Minkowski distances and standardisation for clustering and classification of high dimensional data

Abstract

There are many distance-based methods for classification and clustering, and for data with a high number of dimensions and a lower number of observations, processing distances is computationally advantageous compared to the raw data matrix. Euclidean distances are used as a default for continuous multivariate data, but there are alternatives. Here the so-called Minkowski distances, $L_1$ (city block)-, $L_2$ (Euclidean)-, $L_3$-, $L_4$-, and maximum distances are combined with different schemes of standardisation of the variables before aggregating them. Boxplot transformation is proposed, a new transformation method for a single variable that standardises the majority of observations but brings outliers closer to the main bulk of the data. Distances are compared in simulations for clustering by partitioning around medoids, complete and average linkage, and classification by nearest neighbours, of data with a low number of observations but high dimensionality. The $L_1$-distance and the boxplot transformation show good results.