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From: saswss@hotellng.unx.sas.com (Warren Sarle)
Subject: Re: NN Vs Stats......
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Date: Mon, 16 Jan 1995 02:59:54 GMT
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In article <3fc4ss$4en@scapa.cs.ualberta.ca>, arms@cs.ualberta.ca (Bill Armstrong) writes:
|> ...
|> It might be possible to use piecewise linear functions in place of
|> linear functions in statistics.

Piecewise linear functions are still nonlinear, but the
nondifferentiability make the theory even harder. The properties of
piecewise linear regression models (called switching regression,
change-point regression, etc.) are somewhat peculiar. For example, a
parameter used for estimating a knot location has been found to use
as many as 3 degree of freedom.

|> Then you would have the adaptive
|> logic network as your NN.  Do you know of statistical work that relies
|> on piecewise linear functions as discriminants?  In cases where I have
|> had success, there are indeed few parameters in the model -- probably
|> just a convex function formed by taking the minimum or maximum of
|> several linear functions.

I'm not sure what the exact ALN model is, but if it involves discrete
parameters, like a parameter that says whether to take the max or min,
that would complicate the statistical theory still further.

|> Another question: I have been using piecewise linear surfaces to model
|> fuzzy membership functions.  For example, I have been trying to find
|> a fuzzy set where the measurements in the set indicate normal
|> operation of a piece of equipment.  The idea is that as the equipment
|> gets closer to breakdown, the fuzzy membership in the "operating
|> normally" set decreases from 1 to 0.
|>
|> One could imagine that such a fuzzy set would look something like a
|> Gaussian.  What I have done is used the log of the fuzzy set, since
|> the log of a Gaussian is a paraboloid, which is convex.  This is
|> bounded by several hyperplanes, as any convex set is in the limit,
|> hence it is easy to fit using an ALN.
|>
|> My question: do you know of anyone in statistics using distributions
|> with convex logarithms (I have coined the term "log-convex"
|> distributions -- but the idea may not be original)?  Or does this play
|> any role in fuzzy theory?

I don't know about that.

-- 

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.
