Worked example: anomaly screening and influence

This synthetic example deliberately inserts one atypical continuous path into the final trial.

gaze = simulate_planar_trajectories(
    n_participants=16,
    trials_per_participant=3,
    n_time=61,
    random_state=33,
)

values = gaze.values.copy()
u = gaze.time / gaze.time[-1]
values[-1, :, 0] = np.clip(
    0.10 + 0.80 * np.sin(np.pi * u) ** 2,
    0,
    1,
)
values[-1, :, 1] = np.clip(
    0.90 - 0.70 * u,
    0,
    1,
)
gaze = gaze.with_values(values)

Fit the planned FPCA

fit = fit_mfpca(
    gaze,
    n_components=4,
    scaling="dimension_sd",
)

Screen trajectories for review

review = diagnose_fpca_outliers(
    fit,
    gaze,
    n_components=4,
    score_covariance="robust",
    random_state=33,
)

review.diagnostics.sort_values(
    "score_mahalanobis_sq",
    ascending=False,
).head()

A high score-space distance identifies a trajectory occupying an unusual position in the retained functional score space.

Check participant influence

Because each participant contributes three trials:

influence = leave_one_group_out_fpca_influence(
    gaze,
    group_column="participant_id",
    n_components=3,
    scaling="dimension_sd",
)

influence.summary.sort_values(
    "influence_score",
    ascending=False,
).head()

In the synthetic truth case, the participant containing the deliberately atypical path should have the greatest influence on the component structure.

Visual diagnostics

plot_fpca_outlier_diagnostics(review)
plot_fpca_influence(influence)

The first plot contrasts reconstruction unusualness with score-space unusualness. The second shows how much component structure changes under each group omission.

Interpretation

The correct conclusion is not “delete the most influential participant.”

The correct conclusion is:

This participant merits review because omitting their trials changes the estimated functional covariance structure more than omitting other participants.

Whether that reflects error or scientifically meaningful heterogeneity requires separate evidence.