## 3 Data Processing
### Ship Classification
#### 3.4.2 Fuzzy logic processing  
Once all the previous parameters are estimated, fuzzy logic is used to apply the decision rule. In simple words, fuzzy logic is the theory for which classical binary levels are extended to continuum grades of intermediate memberships [8]. So, the conventional crisp of full-membership and non-membership levels (0 and 1) is overcome to handle imprecise information. The number of levels is known as grades, which are defined by membership functions. Such functions provide the percentage of each input to be a member of each grade. The number of grades, the shape of membership functions and the percentage of overlap are flexible and updated according to the application to develop. They conform what is known as fuzzy \"set\". More than one parameter can be evaluated by fuzzy rules, which outputs can be diversely combined via \"if-then\" statements. In the case of SIDECAR, the normalized RCS of bow (B\({}_{\text{RCS}}\)), middle (M\({}_{\text{RCS}}\)) and stern (S\({}_{\text{RCS}}\)) sections are the inputs to the fuzzy rules. For each one, three grades are considered (low, medium and high) with the membership functions shown in Figure 5. The evaluation of the input parameters with these membership functions provides the percentage of each input to pertain to each grade. The combination of these values according to a set of category rules leads to the percentage of a measured ship signature to be member of the different reference categories. To do that, each reference category has to be associated with a two-grade set with equal membership functions (see Figure 6). The grades of this set are _false_ (no member of the category) and _true_ (member of the category), and they are evaluated with the output of the category rules. Such rules provide decision for the \"false\" and \"true\" grades and are fixed from the relations shown in Table 1.  
The values resulting from the evaluation of the \"false\" and \"true\" rules define the area covered by each membership function of the true/false set. Such area is measured by the Centroid calculation method and is the similarity value between the measured ship signature and the processed category. This value, jointly with the ratio of target dimensions, gives the classification confidence. For a better understanding of how fuzzy logic works, an example for the evaluation of the Oil tanker category is presented (see Figure 6). Let us define the following \"true\" and \"false\" rules where \(\sim\) means the inverse operation and (S\({}_{\text{RCS}}\) IS \(\sim\)high) = (S\({}_{\text{RCS}}\) IS medium) OR (S\({}_{\text{RCS}}\) IS low)  
**Rule 1:** IF (S\({}_{\text{RCS}}\) IS high) AND (M\({}_{\text{RCS}}\) IS low) AND (B\({}_{\text{RCS}}\) IS low) THEN (True)  
**Rule 2:** IF (S\({}_{\text{RCS}}\) IS \(\sim\)high) OR (M\({}_{\text{RCS}}\) IS \(\sim\)low) OR (B\({}_{\text{RCS}}\) IS \(\sim\)low) THEN (False)  
Consider then that the inputs to the fuzzy logic system are S\({}_{\text{RCS}}\) = 0.75, M\({}_{\text{RCS}}\) = 0.2, B\({}_{\text{RCS}}\) = 0.2. So, the percentage to pertain to each grade is (see plots in Figure 6):  
\(\blacksquare\) S\({}_{\text{RCS}}\) IS high = 0.9, S\({}_{\text{RCS}}\) IS medium = 0, S\({}_{\text{RCS}}\) IS low = 0. \(\blacksquare\) M\({}_{\text{RCS}}\) IS high = 0, M\({}_{\text{RCS}}\) IS medium = 0, M\({}_{\text{RCS}}\) IS low = 0.65. \(\blacksquare\) B\({}_{\text{RCS}}\) IS high = 0, B\({}_{\text{RCS}}\) IS medium = 0, B\({}_{\text{RCS}}\) IS low = 0.65.