### When is a Lead Safe in the NBA?

Assuming two NBA teams of equal strength with $$t$$ seconds remaining, what is a safe lead at a prescribed confidence level? Bill James has a safe lead formula for NCAA basketball, and the topic has been addressed by other researchers at various levels of complexity, e.g. Clauset, Kogan and Redner.

I'll present a simple derivation. Start by observing there are about 50 scoring groups per team per game (scoring groups include all baskets and free throws that occur at the same time), with each scoring group worth about two points. Assume scoring events by team are Poisson distributed with parameter $$\lambda = \frac{50\cdot t}{48\cdot 60} = \frac{t}{57.6}$$. Using a normal approximation, the difference of these two distributions is normal with mean 0 and variance $$\sqrt{2}\lambda$$, giving a standard deviation of $$0.1863\sqrt{t}$$.

Using this approximation, what is a 90% safe lead? A 90% tail is 1.28 standard deviations, $$1.28\cdot 0.1863\sqrt{t} = 0.2385\sqrt{t}$$ scoring groups. As a scoring group is about two points, this means a 90% safe lead, assuming a jump ball, is about $$0.477\sqrt{t}$$ points (Clauset et. al. obtained $$0.4602\sqrt{t}$$). For example, a safe lead at halftime is approximately $$0.477 \sqrt{24\cdot 60} = 18.1$$ points.

Next - adjustments for possession arrow and shot clock time; validity of approximation; adjusting for team strengths.

1. Can you explain a bit better what you mean by scoring groups ?

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Mixed Models in R - Bigger, Faster, Stronger

When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models. These are models that contain both fixed and random effects. There are multiple ways of defining fixed vs random random effects, but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and this in turn involves some "shrinkage" towards the mean.

Here's some R code for NCAA ice hockey power rankings using a nested Poisson model (which can be found in my hockey GitHub repository):
model <- gs ~ year+field+d_div+o_div+game_length+(1|offense)+(1|defense)+(1|game_id) fit <- glmer(model, data=g, verbose=TRUE, family=poisson(link=log) ) The fixed effects are year, field (home/away/neutral), d_div (NCAA division of the defense), o_div (NCAA division of the offense) and game_length (number of overtime periods); off…

### Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD