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This term was highly significant, and helped the overall fit of the regression quite a bit. Using the square root of the interaction both makes sense theoretically maintaining a denominator of opportunities or possessions and empirically it is more significant and helps the overall fit of the regression more.

Each player's raw BPM is calculated through the above equation and is then adjusted at the team level. The team's efficiency differential, adjusted for strength of schedule, is known. It is, by definition, the true sum of all players' contributions. The team adjustment is simply a constant added to each player's raw BPM and is the same for every player on the team.

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The constant does 3 things: it adds the intercept to the BPM equation, it adjusts roughly at the team level for things that cannot be captured by the box score primarily defense , and it also adjusts for strength of schedule. The formula for this adjustment looks like this:. Jeremias Engelmann has done extensive work on how lineups behave, and he discovered that lineups that are ahead in a game play worse, while lineups are behind play better — even if the exact same players are playing.

Perhaps it's an effort thing? While the source is unclear, the effect is both significant and linear.

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Therefore, the team adjustment accounts for this effect. At the lineup level things are more tricky, since it is hard to predict how far ahead or behind a lineup typically is in normal game situations. In the postseason, the team adjustment just ensures that 5 times the weighted average of individual BPM scores adds up to the team's raw efficiency differential, without an SOS adjustment this may be tweaked in the future.

Thus you have the final BPM — a box score regression, coupled with a team adjustment. A second regression was run to estimate offensive and defensive BPM based on those values. It was tuned to minimize error on both the offensive and defensive RAPM values simultaneously. A team adjustment is added to the results of this regression to force the team sum to equal the adjusted team offensive rating above or below league average similar to how the team adjustment is done for overall BPM. R 2 is a measure of how well a regression or model fits the data it is built upon.

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A value of 0 means that the model doesn't explain the data at all, while a value of 1. The weighted R 2 results for the BPM regressions are:. RAPM, even a year dataset, has some error associated with it, so even a theoretical "true" measure of performance would still show an R 2 below 1. Perhaps the best way to get a feel for the accuracy of a model is to look at it graphically. Looking at historical results can help gain confidence in the regression's accuracy.

The defensive list is rather repetitive — apparently Ben Wallace was really good at playing defense? Defense is only partially captured by the box score, so elite defenders based on position and communication, like Kevin Garnett and Tim Duncan, will not be properly represented. The regression mathematically accounts for that, pulling all of the estimates closer to average.

These tests include the best measure of a metric's validity: its out-of-sample predictive ability. Neil also recently tested BPM in that way, though the results are not published currently, and it outperformed any other box-score stat, except that it equaled FiveThirtyEight. The fact that this comparison is in sample for BPM inflates its R 2 somewhat, but since this regression has relatively few degrees of freedom vs.

A long and comprehensive discussion on defining this level for the NBA was had at Tom Tango's blog, and is worth a read. Tom Tango is a baseball sabermetrics expert, and one of the originators of the replacement level framework and the Wins Above Replacement methodology common now in baseball. The conclusion was to establish Unlike in major league baseball, players below replacement level do frequently play, primarily for development purposes.

Rookies are frequently below replacement level, but there are no formal minor leagues to act as a development system like major league baseball has, so they end up getting playing time in the NBA in order to develop. Also, some teams tank, and trade for Byron Mullens to help that effort. If one were to define a "replacement level" for offense and defense, it would be Almost all point guards would be well below the The reverse is true of post players.

This yields the number of points the player is producing over a replacement player, per TEAM possessions over an entire season. His VORP, then, would be [8. A player with a VORP of 4. Sometimes good players play few minutes for reasons outside their control, and would be worth more because they should be getting more minutes. Still, for a crude estimate, VORP is valuable. It does measure fairly accurately what a player did produce in terms of value for a given team.

To convert VORP to an estimate of wins over replacement, simply multiply by 2. This translates a player's point differential approximately into wins, using the conversion rate near league-average rather than that in the diminishing returns area of the Pythagorean formula. By this methodology, Michael Jordan in was worth about 32 wins. In reality, he would quickly push the team into the diminishing returns region of the points-to-wins conversion. Here's a look at the top 10 seasons of all time by VORP. This is the best measure of actual value contributed to the team.

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Notice that Michael Jordan played a few more minutes per season than LeBron, thus increasing his overall value. The playoff team efficiency, which is used in the team adjustment portion of the BPM calculation, is derived as follows:. As an example, in San Antonio won the title. That is the same scale as regular season efficiency differential—the Spurs were really dominant. This is reminiscent of Hollinger's playoff ratings. Be aware—players on teams that played more games will have higher VORP values.

A team that swept several rounds may play several games fewer than a team that was taken to seven games a couple of times. While it may be argued that college basketball is somewhat different, there is no easy way to derive BPM coefficients specifically for college basketball, so the NBA coefficients will have to suffice. A rebound is still a rebound The coefficient that could be the most questionable would be the MPG coefficient, because it is unclear if minutes distribution at the college level is based upon the same criteria as at the pro level, and the length and pace of games is different.

Until further information becomes available, all coefficients have been used as-is. VORP, on the other hand, does not make sense for college basketball. VORP is derived based on salaries, and in a consistent market, and is primarily useful in relation to evaluating salaries. In college, on the other hand, every school and conference has widely disparate situations, and since there are no salaries, their is neither a rational method nor strong need for deriving or using VORP.

Data for college basketball is currently available only back to the season. Here are the top 10 seasons in the database, minimum minutes played.

This was a bug fix update to address a couple of issues. The most significant change is to correct the weighting scheme used in the regression. The original BPM regression used a sqrt Poss weighting, which was incorrect. This version corrects the scheme to a number of possessions weighting system, with an additional correction to account for the effect of the prior on the RAPM values. This fix also adjusted the values slightly, causing efficiency and blocks to be valued somewhat more highly and turnovers to be somewhat worse.

Centers and other players with high shooting efficiency and blocks were helped the most by those tweaks such as Anthony Davis , and players with high usage and relatively low efficiency like Russell Westbrook were hurt. The overall BPM gap between those helped the most and hurt the most was about 1. The effects of these corrections are small. This update also includes a revision to how VORP is handled for partial seasons. Previously, partial seasons would show the player's production extrapolated to the full 82 game season—VORP was behaving more as a rate stat.

That has now been changed, so that VORP will now act as a counting stat over the season, with each 1 point being equal to 1 point of season-end team point differential per 82 games. A multitude of variables were evaluated via Excel's regression tools to find the best design for BPM. Here is the best regression: highest adjusted R 2 with every term clearly significant. This is how the variables were finally chosen.

Note, this is without the team adjustment — just raw box score stats vs. Points along the equator have latitudes of zero.


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The north pole has a positive north latitude of 90, and the south pole a negative south latitude of Accordingly, northern-hemisphere locations have positive latitude, and southern-hemisphere locations have negative latitude. Longitude describes how far east a point is, from the prime meridian: an arbitrary line on the earth surface running from pole to pole. The Greenwich Observatory near London, England, has, by definition, a longitude of zero. Latitudes therefore are values in the range [, 90].

Longitudes are values in the range , ]. These values are sometimes expressed in degrees, minutes, and seconds, rather than degrees and decimals. In the Napoleonic era, the meter was first defined so there were ten million of them in the distance from the equator to one of the poles. However the earth bulges a little, so This formula But north or south of the equator, the lines of longitude get closer together, so if you move a degree to the east or west, you move less than The distance you actually move when you go one degree east or west is actually this number of km.

Those of us in some former British colonies use miles. So there are 69 statute miles per degree or 60 nautical miles per degree.