7.6 DISCRETE FUZZY SETS: FUZZINESS, AMBIGUITY, AND CONTRADICTION

The fuzziness measure returns the effective number of fuzzy set members that have

complete fuzziness, that is, grade of membership 0.5. For example, the fuzziness of

{0:5, 0:5, 0:5} is 3; the fuzziness of {0, 0:5, 1} is 1 and the fuzziness of

{0:75, 0:75, 0:25} is 1.5. For the two fuzzy sets just above, the fuzziness is 0.436.

A sounder fuzziness measure that is based on information theory requires normaliza-

tion of grades of membership to a sum of 1:

m

m0i ¼ P i

i mi

! (7:13)

X

(mi log mi À (1 À mi ) À log(1 À mi ))

fuzziness ¼ exp À

i

Note that fuzziness does not measure how decisively the grades of membership in a

fuzzy set point to one and only one member; instead, it measures how sure we are

of the various degrees of membership. For measures of our ability to distinguish

one valid member from others, we have to consider ambiguity.

7.6.2 Ambiguities and Contradictions

We now consider the extent to which more than one member of the output fuzzy set

has a non-zero grade of membership; in other words, the effective number of

members to which the memberships point. Of course, an ambiguity of one is

great; only one member can be considered to be valid.

Ambiguity can be measured in a similar fashion to fuzziness. We present a simple

measure of ambiguity corresponding to the measure of fuzziness in (7.12). For this

simple measure of ambiguity, the ¬rst step is to determine the maximum grade of

membership max(m). We then normalize the original grades of membership to a

maximum of one:

mi

m0i ¼ (7:14)

max(mi )

The total ambiguity is then simply the sum of the normalized grades of membership:

X mi

ambiguity ¼

max(mi )

i

(7:15)

X

(m0i )

ambiguity ¼

Examples of measured fuzziness and ambiguity are given in Figure 7.1. The

ambiguity measure returns the effective number of fuzzy set members that cannot

be distinguished from each other as the best choice. For example, the ambiguity

of {1, 1, 0) is 2; the ambiguity of {0:5, 0:5, 0) is 2; and the ambiguity of

{0:9, 0:1, 0:1) is 1.22. For the two fuzzy sets (7.1, 7.2) the ambiguity is 0.436.

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126 INFERENCE IN A FUZZY EXPERT SYSTEM II

TABLE 7.1 Examples of Fuzziness and Ambiguity

Fuzzy set grades of membership: f0, 1g: fuzziness 0; ambiguity 1

Fuzzy set grades of membership: f0.5, 0.5g: fuzziness 2; ambiguity 2

Fuzzy set grades of membership: f0.25, 0.75g: fuzziness 1; ambiguity 1.333

Let us look again at the two fuzzy sets in Section 7.6:

Fuzzy Set Species Grade of Membership

Setosa 0.024

Versicolor 0.895

Virginica 0.910

Fuzzy Set Speed Grade of Membership

Slow 0.024

Medium 0.895

Fast 0.910

While there is no difference in the mathematics of these two identical sets, their

interpretation and how we handle them is quite different. Note that for a single

sample, valid memberships in the members of fuzzy set species are mutually exclu-

sive; only one correct membership can be assigned to a single sample. If (say) we

have similar high grades of membership in Versicolor and Virginica, we have a

contradiction: They cannot both be true. However, for a single sample, the member-

ships in fuzzy set speed are not mutually exclusive. It is quite likely that a single

speed measurement would not correspond exactly to our concepts of speed as

Slow, Medium, or Fast, and might be (e.g.) half-way between what we think of as

Slow and what we think of as Medium. In that case, since more or less equally

high grades of membership in Slow and Medium would be quite acceptable, we

now have an ambiguity rather than a contradiction.

7.6.3 Handling Ambiguities and Contradictions

Retaining Ambiguities. The action we take with regard to ambiguities and contradic-

tions is quite different. In general, there are several reasons to retain ambiguities and

not try to reduce them to a single valid member. Suppose that in fuzzy set speed, we

had membership of Slow 0.56, Medium 0.52, and Fast 0, and that we had decided to

resolve the multiple grades of memberships by retaining only the largest. We would

now have Slow 0.56, Medium 0, and Fast 0. Now we transmit an inaccurate picture

to later reasoning stages or to the user; if we were then to defuzzify speed, we

would get again quite an inaccurate value. Later rules that should ¬re if (say)

Medium were at least 0.10 would now fail to ¬re, perhaps leading us to a catastro-

phically wrong ¬nal result, such as failing to brake suf¬ciently hard and smashing

into the car ahead of us. Similarly, with classi¬cation problems. It is not at all

unusual to have more than one preliminary classi¬cation with respectable truth

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127

7.7 INVALIDATION OF DATA: NON-MONOTONIC REASONING

values for the same object; and it infrequently happens that the preliminary classi-

¬cation with the highest truth value turns out to be incorrect. So, we need have no

fear of ambiguities; they often lend robustness to a line of reasoning.

Resolving Contradictions. Contradictions, however, should be resolved if possible

when they arise. Of course, the simplest method is to take the highest grade of mem-

bership as the only valid one, and if two grades of membership are identical, poss-

ibly to spin a random number to decide. This can clearly lead us to results that are at

least suspicious if not downright wrong. It is better ¬rst to detect whether the grades

of membership are appreciably contradictory; the measure of ambiguity given in

(7.14) and (7.15) above is one suitable way to do this. If we ¬nd that contradictions

can occur (which is usually the case), we have a choice of several ways in which to

proceed. In any case, we must recognize that the rules we have written so far have

not produced ¬nal results, but have produced preliminary results.

We should always review the rules and membership functions we have used so

far to see if we can produce better preliminary results. It is conceivable, but unlikely,

that this step will solve our problem.

The next step is to see if we can use or acquire additional data to distinguish

among the contradictions. For example, an image analysis program for ultrasound

images of the heart to detect the various heart regions such as left atrium and

right ventricle initially classi¬es the regions based on region area and position in

the image. While this detects a lot of regions correctly, it is virtually guaranteed

to produce contradictions, since the rules to detect a ventricle and merged atrium

and ventricle (mitral valve open) are identical!

The succeeding steps resolve contradictions. For example, suppose a rule has

classi¬ed a region as both left ventricle (LV) and merged left ventricle and left

atrium (LA þ LV). We now look to see if in the same frame a region has been classi-

¬ed as left atrium (LA). If this is true, clearly the classi¬cation as LA þ LV is wrong,

and the classi¬cation as LV is correct. If, however, no region has been classi¬ed as

LA, the classi¬cation as LV is wrong and LA þ LV is correct.

It is true that the rules for determining preliminary results might possibly be

written so as to simultaneously rule out contradictions, but this would make the

rules more dif¬cult to write, to debug, and to maintain, and is not advisable.

If our efforts to resolve a contradiction fail, we should report this to the user (or later

stages of the program) so that the data are available for determining a course of action.