MATLAB Implementation of Linear Discriminant Analysis Algorithm

Linear Discriminant Analysis (LDA) is a classic supervised learning algorithm primarily used for classification tasks and feature dimensionality reduction. Its core principle involves projecting data into a lower-dimensional space where samples from the same class are clustered together while samples from different classes are separated as much as possible.

The MATLAB implementation of LDA follows these key steps: First, compute the mean vectors for each class serving as class centroids using functions like mean() or grpstats(). Then calculate the within-class scatter matrix (measuring intra-class variance) and between-class scatter matrix (measuring inter-class separation) through matrix operations involving covariance calculations. Next, solve the generalized eigenvalue problem using eig() or eigs() functions to obtain optimal projection directions that maximize class separability. Finally, perform classification using the projected data.

During classification, new samples are projected onto the discriminant subspace where class assignment is determined by computing distances to class centroids (typically using Mahalanobis distance for scale-invariant measurements). Important implementation note: LDA requires the within-class scatter matrix to be invertible - when dealing with high-dimensional data, regularization techniques like adding a small identity matrix (e.g., sw_reg = Sw + lambda*eye(size(Sw))) may be necessary to ensure numerical stability.

Typical MATLAB applications of LDA include pattern recognition, biometric classification, and other domains requiring both interpretability and classification performance. Compared to unsupervised methods like PCA, LDA generally achieves superior projection results in classification tasks by leveraging class label information through its supervised learning approach. Key MATLAB functions for implementation may include fitcdiscr() for ready-to-use classifiers or custom implementations using matrix algebra operations.