Fisher Algorithm Source Code

When applying statistical methods to pattern recognition problems, we face the dimensionality challenge. Generally, classification problems in low-dimensional feature spaces are simpler than those in high-dimensional spaces. Therefore, we explore dimensionality reduction techniques. The fundamental approach involves projecting d-dimensional feature space onto a straight line to form one-dimensional space, which is mathematically straightforward to implement. However, the key issue is that originally linearly separable samples may become non-separable after projection.

For linearly separable samples, we can always find a projection direction that maintains linear separability after dimensionality reduction. But how do we determine the projection direction that not only preserves linear separability but also enhances separability? (Meaning maximizing the distance between different classes while minimizing the spread within each class) This is precisely what Fisher Linear Discriminant aims to solve.

The Fisher algorithm implementation typically calculates two critical matrices: the within-class scatter matrix (measuring class compactness) and the between-class scatter matrix (measuring class separation). The optimal projection vector is obtained by maximizing the ratio of between-class to within-class scatter, which can be solved through eigenvalue decomposition. During code implementation, key functions would include computing mean vectors for each class, constructing scatter matrices, and solving the generalized eigenvalue problem to obtain the projection direction.

Therefore, when performing dimensionality reduction, we must consider these factors and adjust data processing according to requirements to achieve optimal results. Practical implementation would involve normalizing features, handling potential singularity issues in scatter matrices, and validating the projection direction through cross-validation techniques.