Clustering via centroids a bag of qualitative values and measuring its inconsistency

It is well understood how to compute the average of a set of numeric values; thus, handling inconsistent measurements is possible. Recently, using confusion, we showed a way to compute the “average,” consensus or centroid of a bag of assertions (made by observers) about a non-numeric property, such as John’s pet. The values of those assertions lie in a hierarchy. Intuitively, such consensus minimizes the discomfort of all observers (of the pet) when they know which of the animals of the bag was selected as the consensus pet. The inconsistency of the bag is such total discomfort divided by the bag’s size. It is a number that tells how far apart the values of the bag are. It should be emphasized that an asserted value obtained by an observer (such as Schnauzer in “the pet was a Schnauzer”) represents not only itself, but all the values from it up to the root of the hierarchy: Schnauzer, dog, mammal, animal, living creature. A bag of dissimilar assertions will have a large inconsistency, which could diminish if the problem at hand allows several centroids to be selected. John could have two pets, and the inconsistency of these two “consensus values” with all observations will be much better (much smaller): one part of the observers will feel little discomfort with one of the centroids; the remaining part will feel little discomfort with the second centroid. This chapter finds the set of centroids of a bag of qualitative values that minimizes the inconsistency of the bag; that is, the total discomfort of all members of the bag will be smallest. These centroids define clusters, that is, subsets of the bag. All observers are equally credible, so differences in their findings arise from perception errors, and from the limited accuracy of their individual findings.