Object-standardised, normalised abundance data

The chord distance (a metric distance) is a geometric measure equivalent to the Euclidean distance calculated on a data set where the objects (rows) have been standardised to have a length of one. The result of this standardisation is the positioning of objects on the surface of a (hyper)sphere with unit radius. The chord distance is the length of the straight line (the chord) connecting these objects.  In contrast to the Euclidean distance, standardisation makes the chord measure asymmetric. The maximum distance value is √2.

This distance is similar to the chord distance; however, the geodesic distance (greater circle distance) between two objects is the length of the arc on the surface of a (hyper)sphere with unit radius, rather than the chord length.

This measure (the one complement of a similarity measure) focuses on whether objects have dissimilar proportions of variable values. Two objects with different absolute abundances across variables will be identical if they have the same relative proportions of abundance values.