A very interesting result: computing payoffs of players, the following is a diagram that shows when it is optimal for a player to shoot a 2 point or a 3-point shot. One sees that it is hardly ever optimal for a player to shoot a 3-point shot, since the region corresponding to 3-point optimality is quite narrow. This can be interpreted as saying that for a 3-point attempt to be optimal, a player’s 2PT% must be roughly equal to his/her 3PT%, which is certainly not the case for the vast majority of even designated 3-point shooters in the NBA!

# Tag: NBA

## What if Michael Jordan Played in Today’s NBA?

It seems that one cannot turn on ESPN or any YouTube channel nowadays without the ongoing debate of whether Michael Jordan is better than Lebron, what would happen if Michael Jordan played in today’s NBA, etc… However, I have not seen a single scientific approach to this question. Albeit, it is sort of an impossible question to answer, but, using data science I will try.

From a data science perspective, it only makes sense to look at Michael Jordan’s performance in a single season, and try to predict based on that season how he would perform in the most recent NBA season. That being said, let’s look at Michael Jordan’s game-to-game performance in the 1995-1996 NBA season when the Bulls went 72-10.

Using neural networks and Garson’s algorithm , to regress against Michael Jordan’s per game point total, we note the following:

One can see from this variable importance plot, Michael’s points in a given game were most positively associated with teams that committed a high number of turnovers followed by teams that make a lot of 3-point shots. Interestingly, there was not a strong negative factor on Michael’s points in a given game.

Given this information, and the per-game league averages of the 2017 season, we used this neural network to make a prediction on how many points Michael would average in today’s season:

Michael Jordan: 2017 NBA Season Prediction: 32.91 Points / Game (+/- 6.9)

It is interesting to note that Michael averaged 30.4 Points/Game in the 1995-1996 NBA Season. We therefore conclude that the 1995-1996 Michael would average a higher points/game if he played in today’s NBA.

As an aside, a plot of the neural network used to generate these variable importance plots and predictions is as follows:

What about the reverse question? What if the 2016-2017 Lebron James played in the 1995-1996 NBA? What would happen to his per-game point average? Using the same methodology as above, we used neural networks in combination with Garson’s algorithm to obtain a variable importance plot for Lebron James’ per-game point totals:

One sees from this plot that Lebron’s points every game were most positively impacted by teams that predominantly committed personal fouls, followed by teams that got a lot of offensive rebounds. There were no predominantly strong negative factors that affected Lebron’s ability to score.

Using this neural network model, we then tried to make a prediction on how many points per game Lebron would score if he played in the 1995-1996 NBA Season:

Lebron James: 1995-1996 NBA Season Prediction: 18.81 Points / Game (+/- 4.796)

This neural network model predicts that Lebron James would average 18.81 Points/Game if he played in the 1995-1996 NBA season, which is a drop from the 26.4 Points/Game he averaged this most recent NBA season.

Therefore, at least from this neural network model, one concludes that Lebron’s per game points would decrease if he played in the 1995-1996 Season, while Michael’s number would increase slightly if he played in the 2016-2017 Season.

## How to Beat the Golden State Warriors

The Golden State Warriors have posed quite the conundrum for opposing teams. They are quick, have a spectacular ability to move the ball, and play suffocating defense. Given their play in the playoffs thus far, all of these points have been exemplified even more to the point where it seems that they are unbeatable.

I wanted to take somewhat of a simplified approach and see if opposing teams are missing something. That is, is their some weakness in their play that opposing teams can exploit, a “weakness in Helm’s deep”?

The most obvious place to start from a data science point-of-view seemed to me to look at**every single shot**the Warriors took as a team this season in each game and compile a grand ensemble shot chart. Using the data from Basketball-reference.com and some data scraping scripts I wrote in R, I obtained the following: Certainly, on the surface, it seems that there is no discernible pattern between made shots and missed shots. This is where the machine learning comes in!

From here, I now extracted the x and y coordinates of each shot and recorded a response variable of “made” or “missed” in a table, such that the coordinates were now predictor variables and the shot classification (made/missed) was the response variable. Altogether, we had 7104 observations. Splitting this dataset up into a 70% training dataset and a 30% test data set, I tried the following algorithms, recording the % of correctly classified observations:

Algorithm |
% of Correctly Predicted Observations |

Logistic Regression |
56.43 |

Gradient Boosted Decision Trees |
62.62 |

Random Forests |
58.54 |

Neural Networks with Entropy Fitting |
62.47 |

Naive Bayes Classification with Kernel Density Estimation |
57.32 |

One sees that that gradient boosted decision trees had the best performance correctly classifying 62.62% of the test observations. Given how noisy the data is, this is not bad, and much better than expected. I should also mention that these numbers were obtained after tuning these models using cross-validation for optimal parameters.

Using the gradient boosted decision tree model, we made a set of predictions for a vast number of (x,y)-coordinates for basketball court. We obtained the following contour plot:

Overlaying this on top of the basketball court diagram, we got:

The contour plot levels denote the **probabilities** that the GSW will make a shot from a given (x,y) location on the court. As a sanity check, the lowest probabilities seem to be close to the 1/2-court line and beyond the three-point line. The highest probabilities are surprisingly along very specific areas on the court: very close the basket, the line from the basket to the left corner, extending up slightly, and a very narrow line extending from the basket to the right corner. Interestingly, the probabilities are low on the right side of the basket, specifically:

A map showing the probabilities more explicitly is as follows (although, upon uploading it, I realized it is a bit harder to read, I will re-upload a clearer version soon!)

In conclusion, it seems that, at least according to a first look at the data, the Warriors do indeed have several “weak spots” in their offense that opponents should certainly look to exploit by designing defensive schemes that force them to take shots in the aforementioned low-probability zones. As for future improvements, I think it would be interesting to add as predictor variables things like geographic location, crowd sizes, team opponent strengths, etc… I will look into making these improvements in the near future.

## So, What’s Wrong with the Knicks?

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:

- The Bulls are following the Knicks’ blueprint for failure and …
- Spike Lee ‘still believes’ in Melo, says time for Phil Jackson to go
- 25 reasons being a New York Knicks fan is the most depressing …
- Carmelo Anthony needs to escape the Knicks
- 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:

,

=

,

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

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

specificallyrelated 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

eachplayer 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 willalwaysstretch the defense to the maximum.

The problem is then stated as follows. Let be the x and y coordinates of player 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 , 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 and player :

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 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 . 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.