Pooled variable scaling for cluster analysis

Motivation Many popular clustering methods are not scale-invariant because they are based on Euclidean distances. Even methods using scale-invariant distances, such as the Mahalanobis distance, lose their scale invariance when combined with regularization and/or variable selection. Therefore, the results from these methods are very sensitive to the measurement units of the clustering variables. A simple way to achieve scale invariance is to scale the variables before clustering. However, scaling variables is a very delicate issue in cluster analysis: A bad choice of scaling can adversely affect the clustering results. On the other hand, reporting clustering results that depend on measurement units is not satisfactory. Hence, a safe and efficient scaling procedure is needed for applications in bioinformatics and medical sciences research. Results We propose a new approach for scaling prior to cluster analysis based on the concept of pooled variance. Unlike available scaling procedures, such as the SD and the range, our proposed scale avoids dampening the beneficial effect of informative clustering variables. We confirm through an extensive simulation study and applications to well-known real-data examples that the proposed scaling method is safe and generally useful. Finally, we use our approach to cluster a high-dimensional genomic dataset consisting of gene expression data for several specimens of breast cancer cells tissue obtained from human patients. Availability and implementation An R-implementation of the algorithms presented is available at https://wis.kuleuven.be/statdatascience/robust/software. Supplementary information Supplementary data are available at Bioinformatics online.

Complex Geometry of the Universal Teichmüller Space. II

We give an alternate and simpler proof of the important theorem stating that all invariant distances on the universal Teichmüller space 𝐓 coincide, and solve for 𝐓 the problem of Kra on isometric embeddings of a disk into Teichmüller spaces.