1  Introduction

1.1 Problem setting

Given repeated measurements of many variables on many subjects, the objective is to model how each variable evolves over time. The motivating application throughout is MRI-derived gray matter volume and density across a large number of brain regions, measured two or three times per subject. The treatment is general and applies to any high-dimensional panel of noisy continuous measurements observed on a sparse, irregular time grid.

Three families of methods are commonly considered:

  1. Differential equations. A generative law for the rate of change is specified and its parameters estimated.
  2. Supervised learning. Future value or change is regressed on present value and covariates, without commitment to a generating process.
  3. Hybrid models. A mechanistic skeleton is combined with a learned component.

Comparing the three is a reasonable objective, but the comparison is informative only if it accounts for a constraint that is frequently left implicit.

1.2 Organizing constraint

The number of timepoints per subject determines which models are identifiable. This is a question of estimability, not accuracy, and it precedes any question of predictive performance.

The limiting case makes the point directly. With exactly two timepoints and no pooling structure across subjects or regions, a first-order ordinary differential equation fitted to the data is algebraically equivalent to a regression of change on baseline. The two arms are the same model, and any measured difference between them reflects optimizer behavior or parametrization rather than modeling assumptions.

For this reason the site is organized by timepoint regime rather than by method. The method chapters describe what each family can represent; the regime chapters state which of those representations are estimable from a given design.

1.3 Structure

  • Foundations (Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6) establishes notation, the degrees-of-freedom accounting used throughout, the measurement error floor below which methods cannot be distinguished, the second design axis of variable count and driver count, and the spatial structure that makes brain-wide parcellations a particular case of it.
  • Method families (Chapter 7, Chapter 8, Chapter 9) presents each family as a sequence of models of increasing parameter count, annotated with minimum viable design requirements.
  • Timepoint regimes (Chapter 10 through Chapter 13) gives the correspondence between design and available methods for \(T = 2\), \(T = 3\), \(T = 4\)–\(6\), and \(T \geq 7\).
  • Comparison design (Chapter 14, Chapter 15) specifies an evaluation protocol and documents common sources of invalid inference.

1.4 Scope

The treatment is methodological and does not advocate for a particular family. In the largest published benchmark of longitudinal biomarker forecasting, the TADPOLE challenge, no algorithmic class dominated, ensemble predictions ranked highest on nearly all tasks, and feature selection was more consequential than algorithm choice (Marinescu et al. 2021). Results obtained in any three-arm comparison should be interpreted against that finding.