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:





Advertisements

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!

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:

optimproblem

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.

 

xyplot1

xyplot2

xyplot3

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:

distances

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 (%)

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