## 5 Basic Concepts for the SMEP Method
### Absolute Position  
As soon as a sensor data is selected during a compression, the SMEP method generates an I-bit position corresponding to the sensor data and updates _I-bit position sequence_ defined in the below, appending its I-bit position to it:  
**Definition 12**. : _Let \(S^{0}\) and \(S^{c}=S_{c,0}\), \(S_{c,1}\),..., \(S_{c,n}\) be a ground sequence and a sensor data sequence compressed from \(S^{0}\) through the c-th round compression. 1-bit position sequence of \(S^{c}\) is defined as a bit string sequence that consists of concatenations of I-bit positions of all sensor data except \(S_{c,0}\) in \(S^{c}\). That is,_  
_I-bit position sequence of \(S^{c}=I-bit\) position(\(S_{c,1}\)) \(\Theta\)I-bit position(\(S_{c,2}\)) \(\Theta\)... \(\Theta\) I-bit position(\(S_{c,n}\))._  
Moreover, the position of the first bit of the I-bit position sequence is defined as 0. From the above definition, note that the I-bit position corresponding to the first sensor data is excluded from I-bit position sequence. In fact, we can remove it because the position of the first sensor data is always fixed as the first for every compression. In Figure 5, I-bit position sequence of \(S^{1}\) is 1100 (= 1 \(\Theta\) 1 \(\Theta\) 0 \(\Theta\) 0) generated by concatenating I-bit positions of \(S_{1,1}\), \(S_{12}\), \(S_{1,3}\), and \(S_{1,4}\) except \(S_{1,0}\). I-bit position sequences of \(S^{1}\), and \(S^{2}\) are 0110 (=01\(\Theta\)10), and 110 by the same way, respectively. Given an I-bit position sequence, \(B_{p}\), of \(S^{c}\), for any sensor data, \(S_{c,k}\), we can get easily the _I-bit position_ of \(S_{c,k}\) by taking a substring from the \(c(k-1)\) th position bit to \(ck-1\) th position bit in \(B_{p}\). So, let us define a function, _BitSubstring_, now:  
**Definition 13**. : _Given an I-bit position sequence, \(B_{p}\), of \(S^{c}\), for any sensor data, \(S_{c,k}\), BitSubstring is a function to take a bit substring from the \(c(k-1)\) th bit to \(ck-1\) th bit in \(B_{p}\), this is, BitSubstring(\(B_{p}\), \(c\), \(k\)) = a bit substring from the \(c(k-1)\) th bit to \(ck-1\) th bit in \(B_{p}\)._  
**Lemma 3**. : _Given an I-bit position sequence, \(B_{p}\), of \(S^{c}\), for any sensor data, \(S_{c,k}\),_  
_I-bit position(\(S_{c,k}\)) = BitSubstring(\(B_{p}\), \(c\), \(k\)) (For the proof refer to Appendix 1). On the other hand, all data in \(S^{c}\) originate from \(S^{\circ}\). In fact, they are sensor data that compete to survive through selections in c time compressions. In order to restore the lost data in competences, for each sensor data in the compressed data sequence its position in the ground sequence, which we call the position the absolute position defined the below, has to be known:_  
**Definition 14**. : _Let \(S_{c,k}\) be a sensor data that originates from \(S_{0,i}\). Then, the absolute position of \(S_{c,k}\) is defined as the position \(i\) of \(S_{0,i}\) in \(S^{\circ}\)._  
The absolute position of a sensor data after c compressions can be known with its I-bit position by the following Lemma (For the proof refer to Appendix 1):  
Figure 5: I-bit position generation process of sensor data in compression interval. **Lemma 4**. : _Let \(S^{0}\) and \(S^{c}=S_{c,0}\), \(S_{c,1}\),..., \(S_{c,n}\) be a ground sequence and a sensor data sequence derived from \(S^{\circ}\) through the \(c\)-th round compression, respectively. Given \(B_{p}\), an I-bit position sequence, for any \(S_{c,k}\) originated from \(S_{0,i}\) in \(S_{0}\), the absolute position \(i\) is computed as follows:_  
\(i=0\)_, if_ \(k=0\)__  
\(i=(k-1)2^{c}+(I\)_-bit position\((S_{c,k}))_{10}+1\)_, if_ \(k\geq 1\)__  
By the two above lemmas, the following theorem is trivially true:  
**Theorem 6**.