Our new article was recently published in The Journal of Geometry and Physics. It is shown that under certain conditions, The Einstein Field Equations have the same form as a fold bifurcation seen in Dynamical Systems theory, showing even a deeper connection between General Relativity and Dynamical Systems theory! (You can click the image below to be taken to the article):
Category: Science
An Equation to Predict NBA Playoff Probabilities
Based on a previous paper I wrote that used machine learning to determine the most relevant factors for teams making the NBA playoffs, I did some further analysis in an attempt to come up with an equation that outputs the probability of an NBA team making the playoffs in a given season.
From the aforementioned paper, one concludes that the two most important factors in determining whether a team makes the playoffs or not is its opponent assists per game and opponent twopoint shots made per game. Based on that, I came up with the following equation:
A plot of this equation is as follows:
A contour plot is perhaps more illuminating:
One can see from this contour plot that teams have the highest probabilities of making the playoffs when their opponent 2point shots and opponent assists are both around 20. In general, we also see that while a team can allow more opponent 2point shots, having a low number of opponent assists per game is evidently the most important factor.
Using this equation, I was able to classify 71% of playoff teams correctly from the last 16 years of NBA data. Even though the playoff classifier developed in the paper mentioned above is more accurate in general, those methods are nonparametric, so, it is difficult to obtain an equation. To get an equation as we have done here, can be extremely useful for modelling purposes and understanding the nature of probabilities in deciding whether a certain team will make the playoffs in a given season. (Also: note that we are using the convention of using 0.50 as the threshold probability, so a probability output of >0.5, is classified as a team making the playoffs.)
Dynamical Systems in Cosmology Lectures
In the final two lectures of my differential equations class , I discussed how Dynamical Systems theory can be used to understand and describe the dynamics of cosmological solutions to Einstein’s field equations. Videos and lecture notes posted below:
Lecture Notes:
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:
Importantly, one can calculate that the mean daily returns, , while the volatility in daily returns, . 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:
,
where 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:
We concluded the following from our simulation. At the end of January 25, 2017, the DJIA was predicted to close at:
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 54day period. We found the following:
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 daytoday volatility of the DJIA!
New Cosmology Paper
New #cosmology paper: https://arxiv.org/pdf/1609.01310.pdf
Using a dynamical systems approach to provide a unifying framework for the AdS, Minkowski, and de Sitter universes. #physics #mathematics #science
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 be the x and y coordinates of player on the court. We wish to solve:
Problems of this type are known as multiobjective 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 Nondominated Sorting Genetic AlgorithmII (NSGAII) 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 xy 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 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 (%) 
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 nonparametric regression shows:
We therefore chose to perform a nonlinear regression using neural networks, for which our structure was:
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%.