Explain linear discriminant analysis in machine learning
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Linear Discriminant Analysis, or LDA, is a powerful technique in machine learning used for both dimensionality reduction and classification. The main goal of LDA is to find projections that maximize the separation between different classes while minimizing the variance within each class.
LDA works by calculating two key matrices. The within-class scatter matrix measures how much the data points vary within each individual class. The between-class scatter matrix measures how far apart the class means are from each other. The goal is to maximize the ratio of between-class scatter to within-class scatter.
LDA finds the optimal projection by maximizing the Fisher criterion, which is the ratio of between-class scatter to within-class scatter. This is solved by finding the eigenvectors of the matrix S inverse W times S B. The resulting projection maximizes class separation while minimizing within-class variance.
LDA reduces dimensionality by projecting the original data onto the discriminant vectors. The maximum number of dimensions in the reduced space is C minus 1, where C is the number of classes. Once projected, the data can be classified using simple methods like nearest neighbors or decision boundaries.
To summarize, Linear Discriminant Analysis is a powerful technique that finds optimal projections to separate classes. It maximizes the Fisher criterion, reduces dimensionality effectively, and serves as both a preprocessing step and classification method in machine learning applications.