   Next: Singular Value Decomposition Up: Algorithms for Principal Component Previous: Algorithms for Principal Component

### Principal subspace and components

Assume that we have  -dimensional data vectors , which form the data matrix = . The matrix is decomposed into (1)

where is a matrix, is a matrix and . Principal subspace methods [6,4] find and such that the reconstruction error (2)

is minimized. There denotes the Frobenius norm, and , , and elements of the matrices , , and , respectively. Typically the row-wise mean is removed from as a preprocessing step.

Without any further constraints, there exist infinitely many ways to perform such a decomposition. However, the subspace spanned by the column vectors of the matrix , called the principal subspace, is unique. In PCA, these vectors are mutually orthogonal and have unit length. Further, for each , the first vectors form the -dimensional principal subspace. This makes the solution practically unique, see [4,2,5] for details.

There are many ways to determine the principal subspace and components [6,4,2]. We will discuss three common methods that can be adapted for the case of missing values.   Next: Singular Value Decomposition Up: Algorithms for Principal Component Previous: Algorithms for Principal Component
Tapani Raiko 2007-09-11