AdaptSPEC-X: Covariate-Dependent Spectral Modeling of Multiple Nonstationary Time Series

Abstract

We present the AdaptSPEC-X method for the joint analysis of a panel of possibly nonstationary time series. The approach is Bayesian and uses a covariate-dependent infinite mixture model to incorporate multiple time series, with mixture components parameterized by a time-varying mean and log spectrum. The mixture components are based on AdaptSPEC, a nonparametric model which adaptively divides the time series into an unknown number of segments and estimates the local log spectra by smoothing splines. AdaptSPEC-X extends AdaptSPEC in three ways. First, through the infinite mixture, it applies to multiple time series linked by covariates. Second, it can handle missing values, a common feature of time series which can cause difficulties for nonparametric spectral methods. Third, it allows for a time-varying mean. Through these extensions, AdaptSPEC-X can estimate time-varying means and spectra at observed and unobserved covariate values, allowing for predictive inference. Estimation is performed by Markov chain Monte Carlo (MCMC) methods, combining data augmentation, reversible jump, and Riemann manifold Hamiltonian Monte Carlo techniques. We evaluate the methodology using simulated data, and describe applications to Australian rainfall data and measles incidence in the United States. Software implementing the method proposed in this article is available in the R package BayesSpec. Supplementary files for this article are available online.

Keywords:

• Locally stationary time series
• Measles
• Multiple time series
• Rainfall
• Reversible jump Markov chain Monte Carlo
• Whittle likelihood

Supplementary Material

The article is accompanied by a set of Appendices in the supplementary material. Appendix A provides details of the conditional distributions necessary for the sampling scheme, and Appendix B expands on the covariance structure used to derive the conditional distribution of the missing values.

Notes

1 Available from GitHub at https://github.com/mbertolacci/BayesSpec/. As of publication, the version of the package on CRAN does not contain AdaptSPEC-X.

2 Available from GitHub at https://github.com/mbertolacci/BayesSpec/. As of publication, the version of the package on CRAN does not contain AdaptSPEC-X.

3 Calculated as $10\times 365.25\times \text{difference}\mathrm{ }\text{in}\mathrm{ }\text{daily}\mathrm{ }\text{average}/(2004-1950)$

4 https://www.tycho.pitt.edu/

Additional information

Funding

E. Cripps and M. Bertolacci were supported by the ARC Industrial Transformation Research Hub for Offshore Floating Facilities which is funded by the Australian Research Council, Woodside Energy, Shell, Bureau Veritas and Lloyds Register (Grant No. 140100012). S. Cripps and E. Cripps were supported by the ARC Industrial Transformation Training Hub, Data Analytics for Resources and Environments (IC190100031), funded by the Australian Government. O. Rosen was supported in part by grants NSF DMS-1512188 and NIH 2R01GM113243-05.