## 3 Analysis and Optimization of HRM_CS Model
### Optimization of HRM_CS Model  
The HRM_CS objective is to provide a data acquisition method with longer lifetime and higher recovery accuracy under the real WSN environment. In this section, three performance metrics, namely, recovery accuracy (Error), communication cost and compression rate are analyzed. Based on this analysis, the optimized model from the two variant models of HRM_CS, _i.e._, HRM_CS1 with BCS and HRM_CS2 with _L1_-minimization can be obtained. Detailed experimental results are shown in Table 1, where the raw AD values denote the values from the card of Analog signals conversion Digital signals in a real WSN scenario GreenOrbs [18].  
First, we investigate recovery accuracy (Error). The max Error is less than 1%, which meets the system demands. Although the HRM_CS1 is a little inferior to the HRM_CS2 with respect to recovery accuracy, HRM_CS1 is superior to the HRM_CS2 with respect to much more stability, _i.e._, the recovery method based on BCS has greater success than \(LI\)-minimization (hereafter called \(L1\)). Moreover, the number \(M\) of head nodes does not have a large effect on the error and variance of recovery in a BCS scenario. This is because BCS estimates the most probable value by maximizing \(a\)_posteriori_ without depending on the particular WSN topology considered.  
Second, we analyze communication cost. Here we compute the total number of packets in the HRM_CS. The main steps are as follows: (1) \(N-M\) nodes in the cluster send their own sensor readings to \(M\) cluster-head nodes with an associated cost \(O(N-M)\); (2) Next, superposition of the signal is carried out at the cluster-head node. In addition, the \(M\) cluster-head nodes transmit their own sensor readings to the sink along a routing path that minimizes the number of transmissions. On this path the packet is not processed but simply forwarded with the longest path \(O(N^{\
icefrac{{}}{{\alpha}}})\); (3) The total cost of delivering packets to the sink from \(M\) cluster-head nodes is \(O(MN^{\
icefrac{{}}{{\alpha}}})\).  
Last, we consider compression rate. From a sink point of view, compression rate is defined as \((1-\frac{M}{N})\times 100\%\). It is related to the number \(M\) of head nodes and increases as \(M\) decreases, where \(N\) is the total number of sensor nodes.  
From the above analysis, it can easily been seen that the WSN based on the HRM_CS has longer lifetime, compared with the classic approaches given by Equations (3) and (4), where the data are not well compressed. In particular, it can also been seen that the HRM_CS1 with \(M=40\) is our optimal choice since the higher compression rate and stability can be gotten at a little cost of recovery error, compared with the HRM_CS2.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Recovery** & \multirow{2}{*}{**Threshold**} & **Number of Cluster** & **Coherence** & **Success Recovery** & **Error** & \multirow{2}{*}{**Variance**} \\
**Method** & & **Heads \(M\)** & \(\mu\) & **Rate (\%)** & & **(\%)** & \\ \hline \(L1\) & 50 & 40 & 1.950 & 20 & 0.262 & 298.748 \\ BCS & 50 & 40 & 1.950 & 100 & 0.327 & 466.902 \\ \(L1\) & 40 & 56 & 1.495 & 20 & 0.247 & 265.394 \\ BCS & 40 & 56 & 1.495 & 100 & 0.329 & 472.804 \\ \(L1\) & 35 & 68 & 1.085 & 25 & 0.224 & 218.526 \\ BCS & 35 & 68 & 1.085 & 100 & 0.326 & 463.506 \\ \(L1\) & 28 & 84 & 0.658 & 33 & 0.250 & 272.964 \\ BCS & 28 & 84 & 0.658 & 100 & 0.326 & 463.182 \\ \(L1\) & 25 & 100 & 0.556 & 60 & 0.209 & 191.134 \\ BCS & 25 & 100 & 0.556 & 100 & 0.326 & 462.999 \\ \(L1\) & 20 & 140 & 0.388 & 100 & 0.222 & 55.810 \\ BCS & 20 & 140 & 0.388 & 100 & 0.326 & 462.787 \\ \hline \end{tabular} Error: recovery error; Variance: variance of raw AD values between original signal and recovery signal; \(L1\): the recovery method based on \(LI\)-minimization; BCS: the recovery method based on BCS.  
\end{table}
Table 1: Performance comparison for HRM_CS.