Efficient Bayesian inference for natural time series using ARFIMA processes

Abstract

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.

The Structure of Climate Variability Across Scales

“…In general, ARFIMA models can also be driven by non-Gaussian (e.g., t-distributed) noise . ARFIMA models are more flexible than fractional Brownian motion since they combine a long-range dependence component with SRD behavior (Beran, 1994;Beran et al, 2013;Franzke et al, 2012;Graves et al, 2015). The R package ARFIMA can be used to estimate ARFIMA models (Veenstra, 2012).…”

“…The CET time series exhibits decadal-scale variations about an instantaneous mean (Franzke, 2009). The observed decadal-scale variability is a visible imprint of the scaling and long-range dependence (e.g., Gil-Alana, 2008;Graves et al, 2015). Intuitively, long-range dependence has the property that spatially coherent anomalies persist for a long time; for example, heat waves or droughts may last for many years (Cook et al, 2015), which is indicative of a decay of serial correlation which is slower than exponential, for example, power law decay.…”

Bayesian Inference for ARFIMA Models

“…The methods developed in this article, implemented using GPUs, make ARFIMA models practical for studying these data. Given the tools available, current approaches are forced to use approximations to the likelihood function in these applications (Hsu and Breidt, ; Graves et al , ). As illustrated in Geweke and Porter‐Hudak () the approximations typically used in these studies, including truncation of lag operators and finite Fourier transforms, can be treacherous precisely because of the long memory inherent in the model.…”

An approximate fractional Gaussian noise model with $$\mathcal {O}(n)$$ O ( n ) computational cost

“…The observed temperatures do have uncertainties (Parker and Horton 2005) and has been revised several times (Manley 1953(Manley , 1974Parker et al 1992). Especially, the measurements up to 1699 have a precision of 0.5 • C, while the precision is 0.1 • C thereafter (Graves et al 2015). We analyse temperatures up to December 2016, which gives a total of n = 4296 observations.…”

“…Still, we do prefer to use PC priors as these represent a principlebased choice of priors (Simpson et al 2017) which also facilitates interpretation of hyperprior parameters (Sørbye et al 2018). The higher-value of the long-memory parameter in this case might be explained by a lack of resolution for the first time period (Graves et al 2015).…”