Logistic Regression in Horse Racing Prediction

Introduction

Logistic Regression is a statistical method for analyzig data in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomus variable (pass/fail, win/lose). In the context of horse racing, logistic regression can be used to predict the probability of a particular horse winning.

Fundamental Concepts

The logitstic function (i.e. sigmoid) is an S-shaped curve: $∀_{x \in \mathbb{R}} : x \in [0,1]$ $$σ(z) = \frac{1}{1+e^{-z}}$$

Such that:

• $σ(z)$ is the output
• $z$ is the input to the function (linear combination of weights and obs. variables).

Model Representation

In logistic regression, typically trying to model the probability that a given input point belongs to certain cases (horse wins). The probability that an outcome will occur is modeled as a linear function of combination of predictor variables.

$$P(Y=1 | X) = \frac{1}{1+\exp(-(β_0 + \beta_1X_1 + ⋯ + \beta_nX_n))}$$

Such that:

• $P(Y=1 | X)$ is the prob. that the horse wins given predictors.
• $X_1, X_2, X_3 ⋯ X_n$ are the features (speed, distance to finish).
• $\beta_0, \beta_1, \cdots \beta_n$ are the coefficients to be learned from the training data.

Learning the Model

The coefficients $\beta_0, \beta_1, \cdots, \beta_n$ are usually learned through a process called Maximum Liklihood Estimation (MLE). The idea is to find the set of parameters that make the observed data most probable.

Application to Horse Racing

Feature Engineering: Some might include:

• Distance to finish: remaining distance each horse has to cover to finish the race.
• Speed: Current speed or average speed of the horse.

Model Fitting

Given a set of race data:

• X: A matrix where each row represents a horse, each column represents a feature of that horse.
• Y: A vector where each entry represents whether or not the horse is in the lead or not (1 or 0).

Prediction

Once the model is fitted, it can predict the winning probability for each horse in a new race. The model will output a probability score between 0 and 1 for each horse, indicating the likelihood of that horse winning the race.

Interpretation

• Coefficients: Positive coefficients increase the log-odds of the response variable (∴ increase the probability) and negative coefficients decrease the log-odds (∴ decrease the probability).
• Odds Ratio: For one-unit increase in the predictor variable, the odds ratio is the factor by which the odds of the outcome increase, or decrease given the coefficient.