Interpretable Functional Principal Component Analysis

Functional principal component analysis (FPCA) is popularly used to explore major sources of variation in a sample of random curves. The major sources of variation in these curves are represented by the estimated functional principal components (FPCs). The intervals with high values of FPCs are interpreted as where sample curves have major variations. However, these intervals are often hard to be identified by naive users, because of the vague definition of "high value". We develop a novel penalty-based method to derive FPCs that are only non-zero on short intervals, and strictly zero on other intervals. Our derived FPCs are easier to be interpreted: those non-zero intervals are where sample curves have major variations. We propose an efficient algorithm to estimate interpretable FPCs using projection deflation. The estimated interpretable FPCs are shown to be strongly consistent and asymptotically normal under mild conditions. Our simulation studies show that our method can obtain more interpretable FPCs than other FPCA methods, while these FPCs explain similar variations of sample curves as FPCs estimated from other methods. Our method is also demonstrated by analyzing real data in two applications.