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Selecting the number of FPCs

Choosing the retained dimension is a model-selection decision, not a cosmetic plotting choice.

A cumulative-variance threshold answers:

How many fitted components summarize a chosen fraction of in-sample functional variance?

Held-out reconstruction cross-validation answers a different question:

How many components are useful for reconstructing trajectories that were not used to estimate the FPCA basis?

Both can be reported. They should not be treated as interchangeable.

Leakage-safe reconstruction CV

cross_validate_fpca_reconstruction() refits FPCA inside every training fold, projects held-out trajectories into that training basis, reconstructs them with each candidate component count, and evaluates integrated RMSE.

cv = cross_validate_fpca_reconstruction(
    gaze,
    candidate_components=(1, 2, 3, 4, 5, 6),
    n_splits=5,
    scaling="dimension_sd",
    cv_unit="group",
    group_column="participant_id",
)

The full dataset is not used to estimate the basis before folds are created.

Do not fit once and score folds afterwards

Fitting FPCA on the full sample before evaluating held-out curves leaks information about the test trajectories into the mean, scaling, and eigenfunctions.

Repeated trials: group the participant

When a participant contributes multiple trials, curve-level K-fold CV can place some of that participant's trials in training and others in testing.

Use grouped CV instead:

cv_unit="group"
group_column="participant_id"

Every participant then belongs to one test fold only.

Use curve-level CV only when curves are genuinely independent at the level relevant to the scientific question.

Inspect before selecting

summary = summarise_fpca_cross_validation(cv)
print(summary)
plot_fpca_cross_validation(cv)

The summary contains:

  • candidate component count;
  • mean fold-level held-out integrated RMSE;
  • SD across folds;
  • SE across folds;
  • number of folds.

The result also retains the curve-to-fold assignments so grouping can be audited.

Selection rules are explicit

Minimum RMSE

k = select_fpca_components_cv(cv, rule="minimum")

This selects the candidate with the smallest mean held-out reconstruction error.

One-standard-error heuristic

k = select_fpca_components_cv(cv, rule="one_se")

This selects the smallest candidate whose mean error is no worse than one standard error above the minimum-error candidate.

The one-SE rule is a parsimony heuristic. It is not a hypothesis test and does not guarantee the scientifically correct dimension.

Cross-validation is not the only criterion

A useful component count should also be considered against:

  • cumulative variance explained;
  • bootstrap component stability;
  • reconstruction curves;
  • component interpretability;
  • sensitivity to registration and preprocessing;
  • the sample size available for stable eigenfunction estimation.

A component can slightly improve reconstruction but remain unstable or scientifically uninterpretable.

Candidate range

Do not search arbitrarily large component counts.

Within each training fold, the centered sample covariance has at most n_train - 1 non-zero sample directions. The API rejects candidate counts above that fold-specific maximum instead of fitting a degenerate direction.

When not to use this workflow

Do not use ordinary common-grid reconstruction CV as a substitute for:

  • sparse-FPCA methods when interpolation creates most of each curve;
  • supervised dimension selection when the scientific target is prediction of an external outcome;
  • phase-aware model selection when timing variation should first be modeled separately;
  • participant-independent CV when repeated trials are clustered within participants.

Interpretation

Lower held-out RMSE indicates better out-of-sample reconstruction under the stated representation, scaling, fold definition, and candidate set.

It does not prove that:

  • the selected components are causally meaningful;
  • every selected FPC is stable;
  • a different preprocessing or registration choice would select the same dimension;
  • the selected dimension is optimal for a downstream regression outcome.

Reporting example

The retained FPCA dimension was evaluated using five-fold participant-grouped reconstruction cross-validation. FPCA, including the training-fold mean and dimension scaling, was re-estimated within each fold. Candidate dimensions of one through six FPCs were compared using held-out integrated RMSE. The pre-specified one-standard-error heuristic selected three FPCs; the minimum-RMSE solution retained four. Component stability and interpretation were examined separately.

Use fpca_cross_validation_reporting_text() for a compact reproducible starting point.

  • cross_validate_fpca_reconstruction()
  • summarise_fpca_cross_validation()
  • select_fpca_components_cv()
  • plot_fpca_cross_validation()
  • fpca_cross_validation_reporting_text()

Methodological context

Li, Wang, and Carroll (2013) emphasize that selecting the number of functional principal components is a distinct methodological problem rather than something resolved automatically by FPCA. The implementation here is intentionally a transparent held-out reconstruction diagnostic; it does not claim to reproduce their information-criterion estimator.

  • Li Y, Wang N, Carroll RJ. Selecting the Number of Principal Components in Functional Data. Journal of the American Statistical Association. 2013;108(504). doi:10.1080/01621459.2013.788980.