Archetypal analysis approximates data by means of mixtures of actual extreme cases (archetypoids) or archetypes, which are a convex combination of cases in the data set. Archetypes lie on the boundary of the convex hull. This makes the analysis very sensitive to outliers. A robust methodology by means of M-estimators for classical multivariate and functional data is proposed. This unsupervised methodology allows complex data to be understood even by non-experts. The performance of the new procedure is assessed in a simulation study, where a comparison with a previous methodology for the multivariate case is also carried out, and our proposal obtains favorable results. Finally, robust bivariate functional archetypoid analysis is applied to a set of companies in the S&P 500 described by two time series of stock quotes. A new graphic representation is also proposed to visualize the results. The analysis shows how the information can be easily interpreted and how even non-experts can gain a qualitative understanding of the data.

Archetype Analysis (AA) is a statistical technique that describes individuals of a sample as a convex combination of certain number of elements called Archetypes, which in turn, are convex combinations of the individuals in the sample. For it's part, Archetypoid Analysis (ADA) tries to represent each individual as a convex combination of a certain number of extreme subjects called Archetypoids. It is possible to apply these techniques to functional data applying a basis expansion function and performing AA or ADA to the weighted coefficients in the basis.This document presents an application of Functional Archetypoids Analysis (FADA) to financial time series. The starting time series consists of daily equity prices of the SP500 stocks. From it, measures of volatility and profitability are generated in order to characterize listed companies. These variables are converted into functional data through a Fourier basis expansion function and bivariate FADA is applied. By representing subjects through extreme cases, this analysis facilitates the understanding of both the composition and the relationships between listed companies. Finally, a cluster methodology based on a similarity parameter is presented. Therefore, the suitability of this technique for this kind of time series is shown, as well as the robustness of the conclusions drawn.