A Bayesian method for the analysis of deterministic and stochastic time series

Abstract

I introduce a general, Bayesian method for modelling univariate time series data assumed to be drawn from a continuous, stochastic process. The method accommodates arbitrary temporal sampling, and takes into account measurement uncertainties for arbitrary error models (not just Gaussian) on both the time and signal variables. Any model for the deterministic component of the variation of the signal with time is supported, as is any model of the stochastic component on the signal and time variables. Models illustrated here are constant and sinusoidal models for the signal mean combined with a Gaussian stochastic component, as well as a purely stochastic model, the Ornstein-Uhlenbeck process. The posterior probability distribution over model parameters is determined via Monte Carlo sampling. Models are compared using the "cross-validation likelihood", in which the posterior-averaged likelihood for different partitions of the data are combined. In principle this is more robust to changes in the prior than is the evidence (the prior-averaged likelihood). The method is demonstrated by applying it to the light curves of 11 ultra cool dwarf stars, claimed by a previous study to show statistically significant variability. This is reassessed here by calculating the cross-validation likelihood for various time series models, including a null hypothesis of no variability beyond the error bars. 10 of 11 light curves are confirmed as being significantly variable, and one of these seems to be periodic, with two plausible periods identified. Another object is best described by the Ornstein-Uhlenbeck process, a conclusion which is obviously limited to the set of models actually tested.