A robust smoothed approach to functional canonical correlation analysis

Abstract

In recent years, data collected in the form of functions or curves received considerableattention in fields such as chemometrics, image recognition and spectroscopy, amongothers. These data are known in the literature as functional data, see [3] for a completeoverview. Functional data are intrinsically infinite–dimensional and, as mentioned forinstance in [4], this infinite–dimensional structure is indeed a source of information. Forthat reason, even when recorded at a finite grid of points, functional observations shouldbe considered as random elements of some functional space more than multivariateobservations. In this manner, some of the theoretical and numerical challenges posed bythe high dimensionality may be solved. This framework led to the extension of someclassical multivariate analysis concepts, such as dimension reduction techniques, to thecontext of functional data, usually through some regularization tool.In this talk, we will focus on functional canonical correlation analysis, where data consistof pairs of random curves and the analysis tries to identify and quantify the relationbetween the observed functions. Under a Gaussian model, [2] showed that the naturalextension of multivariate estimators to the functional scenario fails, motivating theintroduction of regularization techniques which may combine smoothing through apenalty term and/or projection of the observed curves on a finite–dimensional linearspace generated by a given basis, see [1] and [3]. The classical estimators use the Pearsoncorrelation as measure of the association between the observed functions and for thatreason they are sensitive to outliers.To provide robust estimators for the first functional canonical correlation and directions,we will introduce two families of robust consistent estimators that combine robustassociation and scale measures with basis expansion and/or penalizations as a regularization tool. Both families turn out to be consistent under mild assumptions. Wewill present the results of a numerical study that shows that, as expected, the robustmethod outperforms the existing classical procedure when the data are contaminated Areal data example will also be presented.