Distance measures for ordinal attributes

Many clustering methods use distance measures to determine the similarity or dissimilarity between any pair of objects. It is useful to denote the distance between two instances x_i and x_j as: d(x_i,x_j). A valid distance measure should be symmetric and obtains its minimum value (usually zero) in case of identical vectors. The distance measure is called a metric distance measure if it also satisfies the following properties:

When the attributes are ordinal, the sequence of the values is meaningful. In such cases, the attributes can be treated as numeric ones after mapping their range onto [0,1]. Such mapping may be carried out as follows:

where z_i,n is the standardized value of attribute a_n of object i. r_i,n is that value before standardization, and M_n is the upper limit of the domain of attribute a_n (assuming the lower limit is 1).