User:Freemanalfano/sandbox

= Box Plus-Minus = Box Plus-Minus, or BPM, attempts to measure the sum of a basketball player's individual contribution to their respective team while they are on the court. The contribution is based around the estimation of how many points above or below league average a player adds to their team per 100 possessions, where the value of a league average player is 0.0. For example, if a player were to have a +4 BPM, that player's respective team would be, on average, four points per 100 possessions better with that player on the floor than if a league average player were on the floor instead of them. The statistic was created and developed by Daniel Myers, a bridge structural engineer and sports data analyst, and released on Basketball Reference, a website that tracks both advanced and simple box score basketball statistics.

History
BPM was created after years of development in 2014 as an adaptation to Statistical Plus-Minus (SPM), which also attempted to determine a player's estimated contribution to their team per 100 possessions. SPM, however, only used box score statistics as its input, whereas BPM also includes team adjusted efficiency and player positions in its calculation. SPM was developed by Dan Rosenbaum, the current Senior Director of Basketball Strategy for the Detroit Pistons. Rosenbaum also developed Adjusted Plus-Minus, or APM, which seeks to account for the quality of a player's teammates and opponents in determining that player's statistical contribution to their team by using a linear regression.

The original version of BPM used Regularized Adjusted Plus-Minus, or RAPM, as its regression basis. RAPM is an adaptation of Rosenbaum's aforementioned APM, which minimizes the effects of significant outliers by way of a ridge regression. This model had minor issues that were later fixed in February of 2020 in an updated version called "BPM 2.0". Most improvements to BPM 2.0 involve the calculation's regression basis, such as using a more statistically significant database for evaluation, further minimizing the effects of outlier data, and removing minutes per game from the regression calculation.

Player data tracked for BPM goes from the 1973-74 NBA season to present. Myers presumably chose 1973 as the cutoff for determining player's BPMs because that is the year the NBA started tracking both blocks and steals in box score statistics. Between 1974 and 1984, BPM values are calculated as season totals, whereas from 1985 to present, BPM is calculated by summing up players' individual game BPM values for every season.

Calculation
The three statistics, or categories of statistics, used to calculate BPM are (1) player box score statistics per 100 possessions, (2) team adjusted efficiency per 100 possessions, and (3) player positions. The steps for calculating BPM are as follows:

1. Quantitatively determine the player's position and role on offense using the given player's full statistics for a given season.

2. Using the player's calculated position and role on offense, find the coefficient necessary to be used in the calculation for BPM.

3. Find that player's team average per adjusted shot attempt and compare this calculated value to the baseline points per adjusted shot attempt.

4. Add a constant points per adjusted shot attempt for a player's team.

5. Calculate the raw BPM for the player (formula shown below).

Raw BPM = a*ReMPG + b*ORB% + c*DRB% + d*STL% + e*BLK% + f*AST% - g*USG%*TO% + h*USG%*(1-TOV%)*[2*(TS% - TmTS%) + i*AST% + j*(3PAr - Lg3PAr) - k] + l*sqrt(AST%*TRB%)

Where:

ReMPG = Regressed Minutes per Game (slightly altered MPG to avoid instances where a player plays very few games but has high minutes per game)

ORB% = Offensive rebounds divided by offensive rebounding opportunities

DRB% = Defensive rebounds divided by defensive rebounding opportunities

STL% = Steals divided by number of opponent possessions while player is on the floor

BLK% = Blocks divided by number of two-point field goal attempts while player is on the floor

AST% = Assists divided by teammate field goals (excluding FGs made by the player)

USG% = Number of possessions "used" by a player divided by number of possessions while they are on the court (“used” meaning ends in a shot attempt, free throws, or turnover by the player)

TOV% = Turnovers divided by possessions "used" by the player

TS% = Points divided by ( 2 * TSA) where TSA = True Shooting Attempts (FG + 0.44 * FTA)

TmTS% = Team's true shooting

3PAr = Three-point attempts divided by Field goal attempts

Lg3PAr =Three-point attempts divided by Field goal attempts (for entire league)

TRB% = Total rebounds divided by total rebounding opportunities

6. Sum up the raw BPM values for that player's team and compare this to the team's adjusted efficiency per 100 possessions.

7. Add a constant to the raw BPM of every player on that player's team. This is known as the "team adjustment."

8. Add the player's raw BPM to the team adjustment.

The coefficients used to quantitatively determine a player's position and a player's offensive creation position vary between 1.0 and 5.0. 1.0 corresponds to a point guard and creator, whereas 5.0 is a center and receiver. Point guards have a position adjustment constant of about -1.7, creators have an offensive role adjustment constant of -0.86, and receivers have an offensive role adjustment constant of +0.86. These coefficients are calculated based solely on a player's box score statistics with adjusted point totals based on a player's team scoring context. Some notable determinations in the coefficient values include awarding assists at a significantly higher rate for big men than for guards, counting offensive rebounds higher for guards than for other positions, and punishing low-usage players for field goal attempts more so than for high-usage players.

The team adjustment constant is calculated by determining a team's efficiency per 100 possessions, which generally falls around -8 because it also serves as the intercept of the regression. Also factored into this calculation is how often a team plays with a lead or from behind, which has a linear effect on the team's quality of play (teams in the lead are about 0.35 points per 100 possessions worse for every additional point added to their lead).

The regression basis was created using four five-year RAPM regressions from 1996 to 2016. This regression is informed using a Bayesian inference based on team quality and minutes per game.

Values are assigned to both a player's Offensive Box Plus-Minus (OBPM) and their Defensive Box Plus-Minus (DBPM). DBPM is calculated by subtracting a player's OBPM from their total BPM.

BPM Data
The following are the categories for where an NBA player belongs historically based on their BPM for a given season:

+10.0 or higher: Historically great season

+8.0 to +9.9: MVP-caliber season

+6.0 to +7.9: All-NBA-caliber season

+4.0 to +5.9: All-Star-caliber season

+2.0 to +3.9: Above average starter

+0.0 to +1.9: Average starter or above average bench player

-0.1 to -1.9: Average bench player

-2.0 and below: Replacement level player The highest all-time BPM was recorded by LeBron James in 2009 on the Cleveland Cavaliers with a BPM of +13.2 for that season. James and former Chicago Bull Michael Jordan account for 10 of the top 15 highest BPM seasons, including four of the top five. Prior to updating to BPM 2.0, Oklahoma City Thunder guard Russell Westbrook had the highest single-season BPM in 2017 of +15.6; this mark is now 11th all-time. Nine of the top 15 BPM seasons of all time were recorded since 2008, while no top 15 all-time BPM seasons happened prior to Jordan in 1988. Jordan has the highest career BPM with a mark of +9.22, followed by James with a total of +8.93. The ABA leader in career BPM is Julius Erving, whose career mark (+9.19) is more than double the career value of any other ABA player. While Jordan and James have seven of the 10 highest OBPM seasons of all time, neither has a season that is in the top two. Golden State Warriors guard Stephen Curry set the record for OBPM during his MVP season in 2016 when he recorded a mark of +10.4.

Drawbacks/Limitations
Myers states that BPM was made "to intentionally only use statistics that are widely available and available historically." While allowing for cross-historical reference, this nature of BPM means that recently developed advanced basketball statistics, such as Player Impact Plus-Minus (PIPM) and NBA.com's Player Impact Estimate (PIE), are not included for players in the modern era. On the other hand, players before 1973 will only have a simplified BPM calculated for their seasons because neither their block nor steal numbers will be included in their calculations.

By using box score statistics as the basis of calculation, many contributions a player provides on the floor will go unnoticed by BPM. Basketball actions generally viewed as "winning plays," such as setting a pick, boxing out an opposing player, or making a "hockey assist" (a pass that leads to an assist) are not included in a player's box score, and thus are ignored in that player's BPM calculation. However, these contributions should still be reflected in a player's BPM value given that these plays likely contribute to that player's respective team outscoring their opponents while they are on the court.

Although BPM, and OBPM in particular, can capture nearly all the value a player brings on the offensive end of the court because of the extensive nature of offensive box score statistics, there are much fewer (and less accurate) box score statistics for defensive contribution. For example, former Seattle Supersonics guard Nate McMillan recorded the highest single-season DBPM of all time in 1994 (+5.5), but was not awarded the Defensive Player of the Year or named to the All-NBA Defensive First Team. While the defensive impact valued in BPM can be a guide for a player's defensive value, it should not be seen as definitive.

Basketball Reference includes BPM values for college basketball players, but there are no BPM values listed for WNBA players on the site. The reasoning for this exclusion is unknown, considering WNBA games track the same box score statistics as the NBA does, and other advanced metrics are listed for WNBA players on Basketball Reference, such as Player Efficiency Rating, Offensive and Defensive Rating, and Win Shares.

Adaptations of BPM
Basketball Reference converts every player's BPM value into their Value Over Replacement Player (VORP), which estimates each player's individual contribution in comparison with the contribution a league average player would provide. Since BPM determined -2.0 as the value for a replacement level player (see BPM Data), VORP is calculated by subtracting a player's BPM by -2.0 and multiplying that value by the percent of possessions a player played for their team and the team's games divided by 82 (number of regular season games in a normal NBA season). This statistic is similar to the famous and widely used baseball statistic, Wins Above Replacement (WAR), in the sense that they both linearly track what a player's economic value is to their team. Michael Jordan recorded the highest single-season VORP in 1988 with a mark of 12.5, while Jordan and LeBron James account for 11 of the top 15 VORP seasons of all time. James has the highest career VORP of 133.67, while Jordan is second all-time with a value of 116.08.

Sports media conglomerate ESPN created Real Plus-Minus, or RPM, which it uses almost exclusively on its website to determine a player's on-court impact on their team's level of play. Similar to APM, RPM seeks to account for the quality of a player's teammates and competition while they are on the court by using possession data from every game. Like Meyers' BPM, RPM features both an offensive (ORPM) and defensive (DRPM) metric, which are added together to calculate a player's total contribution. RPM was created in 2014 by Jeremias Englemann, a Senior Analyst with the Dallas Mavericks, who received consultation in the development process by Steve Ilardi. Subsequently calculated from this value is RPM Wins, which seeks to estimate the number of wins a player adds to their team for a given season.

The aforementioned advanced basketball statistic Player Impact Plus-Minus (PIPM) expands upon BPM by including a metric that adjusts for luck based on plus-minus data. This metric estimates what a team's Offensive Rating (ORtg) or Defensive Rating (DRtg) should be without variance based on statistically predictive measures that determine these ratings. PIPM was created in 2018 by Jacob Goldstein, a research analyst at Monumental Sports and Entertainment, using the luck-adjusted methodology developed by Nathan Walker. Along with providing improved accuracy for the good or bad luck teams and players experience, PIPM is one of the most accurate metrics at predicting future results. Like both BPM and RPM, PIPM has both an offensive and defensive metric that are combined to calculate a player's overall PIPM. And similar to BPM's VORP and RPM's RPM Wins, PIPM has a Wins Added metric that is calculated from its value to determine how many wins a player contributed to their team that season. This is calculated by converting PIPM's per 100 possessions data to a metric based on minutes played using a Pythagorean wins estimate.