XPost: comp.soft-sys.matlab   
   From: mailbox@dmitry-kazakov.de   
      
   On Sun, 17 Apr 2005 11:42:00 +0100, Fuzzy wrote:   
      
   > "Dmitry A. Kazakov"  wrote in message   
   > news:c1x9d15twfhw$.36vug05uapdd$.dlg@40tude.net...   
   >>   
   >> The point is that the metric does not come from fuzzy. Look how it happens   
   >> in statistic to highlight the pattern:   
   >>   
   >> Variant 1. First you postulate that y=ax+b. Then you postulate that   
   >> observed x's and y's are x+err and y+err, where err's distribution is also   
   >> postulated. From that you find a and b minimizing the probability of error.   
   >>   
   >> Variant 2. You again postulate y=ax+b. Then you say that x's and y's are   
   >> exact, but a and b are random (vary from cow to cow). Further you postulate   
   >> a distribution of a and b and estimate the distribution's parameters to   
   >> minimize the probability of error.   
   >>   
   >> In both cases and all their mixtures the metric is *postulated*: you   
   >> pretend to know/presume/surmise how the cow works (y=ax+b). This knowledge   
   >> does not come to you from the probability theory, it does from the   
   >> farm-yard.   
   >>   
   >> Nothing changes here if you replace one type of uncertainty with another   
   >> and switch to fuzzy. Mathematical apparatus may change, instead of least   
   >> squares you could expect C-norm. But the rest will be same.   
   >>   
   >> What you have described as an example with cows looks like the variant 1.   
   >> With the variant 2 there might be difficulties with its justification. If   
   >> the parameters of the cow are not random, but fuzzy, then what does it   
   >> physically mean? Note to make a prediction you'd describe a herd of all   
   >> cows, not a particular cow. Should we treat it as if a cowboy forgot his   
   >> spectacles and now is guessing what's the cow? [ You cannot get rid of   
   >> randomness here, but you could mix fuzziness in. ]   
   >   
   > No, that was a red herring introduced to imply that some sort of "averaging"   
   > across cows "may" be possible to aid the model formulation. The cowboy always   
   > knows the cow and we are interested in predicting a specific cows response   
   given   
   > the data to a new batch of feed.   
      
   OK.   
      
   > Where I thought fuzzy would help is the possibility of belonging to more   
   > than one set.   
      
   Yes. Moreover, fuzzy allows you to describe non-singleton data, missing and   
   contradictory data. (There is also a price of this flexibility.)   
      
   > The cow would probably have some preference AND dislike for   
   > each batch - as the distribution perhaps may not be normal. Maybe a simple   
   > y=ax+b doesn't cut it, perhaps the response is more sophisticated, may have   
   > non linearities etc.   
      
   Absolutely.   
      
   > This is becuase the estimated value will serve as a parameter in a larger   
   > model - say a cow health model - and the dislike it has for a feed may be   
   > more informative than the preference it has for a feed.   
   >   
   > Either way if we *assume* that the responses are fuzzy, a moot point,   
      
   They usually are.   
      
   > I am   
   > looking for any techniques to be able to model the data as given to predict   
   > a response given new output. I am looking for suitable ways to transform the   
   > historical data to be able to produce a "preference model", given that the   
   > data bay be sparse, incomplete and unevenly distributed. I hoped using the   
   > fuzzy paradigm may facilitate making an "educated guess" when, perhaps we   
   > dont have enough data to use standard techniques.   
      
   Right. This is exactly the application field of fuzzy.   
      
   But I must repeat it again, you need a model. Any choice is based on the   
   model. For example, if you take a fuzzy decision tree and do some training   
   on your fuzzy data set. This assumes that you have chosen the model:   
   training on trees implicitly introduces a distance in the feature space   
   which is then used for generalizations. This distance may fit or not the   
   reality. It is quite easy to build examples where trees perform badly   
   wrong...   
      
   Basically, you can use practically any machine learning technique that   
   supports generalization on the fuzzy case. Though mathematically it might   
   be tricky, but there is nothing new, there cannot be anything new. Fuzzy is   
   not a new method of machine learning, it is a framework which allows you to   
   handle fuzzy data. No more, no less.   
      
   P.S. I am using fuzzy decision trees. Maybe someday I'll publish my fuzzy   
   machine learning framework under public license.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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