Simulation methods for squared Euclidean and Mahalanobis type distances for multivariate data and their application in assessing the uncertainty in hierarchical clustering

This paper extends the Suzuki and Shimodora method [Suzuki R, Shimodora H. Pvclust: an R package for assessing the uncertainty in hierarchical clustering. Bioinformatics. 2006;22:1540–1542] for assessing uncertainty in hierarchical cluster analysis to multivariate datasets and examines the reliability of the simulated cluster probabilities in relation to the simulation method adopted. The extension is applied to squared Euclidean and Mahalanobis-type distances and employs three simulation methods, the Monte-Carlo and bootstrap methods, and a new proposed method, the distance distribution. The distance distribution method is very fast and gives satisfactory predictions for the distance, its standard deviation, skewness, and kurtosis. The performance of the Monte-Carlo method is equally satisfactory, whereas the bootstrap method usually gives acceptable predictions only for the distance. The distance distribution and Monte-Carlo methods give similar cluster probabilities. The bootstrap probabilities are not very different; thus this method can be used in datasets of unknown distance distribution.