From: mailbox@dmitry-kazakov.de   
      
   On 17 Oct 2004 10:21:44 -0700, Ethan Seng wrote:   
      
   > "Dmitry A. Kazakov"  wrote in message   
   news:<8mdg4i6wit4i$.1a0hh81z4ncwh.dlg@40tude.net>...   
   >> On Sat, 16 Oct 2004 16:21:41 -0700, user@domain.invalid wrote:   
   >>   
   >>> Dmitry A. Kazakov wrote:   
   >>>> On 15 Oct 2004 09:28:59 -0700, Ethan Seng wrote:   
   >>>>   
   >>>>>Thank you for your kind assistance and advice. Yes, you are right. My   
   >>>>>problem is about determining the membership functions of input   
   >>>>>variables (specifically C and V) for which the crisp output contains   
   >>>>>(t1, y1), (t2,y2), (t3,y3),..., (tN,yN)  pairs of numerical data. I   
   >>>>>also know beforehand that all variables are related by   
   >>>>>y=(30/V)*exp(-C*t/V). Can you suggest some technical reference from   
   >>>>>which I may find the methodology for determining membership functions   
   >>>>>of C and V using Sup-norm? Once again, thank you in advance for your   
   >>>>>attention.   
   >>>>   
   >>>> I would say first look for a good book on numerical methods. Unfortunately   
   >>>> it is not my field so I can only tell that you have a non-linear   
   >>>> optimization problem. It is also ill defined, because obviously you can   
   fix   
   >>>> C=0 and get Vmin=30/ymax, Vmax=30/ymin, which I presume is definitely not   
   >>>> what you want. So you have to add some additional constraints on how to   
   >>>> vary V and C. Like Vmax-Vmin=Cmax-Cmin or (Vmax-Vmin)*(Cmax-Cmin)->min   
   etc.   
   >>>>   
   >>>> A very brutal and crude method could be to evaluate V and C for each pair   
   >>>> (ti,yi), (tk, yk) and then get max and min for both over all pairs. But I   
   >>>> think there should be better, iterative methods. For example, you fix   
   Cmin,   
   >>>> Cmax then find Vmin, Vmax, then attune C's find new V's etc, a kind of   
   >>>> Newton's process. It is pure numerical methods...   
   >>>   
   >>> Thanks for the reply. Does that mean that there will not be a unique   
   >>> solution to the problem?   
   >>   
   >> Yes. It is ill-defined. So the solution will vary depending on which   
   >> additional assumptions one could take.   
   >>   
   >>> In addition, can I conclude that there is no   
   >>> fuzzy logic component to the solution of this problem?   
   >>   
   >> Fuzziness may still come in consideration by two ways:   
   >>   
   >> 1. The problem is formulated in terms of fuzzy sets.   
   >> 2. The solution is fuzzy due to lack of knowledge how to solve that.   
   >>   
   >> (1) is not the case, but it could be, if for example the error risk   
   >> criteria will be defined in terms of a fuzzy measure.   
   >>   
   >> (2) is also well possible. There are many examples of how uncertainty   
   >> appears where there should be no one. The most notorious coming in mind   
   >> ones are Monte Carlo methods, interval computations, fuzzy control.   
   >>   
   >> ... and, well, fuzzy logic is just a nice abstraction. In the end   
   >> everything is just a numerical problem! (:-)) Membership function is a   
   >> function as any other. It is at our discretion to attribute some semantics   
   >> to it. But it is fuzzy no longer we think it is.   
   >   
   > Once again, thank you for sharing. I was looking at some papers in   
   > regard to neural networks and neurofuzzy. Do you think the following   
   > procedure is appropriate?   
   >   
   > 1. Assume we have M data sets, each with N {(t1, y1), (t2,y2),   
   > (t3,y3),..., (tN,yN)} pairs of numerical data.   
   >   
   > 2. Next, calculate the C's and V's of each data set, i.e., from the   
   > (t1, y1), (t2,y2), (t3,y3),..., (tN,yN) N pairs of numerical data in   
   > that particular data set. We can do so since we know the model   
   > equation (y=(30/V)*exp(-C*t/V)).   
   >   
   > 3. Based on these inputs (C's and V's for each data set), used   
   > neurofuzzy techniques (e.g. anfis in Matlab) to determine the   
   > membership functions for C and V for population.   
      
   Mmm, ANFIS does approximation using least squares, AFAIK. The problem is   
   that the class of basis functions it uses is not Aexp(-bt). It seems that   
   you want to use C,V as the input for ANFIS, what will be the parameter   
   then? The group number? Or C(V)? It looks much like simpler:   
      
   1. take i /= k randomly   
   2. evaluate A and b from (ti,yi), (tk,yk)   
   3. generate statistics by repeating 1 and 2   
   4. estimate mean, dispersion, distribution etc   
   5. convert distribution of probabilities to a distribution of possibilities   
   a-la Dubois-Prade, here you are.   
      
   What about harmonic analysis? If you know approximately the interval where   
   C/V should lie, i.e. the fmax (maximal frequency) then you could   
   approximate your data in the basis exp(-t*k*fmax/M), where k=1..M, using   
   least squares; find the maximum; and then repeat it iteratively narrowing   
   the interval of major frequencies around the maximum. With some imagination   
   one could even claim that appropriately normalized spectrum is a kind of   
   membership function of -C/V. The rest is easy.   
      
   Honestly, I do not much like either because an ill-defined problem becomes   
   good-defined by solely choosing a method to solve it. It is like putting   
   the cart before the horse.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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