Abstract
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.
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Acknowledgements
We thank Thomas Yee for helpful discussions on P-splines. We also thank the New Zealand eScience Infrastructure (NeSI) for their high performance computing facilities, and the Centre for eResearch at the University of Auckland for their technical support. PM’s and RM’s work is supported by Grant 3714568 from the University of Auckland Faculty Research Development Fund and the DFG Grant KI 1443/3-1. RM gratefully acknowledges support by a James Cook Fellowship from Government funding, administered by the Royal Society Te Apārangi. All analysis was conducted in R, an open-source statistical software available on CRAN (cran.r-project.org).
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Maturana-Russel, P., Meyer, R. Bayesian spectral density estimation using P-splines with quantile-based knot placement. Comput Stat 36, 2055–2077 (2021). https://doi.org/10.1007/s00180-021-01066-7
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DOI: https://doi.org/10.1007/s00180-021-01066-7