Linear HPM x3 Quadratic Side Context
Last updated: 2026-08-15
This candidate tests the first nonlinear extension of the Linear HPM x3 Attribution Contract, without using splines.
For each of the ten player-compilable additive feature totals \(X_f(U)\), it fits:
The \(\beta\) component can be compiled into the player prior. The \(\gamma\) component remains explicit lineup context: it permits diminishing returns or threshold-like curvature based on each unit's own completed total, while preserving antisymmetry between the two sides.
Unlike the P-spline models, this has exactly one nonlinear degree of freedom per additive feature. It is the lowest-complexity test of whether aggregate nonlinearity adds predictive value beyond the compiled additive player layer.
Run
uv run nba-train-hpm-x3-linear-quadratic-side-context --through-season 2025-26
uv run nba-run-frozen-multiseason-backtest
The frozen leaderboard will compare this candidate against the equivalent compiled-additive linear baseline and the existing HPM models.
Frozen Result
The completed three-season replay evaluates 2023-24 through 2025-26 using only the state available before each target season. It covers 584,970 regular season possessions (3,284 games) and 39,967 playoff possessions (238 games).
| Cohort | Metric | Canonical Compiled-Linear HPM x3 | Linear plus quadratic side context | Quadratic minus linear |
|---|---|---|---|---|
| Regular season | Possession RMSE | 1.197978 | 1.198002 | +0.000024 |
| Regular season | Possession MAE | 1.141349 | 1.141366 | +0.000017 |
| Regular season | Eligible game RMSE | 14.1051 | 14.1311 | +0.0260 |
| Regular season | Full-game RMSE | 14.3529 | 14.3922 | +0.0393 |
| Regular season | Winner accuracy | 68.13% | 67.36% | -0.77 pp |
| Playoffs | Possession RMSE | 1.192766 | 1.192719 | -0.000047 |
| Playoffs | Eligible game RMSE | 16.6817 | 16.5978 | -0.0838 |
The per-side quadratic terms do not earn promotion: the linear baseline is better on every regular-season aggregate metric. The small playoff advantage is a useful follow-up signal, but it is not sufficient to outweigh the larger regular-season cohort or justify the additional degrees of freedom.
Artifact: artifacts/models/frozen_multiseason_backtest/2023-24_to_2025-26/frozen_multiseason_backtest-2023-24-to-2025-26-20260816T033818Z-07bc6a53.