Functional PCA

PCA and eigenanalysis

Define the covariance function $v(s,t)$ by

Each of those principal component weight functions $\xi_j(s)$ satisfies the equation

for an appropriate eigenvalue $\rho$. The left side is an integral transform $V$ of the weight function $\xi$ defined by

The integral transform is called the covariance operator $V$. Therefore we may also express the eigenequation directly as

where $\xi$ is now an eigenfunction rather than an eigenvector.

An important difference between the multivariate and functional eigenanalysis problems, concerning the maximum number of different eigenvalue-eigenfunction pairs. The counterpart of the number of variables $p$ in the multivariate case is the number of function $x_i$ are not linearly dependent, the operator $V$ will have rank $N-1$, and there will be only $N-1$ nonzero eigenvalues.

Computation

Suppose we have a set of $N$ curves $x_i$, and that preliminary steps such as curve registration and the possible subtraction of the mean curve from each (curve centering) have been completed. Let $v(s,t)$ be the sample covariance function of the observed data. In all cases, convert the continuous functional eigenanalysis problem to an approximately equivalent matrix eigenanalysis task.

Discretizing the functions

A simple approach is to discretize the observed functions $x_i$ to a fine grid of $n$ equally spaced values $s_j$ that span the interval $\cal T$. This produces eigenvalues and eigenvectors satisfying

for $n$-vectors $u$.

The sample variance-covariance matrix $V=N^{-1}X’X$ will have elements $v(s_j,s_k)$ where $v(s,t)$ is the sample covariance function. Given any function $\xi$, let $\tilde \bxi$ be the $n$-vector of values $\xi(s_j)$. Let $w=T/n$ where $T$ is the length of the interval $\cal T$. Then for each $s_j$,

so the functional eigenequation $V\xi=\rho\xi$ has the approximate discrete form

Basis function expansion of the functions

One way of reducing the eigenequation \eqref{eq:8.9} to discrete or matrix form is to express each function $x_i$ as a linear combination of known basis functions $\phi_k$.

Suppose that each function has basis expansion

Write in vector-form,

where the coefficient matrix $\C$ is $N\times K$. In matrix terms, the variance-covariance function is

Define the order $K$ symmetric matrix $\W$ to have entries

or $\W=\int \bphi\bphi’$. For the orthonormal Fourier series, $\W=\I$. Now suppose that an eigenfunction $\xi$ for the eigenequation \eqref{eq:8.9} has an expansion

or in matrix form, $\xi(s) = \bphi(s)’\b$. This yields

and \eqref{eq:8.9} becomes

Since this equation must hold for all $s$, this implies the purely matrix equation

and the constrain $\Vert\xi\Vert=1$ implies that $\b’\W\b=1$. Define $\u=\W^{1/2}\b$, solve the equivalent symmetric eigenvalue problem

and compute $\b=\W^{-1/2}\u$ for each eigenvector.

Two special cases:

• the basis is orthonormal, $\W=\I$
• view the observed functions $x_i$ as their own basis expansions, $\C=\I$.

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