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 “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.



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 . 

    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, \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:


    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:


    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!