  This article improves on existing methods to estimate the spectral density of
stationary and nonstationary time series assuming a Gaussian process prior. By
optimising an appropriate eigendecomposition using a smoothing spline
covariance structure, our method more appropriately models data with both
simple and complex periodic structure. We further justify the utility of this
optimal eigendecomposition by investigating the performance of alternative
covariance functions other than smoothing splines. We show that the optimal
eigendecomposition provides a material improvement, while the other covariance
functions under examination do not, all performing comparatively well as the
smoothing spline. During our computational investigation, we introduce new
validation metrics for the spectral density estimate, inspired from the
physical sciences. We validate our models in an extensive simulation study and
demonstrate superior performance with real data.
