  With the advent of continuous health monitoring via wearable devices, users
now generate their unique streams of continuous data such as minute-level
physical activity or heart rate. Aggregating these streams into scalar
summaries ignores the distributional nature of data and often leads to the loss
of critical information. We propose to capture the distributional properties of
wearable data via user-specific quantile functions that are further used in
functional regression and multi-modal distributional modelling. In addition, we
propose to encode user-specific distributional information with user-specific
L-moments, robust rank-based analogs of traditional moments. Importantly, this
L-moment encoding results in mutually consistent functional and distributional
interpretation of the results of scalar-on-function regression. We also
demonstrate how L-moments can be flexibly employed for analyzing joint and
individual sources of variation in multi-modal distributional data. The
proposed methods are illustrated in a study of association of
accelerometry-derived digital gait biomarkers with Alzheimer's disease (AD) and
in people with normal cognitive function. Our analysis shows that the proposed
quantile-based representation results in a much higher predictive performance
compared to simple distributional summaries and attains much stronger
associations with clinical cognitive scales.
