Functional data analysis

Remarks

The operator $\mathcal C$ is compact, self-adjoint and positive semidefinite, so it has an orthonormal basis of eigenfunctions $\phi_k$ with eigenvalues $\lambda_1 \ge \lambda_2 \ge \dots \ge 0$. Functional principal component analysis estimates these and writes $X = \mu + \sum_k \xi_k \phi_k$ with uncorrelated scores $\xi_k = \langle X - \mu, \phi_k \rangle$ of variance $\lambda_k$. The sample versions are $\hat\mu = \frac1n \sum_i X_i$ and $\hat C(s, t) = \frac1n \sum_i (X_i(s) - \hat\mu(s))(X_i(t) - \hat\mu(t))$.

Example: let $X(t) = \xi \sqrt{2} \sin(\pi t)$ on $[0, 1]$, with $\xi$ a real random variable of mean $0$ and variance $1$. Then $\mu = 0$ and $C(s, t) = 2 \sin(\pi s) \sin(\pi t)$, so $\mathcal C$ has the single nonzero eigenvalue $1$, with eigenfunction $\sqrt{2} \sin(\pi t)$ of norm $1$.

Non-example: a table of the heights of children measured once each, at a single age, is ordinary multivariate data, not functional data, since no observation is a function of $t$.