Logistic Regression

Now we need to understand how Logistic regression works so that we can create a logistic model for the rating against wins. Then we should be able to fit that into Elo's model.

A logistic function is of the form:

f(x) = L / (1 + exp(-k(x - x0)))

Where L is the maximum value, which will be 1 in our case. e is Euler's number. k is the 'logistic growth rate'. Finally, x0 is the x-value of the sigmoid midpoint.

So the only values we need to find are k and x0.

We'll find these values using gradient descent. The process will be similar to how neural networks are trained on their training data. We'll be using all of Premier League results as training data.

I'm not going to explain the process in massive detail as this article would be twice the length. But here is a great article I recommend reading on the topic.

• TODO: Probability of drawing

  • Either a logistic function of absolute rating difference against probability of drawing or a normal distribution of rating difference against probability of drawing will be used
  • Hold out validation will be used to check which model performs best

• TODO: Probability of winning (given a result/not a draw)

  • A logistic function will be used to represent win/loss given that a draw doesn't occur

• Gradient descent

Logistic Regression With R

train <- read.csv('train.csv', header=T, na.string=c(""))

model <- glm(Win ~ ., family = binomial(link = 'logit'), data = train)

Elo Home Advantage

It'll be important to find the Elo home advantage, so that we can correctly train our model on the training data. It's clear in football that the home team has an advantage, but by how much?

In theory, the average rating difference of wins should be 0. The average rating of teams winning when they play at home is 105. Therefore, we can increase the home advantage by 105 for all matches.

• Training our model on Premier League results

• Using our model to predict the outcome of future matches

• Creating a model for goals scored

  • Poisson regression