Why Fuzzy?

Fuzzy logic provides an attractive framework to work in when designing a scheme for the detection of BWERs. A consensus definition of a BWER, especially in a quantitative sense, is not available. The low modeling requirements and high prototyping speed of a fuzzy logic scheme, given that such consensus is lacking, are of great benefit.

As discussed in the previous section, we are faced with imprecise radar information, especially with regard to capping (as it depends on radar elevation scans), gradients (undersampling at large distances from the radar), height and missing data. In addition, the conical nature of a radar scan poses problems of ambiguity when a candidate region occurs at low heights since the 3D structure may not be completely sampled.

Whether a structure is a BWER or not is sometimes not clear. In these instances, meteorologists refer to the radar signatures as ``marginal'' BWERs - a clear invocation of degree of membership.

No single attribute of a BWER provides an adequate discrimination capability. Take, for example, the range of values for the capping of a candidate region shown in Figure 6. The graphs are normalized to have an area of one. If we set the minimum threshold at $32dBZ$, four-fifths of the BWERs will be correctly identified but one-third of the non-BWERs will be falsely identified as BWERs. However, since non-BWERs outnumber BWERs 400 to 1, setting the threshold at $32dBZ$ will lead to about 160 false alarms for every correct BWER detection. Obviously, several factors have to be considered in conjunction. Fuzzy logic is just the framework within which we consider the various attributes before coming to a decision.

Figure 6: The distribution of the average capping (in $dBZ$) of candidate regions in the training cases: (a) BWERs (b) marginal BWERs (c) non-BWERs. The graphs have been normalized to have unit area under the curve.

There are other methods of dealing with uncertainty. Probabilistic approaches, in particular, can be used to deal with ambiguity and dissonance [7]. For example, the probability that a structure is a BWER (the probability that this is event $B$) given the measurements of the various features $A_i$ can be stated as:

$$P(B\vert A_0, A_1, ..., A_{n-1}) = P(B) \prod_{i=0}^{n-1}{\frac{P(A_i\vert B)}{P(A_i)}}$$ (1)

if the occurrence of any feature-element $A_i$ is independent of all other $A_i$'s. Of course, none of the features are truly independent of one another - when using a Bayesian inference approach we should compute conditional probabilities of combinations of the various features.

While using Equation 1 and the Bayesian inference approach is possible, there are pragmatic reasons why we chose the fuzzy logic approach. According to the ``frequentist'' school of thought, the probability $P(A_i\vert B)$ is the frequency of BWERs that have the feature $A_i$. Practical constraints of the rarity of BWERs and the relatively small number of analyzed cases would lead us to estimate this probability on the basis of a few localized BWERs observed on half a dozen occasions. $P(A_i)$, the probability of the feature occuring, is not as affected by the rare nature of BWERs but is contingent on choosing a representative sample of training cases.

We can try to compute the probability of a structure being a BWER given several measurements using Equation 1. For example, $P(B\vert A_3)$ might be the answer to the question ``What is the probability that the structure is a BWER given that the gradient is $5~dBZ.km^{-1}$?''. The same question with regard to possibility is quite easy to answer. While it is a positive gradient, $5~dBZ.km^{-1}$ is not very high. A heuristic membership function might yield a fuzzy membership of 0.5 in the set of structures that have high gradients (in the context of regions that are being tested for being BWERs). In our BWER detection problem, we found [9] that a first guess heuristic choice of membership functions fares comparably to an ``optimal'' choice of membership function. The skill scores were comparable even on the training data sets on which the optimization was done.