Fisher Linear Discriminant: The Most Commonly Used Linear Classifier in Pattern Recognition

The Fisher Linear Classifier is a classic linear classification method in pattern recognition, primarily used for projecting high-dimensional data into a one-dimensional space for effective classification. Its core principle involves finding an optimal projection direction that maximizes separation between different classes while ensuring data within the same class remains closely clustered. In implementation, this typically involves calculating scatter matrices and solving a generalized eigenvalue problem.

The algorithm begins by computing mean vectors for both classes, then determines the optimal projection direction by maximizing the ratio between the between-class scatter matrix and the within-class scatter matrix. This direction maximizes the distance between projected class means while minimizing the variance within each class. After determining the projection direction, classification decisions are made through thresholding. Code implementation often involves matrix operations using libraries like NumPy, where the projection vector w is calculated as w = S_w⁻¹(μ₁ - μ₂), with S_w representing the within-class scatter matrix.

In practical applications, the Fisher classifier calculates sample positions on the projection line and sets an appropriate classification threshold (typically the midpoint between the two class means). By comparing sample projection values against this threshold, class membership is determined. The classification effectiveness can be visually demonstrated by plotting histograms of sample distributions along the projection direction, where ideally two classes form distinct clusters on opposite sides of the threshold. Implementation often includes visualization using matplotlib to plot projection histograms and decision boundaries.

This method is particularly suitable for binary classification problems, offering computational simplicity and clear physical interpretation, making it fundamental for understanding more complex classifiers. However, it's important to note its assumption of similar covariance structures between classes; performance may be limited for non-linearly separable data or multi-modal distributions. The scikit-learn library provides Fisher Discriminant Analysis implementation through LinearDiscriminantAnalysis class, which handles both binary and multi-class scenarios.