Consider n realizations X_1(t), X_2(t), \cdots, X_n(t) of a square-integrable random function X(t), t \in \mathcal{T} with the mean function \mu(t) = \operatorname{E}(X(t)) and the covariance function G(s, t) = \operatorname{Cov}(X(s), X(t)) . Functional principal component analysis provides methods for the estimation of \mu(t) and G(s, t) in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k-th mode of variation of X(t) can be estimated by
