Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 (Functional Data Analysis)

The kernel is the Brownian bridge covariance kernel. If $C_X\varphi =\lambda \varphi$, differentiating the integral equation twice givesThe normalized solutions and eigenvalues are therefore

Let $V=(X_1(t_1),X_1(t_2){)}^T$. The Karhunen–Loève expansion and Gaussianity giveandThe conditional multivariate normal distribution formula yieldsIf $\Sigma$ is singular, the same formula uses its Moore-Penrose pseudoinverse.

Letand estimate it by the average of the two within-sample empirical covariance operators. Let $({\hat{\lambda }}_k,{\hat{\varphi }}_k)$ be its leading empirical eigenpairs and let ${\overline{X}}_n,{\overline{X}}_n^{\ast }$ be the sample means. Use the Two-sample FPCA mean statistic

Under $H_0$, the Hilbert-space central limit theorem giveswhere $G$ is a centered Gaussian random element with covariance $\overline{C}$. Distinct eigenvalues give consistent empirical eigenpairs, up to signs, and the standardized leading scores are independent standard normal variables. HenceRejecting above the $(1-\alpha )$ quantile gives an asymptotic level-$\alpha$ test.

Under a fixed alternative $\delta =\mu -{\mu }^{\ast }$,The test is consistent whenever one retained projection is nonzero. Alternatives orthogonal to the first $K$ eigenfunctions are invisible at fixed $K$. Under local alternatives $\delta =h/\sqrt{n}$, the limit is noncentral chi-squared with noncentrality $\frac{1}{2}{\sum }_{k\le K}⟨h,{\varphi }_k{⟩}^2/{\lambda }_k$.

The square-root distance between covariance operators isWriting $A_i=C_i^{1/2}$, minimizing ${\sum }_i\Vert A_i-A{\Vert }_{\mathrm{HS}}^2$ gives $A=n^{-1}{\sum }_i{A}_i$. Since this average is positive,the square-root barycenter of covariance operators.

Because $C_i=L_i{L}_i^{\ast }$ is a covariance operator,Thus both factors are Hilbert-Schmidt. The product of two Hilbert-Schmidt operators is a trace-class operator, with

The Procrustes distance between covariance operators isFor unitary $R$,The polar decomposition of a bounded operator impliesTherefore

The common eigenbasis givesConsequentlyFor the positive square-root factors, the singular values of $C_2^{1/2}{C}_1^{1/2}$ are $\sqrt{{\lambda }_k{\lambda }_k^{\ast }}$, soThus the distances coincide when the covariance operators commute and share an eigenbasis.

Let $B$ be the integral operator with kernel $\beta$. Since $Y=BX+\epsilon$ and the centered error is independent of $X$,Writing ${\beta }_{lk}=⟨B{\varphi }_k,u_l⟩$ and using $\mathbb{E}[a_k{a}_j]={\lambda }_k{1}_{\{j=k\}}$ givesExpanding the function-on-function linear model kernel in the product basis therefore yields

Replacing ${\varphi }_k$ by $-{\varphi }_k$ also replaces $a_k$ by $-a_k$, so both $\mathbb{E}[a_k{b}_l]$ and ${\varphi }_k(s)$ change sign and their product is unchanged. Replacing $u_l$ by $-u_l$ similarly replaces $b_l$ by $-b_l$. Every summand, and hence $\beta$, is independent of all eigenfunction sign choices.

Applying the regression operator from part i givesThe uncorrelated scores satisfy $\mathrm{Var}(a_k)={\lambda }_k$, while $\mathrm{Var}(Y(t))={\sum }_l{\gamma }_l{u}_l(t{)}^2$. HenceThe sign cancellations from part ii leave every squared numerator term unchanged; changing the sign of $u_l$ also leaves each denominator term unchanged. Thus $R(t)$ is sign invariant.