| Title: |
We consider in turn Principal Components Analysis, Factor Analysis, Independent Components Analysis and Non-linear Factor Analysis. |
| Authors: |
Principal Components Analysis |
| Contributors: |
The Pennsylvania State University CiteSeerX Archives |
| Source: |
http://www.inf.ed.ac.uk/teaching/courses/pmr/docs/fadoc.pdf. |
| Publication Year: |
2007 |
| Collection: |
CiteSeerX |
| Description: |
Principal Components Analysis (PCA) is a well-established linear technique for dimensionality reduction. We consider reducing dimensionality from the data space of dimensionality d with data vector x to a space of dimensionality m. Let the sample data have mean µ and covariance matrix S, and eigenvalues/vectors such that Swj = λjwj. We order the eigenvalues so that λ1 ≥ λ2 ≥. λd> 0. The first derivation of PCA concerns the idea of a “bottleneck ” network architecture. We input a vector x which is transformed into a code vector z of lower dimensionality, and we then expand back up to the original dimensionality. With a linear architecture, this is achieved by a m × d matrix A and a d × m matrix B so that the reconstructed output ˆx is given by ˆx = µ + BA(x − µ). The optimal choice for A is to project into the subspace spanned by the first m principal components of S. The optimal orthogonal projection is given by the matrix A = W T, where W = (w1, w2,., wm) T, and B = W. PCA can also be derived by choosing projections of the data which maximize the variance in the projected space. For example, we first look for a vector a (with a.a = 1) such that the projection of the original data a.(x − µ) has maximal variance. This turns out to be given by |
| Document Type: |
text |
| File Description: |
application/pdf |
| Language: |
English |
| Relation: |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.8299; http://www.inf.ed.ac.uk/teaching/courses/pmr/docs/fadoc.pdf |
| Availability: |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.8299; http://www.inf.ed.ac.uk/teaching/courses/pmr/docs/fadoc.pdf |
| Rights: |
Metadata may be used without restrictions as long as the oai identifier remains attached to it. |
| Accession Number: |
edsbas.A54CF18B |
| Database: |
BASE |