Re: fuzzy relations

Jonathan G Campbell (jg.campbell@ulst.ac.uk)
Tue, 28 Apr 1998 15:48:05 +0200 (MET DST)

Damian F Minihan wrote:
>
> hi!
> Can anyone tell me what a fuzzy relation means in layman's terms. That is,
> given two fuzzy sets you can compose a fuzzy relation between them using,
> say, the cartesian product - but what does this actually represent?

I _think_ -- hesitating because mathematicians mey be listening -- a
cartesian product doesn't have to represent anything, other than
(abstractly) a pair, triple, or generally a tuple.

>
> I mean, you could combine a fuzzy set of lamp-post heights with the fuzzy
> set of the weights of african frogs mathematically - but does it have any
> real interpretation?

Normally, one wouldn't have a fuzzy set of ``lamp-post heights'', but
maybe a fuzzy set ``large lamp-post heights'', of which 2 metres would
probably have a membership zero, and 5 metres and above a mebership of
1.0 -- with a smooth transition 0.0--1.0 somewhere in-between, goues
through 0.5 at 4 metres. In the common fuzzy-rule-based systems -- for
control, or for function approximation, one would have a fuzzy partition
of the universe of (all possible)
``lamp-post heights'': say small, medium, large. Let us say that the
average ``lamp-post height'' is 3 metres, then possibly ``medium
lamp-post height'' would be represented by a triangular membership
function which rises from 0.0 at 2 metres to 1.0 at 3 metres and falls
back to zero at 4 metres.

The same can be done for the weights. Let us say that theese are
partitioned as above, merely swapping kilos for metres.

Now, consider yourself as a seller of odd merchandise. Your best
customer rings up an asks for immediate delivery of a long lamp-post and
a medium weight (singing) African frog; you have a 3 kilo frog (A) who
is so attached to a 4 metre lamp-post that this is the only one he/she
will sing under; you also have a 3.25 kilo frog (B) who will sing only
under a 5 metre lamp-post that he/she has grown attached to. You tell
your custormer that you cannot exactly match his order, he says send the
best you can. What do you do?

Frog A has medium weight membership 1.0, ond his 4 metre lamp-post has
large length membership 0.5. Composite membership of ``long lamp-post
and a medium weight'' = 0.5 (using either product or max. in the
composition).

Frog B has 0.75 (at 3.25) and 1.0 (at 5), so the composite membership is
0.75.

Frog B and the 5 metre lamp-post gets shipped.

>
> Also, am I correct in presuming that Prof. Mamdani is a bloke? I am
> preparing a report on the mamdani implication and I don't want to make any
> fundamental cock-ups.

He is.

Hope this helps.

Jon Campbell

-- 
Jonathan G Campbell Univ. Ulster Magee College Derry BT48 7JL N. Ireland 
+44 1504 375367 JG.Campbell@ulst.ac.uk  http://www.infm.ulst.ac.uk/~jgc/