Principal Component Analysis
Principal components analysis (PCA) is a linear technique used to reduce a high-dimensional dataset to a lower dimensional representations for analysis and indexing. For a dataset P in D-dimensional space with its principal component set Φ, given a point p∈P, its projection on the lower d-dimensional subspace can be defined as: p. Φd, where Φd represents the matrix containing 1st to dth largest principal components in Φ and d < D.
PCA finds a low-dimensional embedding of the data points that best preserves their variance as measured in the high-dimensional input space [1, 2, 3]. It identifies the directions that best preserve the associated variances of the data points while minimize “least-squares” (Euclidean) error measured by analyzing data covariance matrix. The first principal component is the eigenvector corresponding to the largest eigenvalue of the dataset’s co-variance matrix, the second component corresponds to the eigenvector with the...
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