S-estimators for functional principal component analysis

A well-known property of functional principal components is that they provide the best q-dimensional approximation to random elements over separable Hilbert spaces. Our approach to robust estimates of principal components for functional data is based on this property since we consider the problem of robustly estimating these finite-dimensional approximating linear spaces. We propose a new class of estimators for principal components based on robust scale functionals by finding the lower dimensional linear space that provides the best prediction for the data. In analogy to the linear regression case, we call this proposal S-estimators. This method can also be applied to sparse data sets when the underlying process satisfies a smoothness condition with respect to the functional associated with the scale defining the S-estimators. The motivation is a problem of outlier detection in atmospheric data collected by weather balloons launched into the atmosphere and stratosphere.