  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.
