LDA reduces dimensionality by projecting the data onto a lower-dimensional space while maximizing class separability. Here’s how it works:
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Compute the Mean Vectors: For each class in the dataset, LDA calculates the mean vector.
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Compute the Scatter Matrices:
- Within-Class Scatter Matrix: Measures how much the data points within each class scatter around their respective mean.
- Between-Class Scatter Matrix: Measures how much the class means scatter around the overall mean.
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Calculate the Eigenvalues and Eigenvectors: LDA solves the generalized eigenvalue problem for the scatter matrices. The eigenvectors represent the directions of maximum variance, and the eigenvalues indicate the magnitude of variance in those directions.
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Select the Top Eigenvectors: The eigenvectors corresponding to the largest eigenvalues are selected to form a new feature space. The number of eigenvectors chosen is typically one less than the number of classes (n_classes - 1).
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Transform the Data: Finally, the original data is projected onto the new feature space defined by the selected eigenvectors, resulting in a lower-dimensional representation that retains the most discriminative information.
This process allows LDA to effectively reduce dimensionality while enhancing the separation between different classes in the dataset.
