zdefuzzified

Z m(z)dz

The discrete version of this, when the membership functions are given as a

discrete set of values, is to sum the products of the arguments and their grade of

membership, and divide this by the sum of the grades of membership:

P

i zi m(zi )

¼P

zdefuzzified ¼ zdefuzzified (7:9)

i m(zi )

In the case where the membership functions are singletons, equation (7.9) becomes

a very fast and simple calculation.

7.5 NON-NUMERIC DISCRETE FUZZY SETS

In “What?” problems, the output will usually be non-numeric: A classi¬cation such

as a disease in problems of diagnosis, a speci¬c trouble as in problems of trouble

shooting, identi¬cation of an unknown object, or recommendation of a course of

action. There are two convenient ways to represent such an output internally: as a

discrete fuzzy set of possibilities, or as a character string to be presented to the

user. Representation as a discrete fuzzy set has advantages. Fuzzy sets lend them-

selves well to representation of ambiguities and contradictions, and numeric

measures are available for the degrees of fuzziness and ambiguity in a discrete

fuzzy set, helping to evaluate how certain we are that a conclusion we have

reached is unique and supported by the evidence.

There are rules whose syntax seems at ¬rst to be virtually the same as the

typical fuzzy control rule (7.6), but used for classi¬cation, and which do not

involve defuzzi¬cation at all. In such rules, the consequent discrete fuzzy set

is a fuzzy set of classi¬cations, perhaps preliminary classi¬cations. For example,

here is a sample from a simpli¬ed program for classi¬cation of the famous

Iris data of Fisher (1936).

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123

7.5 NON-NUMERIC DISCRETE FUZZY SETS

declare Data

N int PL flt PW flt SL flt SW flt;

declare Iris

N int

PetalL fzset (setosa versicolor virginica)

PetalW fzset (setosa versicolor virginica)

SepalL fzset (setosa versicolor virginica)

SepalW fzset (setosa versicolor virginica)

species fzset (setosa versicolor virginica);

In this case PL, PW, SL, and SW are the measured length and width of petal and

sepal, and PetalL, PetalW, SepaL, and SepalW, are linguistic variables with three

members, one for each of the possible species classi¬cations. Of course, member-

ship functions must be de¬ned; the linguistic term setosa will identify one member-

ship function for petal length, one for petal width, one for sepal length and one for

sepal width. Species is a discrete fuzzy set of classi¬cation; it is non-numeric, and

has no membership functions attached.

Our ¬rst classi¬cation rule fuzzi¬es the data, just as in fuzzy control:

rule block 0 (goal Fuzzify input data)

IF (in Data N = <N> AND PL = <PL> AND PW = <PW> AND SL =

<SL> AND SW =

<SW>)

(in Iris N = <N>)

THEN

fuzzify 2 PetalL <PL>,

fuzzify 2 PetalW <PW>,

fuzzify 2 SepalL <SL>,

fuzzify 2 SepalW <SW>;

Now, we are ready to classify with three rules, one for each candidate species:

rule block 1 (goal classify as setosa)

IF (in Iris PetalL is setosa AND PetalW is setosa AND

SepalL is setosa AND SepalW is setosa)

THEN

in 1 species is setosa;

rule block 1 (goal classify as virginica)

IF (in Iris PetalL is virginica AND PetalW is virginica

AND SepalL is virginica AND SepalW is virginica)

THEN

in 1 species is virginica;

rule block 1 (goal classify as versicolor)

IF (in Iris PetalL is versicolor AND PetalW is versicolor

AND SepalL is versicolor AND SepalW is versicolor)

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124 INFERENCE IN A FUZZY EXPERT SYSTEM II

THEN

in 1 species is versicolor;

When these rules have ¬red, discrete fuzzy set species will hold the grades of

membership assigned to each species. Since the classi¬cations are non-numeric,

no defuzzi¬cation is required, or even possible.

7.6 DISCRETE FUZZY SETS: FUZZINESS, AMBIGUITY, AND

CONTRADICTION

When we desire to output results as a discrete fuzzy set of possibilities, it is most

often the case that more than one fuzzy set member will have an appreciably non-

zero grade of membership. For example, in the iris classi¬cation problem described

above, our output fuzzy set might be

Fuzzy Set Species Grade of Membership

(7:10)

Setosa 0:024

Versicolor 0:895

Virginica 0:910

Or, if we want to describe the speed of a car:

Fuzzy Set Speed Grade of Membership

(7:11)

Slow 0:024

Medium 0:895

Fast 0:910

In (7.10), the classi¬cations are mutually exclusive; the specimen must be one

species only. A plant cannot belong to two species at the same time. We have, there-

fore, a contradiction between Versicolor and Virginica, which we will have to

resolve one way or another.

In (7.11), however, it is quite possible for a car to share the characteristics of speed

Medium and speed Fast. On an expressway a car traveling just below the speed limit

of 65 mph might be considered to be going Fast, but a state trooper would probably

consider that speed to be Medium. We have here not a contradiction, but an ambiguity.

7.6.1 Fuzziness and Ambiguity

We might want to know quantitatively to what extent the members of the fuzzy set fail

to have crisp memberships, either 0 or 1. We now measure fuzziness, the extent to

which a fuzzy set is not crisp. First, we present a very simple measure of fuzziness:

X

fuzziness1 ¼ (1 À abs(2mi À 1)) (7:12)

i

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