Articles | Volume 22, issue 6
Nonlin. Processes Geophys., 22, 679–700, 2015
Nonlin. Processes Geophys., 22, 679–700, 2015

Research article 18 Nov 2015

Research article | 18 Nov 2015

Efficient Bayesian inference for natural time series using ARFIMA processes

T. Graves1, R. B. Gramacy2, C. L. E. Franzke3, and N. W. Watkins4,5,6,7 T. Graves et al.
  • 1URS Corporation, London, UK
  • 2The University of Chicago, Booth School of Business, Chicago, IL, USA
  • 3Meteorological Institute and Center for Earth System Research and Sustainability (CEN), University of Hamburg, Hamburg, Germany
  • 4Centre for the Analysis of Time Series, London School of Economics and Political Science, London, UK
  • 5Centre for Fusion Space and Astrophysics, University of Warwick, Coventry, UK
  • 6Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
  • 7Faculty of Mathematics, Computing and Technology, Open University, Milton Keynes, UK

Abstract. Many geophysical quantities, such as atmospheric temperature, water levels in rivers, and wind speeds, have shown evidence of long memory (LM). LM implies that these quantities experience non-trivial temporal memory, which potentially not only enhances their predictability, but also hampers the detection of externally forced trends. Thus, it is important to reliably identify whether or not a system exhibits LM. In this paper we present a modern and systematic approach to the inference of LM. We use the flexible autoregressive fractional integrated moving average (ARFIMA) model, which is widely used in time series analysis, and of increasing interest in climate science. Unlike most previous work on the inference of LM, which is frequentist in nature, we provide a systematic treatment of Bayesian inference. In particular, we provide a new approximate likelihood for efficient parameter inference, and show how nuisance parameters (e.g., short-memory effects) can be integrated over in order to focus on long-memory parameters and hypothesis testing more directly. We illustrate our new methodology on the Nile water level data and the central England temperature (CET) time series, with favorable comparison to the standard estimators. For CET we also extend our method to seasonal long memory.