c } clusters. We use our CS data aggregating method and calculate the energy consumption of the entire network. The sink node is set at the center of sensing field. Given the number of measurements M = 500, in order to meet the target error 0.1, we change the number of cluster head of the network by changing the transmission radius of nodes. We use the transmission radius R = [50, 30, 25, 22, 18, 14, 11] to represent the number of the cluster head N_{ c } = [10, 50, 100, 200, 300, 400, 500].
With the increase of the number of cluster heads, we represent (a) as the program using Kmeans and random space sparse matrix with △ = 2, (b) as the program using LEACH and random space sparse matrix with △ = 2, (c) as the program using Kmeans and Gaussian random matrix, and (d) as the program using LEACH and Gaussian random matrix.
In addition, with the increase of the number of clusters, we represent (a) as the program using Kmeans and random space sparse matrix with △ = 4, (b) as the program using LEACH and random space sparse matrix with △ = 4, (c) as the program using Kmeans and Gaussian random matrix, and (d) as the program using LEACH and Gaussian random matrix.
The algorithm can be described as intracluster method based on existing methods and the intercluster aggregation based on minimum consumption. The common problem in clustering networks which is the energy balancing during the head selection is well considered by the machine learning process.
The WSNs will inevitably use clustering when the node number is large. It is not a fair comparison between the cluster and noncluster structure in largescale networks, so we adopt the overhead of normalized network transmission based on the relative weight.
From Fig. 14, we can see that optimized compressive sensing data collection program reduces the overhead of normalized network transmission than the unoptimized program.
7 Conclusions
A kind of effective data aggregating method based on compressive sensing in WSN is proposed. The method can effectively reduce the energy consumption of the network. The sink node forwards sparse seed to cluster heads. Within a cluster, the cluster head generates its required measurement matrix according to the received sparse seed and then produces the corresponding measurement values by using random space sparse compressive sensing. Cluster heads forward measurement values to the sink node along the intercluster multihop routing tree from one cluster to another. The sink node reconstructs the original signal by using the corresponding compressive sensing reconstruction algorithm. We analyze the energy consumption of the algorithm in the network, the relationship between the size of cluster head and the energy consumption of intercluster, and the relationship between the size of cluster head and the energy consumption of network. The experimental results show that this method can effectively reduce the energy consumption of the network.
Declarations
Acknowledgements
This research work is supported by the National Natural Science Foundation of China (Grant No. 61571328), Tianjin Key Natural Science Foundation (No.13JCZDJC34600), CSC Foundation (No. 201308120010), Major projects of science and technology in Tianjin (No.15ZXDSGX 00050), training plan of Tianjin University Innovation Team (No.TD125016), major projects of science and technology for their services in Tianjin (No.16ZXFWGX00010, No.17YFZCGX00360), the Key Subject Foundation of Tianjin (15JCYBJC46500), and training plan of Tianjin 131 Innovation Talent Team (No.TD201523).
Funding
The work is partially supported by the following funding: training plan of Tianjin University Innovation Team (No.135025).
Availability of data and materials
The data will not be shared due to confidentiality matters.
Authors’ contributions
DgZ designed the algorithm. TZ wrote this paper. JZ did the experimental tests. YD optimized the algorithm and experiments. XdZ checked the whole paper and figures. All authors read and approved the final manuscript.
Authors’ information
Degan Zhang, Ph.D., graduated from the Northeastern University, China. Now, he is a visiting professor of School of Electronic and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia; professor of Tianjin Key Lab of Intelligent Computing and Novel software Technology, Key Lab of Computer Vision and System, Ministry of Education, Tianjin University of Technology, Tianjin, 300384, China. His research interest includes image processing, service computing, etc.
Ting Zhang, Ph.D., is a member (M) of IEEE in 2012. Now, she is a researcher at Tianjin University of Technology, Tianjin, 300384, China. Her research interest includes WSN, mobile computing, etc.
Jie Zhang (Beijing No.20 High School, Xiaoyingxilu, Haidian District, Beijing 100085, China). His research interest includes image processing, CRN, WSN, and IOT.
Yue Dong, Ph.D., is a researcher at Tianjin University of Technology, Tianjin, 300384, China. Her research interest includes WSN, etc.
Xiaodan Zhang, Ph.D., is a member (M) of IEEE in 2012. Now, she is a researcher at Institute of Scientific and Technical Information of China, Beijing, 100038, China. Her research interest includes WSN, mobile computing, etc.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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