From: mailbox@dmitry-kazakov.de   
      
   On 2 Oct 2006 03:59:45 -0700, Bill Silvert wrote:   
      
   > For some work I am doing on developing fuzzy rules for environmental   
   > systems I found the usual methods for defining fuzzy memberships in   
   > terms of straight line segments awkward. I have therefore been using a   
   > different approach, described below. Since I am not very familiar with   
   > the literature in the field, I would like to know if this is a standard   
   > method and where I can find references to it.   
   >   
   > The approach I am using requires two parameters, a reference value x0   
   > for which the membership is 0.5 and an exponent n which determines the   
   > width of the transition from mu=0 to mu=1. The membership is then given   
   > by   
   >   
   > mu=(x/x0)^n/[1+(x/x0)^n]   
   >   
   > where the ^ sign refers to exponentiation. I use this to define sets   
   > such as "deep water" and "high sulfer concentration". It has the   
   > advantage of generating a smooth curve. For most environmental   
   > variables the value of n is fairly low, around 3 to 6, since the   
   > transition region is quite broad. For the common illustrative set of   
   > "tall men" I use a much higher value of n=20.   
      
   The problem with an exponential membership function is that the   
   corresponding set does not contain any value for certain: forall x mu(x)<1.   
   Similarly, it does not exclude any value. When such sets appear as inputs   
   in inference, it may lead to contradictions in outputs. As an example,   
   consider the proposition "tall men do exist," which were obviously wrong   
   with any mu.   
      
   Another problem is the set of operations, upon which the set of intervals   
   with fuzzy bounds of exponential shapes were closed. [ OK, if you don't   
   care about exact representations of mu, then, again, there could be more   
   numerically efficient and precise representations. ] But the nasty   
   consequence is that you can't get [fuzzy] numbers this way. Your mu's are   
   fuzzified intervals [x0,+oo[ and ]-oo,x0]. Because the set of operations on   
   them is not closed upon intersection, you cannot construct either a finite   
   fuzzy interval (fuzzified [x0,x1]) or a number (fuzzified {x0}).   
      
   > Any other suggestions on how to define membership functions would be   
   > appreciated.   
      
   There is no silver bullet, IMO. But I think that a sort of 3-rd order   
   spline representation in the segments would be better, provided, you wanted   
   to have a smooth curve.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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