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 time warping represents phase variation by composing a function with an increasing endpoint-preserving diffeomorphism of its domain. Registration seeks to separate this phase variation from amplitude variation in the observed functions.

For an absolutely continuous real function $f$, its square-root velocity function is $Q(f)=f^{\prime }/\sqrt{|f^{\prime }|}$ where $f^{\prime }\ne 0$, with value zero where $f^{\prime }=0$. Under an increasing reparameterization $\gamma$, it transforms as $Q(f\circ \gamma )=(Q(f)\circ \gamma )\sqrt{{\gamma }^{\prime }}$.

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.

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.

For two independent samples of size $n$ with a common covariance eigenbasis, the two-sample FPCA mean statistic isUnder equality of means and consistent pooled eigencomponent estimates, it 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 covariance operators $C_1,\dots ,C_n$, their Fréchet mean under the square-root distance isThis follows because taking positive square roots embeds the operators into the Hilbert space of self-adjoint Hilbert-Schmidt operators.

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 scalar-on-function linear model has $Y=\alpha +⟨X,\beta ⟩+\epsilon$, where the response is scalar and the predictor and slope are functions. Basis truncation turns it into a finite-dimensional linear regression.

For basis functions $B_1,\dots ,B_K$, the second-derivative roughness penalty has matrix entries ${\Omega }_{jk}=\int B_j^{\prime \prime }(t)B_k^{\prime \prime }(t)dt$. It is positive semidefinite and satisfies $c^{\mathsf{T}}\Omega c=\Vert ({\sum }_k{c}_k{B}_k{)}^{\prime \prime }{\Vert }_2^2$.

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, $C_{YX}=B{C}_X$.