## 5 Basic Concepts for the SMEP Method
### SMEP Compression Process Overview  
The SMEP proceeds the compression with 2-size consecutive compression interval covering with respect to a sensor data sequence with the size \(2^{m}+1\) for in integer \(m\), \(m\geq 1\). Actually, the size \(2^{m}+1\) or \(2^{m}\) is the same as the length of a part of data queue used for saving and compressing the sensor data generated from a sensor in the \(SDSC\) mode and the sensor data sequence for the compression is a sequence of sensor data in the queue. Depending on the \(2^{m}+1\) size, we get the below theorem (For the proof refer to Appendix C):  
**Theorem 4**: _Let \(S\) and \(C_{s}\) be a sensor data sequence \(S\) and a 2-size compression interval covering of \(S\) and \(C_{s}=(I_{0}\), \(I_{1}\),..., \(I_{l})\), respectively. If the size of \(S\) is \(2^{m}+1\) for an integer \(m\geq 1\), then \(l=2^{m-1}-1\), that is, the number of compression intervals of \(C_{s}\) is \(2^{m-1}\)._  
Whenever a compression round is proceeded, one sensor data in each compression interval is selected and the others are discarded in it. At this time, the discarded sensor data becomes the lost sensor data. Figures 2 and 3 show the compression process on a sensor data sequence in a data queue with the size \(17=2^{4}+1\). As shown in Figure 2, in the \(c\)-th (\(c=1,2,3,\)... ) round compression, sensor data after the c-1 th round compression are selected as follows:  
At first, the first sensor data \(S_{c-1,0}\) is selected as \(S_{c,0}\), unconditionally, where \(S_{c,i}\) means the \(i\)-th sensor data of a sensor data sequence derived through the \(c\)-th round compression. Then, by selecting one sensor data per every compression interval, according to the order between compression intervals in the 2-size compression interval covering, the compression derives a new compressed sensor data sequence. One of both of the 2 nd and the 3 rd sensor data in each interval is selected so that the SMEP method should keep a selection rule, _one per compression interval_, for all compression intervals except the 1 st compression interval. In selection, the \(i\)-th sensor data selected in the previous compression round becomes the \(i\)-th sensor data in the new compressed sensor data sequence. After the compression, the size of the new sensor data sequence shrinks to the half the previous one, this is, \(2^{m-1}+1\), occupying data queue space with the same size, and the \(2^{m-1}+1\) size queue space remains empty. After that, as sensor data generated newly from a sensor are inserted into the remaining queue space, the space becomes full like Figures 2 and 3 so that new sensor data should be compressed with the same way until all sensor data in the full state data queue are derived through \(c\) time compressions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notations** & **Terms** & **Definitions** \\ \hline _BitSubstring_(\(B_{p},c,k\)) & _bit substring from the (\(k-1\)) th bit to \(\alpha\)-\(1\) th bit in \(B_{p}\) & Definition 13 \\ - & _absolute position of \(S_{c,k}\)_ & Definition 14 \\ - & _selection by compression interval pattern with respect to \(I_{c,i}\)_ & Definition 15 \\ - & _selection error of a sensor data_ & Definition 16 \\ \(E_{p}\) & _line interpolation errors of \(S_{p}\)_ & Definition 17 \\ - & _selection by minimum error rule_ & Definition 18 \\ _Zones[\(0..m-1\)]_ & _Zones with respect to the data queue DataQuet[\(0..2^{m}\)]_ & Definition 19 \\ - & _operations on Zond[\(0..m\)-1]_ & Definition 20 \\ _ZoneBpSeq[\(0..m-1\)]_ & _zone bit sequence array_ & Definition 21 \\ \(E_{M,S^{\prime}}\) & _average error rate (AER) of \(S^{\prime}\) with respect to \(S^{0}\) in the compression method \(M\)_ & Definition 22 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Cont.  
When all sensor data of a sensor data sequence in the full state data queue are derived through c time compressions, the \(c\)-th round compression has been completed and the new \(c\)+1 th round compression begins. Figure 3 shows only the first round compression process.