lecture_04

# Assume step 1 has been done problems become direction

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 10/01/2013 for the course FSRM 588 taught by Professor Xiao during the Fall '13 term at Rutgers.

Ask a homework question - tutors are online