MSE-based regularization approach to rank determination in CLS and TLS estimation

The corrected least squares (CLS) approach using an over-determined model is investigated to decide the number of sinusoids in additive white noise. Like the total least squares (TLS) approach, the CLS estimation is different from the ordinary least squares (LS) method in that the noise variance is subtracted from the diagonal elements of the correlation matrix of the noisy observed data. Therefore the inversion of the resultant matrix becomes ill-conditioned and then adequate truncation of the eigenvalue decomposition (EVD) should be done. This paper clarifies how to simultaneously estimate the noise variance and truncate the eigenvalues, since they are mutually dependent. By introducing a multiple number of regularization parameters and determining them to minimize the MSE of the model parameters, we can give an optimal scheme for the truncation of eigenvalues. Furthermore, an iterative algorithm using only observed data is also clarified.