Forward RAPM Calibration

Purpose

This experiment converts the sequential, forward lagged-prior RAPM signal into a team regular-season win estimate. It is intentionally narrow: prior RAPM only. Age, draft status, physical measurements, box-score features, and target-season RAPM adjustments are excluded so the result is a clean first test of whether the player rating scale can be carried across eras.

It is not part of the possession-level Leaderboard. Its target is one team-season win percentage rather than a possession outcome.

Information Boundary

For team \(j\) in season \(t\), player \(i\)'s input \(r_{i,t}^{-}\) is the RAPM estimate frozen at the end of season \(t-1\). A player with no prior gets the established cold-start value of zero.

Prior-centered RAPM has an arbitrary league-wide location and scale: adding a constant to every player coefficient does not change any lineup prediction. Before comparing seasons, each frozen prior is therefore normalized using the previous season's player-exposure-weighted mean \(\mu_{t-1}\) and standard deviation \(\sigma_{t-1}\):

\[ z_{i,t}^{-} = \frac{r_{i,t}^{-} - \mu_{t-1}}{\sigma_{t-1}}. \]

Both reference quantities are fully known before season \(t\). This does not add player side information; it only fixes the otherwise unidentified RAPM coordinate system. The raw prior average remains in the artifact for audit, but the standardized score is the calibration predictor.

The team signal is the realized-stint-seconds-weighted player average:

\[ S_{j,t} = \frac{\sum_i q_{i,j,t} z_{i,t}^{-}} {\sum_i q_{i,j,t}}, \]

where \(q_{i,j,t}\) is the number of seconds player \(i\) appeared in team \(j\)'s reconstructed regular-season lineup stints. Overtime seconds are included naturally. This makes the first version usage-conditional: the rating is forward-looking, while the season's eventual playing-time allocation is observed retrospectively.

The calibration then fits, using only completed seasons before \(t\),

\[ \widehat{W}_{j,t} = \operatorname{clip}_{[0,1]}(\alpha_t + \beta_t S_{j,t}), \]

where \(W_{j,t}\) is regular-season win percentage. Parameters are estimated by weighted least squares, with a team-season weighted by its game count. The clip only prevents impossible probabilities at extreme extrapolations.

The 2025-26 estimate is fit on 1997-98 through 2024-25 team outcomes and evaluated on 2025-26. No 2025-26 win, possession result, or fitted RAPM adjustment enters its calibration slope or intercept.

Evaluation

Every season after a four-season warmup receives an expanding-window forecast. Reported metrics are team-level, with all teams equally weighted:

win-percentage RMSE and MAE;
wins RMSE and MAE, retaining the varying season lengths;
skill relative to predicting a .500 win percentage for every team;
Spearman rank correlation between predicted and observed team win percentage.

The persistent artifacts include team_season_inputs.parquet, all forward_predictions.parquet, season-level metrics, the 2025-26 target predictions, and the final calibration coefficients. The team input table also records prior-exposure fraction, making cold-start reliance visible.

Interpretation

A +1 RAPM point does not receive an assumed, fixed win value. The expanding historical calibration estimates the empirical relationship at each target date. That is useful for comparing team strength across eras, but it is not yet a deployable preseason forecast because it uses realized season minutes.

A later allocation model can replace \(q_{i,j,t}\) with preseason projected minutes or observed-to-date minutes without changing the rating or calibration contracts.

Initial Results

Run forward-calibration-2025-26-20260804T132355Z-7ec43aa9 uses the regular-only forward lagged-prior RAPM run forward-lagged-rapm-2025-26-20260803T203054Z-c627d89d.

The target-season calibration was fit on 833 team-seasons from 1997-98 through 2024-25 and then evaluated on all 30 teams and 2,460 games in 2025-26. Median 2025-26 exposure to a player with a prior was 92.8%; the minimum was 62.5%.

Measure	2025-26 result
Calibration intercept	0.5136
Win percentage per one standardized team RAPM unit	0.3254
Win-percentage RMSE	0.1303
Win-percentage MAE	0.1091
Wins RMSE	10.68
Wins MAE	8.94
Skill vs .500 prediction	38.4%
Spearman team-rank correlation	0.639

The expanding-window backtest covers 25 target seasons from 2001-02 through 2025-26. Its unweighted mean season-level results are 0.0994 win-percentage RMSE, 7.29 wins RMSE, 56.2% skill versus .500, and 0.754 rank correlation. This is a useful historical signal, but it should not yet be compared to Vegas or presented as a preseason model: realized player allocation is doing real work here.