From: mailbox@dmitry-kazakov.de   
      
   On 14 Oct 2004 20:29:46 -0700, Ethan Seng wrote:   
      
   > I am a graduate student looking at ways to apply fuzzy logic and fuzzy   
   > set theory to model uncertainty in biological systems. May I ask   
   > whether it is possible to   
   > obtain fuzzy membership functions of input variables for which no   
   > numerical   
   > data can be obtained or measured a priori? In other words, given only   
   > the numerical values of the output variable as well as the system   
   > model (relating the output to the inputs, say 'y = A*exp(-b*t)', where   
   > A, b and t are the inputs (independent variables), and y is the output   
   > (dependent variable)), can one derive the input membership functions   
   > (i.e., A, b and t) making up the fuzzy inference system? If yes,   
   > please suggest the possible solution(s)?   
      
   So you have a set of (y1, t1), (y2,t2), (y3,t3),..., (yN,tN) and want to   
   find A and b? Is that correct?   
      
   If so, then the question is what else is known about yi, ti, A, b? For   
   example you can take yi, ti crisp and A, b fuzzy. 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 (in some defined [by you]   
   sense). However, here you will get nothing richer than just intervals:   
   A=[Amin, Amax], b=[bmin, bmax]. They are, of course, fuzzy numbers with   
   rectangular membership functions. Mathematically, I guess, it should be   
   just an approximation using Sup-norm. I think that the result may vary   
   depending on further assumptions about how A and b might be related, how   
   their joint membership function looks like. Anyway, you will get nothing,   
   but intervals.   
      
   It should become more interesting and more complicated if yi and ti are   
   also fuzzy. One can consider alpha cuts of yi, ti and reduce the problem to   
   a set of problems like above (each per cut-value).   
      
   Another thing is when you would consider points (yi, ti) rather as   
   evidences of realizations of y(t). The confidence in the evidences can be   
   estimated according to how good [in defined by you sense] the point fits   
   the exponential model. That could be probability, possibility, necessity   
   etc [of your informed choice]. And the goal would be then to find fuzzy A   
   and b, which estimate the larger number of points with higher confidence,   
   i.e. minimize the risk of error [defined in terms of confidence you   
   choose]. To do this, you have to make a *lot* of assumptions about y(t) to   
   define the norm used for the approximation.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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