Structured functional principal component analysis

Abstract

Motivated by modern observational studies, we introduce a class of functional models that expand nested and crossed designs. These models account for the natural inheritance of the correlation structures from sampling designs in studies where the fundamental 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 high-dimensional data. Methods are used in applications including high-frequency accelerometer data for daily activity, pitch linguistic data for phonetic analysis, and EEG data for studying electrical brain activity during sleep.

Keywords: Functional linear mixed model; Functional principal component analysis; Latent process; Multilevel correlation structure; Variance component.

Figure 1

Activity intensity measurements over 5 days for 4 subjects. Original data contain 10 observations per second. This plot shows the average of activity intensity in non-overlapping 15 minute intervals for improved display clarity.

Figure 2

F0-contours for 3 words (triangles for word 2, dots for word 3, and multiplication signs for word 4) spoken by 3 speakers (the lightest color for speaker “a”, a slightly darker color for speaker “c”, and the darkest color for speaker “g”). Each contour was measured at 11 equal distant time points within a vowel (“ə”, “a”, “e”, “i”, or “u”) when a particular word was spoken by one of the eight speakers. Every word was repeated under three different contexts.

Figure 3

The estimated eigenfunctions for three latent processes over 100 simulations when σ 0.5 are shown in gray (we randomly plot 50 out of 100 estimates). The true eigenfunctions are displayed in black curves.=The first- and second-level hierarchies X and U are captured by two sets of trigonometric basis. The third-level process W is polynomial. Within each process, the first few eigenfunctions in correspondence with larger percentages of variance explanation are better estimated than the later eigenfunctions. The eigenfunctions for W are better estimated than X and U because we observe more levels of independent realizations for W.

Figure 4

Principal components for process X_i(t), Z_j(t), and U_ijk(t) using two-way crossed model with sub-sampling (C2s) to analyze the phonetic data. The top row show the first four PCs for the speaker-specific effect X_i(t), while the second row display PCs for word effect Z_j(t). The proportion of variation explained by every principal component within each latent process is listed inside the plotting window. The estimated percentages of total variation explained by the latent processes are shown in front of the rows.

Figure 5

Principal components for process X_i(t), U_ij(t), and W_ijk(t) using three-way nested model (N3) to analyze the accelerometer data. The proportion of variation explained by each PC component is listed in the plots. The top row show the first four PC components for the patient-specific effect X(t), the second row display results for the day-specific effect U(t) and the third row are estimated principal components for hour-specific effect W(t). The proportion of variation explained by each latent process is labeled on the left side.