RKHS-based functional nonparametric regression for sparse and irregular longitudinal data

This paper focuses on sparse and irregular longitudinal data with a scalar response. The predictor consists of sparse and irregular observations on predictor trajectories, potentially contaminated with measurement errors. For this type of data, Yao, Müller, & Wang (2005a) proposed a principal components analysis through conditional expectation (PACE) approach, which is capable of predicting each predictor trajectory based on sparse and irregular observations. Nonparametric functional data analysis provides an attractive alternative due to its high flexibility. Early work includes functional additive models as in Müller & Yao (2008) and Ferraty & Vieu (2006), which are mainly based on kernel smoothing methods. In this work, we propose a new functional nonparametric regression framework based on reproducing kernel Hilbert spaces (RKHS). The proposed method involves two steps. The first step is to estimate each predictor trajectory based on sparse and irregular observations using PACE. The second step is to conduct a RKHS-based nonparametric regression using the estimated predictor trajectories. Our approach shows improvement over existing methods in simulation studies as well as in a real data example.