Functional quantile principal component analysis.

This paper introduces functional quantile principal component analysis (FQPCA), a dimensionality reduction technique that extends the concept of functional principal components analysis (FPCA) to the examination of participant-specific quantiles curves. Our approach borrows strength across participants to estimate patterns in quantiles, and uses participant-level data to estimate loadings on those patterns. As a result, FQPCA is able to capture shifts in the scale and distribution of data that affect participant-level quantile curves, and is also a robust methodology suitable for dealing with outliers, heteroscedastic data or skewed data. The need for such methodology is exemplified by physical activity data collected using wearable devices. Participants often differ in the timing and intensity of physical activity behaviors, and capturing information beyond the participant-level expected value curves produced by FPCA is necessary for a robust quantification of diurnal patterns of activity. We illustrate our methods using accelerometer data from the National Health and Nutrition Examination Survey, and produce participant-level 10%, 50%, and 90% quantile curves over 24 h of activity. The proposed methodology is supported by simulation results, and is available as an R package.