Skip to content

Mathematical reference: audited sparse-FPCA bandwidth selection

This page defines the first 1.1 bandwidth-selection contract for the native univariate sparse-FPCA estimator. It is a tuning procedure around fit_sparse_fpca(); it does not change that estimator's scientific definition or defaults.

Qualified candidate space

The first qualified candidate is the two-dimensional smoothing tuple

\[ h=(h_\mu,h_G), \]

where \(h_\mu\) is the local-linear mean bandwidth and \(h_G\) is the latent-covariance bandwidth.

The diagonal-difference noise bandwidth is not part of \(h\) in A2. If noise_variance_method="diagonal_difference" is used, one analyst-declared \(h_\epsilon\) and one noise_support interval are held fixed across all candidates and folds. If measurement-error variance is externally fixed, neither \(h_\epsilon\) nor noise_support is used.

Training-fold population model

Let fold \(f\) contain training curves \(\mathcal T_f\) and validation curves \(\mathcal V_f\). For candidate \(h=(h_\mu,h_G)\), the native sparse-FPCA population objects are refitted using training curves only:

\[ \widehat\mu_{-f,h}(t), \qquad \widehat G_{-f,h}(s,t), \qquad \widehat\sigma^2_{\epsilon,-f,h}. \]

No observation from \(\mathcal V_f\) is used in those estimates.

If measurement-error variance is externally fixed, that declared variance is retained in every training fold. If diagonal-difference noise estimation is used, the fixed \(h_\epsilon\) and declared support are applied within every training fold, while the noise variance itself is re-estimated from training data.

Held-out whole-curve Gaussian loss

For validation curve \(i\in\mathcal V_f\) observed at native times

\[ T_i=(t_{i1},\ldots,t_{im_i}), \]

write

\[ \mathbf y_i= \begin{bmatrix} Y_i(t_{i1}) & \cdots & Y_i(t_{im_i}) \end{bmatrix}^{\top}, \]

and evaluate the training-fold population model at those native times:

\[ \widehat{\boldsymbol\mu}_{i,-f,h} = \widehat\mu_{-f,h}(T_i), \]
\[ \widehat\Sigma_{i,-f,h} = \widehat G_{-f,h}(T_i,T_i) + \widehat\sigma^2_{\epsilon,-f,h}I_{m_i}. \]

The candidate loss for curve \(i\) is the Gaussian negative log predictive density normalized by the number of observed samples:

\[ \ell_{i,f}(h) = \frac{1}{2m_i} \left[ \log\left|\widehat\Sigma_{i,-f,h}\right| + (\mathbf y_i-\widehat{\boldsymbol\mu}_{i,-f,h})^{\top} \widehat\Sigma_{i,-f,h}^{-1} (\mathbf y_i-\widehat{\boldsymbol\mu}_{i,-f,h}) + m_i\log(2\pi) \right]. \]

The implementation uses stable linear solves. A held-out covariance matrix that is not positive definite or exceeds the declared predictive condition-number limit fails explicitly.

The fold criterion is

\[ L_f(h) = \frac{1}{|\mathcal V_f|} \sum_{i\in\mathcal V_f}\ell_{i,f}(h). \]

Thus validation curves receive equal weight at the fold-aggregation stage; the criterion is not an observation-level iid loss.

Across valid folds, the aggregate criterion is

\[ \overline L(h) = \frac{1}{F_h} \sum_{f\in\mathcal F_h}L_f(h), \]

where \(\mathcal F_h\) is the set of folds for which candidate \(h\) completed successfully and \(F_h=|\mathcal F_h|\).

Eligibility and failures

A candidate is eligible only when

\[ F_h\ge F_{\min}, \]

where min_valid_folds defines \(F_{\min}\). The default is

\[ F_{\min}=K, \]

for \(K\)-fold cross-validation, so every fold must be valid by default.

Failed folds remain in the audit table with a failure code and message. They are not silently removed, replaced by another bandwidth, or converted to a finite loss. The audit retains raw and effective post-support training/validation observation counts.

Every training fold must contain at least three curves, matching the native sparse-FPCA population-fitting contract. Impossible fold designs fail before candidate evaluation.

Selection rule

Among eligible finite candidates,

\[ \widehat h = \operatorname*{arg\,min}_{h}\overline L(h). \]

Ties are resolved deterministically by preferring larger mean and covariance bandwidths lexicographically, then by candidate_id. The tie-break is an implementation rule for reproducibility; it is not an inferential statement about optimal smoothing.

The first tranche does not implement a one-standard-error rule. A two-dimensional bandwidth tuple does not currently have a prespecified scalar complexity ordering analogous to retaining fewer principal components. A 1-SE selector would require such an ordering to be defined scientifically before implementation.

Why the noise bandwidth is held fixed

An exploratory qualification originally included \(h_\epsilon\) in the candidate tuple and used the same marginal predictive loss. In a known-truth scenario with simulated measurement-error variance \(0.0196\), that procedure selected a noise bandwidth whose full-data fit estimated measurement-error variance near \(0.183\), even though the total covariance and leading functional structure were recovered well.

The marginal Gaussian loss depends on

\[ G(T_i,T_i)+\sigma_\epsilon^2 I, \]

so good prediction of the total observation covariance does not identify how that covariance should be divided between the latent process and measurement error. Consequently, A2 does not use this criterion to tune \(h_\epsilon\). Decomposition-specific noise-bandwidth selection requires a separate criterion and qualification.

Resampling unit

Two resampling units are supported:

  • resampling_unit="curve": whole curves are assigned to folds;
  • resampling_unit="group": all curves sharing the declared group_column are assigned to the same validation fold.

For repeated participant/trial data, group resampling is the leakage-safe route when participant-level generalization is intended.

Observation-level random splitting is not part of this method.

Distinction from PACE scoring

The validation loss uses

\[ \widehat G_{-f,h}(T_i,T_i) + \widehat\sigma^2_{\epsilon,-f,h}I, \]

not the PACE score system with score_ridge, and validation observations are not used to estimate held-out PACE scores before computing the criterion.

This separation is intentional: the first bandwidth selector tunes the training-fold population observation model rather than a downstream score reconstruction.

Scope of the selected bandwidths

\(\widehat h\) is the minimizer of the declared criterion over the declared candidate set and fold design. It is not a universal population parameter and should not be described as recovery of a unique "true bandwidth".

Bandwidth choice depends on, among other things:

  • sample size;
  • native observation density/design;
  • measurement noise;
  • smoothing family and kernel;
  • analysis support;
  • candidate grid;
  • resampling unit; and
  • validation criterion.

API and audit objects

Selector: eyetrajectoriespy.sparse_bandwidth_selection.select_sparse_fpca_bandwidths().

Result: SparseFPCABandwidthSelectionResult, retaining assignments, candidate grid, candidate-fold results, held-out curve losses, aggregate candidate summaries, selected bandwidths, status, and provenance.

Reporting helper: sparse_fpca_bandwidth_selection_reporting_text().

The selected mean/covariance values are not applied automatically. They must be supplied explicitly to a later fit_sparse_fpca() call, which continues to record automatic_bandwidth_selection_performed=False. When diagonal-difference noise estimation is used, the same predeclared noise_bandwidth and noise_support should be carried into that final fit.

See the user guide and the native sparse FPCA/PACE guide.