Classification Methods for Remotely Sensed Data, Second Edition

Page 167

where μ(s1) and μ(s2) are membership grades for measures s1 and s2, respectively. The use of minimisation as a weighting method is given by:

The calculation of the strength of rules 2 and 3 is carried out in a similar manner.

In Figure 4.10, the condition given by rule 1 is matched (given membership grade μ) to the degree of 0.2 (for input s1) and 0.6 (for input s2), respectively. After Equation (4.31) is applied (i.e. min{0.2, 0.6}=0.2), the degree of match to the condition in rule 1 is eventually determined as 0.2 (i.e. w1=0.2). Here, a minimisation method (shown in Equation (4.32)) is adopted for calculating the rule strength. Thus, the strength of rule 1 should be truncated (minimised) to 0.2 as shown in the shaded area. Similarly, the condition of rule 2 is partially satisfied to the degree of 0.4, thus rule 2 only contributes 0.4 of the strength (also shown in the shaded area of Rule 2 in Figure 4.10). Since the condition defined by rule 3 is not matched, rule 3 is not triggered.

Once the strength of each rule is determined, all of the triggered rules are then aggregated in terms of the union () operator as defined in Section 4.1.2. The aggregation of rule 1 and rule 2 (Figure 4.10) is expressed as:

where β1 and β2 are defined in Equation (4.31). Since the result of rule aggregation is a membership function, a defuzzification process has to be implemented in order to obtain a deterministic value.

4.4.3 Defuzzification

Several kinds of defuzzification strategies have been suggested in the literature. The most popular methods of defuzzification are the centre-of-gravity and the mean-of-maximum methods (Pedrycz, 1989; Kosko, 1992). A membership function is often represented in terms of discrete data. The centre-of-gravity can be calculated from the following equation:

where n is the number of elements of the sampled membership function, and μ(s) is the membership grade of measurement s.