Phase FPCA and registration sensitivity

Registration changes the time parameterization of a trajectory. In eye tracking, that is scientifically consequential because timing may encode verification latency, hesitation, or strategy.

Treat warping as data

After explicit landmark registration:

registration = register_to_landmarks(
    gaze,
    observed_landmarks,
    reference_landmarks=reference_landmarks,
)

convert the estimated warpings to a functional object:

phase = phase_trajectory_set(
    registration,
    representation="displacement",
)

For displacement functions,

\[ D_i(t)=h_i(t)-t, \]

positive and negative values indicate how the source time differs from the reference time across the trial.

FPCA of phase variation

phase_fit = fit_phase_fpca(
    registration,
    n_components=0.95,
)

Phase FPCA asks how timing deformation itself varies across curves. This is complementary to FPCA on the registered spatial trajectories.

Keep landmark timing too

landmark_table = phase_landmark_frame(registration)

Do not reduce all phase information to a single average warping statistic when the experiment has interpretable landmarks.

Did registration change the scientific structure?

sensitivity = compare_registered_unregistered_fpca(
    registration,
    n_components=3,
    scaling="dimension_sd",
)

registration_sensitivity_frame(sensitivity)

The comparison matches pre- and post-registration FPCs and reports:

• functional shape similarity;
• matched component identities;
• score correlations after sign alignment.

Large changes are not automatically bad. They indicate that phase variation was contributing materially to the unregistered covariance structure.

Recommended reporting

Report both the justification for registration and what changed because of it. When timing is theoretically meaningful, include phase results or a registered-versus-unregistered sensitivity analysis rather than presenting only the aligned curves.