  Long period (LP) earthquakes are common at active volcanoes, and are
ubiquitous at persistently active andesitic and dacitic subduction zone
volcanoes. They provide critical information regarding the state of volcanic
unrest, and their occurrence rates are key data for eruption forecasting. LPs
are commonly quasi-periodic or 'anti-clustered', unlike volcano-tectonic (VT)
earthquakes, so the existing Poisson point process methods used to model
occurrence rates of VT earthquakes are unlikely to be optimal for LP data. We
evaluate the performance of candidate formulations for LP data, based on
inhomogeneous point process models with four different inter-event time
distributions: exponential (IP), Gamma (IG), inverse Gaussian (IIG), and
Weibull (IW). We examine how well these models explain the observed data, and
the quality of retrospective forecasts of eruption time. We use a Bayesian MCMC
approach to fit the models. Goodness-of-fit is assessed using Quantile-Quantile
and Kolmogorov-Smirnov methods, and benchmarking against results obtained from
synthetic datasets. IG and IIG models were both found to fit the data well,
with the IIG model slightly outperforming the IG model. Retrospective
forecasting analysis shows that the IG model performs best, with the initial
preference for the IIG model controlled by catalogue incompleteness late in the
sequence. The IG model fits the data significantly better than the IP model,
and simulations show it produces better forecasts for highly periodic data.
Simulations also show that forecast precision increases with the degree of
periodicity of the earthquake process using the IG model, and so should be
better for LP earthquakes than VTs. These results provide a new framework for
point process modelling of volcanic earthquake time series, and verification of
alternative models.
