Robust functional principal components for sparse longitudinal data

Abstract: In this paper we review existing methods for robust functional principal component analysis (FPCA) and propose a new method for FPCA that can be applied to longitudinal data where only a few observations per trajectory are available. This method is robust against the presence of atypical observations, and can also be used to derive a new non-robust FPCA approach for sparsely observed functional data. We use local regression to estimate the values of the covariance function, taking advantage of the fact that for elliptically distributed random vectors the conditional location parameter of some of its components given others is a linear function of the conditioning set. This observation allows us to obtain robust FPCA estimators by using robust local regression methods. The finite sample performance of our proposal is explored through a simulation study that shows that, as expected, the robust method outperforms existing alternatives when the data are contaminated. Furthermore, we also see that for samples that do not contain outliers the non-robust variant of our proposal compares favourably to the existing alternative in the literature. A real data example is also presented.