MATLAB Implementation of Fisher Algorithm for Multi-Class Classification

MATLAB Implementation of Fisher Algorithm for Multi-Class Classification

Fisher's algorithm (also known as Linear Discriminant Analysis, LDA) is a classical linear classification method widely applied in pattern recognition and machine learning tasks. Its core principle involves finding optimal projection directions that maximize inter-class separation while minimizing intra-class variance.

In multi-class classification problems, Fisher's algorithm extends beyond finding a single optimal projection direction. Instead, it performs dimensionality reduction to map data into a lower-dimensional space while maximizing class separability. MATLAB provides convenient implementation through the `multiclassfisher` function, which handles Fisher discriminant analysis for multiple classes with efficient matrix operations.

Implementation Approach: - Calculate mean vectors for each class and the global mean vector for the entire dataset using MATLAB's mean() and groupstats functions - Compute within-class scatter matrix (Sw) by summing covariance matrices of each class, and between-class scatter matrix (Sb) using class means and global mean - Solve for the projection matrix through generalized eigenvalue decomposition (eig(Sb, Sw)) that maximizes the ratio between-class to within-class scatter - Apply the projection matrix to reduce dimensionality of original data, enabling effective classification with linear classifiers like k-nearest neighbors

In MATLAB's implementation, the `multiclassfisher` function encapsulates these steps with optimized linear algebra operations. Users simply input training data and corresponding class labels to obtain a trained projection model. The model can then transform test data using matrix multiplication (testData * projectionMatrix) for classification predictions.

Extension Considerations: While Fisher's algorithm is simple and effective, it may underperform with non-linearly separable data. Solutions include Kernel LDA implementations using MATLAB's kernel functions, or employing deep learning models with MATLAB's Deep Learning Toolbox. For imbalanced datasets, weighted Fisher discriminant analysis can be implemented by modifying class covariance calculations with sample weighting techniques.