Functional principal component analysis

Functional principal component analysis diagonalizes a compact covariance operator. Its eigenfunctions are principal component functions, and projecting a centered observation onto them gives uncorrelated functional principal component scores.

A principal component function is a normalized eigenfunction ${\varphi }_k$ of a covariance operator, ordered by decreasing eigenvalue ${\lambda }_k$.

The functional principal component score of a centered function $X$ along ${\varphi }_k$ is ${\xi }_k=⟨X,{\varphi }_k⟩$. Its variance is the corresponding covariance eigenvalue ${\lambda }_k$.

The Karhunen–Loève expansion writes a centered square-integrable random function as $X={\sum }_k{\xi }_k{\varphi }_k$ in mean square, where the ${\varphi }_k$ are principal component functions and the uncorrelated scores satisfy $\mathbb{E}{\xi }_k^2={\lambda }_k$.

The functions $h_n(t)=\sqrt{2}\sin ((n-1/2)\pi t)$ give the Brownian covariance eigenbasis on $[0,1]$, with eigenvalues ${\lambda }_n=((n-1/2)\pi {)}^{-2}$. Integrating by parts against $g_n=\sqrt{2}\cos ((n-1/2)\pi t)$ gives independent coefficients ${\xi }_n=\int g_{n}dW$. Pathwise L2 completeness yields $W={\sum }_n\sqrt{{\lambda }_n}{\xi }_n{h}_n$ in L2 and the squared-norm series.