Re: Fuzziness as opposed to Probability

From: Koczy Laszlo (koczy@alpha.ttt.bme.hu)
Date: Fri Aug 03 2001 - 11:31:29 MET DST

  • Next message: predictr@bellatlantic.net: "Re: Fuzziness as opposed to Probability"

    I just want to comment on these issues. My favourite way of looking at
    fuzzy inferece/control methods is that they represent a compromise
    approach to approximation (in a rather user friendly way!), with a
    reasonable computational complexity side as well. The two are of course
    contradictory requirements and while one is optimal, the other might quite
    easily be not acceptable at all. Think about the statements on the
    universal approximation properties that have been proved by Wang, Kosko,
    later Nguyen and Kreinovich, even later by Castro and many others. They
    are very interesting but no one has analysed the computational complexity
    issues (we did so later with Klement and Moser in a special issue of the
    International Journal for General Systems a few years ago), and it come
    out clearlt that these statements are barely applicable from the
    engineering point of view, because arbitrary accuracy in the approximation
    requires unbounded number of rules. This has been well known already in
    the Neural Networks field where Kurkobva stated it very clearly several
    years back, while former results were quite similar to the fuzzy ones I
    mentioned here. Of course, all this has to do with Hilbert's famous (and
    wrong) 13th Conjecture and Kolmogorov's theorem abvout decomposition. In
    the fuzzy context it was alsoi shown by Bauer, Klement, Moser and
    Leikermoser that under certain circumstances the exact construction of
    functions by appropriate fuzzy rule bases was also possible.

    On the other hand, if complexity was bounded, the approximation accuracy
    became not so good. A joint optimum of the two (with cost functions for
    both) were shown to be existing by myself and Zorat in FSS a few years
    back.

    There are amazingly nice properties of some fuzzy reeasoning schemes
    though, they are much better approximators in the sense of uniform and
    stable behaviour than many of the classic interpolation techniques. This
    we have shown in many paopers with Joo, Tikk, Moser and others. And, in
    addition, the scheme is fully transparent for the user.

    I think, this issue is definitely something that critics of the fuzzy
    control/reasoning methods could nag on! SAenseless criticism, based on not
    knowing the mathematical background of fuzzy methiods is not worth of
    commenting. By the way, all fuzzy rule based schemes are known from the
    point of explicit behaviour. (Given first by El Hajjaji and Rachid, then
    by myself and Sugeno.) It is clear why these methods are such as they are
    and what functions they implement!

    Laszlo KOczy

    On Fri, 3 Aug 2001 predictr@bellatlantic.net wrote:

    > Radford Neal wrote:
    > "The whole point of constructing a mathematical formalism for inference
    > is to produce conclusions or decisions that are more reliable than would
    > be produced by unaided human intuition."
    >
    > Will Dwinnell responded:
    > "To me, this is the crux of the matter. I don't know about the fine
    > point that this statement was in reference to, but I think it expresses
    > quite well what I think of as the "engineer's perspective". There is
    > someone who posts frequently online whose signature includes something
    > to the effect that "engineering is making what you want out of what you
    > can get", which seems to be our lot in life as entities travelling an
    > informationally imperfect world.
    >
    > My general question to critics of fuzzy logic in general is: what is
    > wrong with using fuzzy logic if it provides useful results? Please note
    > that I did not write "optimal" or "theoretically satisfying" results.
    > While I have not studied these issues obsessively, I do tend to agree
    > with the fuzzy critics' general complaint that too much has been made of
    > fuzzy logic. On the other hand, people have built fuzzy systems that
    > work, that is, which solve the problems for which they were intended.
    > To me, it seems that issues like whether they could have been built
    > using some other formalism (be it probability or somthing else) are less
    > important than issues of economy and effectiveness."
    >
    > Herman Rubin asked:
    > "Tell me how to get results."
    >
    > I am not sure what you are asking. The construction of fuzzy logic
    > systems is well-described in the literature and I'd refer you to Earl
    > Cox's "The Fuzzy Systems Handbook", but I suspect you're asking about
    > somthing else?
    >
    >
    > Herman Rubin continues:
    > "How does fuzzy logic contribute to getting a consistent scheme of action?
    >
    > Can you elaborate on what you mean by a "consistent scheme of action"?
    >
    >
    > Herman Rubin continues:
    > "Expectation derived from probability does this. Consistent action has
    > been shown to force probability."
    >
    > Herman Rubin, in another message wrote:
    > "Complete a "fuzzy" approach in a consistent way, and only probability
    > can result."
    >
    > If you are asserting that fuzzy logic, if implemented in some
    > appropriate manner must collapse to probability, then you may be right.
    > I don't know. But I am not clear on why this would imply that actual
    > fuzzy logic systems can't work.
    >
    > Will Dwinnell
    > predictor@dwinnell.com
    >
    >
    >
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