Distance Based Methods
J. S. Marron, Ian L. Dryden · 2021
This chapter considers many aspects of distance-based methods of statistical analysis. Several common metrics, i.e. distances, are compared on the basis of relative properties of their Fréchet means. An overview of Multi-Dimensional Scaling shows how this method gives data object representations similar to PCA for the Euclidean metric, and extends the scores visualization aspect of PCA to the case of many other metrics. Several important examples of distances are discussed, such as the Wasserstein, i.e. Earth Mover&s;s, i.e. Mallows, Metric in the context of Functional Data Analysis, and the Procrustes, and Generalized Procrustes approaches to landmark-based shape analysis. Choice of metric is seen to have a major impact in the case of covariance matrices as data objects via visualization of geodesic paths.