On 2007-07-30, Kirk Zurell  wrote:   
      
   > 2. Triangular and trapezoid functions offer partial degrees of truth, so   
   > these offer the real meat of fuzzy calculations. If your input can be   
   > somewhat x and somewhat y at the same time, choose these. If in doubt,   
   > choose these.   
      
   Kirk,   
      
     Allow me to modify the above -- and comment on the use of distribution   
   formulae.   
      
     Triangles and trapezoids have value representing linguistic variables, but   
   only when the underlying semantics match the shapes. Both have straight   
   sides from mu=0 to mu=1 which represents a constant change as the domain is   
   traversed. For the triangle, the maximum truth is at a single point and   
   falls sharply from that on both sides. For the trapezoid, the maximum truth   
   value spans a range of the underlying Universe of Discourse. These may not   
   be appropriate for all variables.   
      
     Consider, for example, what Jerry Mendle calls Type 2 fuzzy sets: those   
   whose membership values are themselves not known with precision. Or, in my   
   world, we consider them to be fuzzy sets of linguistic variables that have   
   no underlying measurement; for example, "Significant," "Beautiful,"   
   "Acceptable."   
      
     Such linguistic variables are much better represented by smooth curves   
   such as the Gaussian (normal curve, bell curve) with S- and Z-shape curves   
   at the ends of the domains. The semantics are that the rate of change of the   
   truth values (membership degree, mu) varies as the domain value varies and   
   that there is a gradual approach and departure from the maximum truth value.   
   In the world of subjective decision making (unlike in fuzzy control   
   applications) these are both more realistic and easier for people to accept   
   than are trapezoids and triangles.   
      
     Both types have a place. It all depends on the linguistic variable and how   
   it can best be expressed mathematically.   
      
   Rich   
      
   --- SoupGate-Win32 v1.05   
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