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Let \(x\) be \(p-\)dimensional random vector. For multivariate data, PCA finds weights \(\beta\) s.t. \(\beta^Tx\) has maximum variance. For functional data, \(\beta\) and \(x\) are functions \(\beta(s)\) and \(x(s)\), the summations become integration
\[\begin{equation} f = \int \beta(s)x(s)ds \end{equation}\]
Let \(\beta_1\) denote the first weight function, which is chosen to maximize \(N^{-1}\sum_i(\int \beta_1x_i)^2\) subject to \(\int\beta_1(s)^2ds=1\).
Another way to define PCA is to find a set of K orthonormal functions s.t. the expansion of each curve in terms of basis functions approximates the curve as closely as possible. The expansion can be written as \[\begin{equation} \hat{x}_i(t) = \sum_{k=1}^K f_{ik}\beta_k(t). \end{equation}\]
A measure of fitting is the integrated squared error \[\begin{equation} ||x_i-\hat{x}_i||^2 = \int [x(s) - \hat{x}(s)]^2 ds \end{equation}\]
Suppose we have a set of \(N\) curves \(x_i\) and column centered. Let \(v(s,t)\) be the sample covariance function of observed data.
Discretize the \(x_i\) to a fine grid of \(n\) equally spaced values \(s_j\). Then we work with \(N\times n\) matrix \(X\).
To reduce the eigenequation \(\int v(s,t)\beta(t)dt = \rho\beta(s)\) to discrete or matrix form , we can express each \(x_i\) as a linear combination of known basis functions \(\phi_k\). Then each function can be written as \[\begin{equation} x_i(t) = \sum_{k=1}^K c_{ik}\phi_k(t), \end{equation}\] or in matrix form \[\begin{equation} x=C\phi, \end{equation}\] where \(C\) is a \(N\times K\) matrix, \(\phi\) is a vector-valued function to have component \(\phi_1,...,\phi_K\).
The covariance matrix is then \[\begin{equation} v(s,t) = N^{-1}\phi(s)^TC^TC\phi(t). \end{equation}\]
Define the order \(K\) symmetric matrix \(W\) as \[\begin{equation} w_{k_1,k_2} = \int \phi_{k_1}\phi_{k_2}. \end{equation}\]
Suppose an eigenfunction \(\beta\) has expansion \[\begin{equation} \beta(s) = \phi(s)^Tb. \end{equation}\]
Then we have \[\begin{equation} \int v(s,t)\beta(t)dt = \phi(s)^TN^{-1}C^TCWb = \rho\phi(s)^Tb. \end{equation}\]
The above equation should hold for all \(s\), so \[\begin{equation} N^{-1}C^TCWb = \rho b \end{equation}\]
Suppose \(\beta\) is the leading principal component, we usually penalize the roughness of \(\beta\), \(PEN_2(\beta) = \int \beta^{''}(t)^2dt\). The penalized sample variance is given by \[\begin{equation} \frac{var \int \beta x_i}{||\beta||^2+\lambda*PEN_2(\beta)}. \end{equation}\]
Suppose that \(\phi_\nu\) is a suitable basis for the space of smooth functions S on \([0,1]\), for example B-splines. We can expand \(x(s)\) as \(x(s) = \sum_\nu c_\nux_\nu(s) = c'\phi(s)\). Let \(c_i\) be the vector of coefficients of the data function \(x_i(s)\) in the basis \(\phi_\nu\). Let \(V\) be the covariance matrix of the vectors \(c_i\). Define \(J\) to be the matrix \(\int\phi\phi'\) whose elements are \(\int \phi_j\phi_k\) and \(K\) the matrix whose elements are are \(\int D^2phi_jD^2\phi_k\), then the penalized sample variance is \[\begin{equation} PCAPSV = \frac{y'JVJy}{y'Jy+\lambda y'Ky} \end{equation}\]
Instead of carrying out our smoothing step within the PCA, we smooth the data first, and then carry out an unsmoothed PCA. This approach to func- tional PCA was taken by Besse and Ramsay (1986), Ramsay and Dalzell (1991) and Besse, Cardot and Ferraty (1997).