Advertisement

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
Article

Abstract

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.

Keywords

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

Notes

Acknowledgements

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 (cran.r-project.org). 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

11222_2017_9796_MOESM1_ESM.txt (2.8 mb)
Supplementary material 1 (txt 2916 KB)
11222_2017_9796_MOESM2_ESM.txt (86 kb)
Supplementary material 2 (txt 85 KB)

References

  1. Aasi, J., et al.: Advanced LIGO. Class. Quantum Gravity 32, 074001 (2015)CrossRefGoogle Scholar
  2. Abadie, J., et al.: All-sky search for gravitational-wave bursts in the second joint LIGO-Virgo run. Phys. Rev. D 85, 122007 (2012)CrossRefGoogle Scholar
  3. Abbott, B.P., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102 (2016a)MathSciNetCrossRefGoogle Scholar
  4. Abbott, B.P., et al.: GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116, 241103 (2016b)CrossRefGoogle Scholar
  5. Abbott, B.P., et al.: GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017a)CrossRefGoogle Scholar
  6. Abbott, B.P., et al.: GW170104: observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118, 221101 (2017b)CrossRefGoogle Scholar
  7. Abbott, B.P., et al.: GW170814: a three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett. 119, 141101 (2017c)CrossRefGoogle Scholar
  8. Abbott, B.P., et al.: GW170608: observation of a 19-solar-mass binary black hole coalescence. Pre-print, arXiv:1711.05578 (2017d)
  9. Acernese, F., et al.: Advanced Virgo: a second-generation interferometric gravitational wave detector. Class. Quantum Gravity 32(2), 024001 (2015)CrossRefGoogle Scholar
  10. Barnett, G., Kohn, R., Sheather, S.: Bayesian estimation of an autoregressive model using Markov chain Monte Carlo. J. Econ. 74, 237–254 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bartlett, M.S.: Periodogram analysis and continuous spectra. Biometrika 37, 1–16 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Brockwell, P.J., Davis, R.A.: Time Series: Theory and Methods, 2nd edn. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  13. Brooks, S.P., Giudici, P., Roberts, G.O.: Efficient construction of reversible jump Markov chain Monte Carlo proposal distributions. J. R. Stat. Soc.: Ser. B (Methodol.) 65, 3–55 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Cai, B., Meyer, R.: Bayesian semiparametric modeling of survival data based on mixtures of B-spline distributions. Comput. Stat. Data Anal. 55, 1260–1272 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Carter, C.K., Kohn, R.: Semiparametric Bayesian inference for time series with mixed spectra. J. R. Soc. Ser. B 59, 255–268 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Chopin, N., Rousseau, J., Liseo, B.: Computational aspects of Bayesian spectral density estimation. J. Comput. Graph. Stat. 22, 533–557 (2013)MathSciNetCrossRefGoogle Scholar
  17. Choudhuri, N., Ghosal, S., Roy, A.: Bayesian estimation of the spectral density of a time series. J. Am. Stat. Assoc. 99(468), 1050–1059 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Christensen, N.: LIGO S6 detector characterization studies. Class. Quantum Gravity 27, 194010 (2010)CrossRefGoogle Scholar
  19. Cogburn, R., Davis, H.R.: Periodic splines and spectral estimation. Ann. Stat. 2, 1108–1126 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Crandell, J.L., Dunson, D.B.: Posterior simulation across nonparametric models for functional clustering. Sankhya B 73, 42–61 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. de Boor, C.: B(asic)-spline basics. In: Piegl, L. (ed.) Fundamental Developments of Computer-Aided Geometric Modeling. Academic Press, Washington (1993)Google Scholar
  22. Earl, D.J., Deem, M.W.: Parallel tempering: theory, applications, and new perspectives. Phys. Chem. Chem. Phys. 7, 3910–3916 (2005)CrossRefGoogle Scholar
  23. Edwards, M.C., Meyer, R., Christensen, N.: Bayesian parameter estimation of core collapse supernovae using gravitational wave signals. Inverse Probl. 30, 114008 (2014)CrossRefzbMATHGoogle Scholar
  24. Edwards, M.C., Meyer, R., Christensen, N.: Bayesian semiparametric power spectral density estimation with applications in gravitational wave data analysis. Phys. Rev. D 92, 064011 (2015)CrossRefGoogle Scholar
  25. Edwards, M.C., Meyer, R., Christensen, N.: bsplinePsd: Bayesian power spectral density estimation using B-spline priors. R package (2017)Google Scholar
  26. Einstein, A.: Approximative integration of the field equations of gravitation. Sitzungsberichte Preußischen Akademie der Wissenschaften 1916(Part 1), 688–696 (1916)zbMATHGoogle Scholar
  27. Gangopadhyay, A.K., Mallick, B.K., Denison, D.G.T.: Estimation of the spectral density of a stationary time series via an asymptotic representation of the periodogram. J. Stat. Plan. Inference 75, 281–290 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., Rubin, D.B.: Bayesian Data Analysis, 3rd edn. Chapman & Hall/CRC, Boca Raton (2013)zbMATHGoogle Scholar
  29. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern. Anal. Mach. Intell. 6, 721–741 (1984)CrossRefzbMATHGoogle Scholar
  30. Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  32. Jara, A., Hanson, T.E., Quintana, F.A., Müller, P., Rosner, G.L.: DPpackage: Bayesian semi- and nonparametric modeling in R. J. Stat. Softw. 40, 1–30 (2011)CrossRefGoogle Scholar
  33. Kirch, C., Edwards, M.C., Meier, A., Meyer, R.: Beyond Whittle: nonparametric correction of a parametric likelihood with a focus on Bayesian time series analysis. Pre-print, arXiv:1701.04846v1 (2017)
  34. Kooperberg, C., Stone, C.J., Truong, Y.K.: Rate of convergence for logspline spectral density estimation. J. Time Ser. Anal. 16, 389–401 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Lenhoff, M.W., Santner, T.J., Otis, J.C., Peterson, M.G., Williams, B.J., Backus, S.I.: Bootstrap prediction and confidence bands: a superior statistical method for the analysis of gait data. Gait Posture 9, 10–17 (1999)CrossRefGoogle Scholar
  36. Liseo, B., Marinucci, D., Petrella, L.: Bayesian semiparametric inference on long-range dependence. Biometrika 88, 1089–1104 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Littenberg, T.B., Cornish, N.J.: Bayesian inference for spectral estimation of gravitational wave detector noise. Phys. Rev. D 91, 084034 (2015)CrossRefGoogle Scholar
  38. Littenberg, T.B., Coughlin, M., Farr, B., Farr, W.M.: Fortifying the characterization of binary mergers in LIGO data. Phys. Rev. D 88, 084044 (2013)CrossRefGoogle Scholar
  39. Macaro, C.: Bayesian non-parametric signal extraction for Gaussian time series. J. Econ. 157, 381–395 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. Macaro, C., Prado, R.: Spectral decompositions of multiple time series: a Bayesian non-parametric approach. Psychometrika 79, 105–129 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Meier, A., Kirch, C., Edwards, M.C., Meyer, R.: beyondWhittle: Bayesian spectral inference for stationary time series. R package (2017)Google Scholar
  42. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)CrossRefGoogle Scholar
  43. Neumann, M.H., Kreiss, J.-P.: Regression-type inference in nonparametric regression. Ann. Stat. 26, 1570–1613 (1998)CrossRefzbMATHGoogle Scholar
  44. Neumann, M.H., Polzehl, J.: Simultaneous bootstrap confidence bands in nonparametric regression. J. Nonparametr. Stat. 9, 307–333 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Perron, F., Mengersen, K.: Bayesian nonparametric modeling using mixtures of triangular distributions. Biometrics 57, 518–528 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  46. Petrone, S.: Random Bernstein polynomials. Scand. J. Stat. 26, 373–393 (1999a)MathSciNetCrossRefzbMATHGoogle Scholar
  47. Petrone, S.: Bayesian density estimation using Bernstein polynomials. Can. J. Stat. 27, 105–126 (1999b)MathSciNetCrossRefzbMATHGoogle Scholar
  48. Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)CrossRefzbMATHGoogle Scholar
  49. Rosen, O., Wood, S., Roy, A.: AdaptSpec: adaptive spectral density estimation for nonstationary time series. J. Am. Stat. Assoc. 107, 1575–1589 (2012)CrossRefzbMATHGoogle Scholar
  50. Rousseau, J., Chopin, N., Liseo, B.: Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian time series. Ann. Stat. 40, 964–995 (2012)CrossRefzbMATHGoogle Scholar
  51. Röver, C.: Student-t based filter for robust signal detection. Phys. Rev. D 84, 122004 (2011)CrossRefGoogle Scholar
  52. Röver, C., Meyer, R., Christensen, N.: Modelling coloured residual noise in gravitational-wave signal processing. Class. Quantum Gravity 28, 015010 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  53. Schwabe, S.H.: Sonnenbeobachtungen im jahre 1843. Astron. Nachr. 21, 233–236 (1843)Google Scholar
  54. Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)MathSciNetzbMATHGoogle Scholar
  55. Sun, J., Loader, C.R.: Confidence bands for linear regression and smoothing. Ann. Stat. 22, 1328–1345 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  56. Swendsen, R.H., Wang, J.S.: Replica Monte Carlo simulation of spin glasses. Phys. Rev. Lett. 57, 2607–2609 (1986)MathSciNetCrossRefGoogle Scholar
  57. Thomson, D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70, 1055–1096 (1982)CrossRefGoogle Scholar
  58. Tonellato, S.F.: Random field priors for spectral density functions. J. Stat. Plan. Inference 137, 3164–3176 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  59. Vitale, S., Congedo, G., Dolesi, R., Ferroni, V., Hueller, M., Vetrugno, D., Weber, W.J., Audley, H., Danzmann, K., Diepholz, I., Hewitson, M., Korsakova, N., Ferraioli, L., Gibert, F., Karnesis, N., Nofrarias, M., Inchauspe, H., Plagnol, E., Jennrich, O., McNamara, P.W., Armano, M., Thorpe, J.I., Wass, P.: Data series subtraction with unknown and unmodeled background noise. Phys. Rev. D 90, 042003 (2014)CrossRefGoogle Scholar
  60. Wahba, G.: Automatic smoothing of the log periodogram. J. Am. Stat. Assoc. 75, 122–132 (1980)CrossRefzbMATHGoogle Scholar
  61. Welch, P.D.: The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15, 70–73 (1967)CrossRefGoogle Scholar
  62. Whittle, P.: Curve and periodogram smoothing. J. R. Stat. Soc.: Ser. B (Methodol.) 19, 38–63 (1957)MathSciNetzbMATHGoogle Scholar
  63. Zheng, Y., Zhu, J., Roy, A.: Nonparametric Bayesian inference for the spectral density function of a random field. Biometrika 97, 238–245 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations