Category: Data Science

The Trump Rally, Really?

Today, The Dow Jones Industrial Average (DJIA) surpassed the 20,000 mark for the first time in history. At the time of the writing of this posting (12:31 PM on January 25), it is actually 20,058.29, so, I am not sure if it will close above 20,000 points, but, nevertheless, a lot of people are crediting this to Trump’s presidency, but I’m not so sure you can do that. First, the point must be made, that it is really the Obama economic policies that set the stage for this. On January 20, 2009, when Obama was sworn in, the Dow closed at 7949.089844 points. On November 8, 2016, when Trump won the election, the Dow closed at 18332.74023. So, during the Obama administration, the Dow increased by approximately 130.63%. I just wanted to make that point.

Now, the question that I wanted to investigate was would the Dow have closed past 20,000 points had Trump not been elected president. That is, assuming that the Obama administration policies and subsequent effects on the Dow were allowed to continue, would the Dow have surpassed 20,000 points.

For this, I looked at the DJIA data from January 20, 2009 (Obama’s first inauguration) to November 08, 2016 (Trump’s election). I specifically calculated the daily returns and discovered that they are approximately normally distributed using a kernel density method:

obamadowpdf

Importantly, one can calculate that the mean daily returns, \mu = 0.00045497596503813, while the volatility in daily returns, \sigma = 0.0100872666938282. Indeed, the volatility in daily returns for the DJIA was found to be relatively high during this period. Finally, the DJIA closed at 18332.74023 points on election night, November 08, 2016, which was 53 business days ago.

The daily dynamics of the DJIA can be modelled by the following stochastic differential equation:

S_{t} = S_{t-1} + \mu S_{t-1} dt + \sigma S_{t-1} dW,

where dW denotes a Wiener/Brownian motion process. Simulating this on computer, I ran 2,000,000 Monte Carlo simulations to simulate the DJIA closing price 53 business days from November 08, 2016, that is, January 25, 2017. The results of some of these simulations are shown below:

djiaclosingvaluesims

We concluded the following from our simulation. At the end of January 25, 2017, the DJIA was predicted to close at:

18778.51676 \pm 1380.42445

That is, the DJIA would be expected to close anywhere between 17398.0923062336 and 20158.94121. This range, albeit wide, is due to the high volatility of the daily returns in the DJIA, but, as you can see, it is perfectly feasible that the DJIA would have surpassed 20,000 points if Trump would not have been elected president.

Further, perhaps what is of more importance is the probability that the DJIA would surpass 20,000 points at any time during this 54-day period. We found the following:

probofexceeding

One sees that there is an almost 20% (more precisely, 18.53%) probability that the DJIA would close above 20,000 points on January 25, 2017 had Trump not been elected president. Since, by all accounts, the DJIA exceeding 20,000 points is considered to be an extremely rare/historic event, the fact that the probability is found to be almost 20% is actually quite significant, and shows, that it is quite likely that a Trump administration actually has little to do with the DJIA exceeding 20,000 points.

Although, this simulation was just for 53 working days from Nov 08, 2016, one can see that the probability of the DJIA exceeding 20,000 at closing day is monotonically increasing with every passing day. It is therefore quite feasible to conclude that Trump being president actually has little to do with the DJIA exceeding 20,000 points, rather, one can really attribute it to the day-to-day volatility of the DJIA!

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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:

distancematrix

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.

Analyzing Lebron James’ Offensive Play

Where is Lebron James most effective on the court?

Based on 2015-2016 data, we obtained from NBA.com the following data which tracks Lebron’s FG% based on defender distance:

lebrondef

From Basketball-Reference.com, we then obtained data of Lebron’s FG% based on his shot distance from the basket:

lebronshotdist

Based on this data, we generated tens of thousands of sample data points to perform a Monte Carlo simulation to obtain relevant probability density functions. We found that the joint PDF was a very lengthy expression(!):

 

Graphically, this was:

lebronjointplot

A contour plot of the joint PDF was computed to be:

lebroncontour

From this information, we can compute where/when LeBron has the highest probability of making a shot. Numerically, we found that the maximum probability occurs when Lebron’s defender is 0.829988 feet away, while Lebron is 1.59378 feet away from the basket. What is interesting is that this analysis shows that defending Lebron tightly doesn’t seem to be an effective strategy if his shot distance is within 5 feet of the basket. It is only an effective strategy further than 5 feet away from the basket. Therefore, opposing teams have the best chance at stopping Lebron from scoring by playing him tightly and forcing him as far away from the basket as possible.

The Relationship Between The Electoral College and Popular Vote

An interesting machine learning problem: Can one figure out the relationship between the popular vote margin, voter turnout, and the percentage of electoral college votes a candidate wins? Going back to the election of John Quincy Adams, the raw data looks like this:

Electoral College Party Popular vote  Margin (%)

Turnout

Percentage of EC
John Quincy Adams D.-R. -0.1044 0.27 0.3218
Andrew Jackson Dem. 0.1225 0.58 0.68
Andrew Jackson Dem. 0.1781 0.55 0.7657
Martin Van Buren Dem. 0.14 0.58 0.5782
William Henry Harrison Whig 0.0605 0.80 0.7959
James Polk Dem. 0.0145 0.79 0.6182
Zachary Taylor Whig 0.0479 0.73 0.5621
Franklin Pierce Dem. 0.0695 0.70 0.8581
James Buchanan Dem. 0.12 0.79 0.5878
Abraham Lincoln Rep. 0.1013 0.81 0.5941
Abraham Lincoln Rep. 0.1008 0.74 0.9099
Ulysses Grant Rep. 0.0532 0.78 0.7279
Ulysses Grant Rep. 0.12 0.71 0.8195
Rutherford Hayes Rep. -0.03 0.82 0.5014
James Garfield Rep. 0.0009 0.79 0.5799
Grover Cleveland Dem. 0.0057 0.78 0.5461
Benjamin Harrison Rep. -0.0083 0.79 0.58
Grover Cleveland Dem. 0.0301 0.75 0.6239
William McKinley Rep. 0.0431 0.79 0.6063
William McKinley Rep. 0.0612 0.73 0.6532
Theodore Roosevelt Rep. 0.1883 0.65 0.7059
William Taft Rep. 0.0853 0.65 0.6646
Woodrow Wilson Dem. 0.1444 0.59 0.8192
Woodrow Wilson Dem. 0.0312 0.62 0.5217
Warren Harding Rep. 0.2617 0.49 0.7608
Calvin Coolidge Rep. 0.2522 0.49 0.7194
Herbert Hoover Rep. 0.1741 0.57 0.8362
Franklin Roosevelt Dem. 0.1776 0.57 0.8889
Franklin Roosevelt Dem. 0.2426 0.61 0.9849
Franklin Roosevelt Dem. 0.0996 0.63 0.8456
Franklin Roosevelt Dem. 0.08 0.56 0.8136
Harry Truman Dem. 0.0448 0.53 0.5706
Dwight Eisenhower Rep. 0.1085 0.63 0.8324
Dwight Eisenhower Rep. 0.15 0.61 0.8606
John Kennedy Dem. 0.0017 0.6277 0.5642
Lyndon Johnson Dem. 0.2258 0.6192 0.9033
Richard Nixon Rep. 0.01 0.6084 0.5595
Richard Nixon Rep. 0.2315 0.5521 0.9665
Jimmy Carter Dem. 0.0206 0.5355 0.55
Ronald Reagan Rep. 0.0974 0.5256 0.9089
Ronald Reagan Rep. 0.1821 0.5311 0.9758
George H. W. Bush Rep. 0.0772 0.5015 0.7918
Bill Clinton Dem. 0.0556 0.5523 0.6877
Bill Clinton Dem. 0.0851 0.4908 0.7045
George W. Bush Rep. -0.0051 0.51 0.5037
George W. Bush Rep. 0.0246 0.5527 0.5316
Barack Obama Dem. 0.0727 0.5823 0.6784
Barack Obama Dem. 0.0386 0.5487 0.6171

Clearly, the percentage of electoral college votes a candidate depends nonlinearly on the voter turnout percentage and popular vote margin (%) as this non-parametric regression shows:

electoralmap.png

We therefore chose to perform a nonlinear regression using neural networks, for which our structure was:

nnetplot

As is turns out, this simple neural network structure with one hidden layer gave the lowest test error, which was 0.002496419 in this case.

Now, looking at the most recent national polls for the upcoming election, we see that Hillary Clinton has a 6.1% lead in the popular vote. Our neural network model then predicts the following:

Simulation Popular Vote Margin Percentage of Voter Turnout Predicted Percentage of Electoral College Votes (+/- 0.04996417)
1 0.061 0.30 0.6607371
2 0.061 0.35 0.6647464
3 0.061 0.40 0.6687115
4 0.061 0.45 0.6726314
5 0.061 0.50 0.6765048
6 0.061 0.55 0.6803307
7 0.061 0.60 0.6841083
8 0.061 0.65 0.6878366
9 0.061 0.70 0.6915149
10 0.061 0.75 0.6951424

One sees that even for an extremely low voter turnout (30%), at this point Hillary Clinton can expect to win the Electoral College by a margin of 61.078% to 71.07013%, or 328 to 382 electoral college votes. Therefore, what seems like a relatively small lead in the popular vote (6.1%) translates according to this neural network model into a large margin of victory in the electoral college.

One can see that the predicted percentage of electoral college votes really depends on popular vote margin and voter turnout. For example, if we reduce the popular vote margin to 1%, the results are less promising for the leading candidate:

Pop.Vote Margin Voter Turnout % E.C. % Win E.C% Win Best Case E.C.% Win Worst Case
0.01 0.30 0.5182854 0.4675000 0.5690708
0.01 0.35 0.5244157 0.4736303 0.5752011
0.01 0.40 0.5305820 0.4797967 0.5813674
0.01 0.45 0.5367790 0.4859937 0.5875644
0.01 0.50 0.5430013 0.4922160 0.5937867
0.01 0.55 0.5492434 0.4984580 0.6000287
0.01 0.60 0.5554995 0.5047141 0.6062849
0.01 0.65 0.5617642 0.5109788 0.6125496
0.01 0.70 0.5680317 0.5172463 0.6188171
0.01 0.75 0.5742963 0.5235109 0.6250817

One sees that if the popular vote margin is just 1% for the leading candidate, that candidate is not in the clear unless the popular vote exceeds 60%.

 

Breaking Down the 2015-2016 NBA Season

In this article, I will use Data Science / Machine Learning methodologies to break down the real factors separating the playoff from non-playoff teams. In particular, I used the data from Basketball-Reference.com to associate 44 predictor variables which each team: “FG” “FGA” “FG.” “X3P” “X3PA” “X3P.” “X2P” “X2PA” “X2P.” “FT” “FTA” “FT.” “ORB” “DRB” “TRB” “AST”   “STL” “BLK” “TOV” “PF” “PTS” “PS.G” “oFG” “oFGA” “oFG.” “o3P” “o3PA” “o3P.” “o2P” “o2PA” “o2P.” “oFT”   “oFTA” “oFT.” “oORB” “oDRB” “oTRB” “oAST” “oSTL” “oBLK” “oTOV” “oPF” “oPTS” “oPS.G”

, where a letter ‘o’ before the last 22 predictor variables indicates a defensive variable. (‘o’ stands for opponent. )

Using principal components analysis (PCA), I was able to project this 44-dimensional data set to a 5-D dimensional data set. That is, the first 5 principal components were found to explain 85% of the variance. 

Here are the various biplots: 


In these plots, the teams are grouped according to whether they made the playoffs or not. 

One sees from this biplot of the first two principal components that the dominant component along the first PC is 3 point attempts, while the dominant component along the second PC is opponent points. CLE and TOR have a high negative score along the second PC indicating a strong defensive performance. Indeed, one suspects that the final separating factor that led CLE to the championship was their defensive play as opposed to 3-point shooting which all-in-all didn’t do GSW any favours. This is in line with some of my previous analyses. 

Optimal Positions for NBA Players

I was thinking about how one can use the NBA’s new SportVU system to figure out optimal positions for players on the court. One of the interesting things about the SportVU system is that it tracks player (x,y) coordinates on the court. Presumably, it also keeps track of whether or not a player located at (x,y) makes a shot or misses it. Let us denote a player making a shot by 1, and a player missing a shot by 0. Then, one essentially will have data in the form (x,y, \text{1/0}).

One can then use a logistic regression to determine the probability that a player at position (x,y) will make a shot:

p(x,y) = \frac{\exp\left(\beta_0 + \beta_1 x + \beta_2 y\right)}{1 +\exp\left(\beta_0 + \beta_1 x + \beta_2 y\right)}

The main idea is that the parameters \beta_0, \beta_1, \beta_2 uniquely characterize a given player’s probability of making a shot.

As a coaching staff from an offensive perspective, let us say we wish to position players as to say they have a very high probability of making a shot, let us say, for demonstration purposes 99%. This means we must solve the optimization problem:

\frac{\exp\left(\beta_0 + \beta_1 x + \beta_2 y\right)}{1 +\exp\left(\beta_0 + \beta_1 x + \beta_2 y\right)} = 0.99

\text{s.t. } 0 \leq x \leq 28, \quad 0 \leq y \leq 47

(The constraints are determined here by the x-y dimensions of a standard NBA court).

This has the following solutions:

x = \frac{-1. \beta _0-1. \beta _2 y+4.59512}{\beta _1}, \quad \frac{-1. \beta _0-28. \beta _1+4.59512}{\beta _2} \leq y

with the following conditions:

constraints1

One can also have:

x = \frac{-1. \beta _0-1. \beta _2 y+4.59512}{\beta _1}, \quad y \leq 47

with the following conditions:

constraints2

Another solution is:

x = \frac{-1. \beta _0-1. \beta _2 y+4.59512}{\beta _1}

with the following conditions:

constraints3

The fourth possible solution is:

x = \frac{-1. \beta _0-1. \beta _2 y+4.59512}{\beta _1}

with the following conditions:

constraints4

In practice, it should be noted, that it is typically unlikely to have a player that has a 99% probability of making a shot.

To put this example in more practical terms, I generated some random data (1000 points) for a player in terms of (x,y) coordinates and whether he made a shot from that distance or not. The following scatter plot shows the result of this simulation:

bballoptim5

In this plot, the red dots indicate a player has made a shot (a response of 1.0) from the (x,y) coordinates given, while a purple dot indicates a player has missed a shot from the (x,y) coordinates given (a response of 0.0).

Performing a logistic regression on this data, we obtain that \beta_0 = 0, \beta_1 = 0.00066876, \beta_2 = -0.00210949.

Using the equations above, we see that this player has a maximum probability of 58.7149 \% of making a shot from a location of (x,y) = (0,23), and a minimum probability of 38.45 \% of making a shot from a location of (x,y) = (28,0).

Basketball Paper Update

Everyone by now knows about this paper I wrote a few months ago: http://arxiv.org/abs/1604.05266

Using data science / machine learning methodologies, it basically showed that the most important factors in characterizing a team’s playoff eligibility are the opponent field goal percentage and the opponent points per game. This seems to suggest that defensive factors as opposed to offensive factors are the most important characteristics shared among NBA playoff teams. It was also shown that championship teams must be able to have very strong defensive characteristics, in particular, strong perimeter defense characteristics in combination with an effective half-court offense that generates high-percentage two-point shots. A key part of this offensive strategy must also be the ability to draw fouls. 

Some people have commented that despite this, teams who frequently attempt three point shots still can be considered to have an efficient offense as doing so leads to better rebounding, floor spacing, and higher percentage shots. We show below that this is not true. Looking at the last 16 years of all NBA teams (using the same data we used in the paper), we performed a correlation analysis of an individual NBA team’s 3-point attempts per game and other relevant variables, and discovered: 


One sees that there is very little correlation between a team’s 3-point attempts per game and 2-point percentage, free throws, free throw attempts, and offensive rebounds. In fact, at best, there is a somewhat “medium” anti-correlation between 3-point attempts per game and a team’s 2-point attempts per game.