RegressedHoops

All-time lists are intended to measure a player's greatness and happen to be one of the most debated topics among NBA fans. These lists are of course a very subjective matter and an easy way to make someone angry at you.  My goal was to create an all-time list solely based on what players did on the basketball court, or in other words their production throughout their career.  The statistic I chose to use in order to do this is known as "Box Plus/Minus" (BPM). Per Basketball-Reference, "Box Plus/Minus (BPM) is a box score-based metric for evaluating basketball players' quality and contribution to the team. It is the latest version of a stat previously called Advanced Statistical Plus/Minus. BPM relies on a player's box score information and the team's overall performance to estimate a player's performance relative to league average. BPM is a per-100-possession stat, the same scale as Adjusted Plus/Minus: 0.0 is league average, +5 means the player is 5 points better than an average player over 100 possessions (which is about All-NBA level), -2 is replacement level, and -5 is really bad." BPM is of a class of statistics which look to combine box-score stats with plus minus (a "best of both worlds" stat). The weights for several box-score based stats in BPM's formulation have been empirically determined by regressing them over Regularized-Adjusted Plus Minus (RAPM). RAPM is a fairly accurate estimate of a player's actual plus/minus impact using play-by-play data and accounting for player's teammates (thereby eliminating the problems and noise with normal plus/minus). BPM is the best box-score metric available today and is more stable than RAPM itself. It is therefore the ideal stat to be used in the creation of an all-time list. BPM is purely a rate-stat and in order to convert it to actual production we must consider the amount of minutes played. This is handled by Value Over Replacement Player (VORP) which converts BPM into what a player's overall contribution to his team is measured against a theoretical "replacement" level player. The formula for VORP is [BPM – (-2.0)] * (% of minutes played)*(team games/82).

I looked at several factors in order to determine a player's "all time score" so a list could be created. First, I calculated at a player's career total VORP in the regular season. Next, what a player is truly made of is defined in the playoffs where the best-of-the-best have shined the brightest. Just like Neil Paine, for each player I made adjustments for the level of competition faced, home-court advantage, and the leverage of each game, and recalculated their VORP for each round of the playoffs. I weight production in each successive round of the playoffs higher (e.g. conference finals matter more than the 1st round) and finally compute the weighted sum to determine a player's career playoff production. Playoff production will obviously be quite a bit lower than regular season production due to the fewer minutes and games; this undervalues some of the greatest players who have had several championship runs, for example. Therefore, I multiplied each player's playoff production by 6. After this adjustment, playoff production is in general slightly more valuable than regular season production. I personally find this to be fair since the playoffs are where teams' offensive and defensive effort are at their highest. Furthermore, the sample size of games played in the playoffs is usually pretty large for most of the greatest players in NBA history, so this is generally not a concern. Finally, I summed a player's regular season production and playoff production and put it on a scale of 1000 which results in their all time score. A score of 1000 can be thought of as the best possible career. Since BPM is not available for players before the 1973-1974 season due to certain box score stats not being recorded, it was calculated using estimates of these stats (this is for players such as Wilt, Jerry West, Oscar Robertson, etc.).



Michael Jordan is by far the best player of all time -- who knew!?  This list will remain constant in the near future with the one exception of Lebron James. Lebron is currently 3rd all time and will likely end up as the 2nd best player ever at his current pace, passing Kareem in the next few years. It's safe to say that Jordan is untouchable. Other active players that rank highly all time include Jason Kidd (26th), Chris Paul (28th), Paul Pierce (31st), Pau Gasol (41st), and Kevin Durant (47th).

Overall, in my opinion, this all time list is as close to the "truth" as you can get since it ranks players solely based on what they have done on the basketball court, using the best metric we have to measure player performance today. Feel free to leave your thoughts and feedback you may have on improving the methodology used in the comments below. 

Recently, I went back and was looking at game logs of Kobe's prime years, namely 2000-2003 and 2005-2010. It was amazing at how consistently good Kobe was on a night to night basis. He is often criticized for things such as never shooting about 47% from the field in a season or being inefficient. This was hard to believe after eye-balling his game logs and then I came across the culprit - the outliers. I calculated that Kobe was roughly prone to about 10 "bad" games in a season in which he shot the ball very poorly, swaying his season's averages quite a bit. This should be no surprise as 82 games (less in his case as he missed a few games here and there) is a small sample size and outliers do make a significant difference. Now, it's no secret Kobe is more prone to these type of games, than say Lebron, because of his style of play. However, these outliers have perhaps painted Kobe's image in a negative manner, which is rather unfair to him in general.

Therefore, I went through each of his prime seasons and removed 10 outliers from every NBA player's game-log to recalculate each their new season averages. Over 85% of the data is still being retained in these situations (I didn't just go and remove all of the games Kobe shot less than 50%). I hypothesized that this is likely a better method of finding out just how good someone is and shows a new perspective of Kobe in his specific case. For the outlier removal, I used a One-Class Support Vector Machine (SVM) (reference: http://scikit-learn.org/stable/modules/outlier_detection.html). The SVM was trained on every player's game logs containing box score data. Since this is an automated machine learning method, the software picked up on different "types" of outliers for each player. For example, an outlier for Kobe may be a game with low FG% while an outlier for a big man may be games with low rebounds. Here are Kobe's updated season averages from 2000-2010 (with 2004 left out as he missed too many games due to his rape case and such):


As you can see, it has a dramatic impact on Kobe's statistics. His averages with 10 outliers removed are absolutely remarkable. Bryant's FG% on average is around 49% during this time period while maintaining a beyond elite TS% in every season as well. One season that stands out to me is his 2007 season in which he averaged 33.31 PPG on a 61.0% TS%; even after outlier removal the only player in NBA history to score at this volume and efficiency was Michael Jordan. Furthermore, after league-wide outlier removal, Kobe led the league in both PER and WS/48 in 2006 and 2007 (a feat accomplished by only a handful of players in NBA history). These results carry over to the playoffs as well. For example, Kobe's approximate PER of 30.88 in the 2009 playoffs, after outlier removal (6 games removed for each player), is the 5th highest ever in a championship run (at least 16 games).

Next, I wanted to see if my hypothesis was correct and if these updated advanced metrics were actually improved compared to before (correlated better to team win %). The method to calculate this, is to compute the minute-weighted average of WS/48 and PER for every team, and regress this upon winning %. This is described by Neil Paine here.  He found that on a 1-year basis WS/48's correlation to wins was 0.694 and PER's correlation to wins was 0.638. I ran the correlation test again with the outlier-removed WS/48 and PER and obtained correlations of 0.726 and 0.654 respectively. This was very exciting because it had a significant improvement in each of the statistics and confirmed my hypothesis. This subsequently means that the outlier removal is not simply artificially inflating stats to make players like Kobe look better, but is doing a better job of explaining wins (and therefore player value).

To conclude, advanced stats do not "hate" Kobe Bryant. When popular advanced metrics such as WS/48 and PER are improved on with outlier removal, he is viewed in a very favorable light by them. In fact, after this, he is one of the very few players to lead the league in WS/48 and PER (Kareem, Moses Malone, Larry Bird, Jordan, Shaq, T-Mac, KG, Dirk, Lebron, and KD are the only others) -- and he did it twice (joining only Kareem, Bird, Jordan, Shaq, and Lebron to do it multiple times). Further experiments would include seeing how the correlation of WS/48 and PER change with respect to the number of outliers removed.

Feel free to leave your thoughts and suggestions in the comments below.

Kobe Bryant, Lebron James, and Tim Duncan are the three best players of this generation (we can count Shaq as part of the previous generation). All three have accomplished enough in their illustrious careers to be considered Top 10 players of all time. Fans constantly debate with each other about who the best player out of the three are. I decided to compare them based on how they've performed in the playoffs, against the best competition, by looking at their on/off court plus-minus impact stats. The advantage plus-minus stats have over box score based metrics (e.g. PER, Win Shares) is that they technically account for everything a player did on the court.

In order to come up with an offensive on/off impact score I combined the following:

1) Team's change in eFG%
2) Team's change in ORTG (offensive rating)
3) Team's change in TOV%

Furthermore, in order to come up with a defensive on/off impact score I combined the following:

1) Opponent's change in eFG%
2) Opponent's change in ORTG (offensive rating)
3) Opponent's change in TOV%
4) Team's change in STL%
5) Team's change in BLK%
6) Opponent's change in TRB%

Here are the results ranked by combined net impact:

The table shows Kobe is the best overall player by this criterion. Each have played thousands of minutes in the playoffs so sample size is not necessarily a concern. Kobe has the largest offensive impact and Duncan, not surprisingly, has the biggest defensive impact.