Principal Component Analysis for Feature Reduction

## The Curse of Dimensionality As the number of features in a model increases, the data requirement for reliable estimation grows exponentially. With 50 features and 3,000 matches, you are in a high-dimensional regime where standard estimation methods struggle. PCA (Principal Component Analysis) reduces dimensionality while preserving most of the information. ## How PCA Works PCA finds the directions in feature space that explain the most variance. These directions are the "principal components" — uncorrelated linear combinations of the original features. Steps: 1. Standardise all features (mean 0, standard deviation 1) 2. Compute the covariance matrix 3. Find eigenvectors (directions) and eigenvalues (variance explained) 4. Keep the top k components that explain 80–90% of total variance The result: k new features (components) that are uncorrelated and together capture most of the information in the original 50 features. ## When to Use PCA in Betting Models PCA is most useful when: - You have many correlated features (team statistics are often highly correlated) - You want to identify the underlying dimensions of team quality - You are feeding features into a machine learning model that benefits from uncorrelated inputs ## A Sports Example Team statistics (goals scored, shots, xG, possession, pass completion, pressing metrics) are highly correlated. PCA might find that: - Component 1 (explains 45% of variance): "Attacking dominance" — high loadings on all attacking metrics - Component 2 (explains 25% of variance): "Defensive solidity" — high loadings on defensive metrics - Component 3 (explains 15% of variance): "Possession control" — high loadings on possession and pass accuracy These 3 components replace the original many features with interpretable, uncorrelated dimensions.