Fractional Gaussian noise: Prior specification and model comparison

TL;DR

This paper proposes a penalised complexity prior for the Hurst exponent in fractional Gaussian noise models, enabling better model comparison and inference in time series analysis, especially in climate regression contexts.

Contribution

It introduces a PC prior for the Hurst exponent, allowing consistent prior use across fGn and AR(1) models, facilitating Bayesian model comparison.

Findings

The PC prior penalizes divergence from white noise.

The same prior applies to fGn and AR(1) models.

Enables Bayesian comparison of noise models in practice.

Abstract

Fractional Gaussian noise (fGn) is a self-similar stochastic process used to model anti-persistent or persistent dependency structures in observed time series. Properties of the autocovariance function of fGn are characterised by the Hurst exponent (H), which in Bayesian contexts typically has been assigned a uniform prior on the unit interval. This paper argues why a uniform prior is unreasonable and introduces the use of a penalised complexity (PC) prior for H. The PC prior is computed to penalise divergence from the special case of white noise, and is invariant to reparameterisations. An immediate advantage is that the exact same prior can be used for the autocorrelation coefficient of a first-order autoregressive process AR(1), as this model also reflects a flexible version of white noise. Within the general setting of latent Gaussian models, this allows us to compare an fGn model component with AR(1) using Bayes factors, avoiding confounding effects of prior choices for the hyperparameters. Among others, this is useful in climate regression models where inference for underlying linear or smooth trends depends heavily on the assumed noise model.