We will presume that each curve is observed on a grid of \(T\) points with \(0\leq t_1<t_2\dots<t_T\leq \tau\).
Thus, the raw data set \((X_1,X_2,\dots,X_n)\) of \(n\) observations will consist of an \(n\) by \(T\) data matrix.
By applying the singular value decomposition, \(X_1,X_2,\dots,X_n\) can be decomposed into \(X = ULR^{\top}\),
where the crossproduct of \(U\) and \(R\) is identity matrix.
Holding the mean and \(L\) and \(R\) fixed at their realized values, there are four re-sampling methods that differ mainly by the ways of re-sampling U.
(a) Obtain the re-sampled singular column matrix by randomly sampling with replacement from the original principal component scores.
(b) To avoid the appearance of repeated values in bootstrapped principal component scores, we adapt a smooth bootstrap procedure by adding a white noise component to the bootstrap.
(c) Because principal component scores follow a standard multivariate normal distribution asymptotically, we can randomly draw principal component scores from a multivariate normal distribution with mean vector and covariance matrix of original principal component scores.
(d) Because the crossproduct of U is identitiy matrix, U is considered as a point on the Stiefel manifold, that is the space of \(n\) orthogonal vectors, thus we can randomly draw principal component scores from the Stiefel manifold.