Dealing with collinearity in large-scale linear system identification using Gaussian regression

We consider estimation of large-scale linear dynamic systems fed with correlated and possibly poor exciting inputs. We develop a Bayesian strategy where any impulse response is a zero-mean Gaussian process with stable spline covariance to encode BIBO stability. We design a variation of Gibbs sampling to deal efficiently with collinearity: beyond considering blocks forming a partition of the parameter space, some other (overlapping) blocks are also updated on the basis of the level of collinearity of the system inputs. Theoretical properties of the algorithm are studied and numerical experiments are included to test it.