17Fuzzy Membership Functions

asymptotic to 0, but not necessarily linear. As an example he offered the function I(short)(<x>) = 1/(1 +x) where x is some quantitative measurement of height:

1

0.8

0.6 Degree of

0.2

0

In Section 14.6 of Chapter 14 we noted that the product Principle of Charity premise for tall heights is false on interpretation SST, and the reason that it is false is that one height in the domain (but not all) is a member of the fuzzy set tall to degree 0. Goguen’s sample function for short does not have a similar problem because 1/(1+x) never reaches 0. So the product Principle of Charity for short heights—( x)( y)((Sx &P Gyx) →P Sy), symbolizing A height that is 1/8 greater than a short height is also short—would have a value strictly greater than 0.

But we have to work to find a specific measure x of heights that gives plausible degrees of membership in short. If we set x to be a height’s excess over 4 7 in 1 units, we have x = 0 for 4 7 , so that 4 7 is short to degree 1 by Goguen’s function— which looks good. But the measure x = 1 for 4 8 , which makes 4 8 short to degree

.5, doesn’t look so good. And if we measure a height’s excess over 4 7 in 1/8 units, the situation gets even worse: x = 8 for 4 8 , which makes that height short to degree 1/9. Moreover, the product Principle of Charity ( x)( y)((Sx &P Gyx) →P Sy) has the value 1/2 here (letting Gyx have the value 1 when y is 1/8 greater than x and the value 0 otherwise)—when v(x) = 4 7 and v(y) = 4 71/8 , the value of (Sx &P Gyx) →P Sy is the ratio 1/2 /1 (and that’s as small as it can get). But 1/2 isn’t “close to true.”

To see that Goguen’s function isn’t that bad, here’s an example of a better way to flesh out the numbers for the short Sorites (and others are certainly possible). Assuming a range of heights from 3 4 to 8 4 , using 1/8 increments, define x for any height h to be h – 3 , expressed as half-foot units. Using Goguen’s function, then, 3 4 is short to degree 1; 3 41/8 is short to degree 48/49; 3 42/8 is short to degree 24/25; 4 4 is short to degree 1/3; 5 4 is short to degree 1/5; 6 4 is short to degree 1/7; 7 4 is short to degree 1/9; and 8 4 is short to degree 1/11. Moreover, the smallest value of the ratio between two successive heights is the ratio between the shortness of 3 41/8 and the shortness of 3 4 , which is 48/49, so that is the smallest value that the conditional can have and is therefore the value that this membership function gives the product Principle of Charity for the predicate short.

Not all vague predicates are best represented with membership functions that are strictly increasing or strictly decreasing. For example, some vague predicates, such as the predicate medium heat as used in the context of cooking (e.g., cook over medium heat for two minutes, stirring constantly), suggest trapezoidal functions

(having trapezoidal shapes when charted). A membership function for medium heat, where the temperature x is measured in degrees Fahrenheit, might be defined as

I(medium heat)(<x>) = 0 if x < 200,

And there are more possibilities. See Pedrycz and Gomide (1998) for a good discussion of the variety of shapes that membership functions can take.

There is another important issue to be addressed in designing membership functions: sometimes degrees of membership are a function of several measurements rather than just one. We’ve considered tallness and shortness to be functions of height and medium-heatedness to be a function of temperature. But consider, for example, the claim that the air on a midsummer day is comfortable. Comfort here is (minimally) a function of temperature and humidity, so in this case fuzzy set membership should be defined as a function of pairs of values consisting of a temperature and a measure of humidity. Another more complicated example requiring several measurements is from Goguen (1998–1999): “A ‘good’ computer should be (at least): cheap; small; fast; reliable; easy to repair; of large storage capacity; inexpensive to run; equipped with good input and output; and very flexible” (p. 352). It is interesting that in this case each of the criteria is itself vague—for instance, what is cheap for a computer? Goguen suggests that we can define a fuzzy membership function over this collection of vague criteria by assigning a weight to each criterion (reflecting that criterion’s relative importance in the concept of a good computer), such that the weights add up to 1, and then adding the weighted membership values under each of the criteria. As an example, given the fuzzy degrees of membership and weights in the chart

the final weighted value for this computer would be (.05 · .8) + (.1 · .9) + (.2 · .5) + (.05 · .3) + (.15 · .8) + (.1 · 1) + (.15 · 1) + (.2 · .9), which is .795—a fairly good computer according to this example.

17.2 Empirical Construction of Membership Functions

Our examples of membership functions characterize 6 7 as a definitely tall height, 310 degrees as close to medium heat, and so on. Where does this information come from? In this text I’ve used my own intuitions to construct examples. But there are more objective ways to construct membership functions. In each case empirical data are collected and then translated into a membership function, perhaps coercing the data to get some specific type of membership curve.

As an example, Max Black proposed a method of establishing what he called a consistency profile for a vague predicate “based on the assumptions that while the vagueness of a word involves variations in its application by the users of the language in which it occurs, such variations must themselves be systematic and obey statistical laws if one symbol is to be distinguished from another” (1937, p. 442). We’ll use the example concept of tallness to explain what a consistency profile is. For each height in our study, we ask each member of some group G of users of the language whether the height should be classified as tall or as not tall (they don’t get a third choice). For each height h, let m(h) be the number of people in G who say that h should be classified as tall and n(h) the number of people in G who say that h should be classified as not tall. Then the consistency profile is the function C(tall) defined as C(tall)(<h>) = m(h) / n(h).1 Now, this profile will give values outside the range [0. .1] and it can also have undefined values (when n(h) = 0), so we need to adjust the profile to produce defined values within the unit interval. Here is one way:

I(tall)(<h>) = m(h) / (m(h) + n(h)),

That is, the percentage of people in G who say that h should be classified as tall. This use of consistency profiles to define fuzzy membership is an example of the horizontal or polling method of determining membership functions. Alternatively, membership values can be determined by direct rating: rather than ask members of G whether a given height h should be classified as tall or as not tall, we ask members of G to tell us the degree to which h is tall. We would then construct a membership function based on the direct ratings by averaging the degrees that members of G gave for each height or by some other method of combining the rankings for each height into a single membership degree.

1Actually, Black went a step further than we have and suggested using the limit that the ratio m(h)/n(h) approaches as the number of heights and the number of members of the ranking group G increases.