Bootstrapping functional data: a study of distributional property of sample eigenvalues

Modern computer technology has facilitated the presence of high-dimensional data, whose graphical representations are curves, images or shapes. Because of the high-dimensionality, a dimension reduction such as functional principal component analysis or singular value decomposition is often employed. By using functional principal component analysis, a set of observed high-dimensional data can be decomposed into functional principal components and their uncorrelated principal component scores, that is, ft(xi) = f̄(xi) + ∞ ∑ k=1 ξt,kφk(xi), t = 1, . . . , n, i = 1, . . . , p, where f̄(xi) is the sample mean, ξt,k is the kth principal component score of observation t, and φk(xi) is the kth functional principal component observed at data point {x1, . . . , xp}. With the aim of investigating distributional property of sample eigenvalues, we present two bootstrap methods for resampling functional data. The difference between these two techniques stem from the difference of resampling principal component scores. The bootstrap procedures can be briefly summarized as follows: 1. Hold the mean f̄(xi) and φk(xi) fixed at their realizations. 2. This step differs between two techniques: (a) For t = 1, . . . , n, generate bootstrap replication {ξ∗ t,1, . . . , ξ∗ t,K} by sampling with replacement from {ξt,1, . . . , ξt,K}. (b) Since each set of the principal component scores follows a standard normal distribution asymptotically, generate bootstrap replication {ξ∗ 1 , . . . , ξ∗ K} by sampling from independent identically distributed (i.i.d) standard multivariate normal distribution. 3. Construct the bootstrap sample {f b 1(xi), . . . , f b n(xi)} from the bootstrapped principal component scores. The proposed two bootstrap procedures are applied to i.i.d functional data in simulation and to dependent functional data in empirical example. As a result, generating principal component scores from N(0, 1) produces more accurate bootstrap accuracy than resampling principal component scores, and thus more accurately mimicking the behavior of sample eigenvalues and the amount of explained variation.