  Motivated by modern observational studies, we introduce a class of functional
models that expands nested and crossed designs. These models account for the
natural inheritance of correlation structure from sampling design in studies
where the fundamental sampling unit is a function or image. Inference is based
on functional quadratics and their relationship with the underlying covariance
structure of the latent processes. A computationally fast and scalable
estimation procedure is developed for ultra-high dimensional data. Methods are
illustrated in three examples: high-frequency accelerometer data for daily
activity, pitch linguistic data for phonetic analysis, and EEG data for
studying electrical brain activity during sleep.
