Re: References on the 'shape' of fuzzy sets?

Martin Brown (mqb@ecs.soton.ac.uk)
Wed, 7 Aug 1996 13:51:23 +0200


Armin Shmilovici wrote:
> > after reading various papers and books on fuzzy logic control, listening to
> > conference presentations, and making the odd experiment myself, I've come to
> > the conclusion that the exact 'shape' of a fuzzy set is much less important
> > than its overall form, size, and position; meaning that one might as well
> > use linear (trapezoidal) functions in cases where these are easier to
> > handle or more efficient in computation (I'm sure there are many
> > exception!)
> Not True --- look
> X.J.Zeng and M.G. Singh, "Approximation Accuracy Analysis of
> Fuzzy Systems as Functions Approximators", IEEE Trans. FS V. 4 N. 1 February
> 1996, pp.44-63
> The conclusions (as I understand them) are:
> 1. Product inferencing is more accurate then min inferencing, since it
> introduces a second order approximation rather than a first order.
> 2. The B-spline basis functions, when used as fuzzy sets will provide the most
> accuracy (when the other conditions are the same).
> 3. Under inaccuracies in the data (e.g. noisy observations), the B-splines are
> not nessecarily the best to use.

OK, since this paper has been cited, Ill respond! When you are using product/sum
fuzzy systems, they generally reduce to a system which is formed from a linear
combination of (normalised) basis functions which!e!te! represent the intersection of your
input terms:
y = \sum_{i=1}^p a_i(x) * w_i
where each a_i is interpreted like
x1 is small AND x2 is medium.
This has been known since 1991/1992 in the context of B-spline fuzzy membership
functions and Gaussians. This is a quite important result 'cause it means that
the shape of the (normalised) input sets is reflected in the decision surface, and
so for the piecewise polynomial B-splines, this results in a multi-variate piecewise
polynomial decision surface whose modelling capabilities and approximation errors
can (and have) been analysed using *standard* spline theory. Results such as being
linear (or even piecewise linear) in one of the inputs, follow on directly from the
definition of B-splines. B-spline neurofuzzy systems are less "pure fuzzy" (Im not
sure what that means), rather they are B-spline networks which have been studied in
numerical analysis for 25 years and can have fuzzy labels attached to their different
parts. The splitting up of the input space into a lattice and ways of reducing this
effect are well-known in the numerical analysis community, and giving this a fuzzy
interpretation is *only* useful if you are going to use the rule-based interpretation
to either initialise or validate the system. In terms of its actual performance for
real-valued input/output modelling, this is determined totally by the form of surface
which depends on the weights w_i and the shape of the fuzzy input sets a_i. Now we've
been using B-splines for about 5 years and often the piecewise linear triangles are the
most conventient (continuous output - piecewise continuous derivative). However, for
recurrent system, we often find that quadratic (k=3) B-splines are better because they
have a continuous first derivative and a piecewise continuous second. Zadeh's PI
quadatic sets (which can be formed from the addition of two quadratic B-spline basis
functions) have a zero gain at their centre (as well as the neighbouring sets) so they
are not appropriate for most modelling and control schemes (if you need a zero gain at
the centre of every set, then they may be appropriate). This is analogous to local
model Gaussian RBF systems (Sugeno models) where an explicit linear model is placed at
the centre of a Gaussian function (if it doesn't overlap too much with its neighbours)
which makes it easier to produce a linear system as a base line solution. So the
message is generally, your system should be capable of modelling linear systems and
the nonlinearities should smoothly switch between them without unnecessary zero gains
or jumps (ie. either quadratic B-splines which are piecewise polynomials, or local
linear model Gaussian functions which are infinitely smooth). These remarks come from
experience and comman-sense reasoning rather than an explicity proof.

Now about the paper you mention, or the several that have been published in the past
year in this area. The authors return to B-splines several times in these papers and
for the remarks you mention about the effect of measurement error on the identification
of the weights. B-spline basis functions (fuzzy sets) are *not* optimal - they have
good numerical properties, they are convenient to design by moving the knots, they
are piecewise polynomials, they are normalised (sum to unity), and for these reasons
we've been looking at them since 1990 as a means for implementing fuzzy sets. However,
they are *not* optimal. The best basis to use depends on the form of the unknown
function and whether this basis satisfies *any* of the fuzzy set properties is unknown.
It really depends on what you mean by optimal. Just because you can bound the modelling
error does not imply optimality. In fact, for some (artificial?) continuous nonlinear
systems, the interplay between max/min and rule confidences etc. will produce a better
model. For an arbitrary (unknown) function, all you can do is talk about flexibility and
smoothess and these are generally competing concepts. Higher order splines are more
flexible and hence they can model the underlying function more accurately, but this
means that they are more susceptable to noise, and vice versa for the lower order
(triangular) splines. However, you can often get away with fewer higher order splines
and this in turn reduces the sensitivity to noise. Its difficult to make meaningful
comparisons between basis of different forms. Bounding second and fourth derivaties
is useful spline theory but to implement it you need to do data fitting and regularisation
to keep your solution smooth especially for basis functions which have a compact support.
You can do this in a constructive manner by adding knots (basis functions) until the
error is less than a pre-specified tolerance and then calculate the regulatisation
coefficient such that the MSE = the tolerance and the decision surface is smoothed.
However, whether or not the added fuzzy sets have any meaning is debatable. Trying
to minimise a MSE/bound the smoothness is not always compatible with giving a
linguistic meaning to your fuzzy system! Its the job of the designer/expert to specify
what they mean by small and to try and intpreted what the odd-shaped B-spline/Gaussian
basis function really means!

Finally, the simplest "fuzzy" system is not a tensor product (AND) of all the inputs,
rather it is the sum (OR) which can be normalised for several complete subnetworks.
This reduces the number of rules to 2n instead of 2^n and (not surprisingly) n-1
parameters are redundant so you effectively model n+1 terms. Sounds like n linear
gain terms + 1 bias term to me!

Abstracts of our papers and some technical reports can be obtained from the publications
part of our web entry at:
http://www-isis.ecs.soton.ac.uk/
and this has some other relevant information (projects, people etc).

As a parting comment, B-splines are a convient formalism for normalised piecewise
polynomial fuzzy sets, however, they are not the only shape.

Martin Brown
ISIS research group,
Southampton University.

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