Forward Conditional Minutes v0.1

This is the first preseason minutes component. It predicts a player's expected NBA minutes per medically available game, before any team-level rotation allocation is imposed.

Forward Player State

For a player-season with \(A_{i,t}\) medically available games and total NBA minutes \(M_{i,t}\), the observed conditional target is:

\[ m_{i,t}^{\mathrm{avail}} = \frac{M_{i,t}}{A_{i,t}}. \]

The model fits a pooled quadratic age baseline on \(\log(1+m_{i,t}^{\mathrm{avail}})\), then carries an exposure-shrunk player state forward. Available games are measurement exposure: a 10-game observation updates the state less than a 70-game observation. First observed seasons shrink to the age baseline; gap years decay the player-specific deviation back toward that baseline.

\[ z_{i,t}^{\mathrm{post}} = \frac{q z_{i,t}^{\mathrm{prior}} + A_{i,t}\log(1+m_{i,t}^{\mathrm{avail}})} {q + A_{i,t}}. \]

There are no target-season box-score, minutes, or availability outcomes in the forward prediction. A cold start receives only the age baseline.

Combine With Availability

Forward Availability v0.2 supplies \(p_{i,t}^{\mathrm{avail}}\). The two independent components first form raw expected totals:

\[ r_{i,t}=82\,p_{i,t}^{\mathrm{avail}}\, \widehat m_{i,t}^{\mathrm{avail}}. \]

Roster Normalization

Raw player totals do not necessarily sum to a team's 19,680 regulation minutes. For each opening roster, the 15 highest raw totals form the rotation, and the model normalizes those totals as:

\[ w_{i,t}=\operatorname{softmax}(\log r_{i,t}) =\frac{r_{i,t}}{\sum_{j\in R_t}r_{j,t}}, \qquad \widehat M_{i,t}=19{,}680\,w_{i,t}. \]

Zero raw weights remain zero. This makes roster minutes conserve exactly while automatically reallocating a manually unavailable player's minutes to teammates.

Planning Controls

The Win Projections page exposes both forward inputs for every player:

\(+/-\), the default NAIL-RAPM projection, which users may override for a scenario;
\(G_{\mathrm{available}}\), projected medically available games out of 82;
(E[\mathrm{MPG}\mid\mathrm{available}]), conditional playing time.

Internally, the page converts the whole-game control to \(P(\mathrm{available}) = G_{\mathrm{available}}/82\). Users may override any of these inputs before recalculating. A \(+/-\) override affects only the local scenario's minute-weighted team strength; it does not alter the minutes allocation or stored NAIL release. The page re-ranks the full roster after each edit, then applies the same top-15 normalization. Thus, setting a current rotation player's availability to zero drops them from the rotation and promotes the next-highest raw projection; a player outside the rotation enters only by rising into the top 15. Every change reallocates the fixed 240 regulation minutes per game across that team rather than adding or removing minutes from the league schedule.

Evaluation

Hyperparameters are selected from completed 2020-21 through 2022-23 seasons, then tested on frozen 2023-24 through 2025-26 seasons. The replay reports conditional-minutes error, combined raw expected-total-minutes error, and an exact team-minute-conservation audit for every historical opening roster.

The selected player-state configuration is full one-season persistence \(\rho=1.0\), a 15 available-game later-update strength, and a 30 available-game initial-state strength.

Frozen metric, mean of three seasons	Age-only control	Forward Conditional Minutes v0.1
Conditional minutes MAE	9.85	6.14
Conditional minutes RMSE	11.25	8.04
Available-game-weighted conditional MAE	9.62	5.45
Available-game-weighted conditional RMSE	11.09	7.11
Raw expected-total-minutes MAE	695.1	505.9
Raw expected-total-minutes RMSE	806.5	632.4

The control uses the same Forward Availability v0.2 probability but replaces the player state with the age baseline. The state model wins every aggregate metric. Every one of the 90 frozen opening-roster team rows sums to exactly 19,680 regulation minutes after normalization.

Artifact: artifacts/rotation/forward_conditional_minutes/forward-conditional-minutes-20260910T020256Z-31d03d6.

This is a prerequisite for the later game-level joint availability and rotation model in Issue #7.