### Lunchtime Sports Science: Fitting a Bradley-Terry Model

Power rankings are game rankings that also allow you to estimate the likely outcome if two opponents were to face each other. One of the simplest of these models is known as the Bradley-Terry-Luce model (or commonly, Bradley-Terry). The idea is that each player $$i$$ is assumed to have an unknown rating $$R_i$$. If players $$i$$ and $$j$$ compete, the probability that $$i$$ wins under this model is expected to be about $\frac{R_i}{R_i + R_j}.$ This model is very popular for hockey and other games; one commonly seen version is called KRACH.

Let's fit a Bradley-Terry model to the current season of NCAA D1 men's hockey. The Frozen Four starts on Thursday, April 11, so you'll get to see how well your predictions do.

You'll need to have R installed. Once R is installed, install the "BradleyTerry2" package that's freely available for R (thanks to Heather Turner and David Firth). To do this, start R and run the following command; you'll have to pick a source.
install.packages("BradleyTerry2")
Next, download two files from my hockey GitHub - R code that fits a basic Bradley-Terry model and a data file containing the NCAA D1 men's hockey game results going back to 1998.

https://github.com/octonion/hockey/blob/master/lunchtime/uscho_btl.R
https://github.com/octonion/hockey/blob/master/lunchtime/uscho_games.csv

Make sure both files are in the same directory and run the R code. That's it, you've built a power ranking using a Bradley-Terry model. You should get output that looks like this:

ability      s.e.
Quinnipiac              1.687042594 0.5678939
Massachusetts-Lowell    1.480098569 0.5872701
Minnesota               1.428503522 0.5638946
Yale                    1.115338226 0.5641414
Miami                   1.114307264 0.5479346
Notre Dame              1.109912670 0.5523657
Boston College          1.091836391 0.5815233
St. Cloud State         1.079314965 0.5573018

The order should be the same as USCHO's KRACH rankings.

How do we use these ability estimates to predict game outcomes? These values are the logarithms of the ratings I've mentioned above, so first apply the exponential to get the rating, then the estimated winning probability is the team's rating divided by the sum of the team and opponent ratings. For the teams in the Frozen Four we get a power rating of $$e^{1.687} = 5.40$$ for Quinnipiac and $$e^{1.079} = 2.94$$ for St. Cloud State, so we estimate the probability of Quinnipiac beating St. Cloud State to be about $\frac{5.40}{5.40+2.94} = 0.65.$ What's your estimate for Massachusetts-Lowell beating Yale?

### 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