Latrinsorm
08-06-2013, 03:39 PM
There are two big ticket advanced stats for NBA players: Player Efficiency Rating (PER) by Hollinger and Win Shares (WS) by Oliver (not Chivalrous) (as far as I know). When I did a lot of stuff with Win Shares recently, it struck me how punitive it seemed to be to failures (missed shots, turnovers, etc.) vs. PER, so I wondered whether we could describe that analytically.
Here's the formula for PER:
uPER = (1 / MP) *
[ 3P
+ (2/3) * AST
+ (2 - factor * (team_AST / team_FG)) * FG
+ (FT *0.5 * (1 + (1 - (team_AST / team_FG)) + (2/3) * (team_AST / team_FG)))
- VOP * TOV
- VOP * DRB% * (FGA - FG)
- VOP * 0.44 * (0.44 + (0.56 * DRB%)) * (FTA - FT)
+ VOP * (1 - DRB%) * (TRB - ORB)
+ VOP * DRB% * ORB
+ VOP * STL
+ VOP * DRB% * BLK
- PF * ((lg_FT / lg_PF) - 0.44 * (lg_FTA / lg_PF) * VOP) ]
factor = (2 / 3) - (0.5 * (lg_AST / lg_FG)) / (2 * (lg_FG / lg_FT))
VOP = lg_PTS / (lg_FGA - lg_ORB + lg_TOV + 0.44 * lg_FTA)
DRB% = (lg_TRB - lg_ORB) / lg_TRB
It's not as bad as it looks. (Also there's a scaling factor afterwards to turn uPER into PER, but because we only care about the break point of gain vs. loss that doesn't matter. You can multiply 0 by any number you want, it's still 0.) If all we care about are FG (including 3P) and FGA (including 3PA), we simplify to:
uPER = (1 / MP) *
[ 3P
+ (2 - factor * (team_AST / team_FG)) * FG
- VOP * DRB% * (FGA - FG)]
factor = (2 / 3) - (0.5 * (lg_AST / lg_FG)) / (2 * (lg_FG / lg_FT))
VOP = lg_PTS / (lg_FGA - lg_ORB + lg_TOV + 0.44 * lg_FTA)
DRB% = (lg_TRB - lg_ORB) / lg_TRB
Technically taking one more or less shot would alter both the team and league values, but the differences there will be minute. The average NBA team in 2013 took 6720 shots, 1/6720 - 1/(6720 + 1) = on the order of 1 hundred-millionth, and obviously the league values will be even less susceptible to variation.
Now, let's consider 3 scenarios:
1. Take a shot and miss. (FGA -> FGA+1)
2. Take a 2 and make it. (FGA -> FGA+1, FG -> FG+1)
3. Take a 3 and make it. (FGA -> FGA+1, FG -> FG+1, 3P -> 3P+1)
If we take the 2013 season for our league values (factor, VOP, DRB%), we get the following delta[uPER]s:
1. -0.758022372
2. 1.642553145
3. 2.642553145
Now that we know what each scenario is worth, we can set up a couple equations...
If I take only 2s, and I want to know what % I need to shoot to break even in uPER and therefore PER, then I just do:
-.758 * miss + 1.643 * make = 0
-.758 * (attempt - make) + 1.643 * make = 0
Dividing through by attempts:
-.758 + .758 * make / attempt + 1.643 * make / attempt = 0
2.401 * FG% = .758
FG% = 31.6%
Obviously, that's a terrible, terrible FG%, but that's all you need to shoot above to get contributions to your PER. (Note: of course not all shots occur in a vacuum, and shooting makes it harder for you to get rebounds, assists, etc.)
If I take only 3s, and want to know the same answer, I do the same math and get 3P% = 22.2%. Again, this is an awful, awful 3P%. These two suggest that volume shooters get a lot more benefit from PER than they should, so if you are a volume shooter with a weak PER (Rudy Gay comes to mind), it suggests that you are really just awful.
.
Well, what about Win Shares? We are only concerned with Offensive Win Shares, which looks like this...
PProd = (PProd_FG_Part + PProd_AST_Part + FTM) * (1 - (Team_ORB / Team_Scoring_Poss) * Team_ORB_Weight * Team_Play%) + PProd_ORB_Part
PProd_FG_Part = 2 * (FGM + 0.5 * 3PM) * (1 - 0.5 * ((PTS - FTM) / (2 * FGA)) * qAST)
PProd_AST_Part = 2 * ((Team_FGM - FGM + 0.5 * (Team_3PM - 3PM)) / (Team_FGM - FGM)) * 0.5 * (((Team_PTS - Team_FTM) - (PTS - FTM)) / (2 * (Team_FGA - FGA))) * AST
PProd_ORB_Part = ORB * Team_ORB_Weight * Team_Play% * (Team_PTS / (Team_FGM + (1 - (1 - (Team_FTM / Team_FTA))^2) * 0.4 * Team_FTA))
...divided by this...
TotPoss = ScPoss + FGxPoss + FTxPoss + TOV
ScPoss = (FG_Part + AST_Part + FT_Part) * (1 - (Team_ORB / Team_Scoring_Poss) * Team_ORB_Weight * Team_Play%) + ORB_Part
FG_Part = FGM * (1 - 0.5 * ((PTS - FTM) / (2 * FGA)) * qAST)
qAST = ((MP / (Team_MP / 5)) * (1.14 * ((Team_AST - AST) / Team_FGM))) + ((((Team_AST / Team_MP) * MP * 5 - AST) / ((Team_FGM / Team_MP) * MP * 5 - FGM)) * (1 - (MP / (Team_MP / 5))))
AST_Part = 0.5 * (((Team_PTS - Team_FTM) - (PTS - FTM)) / (2 * (Team_FGA - FGA))) * AST
FT_Part = (1-(1-(FTM/FTA))^2)*0.4*FTA
Team_Scoring_Poss = Team_FGM + (1 - (1 - (Team_FTM / Team_FTA))^2) * Team_FTA * 0.4
Team_ORB_Weight = ((1 - Team_ORB%) * Team_Play%) / ((1 - Team_ORB%) * Team_Play% + Team_ORB% * (1 - Team_Play%))
Team_ORB% = Team_ORB / (Team_ORB + (Opponent_TRB - Opponent_ORB))
Team_Play% = Team_Scoring_Poss / (Team_FGA + Team_FTA * 0.4 + Team_TOV)
ORB_Part = ORB * Team_ORB_Weight * Team_Play%
FGxPoss = (FGA - FGM) * (1 - 1.07 * Team_ORB%)
FTxPoss = ((1 - (FTM / FTA))^2) * 0.4 * FTA
...which, if you can believe it, is actually worse than it looks, because there's a lot of denominators running around. I started doing Taylor expansions but even I thought that was a little much, so I abandoned the analytical approach and went empirical. I took the 2009 Finals Lakers, whittled their roster to the 8 who played significant minutes (Kobe Bryant, Pau Gasol, Lamar Odom, Trevor Ariza, Derek Fisher, Andrew Bynum, Luke Walton, Jordan Farmar), and saw how their Win Shares changed under the same 3 scenarios:
Player Name Scen 1 Scen 2 Scen 3 FG%
Kobe Bryant -0.652 0.812 1.543 42.96%
Pau Gasol -0.638 0.753 1.471 60.00%
Lamar Odom -0.635 0.730 1.443 54.17%
Trevor Ariza -0.664 0.771 1.529 35.71%
Derek Fisher -0.634 0.746 1.463 50.00%
Andrew Bynum -0.688 0.789 1.580 36.36%
Luke Walton -0.601 0.724 1.390 80.00%
Jordan Farmar -0.682 0.772 1.546 36.84%
Please note that these values are for Marginal Offense, which like uPER is scaled to become Offensive Win Shares, so we can treat it the same way: what must I shoot to net Marginal Offense? (I also included the FG% column because it turns out the effect of missing another shot is worse the worse your FG% already is, with a pretty strong linear correlation of R^2 = .8527.) Please also note that unlike PER, WS tries to factor in how many of our shots are assisted, and by ignoring those team values we are overly crediting our players for creating their own shot, although again one shot either way is very small compared to the team values.
Let's try straight averaging and see where that gets us:
Shooting only 2s, I must shoot .649 / (.649 + .762) = 46%. This isn't an incredible FG%, but it is slightly above the NBA average (45%) which is where it should be, right? I should shoot above average if I want to help my team win.
Shooting only 3s, I must shoot .649 / (.649 + 1.496) = 30%. This is still a pretty miserable 3P% and one that is well below the NBA average (36%), but let's think about it at face value. If I make 30% of my 3s, I'm giving my team 30% * 3 = 0.9 points per shot. If I make 46% of my 2s, I'm giving my team 46% * 2 = 0.92 points per shot. It makes sense that the values would be about where they are. (Also, the Win Shares formula assumes all shots are equally easy to rebound, but we know that 3s are easier to rebound than most 2s, so the required 3P% might be even lower.)
Still, the benchmarks are a lot stricter, which is (part of) why a guy who shoots 43% from the field (Kobe Bryant) is about 18% worse than his teammate Gasol (60% from the field) in WS but 27% ahead in PER for the 2009 Finals.
Here's the formula for PER:
uPER = (1 / MP) *
[ 3P
+ (2/3) * AST
+ (2 - factor * (team_AST / team_FG)) * FG
+ (FT *0.5 * (1 + (1 - (team_AST / team_FG)) + (2/3) * (team_AST / team_FG)))
- VOP * TOV
- VOP * DRB% * (FGA - FG)
- VOP * 0.44 * (0.44 + (0.56 * DRB%)) * (FTA - FT)
+ VOP * (1 - DRB%) * (TRB - ORB)
+ VOP * DRB% * ORB
+ VOP * STL
+ VOP * DRB% * BLK
- PF * ((lg_FT / lg_PF) - 0.44 * (lg_FTA / lg_PF) * VOP) ]
factor = (2 / 3) - (0.5 * (lg_AST / lg_FG)) / (2 * (lg_FG / lg_FT))
VOP = lg_PTS / (lg_FGA - lg_ORB + lg_TOV + 0.44 * lg_FTA)
DRB% = (lg_TRB - lg_ORB) / lg_TRB
It's not as bad as it looks. (Also there's a scaling factor afterwards to turn uPER into PER, but because we only care about the break point of gain vs. loss that doesn't matter. You can multiply 0 by any number you want, it's still 0.) If all we care about are FG (including 3P) and FGA (including 3PA), we simplify to:
uPER = (1 / MP) *
[ 3P
+ (2 - factor * (team_AST / team_FG)) * FG
- VOP * DRB% * (FGA - FG)]
factor = (2 / 3) - (0.5 * (lg_AST / lg_FG)) / (2 * (lg_FG / lg_FT))
VOP = lg_PTS / (lg_FGA - lg_ORB + lg_TOV + 0.44 * lg_FTA)
DRB% = (lg_TRB - lg_ORB) / lg_TRB
Technically taking one more or less shot would alter both the team and league values, but the differences there will be minute. The average NBA team in 2013 took 6720 shots, 1/6720 - 1/(6720 + 1) = on the order of 1 hundred-millionth, and obviously the league values will be even less susceptible to variation.
Now, let's consider 3 scenarios:
1. Take a shot and miss. (FGA -> FGA+1)
2. Take a 2 and make it. (FGA -> FGA+1, FG -> FG+1)
3. Take a 3 and make it. (FGA -> FGA+1, FG -> FG+1, 3P -> 3P+1)
If we take the 2013 season for our league values (factor, VOP, DRB%), we get the following delta[uPER]s:
1. -0.758022372
2. 1.642553145
3. 2.642553145
Now that we know what each scenario is worth, we can set up a couple equations...
If I take only 2s, and I want to know what % I need to shoot to break even in uPER and therefore PER, then I just do:
-.758 * miss + 1.643 * make = 0
-.758 * (attempt - make) + 1.643 * make = 0
Dividing through by attempts:
-.758 + .758 * make / attempt + 1.643 * make / attempt = 0
2.401 * FG% = .758
FG% = 31.6%
Obviously, that's a terrible, terrible FG%, but that's all you need to shoot above to get contributions to your PER. (Note: of course not all shots occur in a vacuum, and shooting makes it harder for you to get rebounds, assists, etc.)
If I take only 3s, and want to know the same answer, I do the same math and get 3P% = 22.2%. Again, this is an awful, awful 3P%. These two suggest that volume shooters get a lot more benefit from PER than they should, so if you are a volume shooter with a weak PER (Rudy Gay comes to mind), it suggests that you are really just awful.
.
Well, what about Win Shares? We are only concerned with Offensive Win Shares, which looks like this...
PProd = (PProd_FG_Part + PProd_AST_Part + FTM) * (1 - (Team_ORB / Team_Scoring_Poss) * Team_ORB_Weight * Team_Play%) + PProd_ORB_Part
PProd_FG_Part = 2 * (FGM + 0.5 * 3PM) * (1 - 0.5 * ((PTS - FTM) / (2 * FGA)) * qAST)
PProd_AST_Part = 2 * ((Team_FGM - FGM + 0.5 * (Team_3PM - 3PM)) / (Team_FGM - FGM)) * 0.5 * (((Team_PTS - Team_FTM) - (PTS - FTM)) / (2 * (Team_FGA - FGA))) * AST
PProd_ORB_Part = ORB * Team_ORB_Weight * Team_Play% * (Team_PTS / (Team_FGM + (1 - (1 - (Team_FTM / Team_FTA))^2) * 0.4 * Team_FTA))
...divided by this...
TotPoss = ScPoss + FGxPoss + FTxPoss + TOV
ScPoss = (FG_Part + AST_Part + FT_Part) * (1 - (Team_ORB / Team_Scoring_Poss) * Team_ORB_Weight * Team_Play%) + ORB_Part
FG_Part = FGM * (1 - 0.5 * ((PTS - FTM) / (2 * FGA)) * qAST)
qAST = ((MP / (Team_MP / 5)) * (1.14 * ((Team_AST - AST) / Team_FGM))) + ((((Team_AST / Team_MP) * MP * 5 - AST) / ((Team_FGM / Team_MP) * MP * 5 - FGM)) * (1 - (MP / (Team_MP / 5))))
AST_Part = 0.5 * (((Team_PTS - Team_FTM) - (PTS - FTM)) / (2 * (Team_FGA - FGA))) * AST
FT_Part = (1-(1-(FTM/FTA))^2)*0.4*FTA
Team_Scoring_Poss = Team_FGM + (1 - (1 - (Team_FTM / Team_FTA))^2) * Team_FTA * 0.4
Team_ORB_Weight = ((1 - Team_ORB%) * Team_Play%) / ((1 - Team_ORB%) * Team_Play% + Team_ORB% * (1 - Team_Play%))
Team_ORB% = Team_ORB / (Team_ORB + (Opponent_TRB - Opponent_ORB))
Team_Play% = Team_Scoring_Poss / (Team_FGA + Team_FTA * 0.4 + Team_TOV)
ORB_Part = ORB * Team_ORB_Weight * Team_Play%
FGxPoss = (FGA - FGM) * (1 - 1.07 * Team_ORB%)
FTxPoss = ((1 - (FTM / FTA))^2) * 0.4 * FTA
...which, if you can believe it, is actually worse than it looks, because there's a lot of denominators running around. I started doing Taylor expansions but even I thought that was a little much, so I abandoned the analytical approach and went empirical. I took the 2009 Finals Lakers, whittled their roster to the 8 who played significant minutes (Kobe Bryant, Pau Gasol, Lamar Odom, Trevor Ariza, Derek Fisher, Andrew Bynum, Luke Walton, Jordan Farmar), and saw how their Win Shares changed under the same 3 scenarios:
Player Name Scen 1 Scen 2 Scen 3 FG%
Kobe Bryant -0.652 0.812 1.543 42.96%
Pau Gasol -0.638 0.753 1.471 60.00%
Lamar Odom -0.635 0.730 1.443 54.17%
Trevor Ariza -0.664 0.771 1.529 35.71%
Derek Fisher -0.634 0.746 1.463 50.00%
Andrew Bynum -0.688 0.789 1.580 36.36%
Luke Walton -0.601 0.724 1.390 80.00%
Jordan Farmar -0.682 0.772 1.546 36.84%
Please note that these values are for Marginal Offense, which like uPER is scaled to become Offensive Win Shares, so we can treat it the same way: what must I shoot to net Marginal Offense? (I also included the FG% column because it turns out the effect of missing another shot is worse the worse your FG% already is, with a pretty strong linear correlation of R^2 = .8527.) Please also note that unlike PER, WS tries to factor in how many of our shots are assisted, and by ignoring those team values we are overly crediting our players for creating their own shot, although again one shot either way is very small compared to the team values.
Let's try straight averaging and see where that gets us:
Shooting only 2s, I must shoot .649 / (.649 + .762) = 46%. This isn't an incredible FG%, but it is slightly above the NBA average (45%) which is where it should be, right? I should shoot above average if I want to help my team win.
Shooting only 3s, I must shoot .649 / (.649 + 1.496) = 30%. This is still a pretty miserable 3P% and one that is well below the NBA average (36%), but let's think about it at face value. If I make 30% of my 3s, I'm giving my team 30% * 3 = 0.9 points per shot. If I make 46% of my 2s, I'm giving my team 46% * 2 = 0.92 points per shot. It makes sense that the values would be about where they are. (Also, the Win Shares formula assumes all shots are equally easy to rebound, but we know that 3s are easier to rebound than most 2s, so the required 3P% might be even lower.)
Still, the benchmarks are a lot stricter, which is (part of) why a guy who shoots 43% from the field (Kobe Bryant) is about 18% worse than his teammate Gasol (60% from the field) in WS but 27% ahead in PER for the 2009 Finals.