Question

Proximity is typically defined between a pair of objects.
(a) Define two ways in which you might define the proximity among a group of objects.

(b) How might you define the distance between two sets of points in Euclidean space?

(c) How might you define the proximity between two sets of data objects? (Make no assumption about the data objects, except that a proximity
measure is defined between any pair of objects.)

Answer 1

a) Two examples are the following:

(i) based on pairwise proximity, i.e.,minimum pairwise similarity or maximum pairwise dissimilarity, or

(ii) for points in Euclidean space compute a centroid (the mean of all the points—see Section8.2) and then compute the sum or average of the distances of the points to the centroid.

b) One approach is to compute the distance between the centroids of the two sets of points. The centroid is simply the vector sum of all the Euclidean vectors scaled by their number.

c)  One approach is to compute the average pairwise proximity of objects in one group of objects with those objects in the other group. Other approaches are to take the minimum or maximum proximity of all such pairs.