Shrinkage widely linear recursive least squares (SWL-RLS) and its improved version called structured shrinkage widely linear recursive least squares (SSWL-RLS) algorithms are proposed in this paper. By using the relationship between the noise-free a posterior and a priori error signals, the optimal forgetting factor can be obtained at each snapshot. In the implementation of algorithms, due to the a priori error signal known, we still need the information about the noise-free a priori error which can be estimated with a known formula. Simulation results illustrate that the proposed algorithms have faster convergence and better tracking capability than augmented RLS (A-RLS), augmented least mean square (A-LMS) and SWL-LMS algorithms.

In this article, we considered the asymptotic expectations of the prediction error and the fitting error of a regression model, in which the component functions are chosen from a finite set of orthogonal functions. Under the least squares estimation, we showed that the asymptotic bias in estimating the prediction error based on the fitting error includes the true number of components, which is essentially unknown in practical applications. On the other hand, under a suitable shrinkage method, we showed that an asymptotically unbiased estimate of the prediction error is given by the fitting error plus a known term except the noise variance.