In this chapter, we study the effectiveness of our proposed clustered compressive sensing and evaluate the performance by calculating the reconstruction error between the original signal and the recovered signal. We consider a wireless sensor network with N sensor nodes is employed in a square region of side D units, which is evenly divided into N small square grids. All of the N sensor nodes are uniformly distributed in this region so that each grids has only exactly one sensor node. Each of the N sensor nodes could only communicate with all other sensor nodes in a circular range of radius R units. Since RD, each of the N sensor nodes transmits its data packets through multi-hop connections. All of the data packets will be concentrated to the processing server eventually. The wireless base station is placed in the center of this region and connects with the server through cable connection. The server is in charge of data storing and processing including data clustering and principal component analysis.
The input signal is a simulated temperature distribution in a square region which is separated into four regions. The simulated temperature distribution in each sub-regions is different to each other. In each sub-sub-regions, the simulated temperature is a correlated. The simulated temperature generation procedure is executed as the following steps:
- 22 -
1) we start from a
2 2
DD matrix Hg, g 1, 2,3, 4 with
2
4
D entries, where entries
hi j, |i j | 5 15,
are generated from continuous uniformdistribution;
2)
T
G is obtained fromH
G by inversing discrete cosine transformation;3) sampling the data correspond to the positions of all N sensor nodes, thus we simulated temperature signal
x
t;- 23 -
4) finally, in order to verify the robustness of our proposed methodology, we add an i.i.d.
random Gaussian noise w
0,2
into all entries of signalx
t.The simulated temperature model is shown in Figure 5.
To implement our proposed clustered compressive sensing methodology, we alternate two phases as below. One is training phase and the other one is monitoring phase. In training phase, first of all, T samples of each sensor nodes are collected into the server. Next, all
Figure 5 the temperature distributions model
- 24 -
of the N sensor nodes are divided into G clusters according to their positions because we consider that the readings of those adjacent sensor nodes are similar substantially. We allocate the collected data into clusters according to their cluster indices. Then, each cluster apply the principal component analysis to obtain its sample mean and sample covariance.
And finally, merge those sample mean into a longer vector and form those sample covariance into a diagonal matrix.
Subsequently, in monitoring phase, we randomly select M active sensor nodes from N sensor nodes. The data of the other NM sensor nodes is reconstructed from the subset of input signal by using the sample mean and sample covariance which is calculated in training phase.
To evaluate the performance of proposed methodology, we consider an indicator such as the reconstruction error. simulations have been performed under the following platform: MATLAB R2011b on a computer with Intel Core i5 661 3.33GHz CPU, 8GB RAM, and Windows 7.
- 25 -
Scenario 1
In Scenario 1, we consider that there is only one distribution of the input signal and the number of sensor nodes N100 in the wireless sensor network. The simulated temperature model is shown in Figure 6. Figure 7 shows that the relationship between reconstruction error and the number of active nodes. Since only one distribution in this region, choose one cluster is the best idea. When choose more than one cluster, the sparsity level is getting higher. Thus, for higher the sparsity level, we must need more measurements of active sensor nodes. Therefore, supposed the number of active
Figure 6 the simulated temperature model for one distribution
- 26 -
sensor nodes is the same, choose four clusters has the highest reconstruction error, on the other hand, choose one cluster has the lower reconstruction error. As the result, it is not necessary that allocating those sensor into clusters when the environment is follow simple distributions.
Figure 7 reconstruction error versus the number of active nodes (Scenario 1)
- 27 -
Scenario 2
In scenario 2, we consider that there are four distributions of the input signal. The input signal is shown in Figure 5, and the number of sensor nodes N100. In Figure 9, we can clearly understand that for using four clusters, each of the distribution might have the best fit. Obviously, the reconstruction error for four clusters is the smallest.
Figure 8 the sparsity level versus the number of clusters (Scenario 1)
- 28 -
Thus, the sparsity must be the smallest as shown in Figure 10 and when the number of cluster decrease, the higher the sparsity level need.
Figure 9 Reconstruction error versus the number of active nodes (Scenario2)
- 29 -
However, according to Equation (2.5) and the restricted isometry property, we must need at least 3K measurements, thus the signal could be recovered perfectly. In
44
K case, the necessarily measurement M 132, therefore, using N100 is not enough to completely reconstruct the original signal. Furthermore, we employ
400
N sensor nodes into this region. The result is shown in Figure 11 and the sparsity level is shown in Figure 12. As we can see that the sparsity level become
Figure 10 The sparsity level versus the number of clusters (Scenario 2)
- 30 -
160
K , according to the restricted isometry property, N 400 is still not high enough to perfectly recover the original signal. but the total reconstruction error is lower than N 100, as the result, supposed that we could endure the temperature has the mean square error is equal to 0.1, we could choose four clusters for only need
Figure 11 Reconstruction error versus the number of active nodes (N 400)
- 31 -
about 350 active sensor nodes, on the contract side, if we choose one cluster, we need near 370 active nodes.
Figure 12 the sparsity level versus the number of clusters (N 400)
- 32 -
Finally, for a multi-environment region, it is hard to find the sparsest representation of the original by using principal component analysis, but in a monotonous environment, it is quiet easier than multi-environment. Although, we could not perfectly recover the original signal, we still could regard the approximation signal as a reference when the reconstruction error is satisfied the minimum precision requirement.
- 33 -