Regularized spectrum estimation in spaces induced by stable spline kernels

We introduce a new kernel-based nonparametric approach to estimate the second-order statistics of scalar and stationary stochastic processes. The correlations functions are assumed to be summable and are modeled as realizations of zero-mean Gaussian processes using the recently introduced Stable Spline kernel. In this way, information on the decay to zero of the functions to reconstruct is included in the estimation process. The overall complexity of the proposed algorithm scales linearly with the number of available samples of the processes. Numerical experiments show that the method compares favorably with respect to classical nonparametric spectral analysis approaches with an oracle-type choice of the parameters.