I wrote an extensive script in R that takes the most recent data available for the number of new/confirmed COVID-19 cases per day by location and computes the probability using statistical learning that a selected location will observe a new COVID-19 case. You can access the dashboard by clicking the image below: (Beneath the screenshot are further examples of possible selections.)

# Category: Mathematics

## A Problem With Offensive Rating

*Abstract: It is shown that the standard/common definition of team offensive rating/offensive efficiency implies that a team’s offensive rating increases as its opponent’s offensive rebounds increase, which, in principle, should not be the case.*

Over the past number of years, the *advanced* metric known as Offensive Rating has become the standard way of measuring a basketball team’s offensive efficiency. Broadly speaking, it is defined as points scored per 100 possessions. Specifically, for **teams**, it is defined as (See: https://www.basketball-reference.com/about/ratings.html and https://www.nbastuffer.com/analytics101/possession/ AND https://fansided.com/2015/12/21/nylon-calculus-101-possessions/):

There is a significant issue with this definition as I now demonstrate. Let us compute the partial derivative of this expression with respect to OppORB, we easily obtain:

As the denominator is always positive, we would like to examine the numerator. The numerator is always negative due to physical constraints (i.e., can’t have negative points or rebounds!) and if **OppFG < OppFGA, **which makes intuitive sense. It is only positive if **OppFG > OppFGA, **which logically cannot happen. Therefore, this numerator is always negative (except for the rare case when OppFG = OppFGA of course), which means that the entire partial derivative is positive.

This means that a team’s offensive rating / offensive efficiency increases as it’s opponent’s offensive rebounds increase. Intuitively, this shouldn’t be the case. If your opponent has a high number of offensive rebounds, this should give you less possessions, and put pressure on you to score, thus resulting in less points overall. The problem is that the more general definition of offensive efficiency is 100*(Points Scored)/(Possessions), which is obviously maximized when possessions is minimized. The problem of course, is that the more detailed definition of possessions implies that this minimization of possessions occurs at the cost of maximizing opponent offensive rebounds, which intuitively should not be the case.

## NBA Analytics Dashboard

Here is an embedded dashboard that shows a number of statistical insights for NBA teams, their opponents, and individual players as well. You can compare multiple teams and players. Navigate through the different pages by clicking through the scrolling arrow below. (The data is based on the most recent season “per-game” numbers.)

(If you cannot see the dashboard embedded below for whatever reason, click here to be taken directly to the dashboard in a separate page.)

## The Probability of An Illegal Immigrant Committing a Crime In The United States

Trump has once again put The U.S. on the world stage this time at the expense of innocent children whose families are seeking asylum. The Trump administration’s justification is that:

I decided to try to analyze this statement quantitatively. Indeed, one can calculate the probability that an illegal immigrant will commit a crime *within* The United States as follows. Let us denote crime (or criminal) by *C*, while denoting illegal immigrant by *ii*. Then, by Bayes’ theorem, we have:

It is quite easy to find data associated with the various factors in this formula. For example, one finds that

Putting all of this together, we find that:

That is, the probability that an illegal immigrant will commit a crime (of any type) while in The United States is a very low 11.35%.

Therefore, Trump’s claim of “tremendous amounts of crime” being brought to The United States by illegal immigrants is incorrect.

Note that, the numerical factors used above were obtained from:

- https://www.justice.gov/opa/pr/departments-justice-and-homeland-security-release-data-incarcerated-aliens-94-percent-all
- https://www.washingtontimes.com/news/2017/aug/1/immigrants-22-percent-federal-prison-population/
- https://en.wikipedia.org/wiki/Incarceration_in_the_United_States

## The Risk of The 3-Point Shot

As more and more teams are increasing the number of threes they attempt based on some misplaced logical fallacy that this somehow leads to an efficient offense, we show below that it is in fact in a team’s opponent’s interest for a team to attempt as many three point shots as possible.

Looking at this season’s data, let us examine two things. The first thing is the number of points a team’s opponent is expected to score for every three-point shot the other team attempts. We discovered that remarkably, the number of points obeys a lognormal distribution:

This means that for every three point shot your team attempts, the opposing team is expected to score

which comes out to about 3.7495 points. So, for every 3PA by a team, the opponent is expected to score more than 3 points based on the most recent NBA data. Keeping that in mind, we see also by integrating above that there is a 99.99% probability that the opponent will score more than 2 points for every 3PA by a team, and a 93.693% probability that the opponent will score more than 3 points for every single 3PA by the other team.

This would suggest a significant breakdown of defensive emphasis in the “modern-day” NBA where evidently teams are just interested in playing shot-for-shot basketball, but in a very risky way that is not optimal.

The work so far covered just three-point attempts, but, what are the effects of *missing* a three-point shot? The number of opponent points per a three-point miss also remarkably obeys a lognormal distribution:

Therefore, for every three-point shot your team misses, the opposing team is expected to score:

which comes out to about 5.87345 points. This identifies a remarkable risk to a team missing a three-point shot. This computation shows that one three-point shot miss corresponds to about 6 points for the opposing team! Looking at probabilities by integrating the density function above, one can show that there is a 99.9999% probability that the opposing team would score more than two points for every three-point miss, a 99.998% probability that the opposing team would score more than three points for every three-point miss, a 99.583% probability that the opposing team would score more than four points for every three-point miss, and so on.

What these calculations demonstrate is that gearing a team’s offense to focus on attempting three-point shots is remarkably risky, especially if a team misses a three-point shot. Given that the average number of three-point attempts is increasing over the last number of years, but the average number of makes has relatively stayed the same (See this older article here: https://relativitydigest.com/2016/05/26/the-three-point-shot-myth-continued/), teams are exposing themselves to greater and greater risk of losing games by adopting this style of play.

## New Article Published in Journal of Geometry and Physics

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):

## 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 two-point 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 2-point shots and opponent assists are both around 20. In general, we also see that while a team can allow more opponent 2-point 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 non-parametric, 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.)*