Recovering the underlying trajectory from sparse and irregular longitudinal data

Abstract

In this article, we consider the problem of recovering the underlying trajectory when the longitudinal data are sparsely and irregularly observed and noise‐contaminated. Such data are popularly analyzed with functional principal component analysis via the principal analysis by conditional estimation (PACE) method. The PACE method may sometimes be numerically unstable because it involves the inverse of the covariance matrix. We propose a sparse orthonormal approximation (SOAP) method as an alternative. It estimates the optimal empirical basis functions in the best approximation framework rather than eigen‐decomposing the covariance function. The SOAP method avoids estimating the mean and covariance function, which is challenging when the assembled time points with observations for all subjects are not sufficiently dense. The method also avoids the inverse of the covariance matrix, hence the computation is more stable. It does not require the functional principal component scores to follow the Gaussian distribution. We show that the SOAP estimate for the optimal empirical basis function is asymptotically consistent. The finite‐sample performance of the SOAP method is investigated in simulation studies in comparison with the PACE method. Our method is demonstrated by recovering the CD4 percentage curves from sparse and irregular data in a multi‐centre AIDS cohort study.

Nonlinear prediction of functional time series

“…It is common in applications that individual curves are not fully observed. Nie et al (2022) proposed a sparse orthonormal approximation method to recover the underlying trajectories from sparse and irregular functional data.…”

Automatic search intervals for the smoothing parameter in penalized splines

“…Penalized splines are flexible and popular tools for estimating unknown smooth functions. They have been applied in many statistical modeling frameworks, including generalized additive models (Spiegel et al 2019;Cao 2012), single-index models (Wang et al 2018), functional single-index models (Jiang et al 2020), generalized partially linear single-index models (Yu et al 2017), functional mixed-effects models (Chen et al 2018;Cao and Ramsay 2010;Liu et al 2017), survival models (Orbe and Virto 2021;Bremhorst and Lambert 2016), trajectory modeling for longitudinal data (Koehler et al 2017;Andrinopoulou et al 2018;Nie et al 2022), additive quantile regression models (Muggeo et al 2021;Sang and Cao 2020), varying coefficient models (Hendrickx et al 2018), quantile varying coefficient models (Gijbels et al 2018), spatiotemporal models (Minguez et al 2020;Goicoa et al 2019), spatiotemporal quantile and expectile regression models (Franco-Villoria et al 2019;Spiegel et al 2020).…”