Statistical analysis of effective singular values in matrix rank determination

A major problem in using SVD (singular-value decomposition) as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end, conference regions are derived for the perturbed singular values of matrices with noisy observation data. The analysis is based on the theories of perturbations of singular values and statistical significance test. Threshold bounds for perturbation due to finite-precision and i.i.d. random models are evaluated. In random models, the threshold bounds depend on the dimension of the matrix, the noisy variance, and predefined statistical level of significance. Results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered. Various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given.