From: earlcox@earlcoxreports.com   
      
   Ordinarily, if you take a fuzzy term set for a concept and drop a plumb line   
   from any point along the universe of discourse, the line should pass through   
   only two fuzzy sets. As an example, if you decompose Height into Short,   
   Medium, and Tall, with triangular or bell-shaped fuzzy sets using a 50%   
   overlap, then Short and Medium overlap to the left, and Medium and Tall   
   overlap to the right. The sum of the degrees of membership of the points   
   along the plumb line is usually [1.0]. Thus, Short gives way to Medium in a   
   smooth fashion -- when the degree of membership in Medium is [1] then the   
   degree of membership in both Short and Tall is [0]. In most practical   
   applications (that are not highly non-linear)  this makes semantic sense,   
   for while fuzzy logic does not generally obey the Laws of Excluded Middle   
   and Non-Contradiction, it generally does respect these laws for the special   
   cases of complete membership (through the Extension Principle). Thus if   
   someone is completely Tall (m(x:Tall)=1) then it ordinarily doesn't make   
   sense to attribute them partial membership in another set that is   
   semiotically a component of the overriding model variable.   
      
   If we now overlay Short and Medium with a third fuzzy set, Lanky, that   
   extends from one-third overlap on Short and on-third overlap on Medium, we   
   have a situation where a plumb line dropped over this region intersects   
   three points, the sum of which may exceed [1]. Thus a person could be Short   
   to degree [.4], Medium to degree [.6], and Lanky to degree [.3]. Note that   
   there is nothing particularly wrong with this decomposition. But it does   
   tend to violate our sense of uniformity in model semantics. But this a   
   violation in principle and not a structural violation of the fuzzy model   
   itself. Both Kosko's Standard Additive Method and the Mandami   
   minimum-correlation method work perfectly well on such models. This   
   "over-loading" of semantics is often necessary, in fact, in many complex   
   business models involving risk assessment, fraud detection, and various   
   types of time-series based pattern analyses that exhibit non-linear   
   behaviors.   
      
   Just a quick note,   
   Earl   
      
      
   "Rich Shepard"  wrote in message   
   news:slrnbvtef4.hhk.rshepard@salmo.appl-ecosys.com...   
   > On 2004-01-09, tarmat  wrote:   
   >   
   > > I've heard that it's a bad idea to have three or more member functions   
   > > overlapping (Yen and Langari).   
   > >   
   > > Can you explain why?   
   >   
   >   If I understand you correctly, the answer is "no".   
   >   
   >   Consider the linguistic variable "Height". It could have terms (fuzzy   
   > sets) including very short, short, normal, tall, very tall. By definition,   
   > an instance (e.g., a person) can have a partial membership in more than   
   one   
   > fuzzy set simultaneously. The only way to fuzzify a height of, say, 170 cm   
   > might be to say that it is .9 in the set 'normal' and .1 in the set   
   'tall'.   
   > This requires overlap.   
   >   
   >   What I've learned is that in most (many, at least) situations a 50%   
   > overlap is a good starting point if not the most the most meaningful   
   amount   
   > of overlap.   
   >   
   > HTH,   
   >   
   > Rich   
   >   
   > --   
   > Dr. Richard B. Shepard, President   
   > Applied Ecosystem Services, Inc. (TM)   
   >    
      
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