lecture_04

Assume step 1 has been done problems become direction

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Unformatted text preview: X . D is a p × p diagonal matrix, with diagonal entries d1 ≥ d2 ≥ · · · ≥ dp ≥ 0, which are called singular values of X . v 1 , . . . , v p are eigenvectors of the matrix X T X , corresponding to the eigenvalues d2 ≥ d2 ≥ · · · ≥ d2 . p 1 2 u1 , . . . , up are eigenvectors of the matrix XX T , corresponding to the eigenvalues d2 ≥ d2 ≥ · · · ≥ d2 . The rest eigenvalues of XX T p 1 2 are all zero. PCA: Solution Fix V , we must have η i = V T xi , ˆ q and the problem is reduced to N xi − V q V T xi 2 . q min Vq i=1 Let X be the N × p matrix whose rows are xT , . . . , xT . Compute 1 N the singular value decomposition (SVD) of X X = U DV T . ˆ For each 1 ≤ q ≤ p, the solution V q consists of the first q columns of V . vm z m = X vm = dm um vm m-th principal direction m-th principal component loadings of the m-th principal component Applications Visualization. Compression. Computation. Clearer patterns in lower dimension. Anomaly detection. Remove redundancy. Face recognition and matching. Microarray analysis. Web link analysis. 3 0 1 2 Variances 200 100 0 Variances 300 400 4 Asset Excess Returns Comp.1 Comp.3 Comp.5 Comp.7 Comp.9 Comp.1 Comp.3 Comp.5 Comp.7 Comp.9...
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