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From: saswss@hotellng.unx.sas.com (Warren Sarle)
Subject: Re: fuzzy logic and probability
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Date: Sat, 13 Jul 1996 21:42:39 GMT
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In article <4s2hv8$k38@sjx-ixn6.ix.netcom.com>, jdadson@ix.netcom.com(Jive Dadson) writes:
|> ...
|> There is a relationship in the sense that membership functions have
|> been used by the probability folks for a long time. They call them
|> "kernel" functions. They are not probability densities though. They
|> serve to establish continuity through a kind of interpolation.

A kernel function can certainly be viewed as a fuzzy set, but kernel
density estimation, regression, etc. generally place a kernel at each
training point. That's a very different approach from fuzzy systems,
where a relatively small number of fuzzy sets are used.  If you use a
small number of kernel functions, you've got a radial-basis-function
neural net. A Gaussian RBF net is indeed an adaptive fuzzy system, but
all the rules involve conjunctions of all the inputs, which again is not
the usual approach in fuzzy systems.  There is nothing to prevent you
from constructing an RBF net in which some hidden units are connected to
proper subsets of the inputs, but I haven't actually seen anybody do
that. Anyhow, the terminology for these things is rather fuzzy.

What you will find in the statistical literature that is indisputably
an adaptive fuzzy system is tensor-product B-spline regression. See:

   Dierckx, P. (1995), <cite>Curve and Surface Fitting with Splines,</cite>
   Oxford: Clarendon Press.

   Brown, M., and Harris, C. (1994), <cite>Neurofuzzy Adaptive 
   Modelling and Control,</cite> NY: Prentice Hall.


|> Of course fuzzy isn't just used to calculate probabilities. It is also
|> used for arbitrary functions. Again the probability people
|> have been there and done that. The name of the technique is
|> "Watson/something". I never can rememeber the the other man's name.
|> Starts with D.

Maybe it has a "d" in it? As in Nadaraya-Watson kernel regression?

   Nadaraya, E.A. (1964) "On estimating regression", Theory Probab.
   Applic. 10, 186-90.

   Watson, G.S. (1964) "Smooth regression analysis", Sankhya,
   Series A, 26, 359-72.

There are other types of kernel regression, such as Gasser-Muller,
but Nadaraya-Watson is what's generally used for multidimensional
data.

-- 

Warren S. Sarle       SAS Institute Inc.   The opinions expressed here
saswss@unx.sas.com    SAS Campus Drive     are mine and not necessarily
(919) 677-8000        Cary, NC 27513, USA  those of SAS Institute.
