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 classification problem described above, our output fuzzy set might be

Fuzzy Set Species Grade of Membership

Setosa 0:024

Versicolor 0:895

Virginica 0:910

(7:10)

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

Fuzzy Set Speed Grade of Membership

Slow 0:024

Medium 0:895

Fast 0:910

(7:11)

In (7.10), the classifications 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:

fuzziness1¼X

i

(1abs(2m_i1)) (7:12)

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:

m⁰_i¼ m_i

Note thatfuzzinessdoes 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 considerambiguity.

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 first step is to determine the maximum grade of membership max(m). We then normalize the original grades of membership to a maximum of one:

m⁰_i¼ m_i

max(m_i) (7:14)

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

ambiguity¼X

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.

7.6 DISCRETE FUZZY SETS: FUZZINESS, AMBIGUITY, AND CONTRADICTION 125

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 arenot 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 anambiguityrather 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 fire if (say) Medium were at least 0.10 would now fail to fire, perhaps leading us to a catastro-phically wrong final result, such as failing to brake sufficiently hard and smashing into the car ahead of us. Similarly, with classification problems. It is not at all unusual to have more than one preliminary classification with respectable truth

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

values for the same object; and it infrequently happens that the preliminary classi-fication 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 memmem-bership 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 first 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 find 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 final results, but have producedpreliminary 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 classifies 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 classified a region as both left ventricle (LV) and merged left ventricle and left atrium (LAþLV). We now look to see if in the same frame a region has been classi-fied as left atrium (LA). If this is true, clearly the classification as LAþLV is wrong, and the classification as LV is correct. If, however, no region has been classified as LA, the classification as LV is wrong and LAþLV 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 difficult 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.

7.7 INVALIDATION OF DATA: NON-MONOTONIC REASONING