Statistics and Computing

, Volume 29, Issue 1, pp 67–78 | Cite as

Bayesian nonparametric spectral density estimation using B-spline priors

  • Matthew C. EdwardsEmail author
  • Renate Meyer
  • Nelson Christensen


We present a new Bayesian nonparametric approach to estimating the spectral density of a stationary time series. A nonparametric prior based on a mixture of B-spline distributions is specified and can be regarded as a generalization of the Bernstein polynomial prior of Petrone (Scand J Stat 26:373–393, 1999a; Can J Stat 27:105–126, 1999b) and Choudhuri et al. (J Am Stat Assoc 99(468):1050–1059, 2004). Whittle’s likelihood approximation is used to obtain the pseudo-posterior distribution. This method allows for a data-driven choice of the number of mixture components and the location of knots. Posterior samples are obtained using a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm, and mixing is improved using parallel tempering. We conduct a simulation study to demonstrate that for complicated spectral densities, the B-spline prior provides more accurate Monte Carlo estimates in terms of \(L_1\)-error and uniform coverage probabilities than the Bernstein polynomial prior. We apply the algorithm to annual mean sunspot data to estimate the solar cycle. Finally, we demonstrate the algorithm’s ability to estimate a spectral density with sharp features, using real gravitational wave detector data from LIGO’s sixth science run, recoloured to match the Advanced LIGO target sensitivity.


B-spline prior Bernstein polynomial prior Whittle likelihood Spectral density estimation Bayesian nonparametrics LIGO Gravitational waves Sunspot cycle 



We thank Claudia Kirch, Alexander Meier, and Thomas Yee for fruitful discussions, and Michael Coughlin for providing us with the recoloured LIGO data. 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. NC’s work is supported by National Science Foundation Grant PHY-1505373. All analysis was conducted in R, an open-source statistical software available on CRAN ( We acknowledge the following R packages: Rcpp, Rmpi, bsplinePsd, beyondWhittle, splines, signal, bspec, ggplot2, grid and gridExtra. This paper carries LIGO Document No. LIGO-P1600239.

Supplementary material

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Supplementary material 1 (txt 2916 KB)
11222_2017_9796_MOESM2_ESM.txt (86 kb)
Supplementary material 2 (txt 85 KB)


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Matthew C. Edwards
    • 1
    • 2
    Email author
  • Renate Meyer
    • 1
  • Nelson Christensen
    • 2
    • 3
  1. 1.Department of StatisticsUniversity of AucklandAucklandNew Zealand
  2. 2.Physics and AstronomyCarleton CollegeNorthfieldUSA
  3. 3.Artemis, Université Côte d’Azur, Observatoire de Côte d’Azur, CNRSNiceFrance

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