Definitions, formulas and limits for every metric in the LUMA catalog: ARC, Box+, Impact, RAPM, SHAI, LOAD, xPTS and 3 more. The interactive explorer above shows one entry at a time; the full catalog is listed here.
Possession
An interval of team control, ending in a field-goal attempt, a turnover, or the last free throw of a set. Estimated from the box score, where the coefficient on free-throw attempts approximates the share of them that end a possession.
formula FGA + 0.44 × FTA + TOV − ORB
limits An offensive rebound extends control. The estimator counts the extension as the same possession.
Per 100
A quantity divided by possessions and scaled to one hundred of them. League pace differs by season and by team, so per-game totals are not comparable across either. The default denominator throughout LUMA.
formula 100 × x / POSS
True Shooting
Points scored per scoring attempt, counting three-point field goals and free throws at their realized value rather than treating every attempt alike.
formula PTS / (2 × (FGA + 0.44 × FTA))
limits Assigns the whole value of an assisted field goal to the shooter.
Relative True Shooting
True Shooting measured against the league in the same season.
formula TS − lgTS
Playmaking True Shooting
True Shooting computed over the shots a player creates for teammates instead of the shots he takes, which prices creation on the same scale as scoring.
limits Depends on potential-assist tracking. Unavailable before 2014.
Usage
The share of team possessions a player terminates while on the floor.
formula 100 × (FGA + 0.44 × FTA + TOV) × (Tm MP / 5) / (MP × (Tm FGA + 0.44 × Tm FTA + Tm TOV))
Z-score
Distance from the pool mean, expressed in standard deviations, which renders quantities on different scales additive.
formula (x − μ) / σ
RAPM
Regularized adjusted plus-minus. The game is divided into stints, being the intervals during which the ten players on the floor do not change. The model solves for the set of player coefficients that best explains every stint at once. Because each player appears alongside many teammates and against many opponents, the solution separates the player from his context.
formula argmin ‖Xβ − y‖² + α‖β‖²
limits Requires many possessions. Within a single playoff run it cannot fully separate a player from the lineup he plays in.
Regularization
The penalty term in ridge regression. It shrinks each coefficient toward zero in inverse proportion to the possessions supporting it.
Replacement level
The production of a freely available player. Value is stated as the difference above it, so zero denotes replacement rather than zero production.
Window
The number of seasons blended into one estimate, weighted toward the most recent by exponential decay. Two years is the default.
limits A longer window reduces variance and lags genuine change.
Percentile
Rank within the qualified pool. The pool is set by the pool flag; changing the filter changes the percentile.
Expectation
What an average player would produce from the same shot profile. LUMA prices every attempt across 707 bins of location and type, so a metric described as “above expectation” is measured against that baseline, not against a league average rate.
Adjusted Regularized Contribution (ARC v3) ch_arc
ARC units — sd 3.00 per season · ARC
A decomposable impact metric that blends regularized adjusted plus-minus (IMPACT) with the expected value (EV) of a player's box score actions (BOX+).
estimator Two halves.
IMPACT — ridge-regularized adjusted plus-minus, solved by LSQR over lineup stints of at least 15 seconds, alpha 3000, on counted possessions (FGA + 0.44 FTA + TOV − OREB); a stint whose possessions cannot be counted is dropped, never clock-estimated. Rows are weighted by possessions × 0.5^(n/1.3). The solve is anchored to a box prior at 0.25×, and the defensive side carries a team-season fixed effect at penalty 100 so scheme is absorbed by the team column before it is charged to whoever was on the floor.
BOX+ — every recorded action priced at its expected point value, measured on 2016-2025 play-by-play and applied to all 29 seasons.
formula ARC = [ 0.65 · z(Impact) + 0.35 · z(Box+) ] · s, s set so sd(ARC) = 3.00
blend 0.65 / 0.35 by weight, 0.68 / 0.32 by variance — the halves correlate 0.61, so each weight carries part of the other’s covariance. Only 9% of ARC’s spread is box-score information the Impact solve does not already hold.
scale Zero is the value of an average possession, not of an average player: each component is centred on a mean taken over players weighted by the possessions they played, and ARC is the sum of the centred halves, so every decomposition survives centring unforced. Playing time concentrates on better players, so the unweighted player mean falls below zero — −1.15 across the 578 players of 2025-26. The sd 3.00 scale is fitted on the qualified pool and applied to everyone.
reads One point of ARC is a third of a standard deviation. Nine of the 393 qualified in 2025-26 cleared +6, and the qualified average is −0.49 because better players absorb more possessions.
sums from O-ARC · D-ARC
board Adjusted Regularized Contribution (ARC v3) leaderboard
O-ARC ch_oarc
Offensive Adjusted Regularized Contribution
ARC units — sd 3.00 per season · ARC
The component of ARC attributable to possessions on which a player’s team has the ball.
ARC’s blend applied to the offensive inputs alone — offensive Impact and the offensive terms of Box+.
formula O-ARC = s [ 0.65 (O-Impact − mean O-Impact)/sd(Impact) + 0.35 (O-Box+ − mean O-Box+)/sd(Box+) ], s = 3.00/sd(blend)
blend 0.65 / 0.35 by weight, as ARC. By variance, 0.61 / 0.39.
scale Zero is the average possession, not the average player.
part of Adjusted Regularized Contribution (ARC v3)
board O-ARC leaderboard
D-ARC ch_darc
Defensive Adjusted Regularized Contribution
ARC units — sd 3.00 per season · ARC
The component of ARC attributable to possessions on which the opponent has the ball, counted as points prevented.
ARC’s blend applied to the defensive inputs alone — defensive Impact and the defensive terms of Box+.
formula D-ARC = s [ 0.65 (D-Impact − mean D-Impact)/sd(Impact) + 0.35 (D-Box+ − mean D-Box+)/sd(Box+) ], s = 3.00/sd(blend)
blend 0.65 / 0.35 by weight, as ARC. By variance, 0.96 / 0.04.
scale Zero is the average possession, not the average player. Spread is sd 1.31 against O-ARC’s 2.52.
part of Adjusted Regularized Contribution (ARC v3)
board D-ARC leaderboard
BOX+ ch_boxvalue
points per 100 possessions raw, ARC units normalised — the site shows the normalised series, where BOX+ is the 0.35 half of the blend · ARC · Box+
The expected point value (EPV) of what a player’s box score records.
Every recorded action is priced at the consequence measured for it in play-by-play, summed and divided by possessions.
formula Box+ = ( PTS + 0.727 AST − 2.052 TOV + 1.080 OREB + 2.138 STL + 0.579 BLK + 0.100 DREB − 0.862 PF + charge ) per 100 possessions, × poss / (poss + 2500)
calibration Measured on 2016-2025 play-by-play, pooled and season-independent, then applied to all 29 seasons — one box channel across the history rather than a split at 2016. Assist +0.727 is the pooled surplus of points created over an ordinary terminal attempt, divided by actual assists. Turnover −2.052 is the forfeited attempt (0.993) plus the opponent’s next one (1.059). Steal +2.138 is 0.993 plus 1.145, the higher value of the attempt that follows a live-ball turnover. Block +0.579 follows from a 0.583 defensive recovery rate, offensive rebound +1.080 from continuation after a miss, personal foul −0.862 from the decade’s observed free-throw cost collapsed onto the official PF count.
charge 3.0 (TS − league TS) × true-shooting attempts per 100 × (usage / league usage) × tsa / (tsa + 150). It pays scoring above league efficiency in proportion to volume, and the last factor discounts thin shooting samples.
shrink poss / (poss + 2500), so a partial season reads as a partial season rather than as an extreme one.
scale A sum, not a rating: zero is an empty stat line, not an average player. The qualified mean in 2025-26 is +12.56 raw.
Both series are published for every node — raw in points per 100, normalised into ARC units by an affine per-season map whose scalars are fitted from the board itself, so the tree cannot drift from the rating. On the 2025-26 board the normalised half is c_box = 0.1639 BOX+ − 2.2918, r = 0.999997; the constants are per season.
identity Offensive BOX+ + Defensive BOX+ = BOX+. Where a player has no split, both halves are absent rather than estimated.
limits Records only what the box sheet records. Screening, spacing and rotation enter through Impact.
sums from Offensive BOX+ · Defensive BOX+ · Box attribution
board BOX+ leaderboard
Offensive BOX+ ch_obox
points per 100 possessions · ARC · Box+
The share of BOX+ produced on a team’s own possessions.
estimator A proration, not a second pricing. The ratio comes from the v2 box formula, which solved the two sides explicitly (PTS, AST, OREB, TOV and the miss penalties against STL, BLK, DREB); that ratio is then applied to the v3 action-EV total, so the halves sum to the BOX+ the board publishes rather than to a number from the previous version.
formula Offensive BOX+ = BOX+ × oShare, oShare = oboxRaw / (oboxRaw + dboxRaw)
scale A share of that sum, on the same scale.
part of BOX+
board Offensive BOX+ leaderboard
Defensive BOX+ ch_dbox
points per 100 possessions · ARC · Box+
The share of BOX+ produced on the opponent’s possessions.
formula Defensive BOX+ = BOX+ − Offensive BOX+
scale A share of that sum, on the same scale.
part of BOX+
board Defensive BOX+ leaderboard
IMPACT ch_impactvalue
points per 100 possessions — shown in ARC units on the board, where it is the 0.65 half of the blend · ARC · Impact
A box-priored regularized adjusted plus-minus.
The team point differential per 100 possessions attributable to a player once the teammates and opponents he shares the floor with are solved out. Every stint is one row, recording the ten players on the floor and the margin over the possessions counted in it, and the whole window is solved at once — a player separates from his lineups because he appears in many of them.
estimator Ridge, penalty 3000, solved by LSQR over stints of at least 15 seconds. Possessions are counted from the play-by-play (FGA + 0.44 FTA + TOV − OREB); a stint whose possessions cannot be counted is dropped, never estimated from the clock. Rows are weighted by possessions, and in multi-year windows by a season decay of 0.5^(n/1.3).
The solve is anchored to a box prior at 0.25×: the target is the stint margin less 0.25 times the on-court OBPM/DBPM difference, and the same quarter of a player’s own box rating is added back to his coefficient afterwards. Defence additionally carries a team-season fixed effect at a looser penalty, so scheme is absorbed by the team column before it is charged to whoever was on the floor.
scale Zero is the average possession, not the average player. The prior does not dominate: its term carries sd 0.73 against Impact’s 2.02, and shipped Impact correlates 0.96 with the same solve run without it.
identity Offensive + Defensive = Impact, exactly, across 8,551 player-seasons. Possession value + Shot value = Impact, also exactly, stated on this channel basis.
why Plus-minus asks how a team does with a player on the floor, but a player never plays alone, so his number carries his teammates and opponents with it. This solve separates them: every stint is a row, and it works out how much of the result belongs to each of the ten players on the floor. Each player starts from an estimate built off his box score — the box prior — and moves as the lineup evidence comes in. A ridge penalty guards against low-sample performances, so it takes more possessions of evidence to move a player further from that starting estimate.
limits Requires many possessions. Within a single postseason it cannot fully separate a player from his lineup.
sums from Offensive IMPACT · Defensive IMPACT
board IMPACT leaderboard
Offensive IMPACT ch_oimpact
points per 100 possessions · ARC · Impact
The half of IMPACT earned while a player’s team has possession.
Team points per 100 possessions added while he is on the floor, net of the teammates and opponents he shares it with.
estimator The offensive coefficient of the Impact solve.
formula Offensive impact = oTS + oTOV + oREB
scale Zero is the average possession.
identity Offensive + Defensive = Impact. oTS + oTOV + oREB reproduces Offensive Impact to 0.001 across 8,551 player-seasons.
part of IMPACT
sums from oTS · Ball security (O) · Off. rebounding (O)
board Offensive IMPACT leaderboard
Defensive IMPACT ch_dimpact
points per 100 possessions · ARC · Impact
The half of IMPACT earned while the opponent has possession.
Opponent points per 100 possessions prevented while he is on the floor, net of the teammates and opponents he shares it with. Positive is better.
estimator The defensive coefficient of the Impact solve.
Defence carries one extra column the offensive side does not: a fixed effect for every team-season, penalised at 100 against the players’ 3000. The looser penalty lets the team column take shared variance first, so a scheme is charged to the team that runs it rather than split among whoever was standing in it. The columns were checked against real team defence — the fitted effect correlates −0.9236 with points allowed per 100 relative to league average, across 90 team-seasons, confirming the sign convention that positive is good defence.
formula Defensive impact = dTS + dTOV + dREB
scale Zero is the average possession.
why Without the team column a defensive rating does not belong to the player. Take the 65 players who changed teams between 2020-23 and 2023-26 and predict each one’s later defence twice: from his own earlier rating, and from his new teammates’. At the shipped penalty without the fixed effect those predict equally well — 0.1750 from himself, 0.1742 from his new teammates. The rating was a team’s rating wearing a player’s name. With the fixed effect his own past predicts at 0.2438 and his new teammates at 0.0118, a ratio above 20.
The correction is defence-only. Offence clusters by team less to begin with, and there the team and the player are genuinely the same thing — Jokic is Denver’s offence, and stripping that out would be removing the player, not the scheme.
limits A defender whose value IS his team’s scheme has that credit absorbed by the team column. The fixed effect cannot separate a defensive system from the player who makes it work, and it does not try to; it removes team defence from every player on the roster equally.
part of IMPACT
sums from dTS · Turnovers forced (D) · Def. rebounding (D)
board Defensive IMPACT leaderboard