A note on a fuzzy measure of typicality

In many applications of the emerging discipline of intelligent information analysis the issue of determining a typical (prototypical) object from a collection of objects arises. This question is of particular importance in development of knowledge discovering systems.1 The idea of a typical value also plays an important role in commonsense reasoning systems where typical values are often used as default values.2 The usual context in which we desire to determine a typical value involves a situation in which we have a collection of observations and are using these observations as a basis to try find if there is some typical value characterizing this collection of observations. Informally speaking a typical value is a value that is the same or very similar to most of the observations in the data we are trying to typify. In the following we shall conjecture an approach to validating typical values based on a collection of data. This approach will allow us to provide a measure of typicality for any suggested typical element. Furthermore, rather than restricting typical values to being crisp elements we shall allow for typical values to be linguistic values represented as fuzzy subsets. We note that the issue of typicality has been investigated by other researchers such as Yager,3–5 Zadeh,6 Dubois and Prade,7 and Kandel.8,9 A closely related concept, in purely numeric domains, is the mean value. However, we note a fundamental problem with any attempt at using the mean to determine the typical values of a collection of data. First we note that a mean value need not be a typical value. Consider a data set consisting of 20 readings, 10 of which are 100 and the other 10 are zero. In this case the mean value is 50, however we note this is not a typical value. The reason for it not being a typical value is that it is not similar to most of the elements in the data. As a matter of fact this example illustrates an important characteristic of typical values, one