Singular Value Decomposition Factor Analysis

Factor analysis is a multivariate statistical method for extracting the information content from a data set. In the Olis case we have a large matrix consisting of absorbance etc. as function of both wavelength and time. Factor analysis, in the Olis case singular value decomposition (SVD), reduces this to 3 smaller matrices that contain all the information present in the data. The kinetic information is contained in a kinetic eigenvector matrix. Experience has shown that only a small number of eigenvectors are needed to describe the data. Olis uses 6. The various rate processes are stored over a small number of eigenvectors and a kinetic model may be fitted to those to extract the rate constants. Similarly the spectral information is stored in a set of 6 eigenvectors. Finally a vector of singular values is produced. Typically these start off at a high value and fall off rapidly to a value near zero. The number of singular values above zero is an indication of the number of species significantly present and is useful for model selection.

A representation of the original data may be obtained by a suitable matrix multiplication of the 2 eigenvector sets of vectors and the singular values. This reconstruction is the original less most of the noise. Thus the processes of SVD and reconstruction smooth the data. The 3 matrices are commonly used to store the information in the data in a much more compact form than the original; about 30 times smaller. It is also much easier to do the global kinetic fit on the compact kinetic set than the original large data set. Thus factor analysis cleans up the data, affords more compact storage, and greatly reduces the scale of the global fitting problem.

SVD is one of several methods for factor analysis and is generally recognized as being the mathematically most robust. The Olis modifications to SVD in no way modify the way in which it is used or the conclusions drawn from it. The only difference is a large gain in speed.