Multilevel FPCA

Repeated experiments naturally have participant → trial → time structure:

$ \mathbf G_{ij}(t) = \boldsymbol\mu(t) + \mathbf U_i(t) + \mathbf V_{ij}(t). $

U_i(t) represents participant-level deviation and V_{ij}(t) within-participant trial deviation.

result = fit_multilevel_fpca(
    gaze,
    participant_column="participant_id",
    participant_components=0.90,
    trial_components=0.90,
    scaling="dimension_sd",
)

Participant FPCs characterize stable between-person functional differences. Trial FPCs characterize deviations around each participant's own mean trajectory.

Note

The first release is a transparent two-level functional ANOVA decomposition followed by separate FPCAs, not a full probabilistic functional mixed model.

The actual decomposition used before the two separate FPCAs is

\[ \mathbf U_i(t) = \overline{\mathbf G}_{i\cdot}(t)-\boldsymbol\mu(t), \qquad \mathbf V_{ij}(t) = \mathbf G_{ij}(t)-\overline{\mathbf G}_{i\cdot}(t). \]

See the mathematical reference.