Fuzzy Membership Functions

The ``features'' in the weather detection algorithm or its rule base translate into the GA's genes. Thus, each gene represents a fuzzy set. For example, the first gene in the BWER algorithm represents the attribute of a candidate region that the reflectivity value within a BWER is low. The algorithm designer does not need to specify hard thresholds. It is the job of the GA to sort out how low ``low'' really is. Naturally, the fuzzy membership functions constrain the GA - the GA cannot change the form of the solution, only its degree. Thus, this method of combining a fuzzy rule base with a genetic algorithm is useful when there is a lot of empirical knowledge that can go into the algorithm, but no extensive statistical information.

Associated with each gene is a membership function based on the value of a single feature. There are three kinds of membership functions in the BWER algorithm - an attribute of a candidate should be low, should be high or should be about some value. All the empirical knowledge about BWERs can be described using this family of membership functions. For each of the function types, two parameters need to be specified. These parameters, $x_1$ and $x_2$ in Fig. 3, along with the shape of the membership function, describe the fuzzy set completely. For any value of the attribute, the degree to which the property holds can be obtained by looking at the graph. If $x_1$ and $x_2$ change, the extent to which the property holds for various candidates will change.

Figure 3: $x_1$ and $x_2$ for each feature in the rule base are set by the GA as a result of optimization. Given any input $x$, depending on the rule associated with that feature, a fuzzy value $FzVal$ is returned.

For example, if the minimum radar reflectivity value within a candidate region is $30~dBZ$, then if $x_1$ and $x_2$ are $10~dBZ$ and $40~dBZ$ respectively, the minimum reflectivity value within the region is ``low'' to the extent $0.33$, as can be ascertained from Fig. 3a. Had $x_1$ and $x_2$ changed, to $20~dBZ$ and $30~dBZ$, then the same property for the same region would be satisfied to a $1.0$-degree extent. Had the membership function corresponded to the idea that the minimum reflectivity value is ``high'' (with $x_1$ and $x_2$ $10~dBZ$ and $40~dBZ$ respectively), then the rule would have been satisfied to a $0.67$-degree extent (from Fig. 3b). If the property were that the minimum reflectivity value is neither too high nor too low (an ``is about'' condition, see Fig. 3c), it would have been satisfied to a 0.5-degree extent.

The genetic algorithm will change the values of $x_1$ and $x_2$ associated with each gene while keeping the forms of the membership functions for each feature constant to come up with different chromosomes. The better fit chromosomes will be those that have values of $x_1$ and $x_2$ that correspond to the BWERs in the verification database. Of course, the GA will attempt to fit the $x_1$'s and $x_2$'s for all the fuzzy sets together in an attempt to raise the skill of the algorithm.

For each candidate region that the BWER detection algorithm should classify, the value of each feature is computed. Using that value, the extent to which various fuzzy sets hold are computed. First, fuzzy sets that depend on a single feature are computed. Then, fuzzy rules that use several features are evaluated. Fuzzy values are combined using the standard methods and operators introduced by Zadeh (1965) and studied by Voxman and Goetschel (1983). The inverse, $\bar{A}$, of a set, $A$, has the membership function $f_{\bar{A}}(a) = 1-f_A(a)$ where $f_A(a)$ is the membership of element $a$ in the set A. The membership of an element in the union of two sets ($A \cup B$) is given by the maximum of memberships in the two sets ($f_A \vee f_B$). This represents a logical OR. The membership of an element in the intersection of two sets ($A \cap B$) is given by the minimum of the two sets' memberships ( $f_A \wedge f_B$) and represents a logical AND.

For example, one of the rules in the BWER rule base is that the candidate region is said to have a good two-dimensional profile if the minimum reflectivity value in the region is low or the bounding pixels have a high reflectivity value. If the candidate region has a minimum reflectivity value inside of $25~dBZ$ and is bounded by pixels having an average bounding value of $45~dBZ$, then if the $x_1$'s and $x_2$'s for the two rules are $20~dBZ, 40~dBZ, 40~dBZ$ and $60~dBZ$ respectively, the first rule (that the reflectivity value inside the region is low) is satisfied to the extent $0.75$ and the second rule is satisfied to the extent $0.25$. Since a logical OR is given by the maximum of the individual fuzzy values, the combination of the two rules holds to the extent $0.75$. Both the rules (a logical AND) would hold to only a $0.25$-degree extent.

The fuzzy rules are aggregated and a confidence estimate that the candidate region is a BWER is arrived at. This is the output of the BWER algorithm for that candidate region. This is done for each candidate region in the verification database. The set of candidates which have been deemed BWERs is then compared with the truthed BWER set.