NEC Research Institute Technical Report 1991/1992/2002

Normalized Forms for Two Common Metrics

Peter N. Yianilos

Abstract: We show that symmetric set difference and Euclidian distance have particular $[0-1]$ {\em normalized} forms that remain metrics.

The first of these $|A \triangle B| / |A \cup B|$ is easily established and generalizes to measure spaces.

The second applies to vectors in $\bbR^n$ and is given by $\|X-Y\|/(\|X\|+\|Y\|)$. That this is a metric is more difficult to demonstrate and is true for Euclidian distance (the $L_2$ norm) but for no other integral Minkowski metric. The short and elegant proof we give is due to David Robbins and Marshall Buck \cite{RB93}.

We also explore a number of variations.

Keywords: Metric Space, Distance Function, Similarity Function/Coefficient, Euclidian Distance, Association, Clustering, Vector Quantization, Pattern Recognition, Statistical Methods.