Kernel covariance series smoothing

In this paper, we provide a new viewpoint of sequential random processes of the kind F(x), where x is a multivariate vector of covariates, in terms of a smoothing operation governed by a covariance function. By exploiting the eigenvalues and eigenvectors of the covariance function, we represent the smooth function in terms of an orthogonal series over basis functions where the basis function weights depend on the structure of the eigenfunctions with respect to the process F (x). This enables regression using smoothing based on series truncation and low-rank approximation of the covariance matrix. We show that our proposed method compares favorably both to Gaussian process regression, and to Nadaraya-Watson kernel smoothing.