From: mailbox@dmitry-kazakov.de   
      
   On 31 Oct 2004 15:36:01 -0800, Ethan Seng wrote:   
      
   > Thanks again for your reply. May I know the meaning of one of your   
   > previous statements: "all data are split into weighted intervals"? How   
   > was the data split into weighted intervals? Can you provide an example   
   > of how this will work?   
      
   If you have a [membership] function f:R->[0,1], then you can represent it   
   using piecewise constant functions:   
      
   f(x) = Sup Wk   
       k | x in Ik   
      
   where Ik are intervals. If f is monotonically ascending on ]-oo, x0] and   
   monotonically descending on [x0, +oo[, i.e. when it looks like a hill, bell   
   etc. Then intervals Ik and weights Wk can be obtained as:   
      
   for all alpha in [0,1], Ialpha = {x | f(x) >= alpha}; Walpha = alpha.   
      
   The intervals Ialpha are nested. When number of alphas is finite, this   
   becomes an approximation. Going intuitionistic one could also have upper   
   and lower set, but technically only the upper set is interesting.   
      
   The advantage of this representation over, say, piecewise linear-, or   
   bell-functions, is that fuzzy arithmetic is pretty easy to implement. It is   
   based on interval arithmetic, which is quite well explored. So it   
   guarantied that the result of any operation on upper sets of arguments will   
   be also an upper set. No matter that floating-point arithmetic is   
   imprecise.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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