# Robust principal components for irregularly spaced longitudinal data

Consider longitudinal data x_ij, with i=1,...,n and j=1,...,p_i, where x_ij is the j-th observation of the random function X_i( .) observed at time t_j. The goal of this paper is to develop a parsimonious representation of the data by a linear combination of a set of q smooth functions H_k( .) (k=1,..,q) in the sense that x_ij≈μ_j+∑_k=1^qβ_kiH_k( t_j) , such that it fulfills three goals: it is resistant to atypical X_i's ('case contamination'), it is resistant to isolated gross errors at some t_ij ('cell contamination'), and it can be applied when some of the x_ij are missing ('irregularly spaced' ---or 'incomplete'-- data). Two approaches will be proposed for this problem. One deals with the three goals stated above, and is based on ideas similar to MM-estimation (Yohai 1987). The other is a simple and fast estimator which can be applied to complete data with case- and cellwise contamination, and is based on applying a standard robust principal components estimate and smoothing the principal directions. Experiments with real and simulated data suggest that with complete data the simple estimator outperforms its competitors, while the MM estimator is competitive for incomplete data. Keywords: Principal components, MM-estimator, longitudinal .data, B-splines, incomplete data.

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