Month: February 2017

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)

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    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:

    img_4304

    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:

    tovpgameplot1

     

    This PDF has the approximate functional form:

    P(oTOV) =

    knicksotovg

    Therefore, by computing:

    \int_{A}^{\infty} P(oTOV) d(oTOV),

    =

    knicksotoverfc,

    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