\"Size\" and \"shape\" in the measurement of multivariate proximity

1. Ordination and clustering methods are widely applied to ecological data that are nonnegative, for example species abundances or biomasses. These methods rely on a measure of multivariate proximity that quantifies differences between the sampling units (e.g. individuals, stations, time points), leading to results such as: (i) ordinations of the units, where interpoint distances optimally display the measured differences; (ii) clustering the units into homogeneous clusters; or (iii) assessing differences between pre-specified groups of units (e.g., regions, periods, treatment-control groups). 2. These methods all conceal a fundamental question: To what extent are the differences between the sampling units, computed according to the chosen proximity function, capturing the \"size\" in the multivariate observations, or their \"shape\"? \"Size\" means the overall level of the measurements: for example, some samples contain higher total abundances or more biomass, others less. \"Shape\" mea...