12  Intermediate designs

Four to six observations per subject.

12.1 What changes

Three to five first differences and two to four second differences per subject become available. The practical consequences:

• Subject-level shape parameters stop being prior-dominated. Random effects on a curvature or asymptote parameter are constrained by the subject’s own data. The shrinkage diagnostic should be reported to confirm this rather than assumed.
• Smoothing-based estimators become usable. Collocation and gradient matching avoid repeated numerical integration and make larger coupled systems computationally tractable.
• Model selection among functional forms becomes meaningful. Distinguishing exponential from logistic decline is supported by within-subject evidence rather than only by pooled cross-sectional shape.
• Hybrid neural components acquire within-subject constraint. H3 becomes defensible rather than merely fittable, though regularization remains necessary.

12.2 Recommended additions to the sparse protocol

Retain the structure of Chapter 11 and add:

1. Functional form comparison within the mechanistic arm, reported with information criteria and with out-of-sample performance, not with in-sample likelihood alone.
2. Relaxation of the coupling constraint. A low-rank or network-partitioned \(\mathbf{A}\) becomes estimable where a general \(\mathbf{A}\) still does not.
3. Subject-level parameter analysis. Estimated rates and onsets can now be treated as derived phenotypes and related to covariates.
4. Multi-horizon evaluation. Train on the first \(k\) visits, predict visits \(k+1, \dots, T\), varying \(k\). This produces a forecast-horizon curve that separates the arms more informatively than a single held-out visit.

12.3 Evaluation

The held-out-visit design generalizes to a rolling-origin evaluation: for each cutoff \(k\), fit on visits \(1{:}k\) and evaluate on all later visits. Report error as a function of horizon. The expected signature is that supervised and mechanistic arms converge at short horizons and separate as the horizon extends, with the mechanistic and hybrid arms degrading more slowly.

This is also the first regime in which autoregressive application of the supervised arm can be evaluated fairly against a differential equation, since both are being asked to propagate over several intervals.

12.4 Remaining limitations

A general coupling matrix over a large parcellation remains out of reach. Per-subject system identification remains out of reach. Claims about mechanism still rest substantially on the structural prior rather than on the data.