Title: Unbiased Mixed Variables Distance

Author: Michel van de Velden

Abstract:
Distance between observations for mixed variables (that is, variables that are
either numerical, binary, ordinal or categorical) are of importance in various
statistical applications. For example, visualization methods and cluster
analysis but also classification methods like K-Nearest Neighbours, require an
appropriate definition of distance. Gower's general coefficient of similarity
provides a popular and elegant way to derive distances for mixed variables.
However, Gower's proposal, using  basic settings, leads to a distance in which
the variable types influence the contribution of individual variables to the
overall distance. Although this problem appears to be known, solutions appear
to be ad-hoc, and a systematic overview of properties of Gower's proposal does
not appear to exist. In this paper, we do provide such an overview. Moreover,
to overcome the problem, we define unbiased mixed variables distance as a
distance for which the variable specific contributions are not influenced by
measurement units or scales. We show how such distances can be constructed,
and use it to propose a highly adaptable unbiased mixed variables distance
that can easily be implemented and customized.