number of membership functions...
Sat, 6 Jun 1998 19:18:54 +0200 (MET DST)

Hasan wrote ;

>My first question:
>I remember reading a passage about the number of
>membership functions (MFs) that should be used
>(or recommended) during fuzzy reasoning, in Constantin
>von Altrock's book. The recommended number was about
>(if I'm not wrong) 7 for each variable. The reason
>for using such a low number was that humans can use
>only few linguistic terms simultaneously during their
>reasoning or decision making processes.
>Wouldn't using more MFs increase the precision of the
>fuzzy system? Suppose we are provided with sufficient
>amount of (training) data to generate a rule-base that
>could cover tens (or hundreds) of MFs. Then why limit
>ourselves with few MFs?

On theory basis, it is expected that when you increase the number of mfs, precision should increase till the point where all mfs are singletons. Then you loose ;
- the compact caracter of a fuzzy system. You obtain a different behavior for every point of your univers, which becomes a huge big classical expert rule base. Take a look to Kosko's truck and trailer which is controlled with only 35 rules !
- too much precision means too much rules to branch. Contrary to a classical expert system, a fuzzy system evokes all the rules all the time. You increase the dimension of rule base, you get a bigger base with much more rules to activate all the times. Most probably the systems reaction time will be affected.
- finally too much precision is contrary to the overall generic caracter of fuzzy logic. Increase the precision, but then for the inference you have min-max, which is very rough.

Actually, part of these questions have been covered within a chapter of my Ph.D. thesis. If you want to discuss more, e-mail me.

>Second question:
>Fuzzy logic is always referred as the technique which
>can deal with linguistic uncertainties. That's why the
>MFs are represented by linguistic terms. This makes it
>easy to incorporate linguistic rules obtained from a
>human expert.
>However, in cases where only numerical data is used to
>generate rules, is it still necessary to label each MF
>by a linguistic term? What if a large number (say 50)
>of MFs is used? Wouldn't it be inconvenient to use a
>separate linguistic term for each MF? Could "numbers"
>instead of linguistic terms be used? In that case a fuzzy
>rule would look like:
>If x1 is "13" and x2 is "35" then y is "46"

As the name implies, a mf is a label. Label is what you prefer to name. If you prefer to stay coherent with human linguistic code, you better use natural - expected - labels. On the contrary, if you prefer to create your own metalinguistic system within your fuzzy structure, whay not, as long as you use the same mathematical basis for reasoning ?

Hope you get the answers,
Kamil Murat Eksioglu
Sherbrooke University, QC, Canada