Bayesian spectral density estimation using P-splines with quantile-based knot placement

This article proposes a Bayesian approach to estimating the spectral density of a stationary time series using a prior based on a mixture of P-spline distributions. Our proposal is motivated by the B-spline Dirichlet process prior of Edwards et al. (Stat Comput 29(1):67–78, 2019. https://doi.org/10.1007/s11222-017-9796-9) in combination with Whittle’s likelihood and aims at reducing the high computational complexity of its posterior computations. The strength of the B-spline Dirichlet process prior over the Bernstein–Dirichlet process prior of Choudhuri et al. (J Am Stat Assoc 99(468):1050–1059, 2004. https://doi.org/10.1198/016214504000000557) lies in its ability to estimate spectral densities with sharp peaks and abrupt changes due to the flexibility of B-splines with variable number and location of knots. Here, we suggest to use P-splines of Eilers and Marx (Stat Sci 11(2):89–121, 1996. https://doi.org/10.1214/ss/1038425655) that combine a B-spline basis with a discrete penalty on the basis coefficients. In addition to equidistant knots, a novel strategy for a more expedient placement of knots is proposed that makes use of the information provided by the periodogram about the steepness of the spectral power distribution. We demonstrate in a simulation study and two real case studies that this approach retains the flexibility of the B-splines, achieves similar ability to accurately estimate peaks due to the new data-driven knot allocation scheme but significantly reduces the computational costs.