From: mailbox@dmitry-kazakov.de   
      
   On 25 Oct 2004 10:38:28 -0700, Ethan Seng wrote:   
      
   > Thank you for your reply. Previously, you had stated that - given my   
   > problem statement - 'the goal is then approximate y(t) from the class   
   > of exponent functions keeping parameters A and b as much certain as   
   > possible, i.e. to find A and b with the membership functions as close   
   > to singletons as possible. However, here you will get nothing richer   
   > than just intervals: A=[Amin, Amax], b=[bmin, bmax].' Do you mind   
   > explaining why it will only be possible to obtain rectangular   
   > membership functions?   
      
   Without any additional assumptions, one cannot differentiate the solution   
   set finer than crisp.   
      
   For A and b to be intervals means: if I take any A from [Amin, Amax], then   
   for any xi I can find some b in [bmin, bmax], such that yi=A*exp(b*xi). If   
   neither yi nor xi are fuzzy, then there is nothing beyond that. Either yi   
   is equal to A*exp(b*xi) or not. So either A and b are in the solution set   
   or out. This is why it is rectangular (exp is monotonic and contiguous, so   
   the solution set cannot be disjoint, you also said that A and b are   
   independent, so only intervals).   
      
   You have to reformulate it, if you feel that there should be something   
   else. The most obvious way is to move from interpolation (all data are   
   exact) to approximation (the data are uncertain). All regression methods   
   follow this way. But this would require additional assumptions. For   
   example, that (xi, yi) are realizations of a stochastic/fuzzy process, that   
   error risk is squared etc.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)