## 5 Basic Concepts for the SMEP Method
### Selection by Compression Interval Pattern Rule  
For a compression interval, \(I_{c}=[i,I+2]_{c}=S_{c,i}\), \(S_{c,i+1}\), \(S_{c,i+2}\) with size 2, we can have one of nine cases depending on these sensor data values as follows:  
Each of the above cases shows its own pattern for \(S_{i}\), \(S_{i+1}\), \(S_{i+2}\) as shown in Figure 6.  
Figure 6: Nine patterns of sensor data in 2-size compression interval. Assume that \(S^{c}\) is a sensor data sequence in a data queue for compression and \(C_{S^{c}}\) = (\(I_{c,1}\), \(I_{c,2}\),..., \(I_{c,m}\)) is a 2-size compression interval covering with respect to \(S^{c}\). Then, we define the _selection by compression interval pattern_ as follows:  
**Definition 15**. : _Given a compression interval \(I_{c,i}\) = \(S_{c,i}\), \(S_{c,i+1}\), \(S_{c,i+2}\) in \(C_{S^{c}}\) with respect to \(Sc\), the selection by compression interval pattern with respect to \(I_{c,i}\) is defined as the selection of a sensor data among \(Sc\),\(i\), \(Sc\),\(i\)+1, \(Sc\),\(i\)+2 according to the following selection by pattern rule:_  
_Selection by Pattern Rule: If Pattern 1 or Pattern 2 or Pattern 3 is true among nine patterns, then \(S_{c,i+2}\) is selected, otherwise \(S_{c,i+1}\) is selected._  
Simply, we will call the _selection by compression interval pattern_ rule the _pattern selection_ rule or _selection by pattern_ rule. The _selection by pattern_ rule is derived not from noble theoretical results to minimize the errors between every lost sensor data and its corresponding original sensor data in the several compressions, but from empirical results through several experiments with several sets of real-field sensor data. Thus, the compression to use only the _pattern selection_ method tends occasionally to show worse performance than the _CQP_ method and others. For this reason, we have devises a method mixed up with the _pattern selection_ method and the _selection by minimum error_ method in Section 5.5. On the other hand, in two consecutive compression intervals of 2-size compression  
From now on, we will regard the selection error as the line interpolation error. In the definition, the absolute position for each of all sensor data is used for calculating line interpolation errors. We have a valid reason why we have to use the absolute positions of sensor data in calculating the line interpolation errors: Consider three or more consecutive sensor data to take one or more compressions. They seem likely to have the same distance unit 1 each other at glance. Generally, however, there can be big differences in their absolute positions. For example, consider in Figure 2 three sensor data, \(S_{2,0}\), \(S_{2,1}\) and \(S_{2,2}\), where they are consecutive each other with the same 1 unit distance in the 2 nd round compressed sensor data sequence. Their corresponding original sensor data, however, are \(S_{0,0}\), \(S_{0,2}\) and \(S_{0,7}\) and the distance units between them are different with 2 and 5 between \(S_{0,0}\) and \(S_{0,2}\) and between \(S_{0,2}\) and \(S_{0,7}\), respectively. Exceptionally, the SMEP uses \(S_{c,0}\) for the predetermined sensor data \(S_{c,i}\) in the definition so to select one of \(S_{c,1}\) and \(S_{c,2}\) for \(l_{c,0}\). The reason to do so is because there exists no interval for predetermined sensor data in its left-hand side but if any interval can exit the unique sensor data is the very \(S_{c,0}\) that can be the left boundary sharable sensor data of \(l_{c,0}\). The SMEP executes the _selection by minimum error_ rule for each \(I_{2k}\) and the _selection by pattern_ rule for \(I_{2k+1}\) in \(C_{S_{c}}\), for \(k\)=0, 1,...,\(|C_{S_{c}}|\). Therefore, in the definition, the real compression intervals for the SMEP method to apply the _selection by minimum error_ rule to the compression are \(I_{2k}\) for \(k\) = 0, 1,...,\(|C_{S_{c}}|\). Now, focusing on \(I_{2k}\) (\(k\) = 0, 1,...,\(|C_{S_{c}}|\)), we investigate some properties for minimizing the selection error. **Lemma 5**.