The “Interference” of Phil Jackson

By: Dr. Ikjyot Singh Kohli

So, I came across this article today by Matt Moore on CBSSports, who basically once again has taken to the web to bash the Triangle Offense. Of course, much of what he claims (like much of the Knicks media) is flat-out wrong based on very primitive and simplistic analysis, and I will point it out below. Further, much of this article seems to motivated by several comments Carmelo Anthony made recently expressing his dismay at Jeff Hornacek moving away from the “high-paced” offense that the Knicks were running before the All-Star break:

“I think everybody was trying to figure everything out, what was going to work, what wasn’t going to work,’’ Anthony said in the locker room at the former Delta Center. “Early in the season, we were winning games, went on a little winning streak we had. We were playing a certain way. We went away from that, started playing another way. Everybody was trying to figure out: Should we go back to the way we were playing, or try to do something different?’’

Anthony suggested he liked the Hornacek way.

“I thought earlier we were playing faster and more free-flow throughout the course of the game,’’ Anthony said. “We kind of slowed down, started settling it down. Not as fast. The pace slowed down for us — something we had to make an adjustment on the fly with limited practice time, in the course of a game. Once you get into the season, it’s hard to readjust a whole system.’’

First, it is well-known that the Knicks have been implementing more of the triangle offense since All-Star break. All-Star Weekend was Feb 17-19, 2017. The Knicks record before All-Star weekend was amusingly 23-34, which is 11 games below .500 and is nowhere mentioned in any of these articles, and is also not mentioned (realized?) by Carmelo. 

Anyhow, the question is as follows. If Hornacek was allowed to continue is non-triangle ways of pushing the ball/higher pace (What Carmelo claims he liked), would the Knicks have made the playoffs? Probably not. I claim this to be the case based on a detailed machine-learning-based analysis of playoff-eligible teams that has been available for sometime now. In fact, what is perhaps of most importance from this paper is the following classification tree that determines whether a team is playoff-eligible or not:


So, these are the relevant factors in determining whether or not a team in a given season makes the playoffs. (Please see the paper linked above for details on the justification of these results.)

Looking at these predictor variables for the Knicks up to the All-Star break.

  1. Opponent Assists/Game: 22.44
  2. Steals/Game: 7.26
  3. TOV/Game: 13.53
  4. DRB/Game: 33.65
  5. Opp.TOV/Game: 12.46

Since Opp.TOV/Game = 12.46 < 13.16, the Knicks would actually be predicted to miss the NBA Playoffs. In fact, if current trends were allowed to continue, the so-called “Hornacek trends”, one can compute the probability of the Knicks making the playoffs:


From this probability density function, we can calculate that the probability of the Knicks making the playoffs was 36.84%. The classification tree also predicted that the Knicks would miss the playoffs. So, what is being missed by Carmelo, Matt Moore, and the like is the complete lack of pressure defense, hence, the insufficient amount of opponent TOV/G. So, it is completely incorrect to claim that the Knicks were somehow “Destined for glory” under Hornacek’s way of doing this. This is exacerbated by the fact that the Knicks’ opponent AST/G pre-All-Star break was already pretty high at 22.44.

The question now is how have the Knicks been doing since Phil Jackson’s supposed interference and since supposedly implementing the triangle in a more complete sense? (On a side note, I still don’t think you can partially implement the triangle, I think it needs a proper off-season implementation as it is a complete system).

Interestingly enough, the Knicks opponent assists per game (which, according to the machine learning analysis is the most relevant factor in determining whether a team makes the playoffs) from All-Star weekend to the present-day is an impressive 20.642/Game. By the classification tree above, this actually puts the Knicks safely in playoff territory, in the sense of being classified as a playoff team, but it is too little, too late.

The defense has actually improved significantly with respect to the key relevant statistic of opponent AST/G. (Note that, as will be shown in a future article, DRTG and ORTG are largely useless statistics in determining a team’s playoff eligibility, another point completely missed in Moore’s article) since the Knicks have started to implement the triangle more completely.

The problem is that it is obviously too little, too late at this point. I would argue based on this analysis, that Phil Jackson should have actually interfered earlier in the season. In fact, if the Knicks keep their opponent Assists/game below 20.75/game next season (which is now very likely, if current trends continue), the Knicks would be predicted to make the playoffs by the above machine learning analysis. 

Finally, I will just make this point. It is interesting to look at Phil Jackson teams that were not filled/packed with dominant players. As the saying goes, unfortunately, “Phil Jackson’s success had nothing to do with the triangle, but, because he had Shaq/Kobe, Jordan/Pippen, etc… ”

Well, let’s first look at the 1994-1995 Chicago Bulls, a team that did not have Michael Jordan, but ran the triangle offense completely. Per the relevant statistics above:

  1. Opp. AST/G = 20.9
  2. STL/G = 9.7
  3. AST/G = 24.0
  4. Opp. TOV/G = 18.1

These are remarkable defensive numbers, which supports Phil’s idea, that the triangle offense leads to good defense.



NCAA March Madness 2017 Predictions

By: Dr. Ikjyot Singh Kohli

Update: March 18, 2017: In a stunning upset, Wisconsin just beat Villanova. It is easy to see why this happened based on the factor relevance diagram below. To win games, Villanova has relied heavily on moving the ball, while Wisconsin has relied heavily on opposing assists! Wisconsin had a minor 5 assists in the whole game today, great defense by them.




Original Article: March 16, 2017

So, I’m a bit late this year with these, but, it’s only the first day of the tournament as I write this (teaching 2 courses in 1 semester tends to take up A LOT of one’s time!). Anyways, I tried to use Machine Learning methodologies such as neural networks to make predictions on who is going to win the NCAA tournament this year.

To do this, I trained a neural network model on the last 17 seasons of NCAA regular-season team data.

The first thing that I found was what are the most relevant predictor variables in a team’s NCAA championship success:

  1. Free Throws Made : 99.99% relevance
  2. Opponent Assists : 55.86% relevance
  3. Opponent Field Goal Attempts : 31.44% relevance
  4. Free Throws Attempted : -83.13% relevance
  5. Opponent Field Goals Made: -69.2% relevance

It is interesting that the most important factor in deciding whether or not a team wins the NCAA tournament is actually free throw percentage. In other words, schools that have a knack for shooting a high free throw percentage seem to have the highest probability of winning the NCAA tournament. (Point 1 and Point 4 in the list above translates to having a high free throw percentage.) Obviously, with a neural network the relationship between these predictors and the output is not necessarily linear, so other factors could play a strong role as well.

The neural network structure used looked like this:

Now, for the results:

School Name

Probability of Winning Tournament

Villanova 0.9294916774
Gonzaga 0.8076801
Baylor 0.716319
Arizona 0.5516670309
Duke 0.005617711
Saint Mary’s 0.0048923492
Wichita St. 0.001208123
Purdue 0.001180955
SMU 0.0008327729
North Carolina 0.0006080225
UCLA 0.0003794108
S. Dakota St. 0.0003186754
Oregon 0.0002288606
Princeton 0.0002107522
Wisconsin 0.000206285
Northwestern 0.0001878604
Cincinnati 0.0001875887
Marquette 0.0001828106
Virgnia 0.0001532999
Kent St. 0.0001353252
Miami 0.0001338989
Fla. Gulf Coast 0.0001308963
Vermont 0.0001288239
Notre Dame 0.0001278009
Minnesota 0.0001277032
New Mexico State 0.0001276369
USC 0.0001274456
Middle Tenn. 0.0001268802
Florida 0.0001265646
Texas Southern 0.0001265547
Xavier 0.0001264269
Vanderbilt 0.0001262982
Michigan 0.0001261976
East Tenn. St. 0.0001261878
Nevada 0.0001261331
Butler 0.0001260504
Louisville 0.0001260042
Troy 0.0001259668
Dayton 0.0001259567
Arkansas 0.0001259387
Michigan St. 0.0001259298
Oklahoma St. 0.0001259287
Winthrop 0.0001259213
Iona 0.0001259197
Jacksonville St. 0.0001259174
Creighton 0.0001259092
West Virginia 0.0001259032
North Carolin-Wilmington 0.0001259012
Northern Ky. 0.0001259000
Kansas 0.0001258950
Iowa St 0.0001258950
Bucknell 0.0001258945
Florida St 0.0001258939
Kentucky 0.0001258939
Virginia Tech 0.0001258938
Seton Hall 0.0001258937
Maryland 0.0001258936
North Dakota 0.0001258936
South Carolina 0.0001258935
Rhode Island 0.0001258934
Kansas St. 0.0001258933
Mount St. Mary’s 0.0001258932
VCU 0.0001258931
UC Davis 0.0001258929

This neural network model predicts that the team with the highest probability of winning the NCAA tournament this year is Villanova with a 92.94% chance of winning, followed by Gonzaga with a 80.77% chance of winning, Baylor with a 71.63% chance of winning, and Arizona with a 55.16% chance of winning.

Mathematics Behind The Triangle Offense

It was pointed out to me recently that a few of the articles I have written describing the detailed geometric structure behind the triangle offense is scattered in various places around my blog, so here is a list of the articles in one convenient place: 

  • The Mathematics of Filling the Triangle (First article) 
  • Group Theory and Dynamical Systems Theory Behind The Triangle Offense 
  • A Demonstration That The Triangle Offense is the most efficient/optimal way for 5 players to space the floor.
  • By: Dr. Ikjyot Singh Kohli (About the Author)

    1. The Mathematics of Filling the Triangle (First article) 
    2. Group Theory and Dynamical Systems Theory Behind The Triangle Offense 
    3. A Demonstration That The Triangle Offense is the most efficient/optimal way for 5 players to space the floor.

    By: Dr. Ikjyot Singh Kohli (About the Author)

    So, What’s Wrong with the Knicks?

    By: Dr. Ikjyot Singh Kohli

    As I write this post, the Knicks are currently 12th in the Eastern conference with a record of 22-32. A plethora of people are offering the opinions on what is wrong with the Knicks, and of course, most of it being from ESPN and the New York media, most of it is incorrect/useless, here are some examples:

    1. The Bulls are following the Knicks’ blueprint for failure and …
    2. Spike Lee ‘still believes’ in Melo, says time for Phil Jackson to go
    3. 25 reasons being a New York Knicks fan is the most depressing …
    4. Carmelo Anthony needs to escape the Knicks
    5. Another Awful Week for Knicks

    A while ago, I wrote this paper based on statistical learning that shows the common characteristics for NBA playoff teams. Basically, I obtained the following important result:


    This classification tree shows along with arguments in the paper, that while the most important factor in teams making the playoffs tends to be the opponent number of assists per game, there are paths to the playoffs where teams are not necessarily strong in this area. Specifically, for the Knicks, as of today, we see that:

    opp. Assists / game : 22.4 > 20. 75, STL / game: 7. 2 < 8.0061, TOV / game : 14.1 < 14.1585, DRB / game: 33.8 > 29.9024, opp. TOV / game: 13.0 < 13.1585.

    So, one sees that what is keeping the Knicks out of the playoffs is specifically pressure defense, in that, they are not forcing enough turnovers per game. Ironically, they are very close to the threshold, but, it is not enough.

    A probability density approximation of the Knicks’ Opp. TOV/G is as follows:



    This PDF has the approximate functional form:

    P(oTOV) =


    Therefore, by computing:

    \int_{A}^{\infty} P(oTOV) d(oTOV),



    where Erfc is the complementary error function, and is given by:

    erfc(z) = \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-t^2} dt


    Given that the threshold for playoff-bound teams is more than 13.1585 opp. TOV/game, setting A = 13 above, we obtain: 0.435. This means that the Knicks have roughly a 43.5% chance of forcing more than 13 TOV in any single game. Similarly, setting A = 14, one obtains: 0.3177. This means that the Knicks have roughly a 31.77% chance of forcing more than 14 TOV in any single game, and so forth.

    Therefore, one concludes that while the Knicks problems are defensive-oriented, it is specifically related to pressure defense and forcing turnovers.


     By: Dr. Ikjyot Singh Kohli, About the Author

    Basketball Machine Learning Paper Updated 

    I have now made a significant update to my applied machine learning paper on predicting patterns among NBA playoff and championship teams, which can be accessed here: arXiv Link . 

    Basketball – A Game of Geometry

    In a previous post, I described the most optimal offensive strategy for the Knicks based on developing relevant joint probability density functions.

    In this post, I attempt a solution to the following problem:

    Given 5 players on the court, how can one determine (x,y) coordinates for each player such that the spacing / distance between each player is maximized. Thus, mathematically providing a solution in which the arrangement of these 5 players is optimal from an offensive strategy standpoint. The idea is that such an arrangement of these 5 players will always stretch the defense to the maximum.

    The problem is then stated as follows. Let (x_i, y_i) be the x and y coordinates of player i on the court. We wish to solve:


    Problems of this type are known as multi-objective optimization problems, and in general are quite difficult to solve. Note that in setting up the coordinate system for this problem, we have for convenience placed the basket at (x,y) = (0,0), i.e., at the origin.

    Now, for solving this problem we used the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) in the MCO package in R.

    In general, what I found were that there are many possible solutions to this problem, all of which are Pareto optimal. Here are some of these results.





    Here are some more plots of of player coordinates clearly showing the origin point (which as mentioned earlier, is the location of the basket):

    Each plot above shows the x-y coordinates of players on the floor such that the distance between them is a maximum. Thus, these are some possible configurations of 5 players on the floor where the defense of the opposing team would be stretched to a maximum. What is even more interesting is that in each solution displayed above, and indeed, each numerical solution we found that is not displayed here, there is at least one triangle formation. It can therefore be said that the triangle offense is amongst the most optimal offensive strategies that produces maximum spacing of offensive players while simultaneously stretching the defense to a maximum as well. Here is more on the unpredictability of the triangle offense and its structure. 

    Based on these coordinates, we obtained the following distance matrices showing the maximum / optimal possible distance between player i and player j:


    Above, we show 5 possible distance matrices out of the several generated for brevity. So, one can see that looking at the fifth matrix for example, players are at a maximum and optimal distance from each other if for example the distance between player 1 and 2 is 9.96 feet, while the distance between player 3 and 4 is 18.703 feet, while the distance between player 4 and 5 is 4.96 feet, and so on.

    The Most Optimal Strategy for the Knicks

    In a previous article, I showed how one could use data in combination with advanced probability techniques to determine the optimal shot / court positions for LeBron James. I decided to use this algorithm on the Knicks’ starting 5, and obtained the following joint probability density contour plots:

    One sees that the Knicks offensive strategy is optimal if and only if players gets shots as close to the basket as possible. If this is the case, the players have a high probability of making shots even if defenders are playing them tightly. This means that the Knicks would be served best by driving in the paint, posting up, and Porzingis NOT attempting a multitude of three point shots.

    By the way, a lot of people are convinced nowadays that someone like Porzingis attempting 3’s is a sign of a good offense, as it is an optimal way to space the floor. I am not convinced of this. Spacing the floor geometrically translates to a multi-objective nonlinear optimization problem. In particular, let (x_i, y_i) represent the (x-y)-coordinates of a player on the floor. Spreading the floor means one must maximize (simultaneously) each element of the following distance metric:


    subject to -14 \leq x_i \leq 14, 0 \leq y_i \leq 23.75. While a player attempting 3-point shots may be one way to solve this problem, I am not convinced that it is a unique solution to this optimization problem. In fact, I am convinced that there are a multiple of solutions to this optimization problem.

    This solution is slightly simpler if one realizes that the metric above is symmetric, so that there are only 11 independent components.