Multilevel functional quantile principal component analysis

Accelerometer studies typically include repeated 24-h observations over several days and more recently have collected information on participants for weeks, months, or years. Meanwhile, there is growing evidence to suggest that components of physical activity beyond average or expected values are important for quantifying and understanding activity. Motivated by these emerging aspects, we introduce Multilevel Functional Quantile Principal Component Analysis (MFQPCA), a novel dimension-reduction technique that extends Functional Quantile Principal Component Analysis to hierarchical functional data. We examine accelerometry data collected for 4 to 9 months from congestive heart failure patients, where daily activity profiles are nested within individuals and exhibit substantial variability: participants often differ in the timing and intensity of physical activity behaviors across days and from each other, and capturing information beyond expected-value curves is necessary for a robust quantification of diurnal patterns. To this end, MFQPCA decomposes quantile-specific variability into between-participant and within-participant components, estimating shared patterns and corresponding scores at each level and quantile of interest. Our method robustly captures complex distributional features in hierarchical data, disentangles between- and within-participant sources of variability at different quantile levels, and facilitates the study of longitudinal changes in these distributions. We produce participant- and participant-day-level 10%, 30%, 50%, 70% and 90% quantile curves over 24 h, revealing that day-to-day variability dominates sedentary periods, whereas between-participant differences grow in vigorous activity. An efficient alternating minimization algorithm is implemented in the open-source R package FunQ, making MFQPCA readily applicable to a wide range of hierarchical functional data settings.