Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

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target protein sequence from biological point of view using a fuzzy membership function as an alternative measure, e.g., A (S) ⎧ 0 if Eval > 0.1 = ⎨ ⎩sim(S,S' ) else where sim(S,S’) is the sequence identity of the alignment, i.e., the percentage of the identical amino acids in the alignment. 0.8 0.6 0.4 0.2 1 LOW MEDIUM HIGH 0 0 0.2 0.4 0.6 0.8 1 LOW MEDIUM HIGH 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 (a) (b) LOW MEDIUM HIGH 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 (c) Figure X.1 Examples of common fuzzy membership functions where X=[0,1]. (a) triangular, (b) trapezoidal, and (c) smooth quadratic functions. See Example X.xx and problems X.x and X.xx
(a) (b) Figure X.2. Practical membership function generators from a nursing application in having children assess their level of pain for medication dosage. In (a), the child slides a bar to the place on the visual scale that best represents her level of pain. The nurse can read off the analog “pain membership” value from the reverse side. (b) This is for younger children and demonstrates a similar, though discrete, version of pain membership. These data are the property of McNeil Consumer Healthcare and McNeil maintains sole publication and dissemination rights. All fuzzy set theory is based on the concept of a membership function. Where do these membership functions come from? In many cases, they are defined as in the two examples above – common sense definitions that convey some linguistic expression. More generally, they come from expert knowledge directly or they can be derived from questionnaires, heuristics, etc. This is a human-centric view and is certainly open to debate. In many cases, the membership functions take on specific functional forms like triangular, trapezoidal, S-functions, pi-functions, sigmoids, and even Gaussians for convenience in representation and computation. Pi-functions and S-functions are constructed from quadratic functions “pieced together” to make smooth curves. Figure X.x displays several common fuzzy membership functions. Alternately, membership functions (or the parameters of the specific equation forms) can be learned from training data, much as probability density functions are learned. Some fuzzy clustering algorithms naturally produce membership functions as their output. A neural network, given proper input/output training data, also acts as a membership function for new input. One of our favorite practical membership functions comes from the field of pediatric nursing. A child is asked to slide the bar to a position indicative of his or her level of pain in Figure 2.3(a). The color and
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Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

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