## 5 Basic Concepts for the SMEP Method
### I-bit Position and Generation  
Figure 4 shows the compression process in the aspect of the merging of two intervals. Compression is conducted by selecting just one from sensor data in a compression interval that corresponds to the merging of two sequent compression intervals of the base sequence. For example, in Figure 4, \(S_{1,3}\), a sensor data, is selected as the one from a compression interval, \(I_{0,2}=[4,6]_{0}\). Here, \(I_{c,i}\) means the \(i\)-th position compression interval in the 2-size compression interval covering on a sensor data sequence derived though the \(c\)-th round compression. Additionally, \([p,q]_{k}\) represents an compression interval, \([p,q]\) on a sequence derived through the \(c\)-th round compression. Shortly, \((I_{0,2},S_{1,3})\) in the figure means the sensor data \(S_{1,3}\) is selected in the compression interval \(I_{0,2}\). For further description, we define the term _cover_:  
Figure 3: The first round compression process. Figure 2: Sensor data selection in the 1 st, 2 nd and 3 rd round compressions. **Definition 9**. : _Let \(S_{i}\) and \(I\) be a sensor data and a compression interval with respect to a sensor data sequence, respectively. \(S_{i}\) covers \(I\) if and only if \(S_{i}\) is selected in \(I\)._  
In Figure 4, \(S_{2,2}\) covers \(I_{1,1}\) and it is selected one of \(S_{1,3}\) and \(S_{1,4}\) to cover \(I_{0,2}\) and \(I_{0,3}\), respectively. Note that \(S_{2,2}\) covers \(I_{0,2}\bullet I_{0,3}\) in this figure. We define the term _cover_ on the view of compression interval:  
**Definition 10**. : _Given two compression intervals \(I_{c\_2j}\) and \(I_{c\_2j+1}\) and a selected sensor data \(S_{c,k}\) to cover \(I_{c-1,k}\) for \(c\geq 2\), \(I_{c-1,k}\) covers \(I_{c-2j}\bullet I_{c-2,j+1}\) if and only if \(S_{c,k}\) covers \(I_{c-2j}\bullet I_{c-2,j+1}\)._  
According to the definition, \(I_{1,1}\) covers \(I_{0,2}\bullet I_{0,3}\) in Figure 4. Moreover, \(I_{2,0}\) covers \(I_{1,0}\bullet I_{1,1}\) and \(I_{1,0}\) and \(I_{1,1}\) also cover \(I_{0,0}\bullet I_{0,1}\) and \(I_{0,2}\bullet I_{0,3}\). Since \(S_{3,1}\) covers \(I_{2,0}\) and \(S_{3,1}\) covers \(I_{0,0}\bullet I_{0,1}\bullet I_{0,2}\bullet I_{0,3}\)\(I_{2,0}\), \(I_{2,0}\) also covers \(I_{0,0}\bullet I_{0,1}\bullet I_{0,2}\bullet I_{0,3}\)\(I_{2,0}\). For the generalization of this property, we introduce the below Lemma 1, Lemma 2 and Theorem 5. For their proofs refer to Appendix H, Appendix I, and Appendix D, respectively. **Lemma 1**. : _Let \(S\) and \(S^{c}=S_{c,0}\), \(S_{c,1}\),..., \(S_{c\_2}{}^{m}\) for integer \(m\geq 0\) be a ground sequence and a sensor data sequence derived through the \(c\)-th round compression on \(S\), respectively. Let \(C_{s}{}^{c}=(I_{c,0}\), \(I_{c,1}\),..., \(I_{c\_2}{}^{m-1}-1)\) be a 2-size compression interval covering. Then, in the SMEP method, if for \(k\geq 0\), \(I_{c,k}\) covers \(I_{c-1,2k}\bullet I_{c-1,2k+1}\)._  
**Lemma 2**. : _Let \(S\) and \(S^{c}=S_{c,0}\), \(S_{c,1}\),..., \(S_{c\_2}{}^{m}\) for integer \(m\geq 0\) be a ground sequence and a sensor data sequence derived through the \(c\)-th round compression on \(S\), respectively. Let \(C_{s}{}^{c}=(I_{c,0}\), \(I_{c,1}\),..., \(I_{c\_2}{}^{m-1}-1)\) be a 2-size compression interval covering on \(S^{c}\). Then, in the SMEP method, if for \(i\geq 0\) and \(j\geq 1\)\(S_{c,i}\) covers \(I_{c-1,j}\) then \(j=i-1\) and \(I_{c-1,j}\) covers \(I_{c-2,2i(i-1)}\bullet I_{c-2,2i-1}\)._  
**Theorem 5**. : _Let \(S\) and \(S^{c}=S_{c,0}\), \(S_{c,1}\),..., \(S_{c\_2}{}^{m}\) for integer \(m\geq 0\) be a ground sequence and a sensor data sequence derived through the \(c\)-th round compression, respectively, based on \(S\).