FPCA and MFPCA¶

Functional PCA finds dominant modes of variation in curves rather than isolated scalar features.

Multivariate planar gaze¶

For \(G_i(t)=[x_i(t),y_i(t)]^\top\):

fit = fit_mfpca(gaze, n_components=0.95, scaling="dimension_sd")

The core estimator uses trapezoidal quadrature weights so unequal common-grid spacing contributes according to elapsed time.

Scaling¶

scaling="none" preserves native relative channel variance. scaling="dimension_sd" equalizes integrated variance across functional dimensions. Neither is universally correct.

Component retention¶

An integer retains a fixed count. A float in (0,1) retains enough components to reach the requested in-sample variance fraction.

When the retained dimension itself is under study, use held-out reconstruction cross-validation. For repeated trials, group folds by participant so trials from one participant cannot appear in both training and test data.

Variance thresholds and held-out reconstruction answer different questions and can be reported side by side.

Interpretation¶

Inspect mean ± one or two score-SD component trajectories. For planar gaze, interpret x and y jointly.

Reconstruction¶

reconstruct_fpca() supports sensitivity checks: compare low-dimensional reconstructions with the original paths to understand what the retained representation preserves.

Mathematical contract¶

The implementation performs PCA after quadrature weighting the centered functional observations. With optional dimension scaling \(s_d\), the weighted representation is

\[ Z_{i,m,d} = \frac{G_{id}(t_m)-\widehat\mu_d(t_m)}{s_d}\sqrt{w_m}. \]

Reconstruction with \(K\) retained FPCs is

\[ \widehat{\mathbf G}^{(K)}_i(t) = \widehat{\boldsymbol\mu}(t) + \sum_{k=1}^{K} \widehat\xi_{ik}\widehat{\boldsymbol\phi}_k(t). \]

See the mathematical reference for the exact trapezoidal weights, scaling, projection, and loading back-transformation used by the package.