Implementation of Fisher Linear Discriminant for Classification Tasks

Fisher Linear Discriminant (FLD) is a classic linear dimensionality reduction method particularly effective for binary classification problems. Its core algorithm seeks an optimal projection direction that maximizes inter-class separation while minimizing intra-class variance. The mathematical implementation calculates this by maximizing the ratio between between-class scatter and within-class scatter matrices, typically solved through eigenvalue decomposition. Key functions would involve computing mean vectors per class, constructing scatter matrices, and solving the generalized eigenvalue problem to obtain the projection vector.

For two-dimensional classification results, the implementation projects sample coordinates onto the optimal direction, mapping data points to one-dimensional space. A threshold value (often calculated as the midpoint between class means) then performs binary classification. To visualize results, code can plot decision boundaries in the original 2D space as lines perpendicular to the projection direction, clearly separating both classes. Programming this involves calculating projected coordinates, determining the optimal threshold, and generating perpendicular line equations based on the projection vector.

For three-dimensional data, FLD remains applicable by projecting data onto one-dimensional space for classification. Visualization techniques can display sample distributions in 3D space with classification boundaries represented as planes (separation hyperplanes). The implementation extends the 2D approach by computing 3D projection vectors and generating plane equations normal to the projection direction. This dimensionality reduction approach helps assess data separability while providing intuitive classifier performance evaluation through 3D plotting libraries.

Fisher Linear Discriminant serves as both a low-dimensional classification/visualization tool and a scalable solution for high-dimensional data. Its computational efficiency comes from linear algebra operations that can be optimized using matrix computation libraries, making it a fundamental method in pattern recognition and machine learning pipelines.