Re: Obtaining fuzzy membership (was Re: Thomas' Fuzziness and Probability)

From: john v verkuilen (jayv@uiuc.edu)
Date: Sun Aug 12 2001 - 13:39:53 MET DST

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    Robert Ehrlich <bobehrlich@home.com> writes:

    >I have used fuzzy k-means. Aside from the ever present cluster validity problem
    >it seems to seve the purpose. Among other things I used it for a pixel by pixle
    >classification of multispectral images. The nice thing was that the memberships
    >could be expressed a functions of the original image.

    Interesting. I suspect that different methods will develop in application
    areas to fit the sorts of data found there. In my case, datasets are usually
    relatively wide (several variables) and shallow (moderate number of cases).
    Each fuzzy set summarizes the variability in several (three to six) categorical
    items, which can be analyzed subsequently. Dual scaling is designed for
    this sort of situation already. To quote:

    "The notion of 'maturity' plays a rather special role in physical
    anthropometry. Most measurements of growing children, though possibily
    difficult to make in a standard and reproducible manner, do not present
    problems of definition. Maturity, on the other hand, though its general
    meaning is fairly clear, does not possess an obvious definition and in
    particular it is not obvious how it should be measured. What are available
    in practice are a large number of attributes of a growing child which pass
    through several well-defined stages or categories as the child's maturity
    increases...; these include secondary sex characteristics, the teeth and the
    bones of various joints including the wrist and the knee. For any one of these
    attributes it is possible to make unequivocal comparisons between two children:
    child A is more mature than child B for a specific attribute if he, or she,
    exhibits a later occurring stage of that attribute. In practice, of course,
    if child A is more mature than child B for one attribute, he is usually so
    for many others and it is just this fact that leads us to the notion of a
    single underlying maturity value for each child, a value which is reflected
    in the stages of all the different attributes."

    M. J. R. Healy and H. Goldstein (1976). "An approach to the scaling of
    categorized attributes," Biometrika, 63(2), pp. 219-229.

    Fuzzy really only changes some identification constraints and, of course,
    the interpretation of the scores generated.

    I do wish there was more attention payed to this topic--I think in my own
    area (social science) this is one of the barriers to more widespread adoption
    of fuzzy methods.

    Jay

    --
    J. Verkuilen						jayv@uiuc.edu
    "Depend upon it, sir, when a man knows he is to be hanged in a fortnight, it
    concentrates his mind wonderfully."  --Dr. Samuel Johnson
    Dissertation pages written:  62 
    

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