Confidence regions for perturbed singular values in system identification

A major problem in using SVD as a tool in determining the effective rank of a perturbed matrix, is that of distinguishing between significant small and insignificant large singular values. In this paper we derive confidence regions for the perturbed singular values of matrices with noisy observation data. The analysis is based on the perturbation theory of singular values and classical significance testing. The threshold bounds depend on the dimension of the matrix, the noise variance and a predefined statistical level of significance. The results are applied to the problem of determining the effective order of a linear system from the approximate rank of a sample autocorrelation matrix. Numerical examples are given.