Functional archetype and archetypoid analysis

Abstract

Archetype and archetypoid analysis can be extended to functional data. Each function is approximated by a convex combination of actual observations (functional archetypoids) or functional archetypes, which are a convex combination of observations in the data set. Well-known Canadian temperature data are used to illustrate the analysis developed. Computational methods are proposed for performing these analyses, based on the coefficients of a basis. Unlike a previous attempt to compute functional archetypes, which was only valid for an orthogonal basis, the proposed methodology can be used for any basis. It is computationally less demanding than the simple approach of discretizing the functions. Multivariate functional archetype and archetypoid analysis are also introduced and applied in an interesting problem about the study of human development around the world over the last 50 years. These tools can contribute to the understanding of a functional data set, as in the classical multivariate case.