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This study presents the Functional Quantile Principal Component Analysis (FQPCA). This methodology draws on the probabilistic approach for PCA proposed by \cite{Bishop1999} and extends the functional PCA to the quantile regression framework \cite{koencker1978}. This results in a model that describes the full curve-and time-specific probability distribution that underlies individual measurements, estimating smooth, curve-specific quantile functions that are dependent on a set of principal components. The median can be seen as a robust alternative of the mean provided by traditional FPCA, while other quantiles give a more complete understanding of the subject- and time- specific data distribution, and may be particularly useful when distributions are skewed, heteroscedasti or vary across subjects. The necessity for this methodology is demonstrated by our illustrative example: we examine the physical activity level of over 3600 individuals in a single day using accelerometer data from the National Health and Nutrition Examination Survey (NHANES) and are able to compare information from different quantile levels. The proposed methodology is available as a package in R programming language.

Accelerometer data, Functional data, Quantile regression, PCA