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Principal component analysis (PCA)

The algorithm successively generates principal components (PC): The first PC is the projection direction that maximizes the variance of the projected data. The second PC is the projection direction that is orthogonal to the first PC and maximizes the variance of the projected data. Repeat until k‐orthogonal lines are obtained (Figure 2.15).

The projected position of a point on these lines gives the coordinates in k‐dimensional reduced space.

Steps in PCA: (i) Compute covariance matrix ∑ of the dataset S, (ii) calculate the eigenvalues and eigenvectors of ∑. The eigenvector with the largest eigenvalue λ₁ is the first PC. The eigenvector with the kth largest eigenvalue λ_k is the kth PC. λ_k/∑_i λ_i = proportion of variance captured by the kth PC.

Figure 2.15 Successive data projections.

The full set of PCs comprises a new orthogonal basis for the feature space, whose axes are aligned with the maximum variances of the original data. The projection of original data onto the first k PCs gives a reduced dimensionality representation of the data. Transforming reduced dimensionality projection back into the original space gives a reduced dimensionality reconstruction of the original data. Reconstruction will have some error, but it can be small and often is acceptable given the other benefits of dimensionality reduction. Choosing the dimension k is based on ∑_{i = 1,k} λ_i/∑_{i = 1,S} λ_i > β[%], where β is a predetermined value.