### Lunchtime Sports Science: Cracking a New Sport

This is the first and what will be a series of relatively short pieces on sports analytics. I'll be using a variety of sports for examples, including both team sports and single-player sports, and I'll also make my code available through my GitHub account.

Here are my recommended tools. If you're unfamiliar with some of these, don't worry. You'll pick them up as you go along, and they form a powerful suite that will keep you on the cutting edge even as a professional data scientist.
1. Hardware - Ideally you want at least 4GB of RAM for larger data sets, but you'll be able to do high-level analysis with almost any modern computing hardware.
2. Linux operating system - You can certainly do top-notch data analysis using any operating system, but Linux is an excellent (and free) working environment. There are a variety of ways to install and use Linux, but I'd recommend Ubuntu's Windows installer. This will allow you to easily install Ubuntu alongside Windows, and it also makes it easy to uninstall Ubuntu later (if you choose). Ubuntu is just one of the (many) Linux distributions available, but it's very popular and well supported.
3. R programming language - R is a powerful statistical programming language, and it has thousands of available packages available. If you're using Ubuntu, installing R is simple - sudo apt-get install r-base. That's it!
4. Python programming language -Python is a powerful and relatively easy to use programming language. One of the most common tasks for sports analytics is web scraping, and Python is an excellent choice thanks to libraries such as Mechanize, Beautiful Soup and lxml. It's also a great language for data cleansing. Installing Python under Ubuntu - sudo apt-get install python.
5. PostgreSQL database server - There are many ways to store and analyze data sets, but a dedicated relational database server is necessary tool for high-level analytics. PostgreSQL is my personal recommendation, but there are other good choices (such as MySQL). PostgreSQL is free, fast, powerful and has a huge variety of procedural languages available (including R and Python). Installing PostgreSQL under Ubuntu - sudo apt-get install postgresql-9.2.
6. GitHub account - Setting up a GitHub account is free and will allow you to automatically following any changes to various sports analytics GitHub projects (such as mine). Later, you can set up your own repositories if you'd like to share your own work with other people. Don't forget to install git under Ubuntu - sudo apt-get install git.
7. We'll start with analyzing hockey. If you'd like to take a look at some of my hockey code and data that I've scraped, you can find them in my hockey GitHub repository. If you've set up Ubuntu and have installed git you can execute the command git clone https://github.com/octonion/hockey.git to make a local copy of my repository.
Here's a basic outline for tackling a new sport.
1. Understand how teams win - Build a model to project the likely outcome for a team when facing a particular opponent. Example - Krach for hockey (which is based on the Bradley-Terry model).
2. Understand how teams score - Build a model to project how many goals/points/runs teams are likely to score or allow when facing a particular opponent. Exampe - Poisson distribution and hockey.
3. Relate the two - Characterize the relationship between scoring and winning or losing. Example - Pythagorean win expectation.
4. Understand how players score/prevent scoring - Determine which aspects of player performance impact team offense and defense and by how much.
5. Understand player contribution to winning/losing - This is nearly automatic once you understand the relationship between team offense/defense and team winning/losing.
In my next article we'll build a basic power ranking model for hockey to predict likely game outcomes.

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### 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…

### What's the Value of a Win?

In a previous entry I demonstrated one simple way to estimate an exponent for the Pythagorean win expectation. Another nice consequence of a Pythagorean win expectation formula is that it also makes it simple to estimate the run value of a win in baseball, the point value of a win in basketball, the goal value of a win in hockey etc.

Let our Pythagorean win expectation formula be $w=\frac{P^e}{P^e+1},$ where $$w$$ is the win fraction expectation, $$P$$ is runs/allowed (or similar) and $$e$$ is the Pythagorean exponent. How do we get an estimate for the run value of a win? The expected number of games won in a season with $$g$$ games is $W = g\cdot w = g\cdot \frac{P^e}{P^e+1},$ so for one estimate we only need to compute the value of the partial derivative $$\frac{\partial W}{\partial P}$$ at $$P=1$$. Note that $W = g\left( 1-\frac{1}{P^e+1}\right),$ and so $\frac{\partial W}{\partial P} = g\frac{eP^{e-1}}{(P^e+1)^2}$ and it follows \[ \frac{\partial W}{\partial P}(P=1) = …

### 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…