Functional data analysis

Functional data analysis treats each observation as a function or another infinite-dimensional object. Means, covariance operators, spectral coordinates, and regression operators replace their finite-dimensional vector and matrix counterparts.

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 $\mathbb{E}{\xi }_k^2={\lambda }_k$.

A functional mean test tests whether the mean of a Hilbert-space-valued random variable is zero. The squared-norm statistic $n\Vert \overline{X}{\Vert }^2$ has a weighted chi-squared limit under the null, while an FPCA test standardizes and truncates the coordinates.

For covariance eigenpairs $({\lambda }_k,{\varphi }_k)$, the $K$-coordinate FPCA mean statistic is $n{\sum }_{k=1}^K⟨\overline{X},{\varphi }_k{⟩}^2/{\lambda }_k$. Under a zero-mean null and standard estimation conditions, its plug-in version converges to a chi-squared distribution with $K$ degrees of freedom.

A covariance-operator distance compares positive trace-class operators while respecting either their linear embedding in operator space or a chosen factorization geometry.

The linear distance between covariance operators is $d_L(C_1,C_2)=\Vert C_1-C_2{\Vert }_{\mathrm{HS}}$.

The square-root distance is $d_R(C_1,C_2)=\Vert C_1^{1/2}-C_2^{1/2}{\Vert }_{\mathrm{HS}}$.

For factorizations $C_i=L_i{L}_i^{\ast }$, the Procrustes distance is $d_P(C_1,C_2)={\inf }_R\Vert L_1-L_{2}R{\Vert }_{\mathrm{HS}}$, where $R$ ranges over unitary operators on the factor space. Equivalently,

A functional linear model relates functional or scalar responses to functional predictors through a linear operator.

A function-on-function linear model has $Y(t)=\int \beta (t,s)X(s)ds+\epsilon (t)$. If the centered error is independent of $X$, then $\mathbb{E}[Y\mid X]$ is the regression operator applied to $X$.

For centered Hilbert-space random variables $X$ and $Y$, the cross-covariance operator is $C_{YX}h=\mathbb{E}[⟨X,h⟩Y]$. In the functional linear model $Y=BX+\epsilon$ with independent centered error, $C_{YX}=B{C}_X$.