## 5 Performance Comparison  
To extensively validate our HRM_CS, we focus on recovery error, energy consumption, and lifetime, compared with other classic WSNs [3,13,25]. As for the limited computational power and communication bandwidth in WSNs, the lifetime of WSNs is the most important metric. * Energy Consumption  
As the analysis in the previous section shows, compared with the traditional clustering method, the energy consumption of the HRM_CS is substantially reduced by introducing compressive sampling. To analyze the energy consumption, we focus on the sending and receiving phase although there are multiple other phases, such as the sensor computation during the data acquisition. We adopt the energy model given as Equations (6) and (7) in Section 3. For further details refer to [4,26]. First, let us consider the effects of preprocessing on \(LI\)-minimization recovery [27]. Figure 7a shows the referenced values after subtracting the minimum value from the original signal. Figure 7b shows the recovery effects with preprocessing and applying a DCT, while Figure 7c shows the recovery effects without preprocessing but applying a DCT. Figure 7d depicts the recovery effects without preprocessing and using PCA. From this experiment, we find that the following results: (1) the recovery error in Figure 7b is smaller than those in Figure 7c,d; (2) the recovery errors in Figure 7c,d are of the same magnitude, however, from the viewpoint of varying trends, the method using PCA in Figure 7d is inferior to that using DCT in Figure 7c since the DCT transform can characterize the original signal using more coefficients with low and middle frequency. Next, we consider the performance of variants of the B HRM_CS model. Figure 8 shows the experimental results of the HRM_CS1, _i.e._, jointly using CS as the recovery algorithm and LEACH as the hierarchical routing method. Figure 9 depicts the experimental results with high recovery accuracy for the HRM_CS2, _i.e._, combining \(LI\)-minimization as the recovery algorithm and LEACH as the hierarchical routing method. Under the HRM_CS2 scenario, if there are too few 1's when creating the measurement matrix, _i.e._, there are too few nodes to participate in the measurement, it will lead to failure during the recovery operation since the coherence between the measurement and sparsity matrices increases, as shown in Figure 10. Based on the above experiments, as for the stability of information recovery, we find that the applying CS naively like \(LI\)-minimization may be not proper under the real scenario of WSNs and HRM_CS1 is superior to HRM_CS2. Figure 7: Effects of preprocessing for recovery. Figure 8: Recovery based on HRM_CS1. Figure 10: Recovery failure based on HRM_CS2. Figure 9: Recovery based on HRM_CS2. In Figure 11, we demonstrate the recovery accuracy by comparing the HRM_CS1 with the GM_CS[13] based on the same recovery method, BCS. Figure 11 depicts the experimental results for the original and recovered signals, for which the error is less than 1 \({}^{\circ}\)C. Furthermore, the mean error of our HRM_CS1 0.0101 is slightly greater than the mean error of the GM_CS 0.0084 while HRM_CS1 has much longer lifetime than that of the GM_CS, as shown in Figure 13.  
As for the number of measurements \(M\), _i.e._, the number of clusters in HRM_CS model, it is a vital parameter and has an influence on the lifetime of WSNs. Based on the basic compressive sensing theory [5], we know that \(M\geq 4K\), \(K\) is the number of significant components in original signal \(X\) while \(K\) varies with the different threshold value during the process of signal, which can be seen in Table 2. As a result, it can be seen from the Figure 12 that the lifetime of WSNs with different number of clusters also take the corresponding changes and the maximal lifetime is obtained at \(M=40\). In the last graph, Figure 13, we show the impact of different data acquisition methods on the lifetime of WSNs.