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

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.

💡 Research Summary

This paper presents a comprehensive and general Bayesian framework for modeling univariate time series data assumed to be drawn from a continuous stochastic process. The method is notable for its flexibility: it accommodates arbitrarily sampled data, incorporates measurement uncertainties for both the time and signal variables under arbitrary (non-Gaussian) error models, and supports any deterministic model for the signal’s mean variation combined with any stochastic model for the intrinsic variability.

The core of the method lies in decomposing the time series model into probabilistic components. The likelihood for an observed data point, consisting of a measured time and signal with associated uncertainties, is computed by marginalizing over the unknown true time and true signal. This involves a double integral over the product of a measurement noise model and the generative time series model. The overall likelihood for a dataset is the product of the likelihoods for individual, independently measured events. For computational efficiency, the paper provides approximations (detailed in an appendix) to simplify this integral under common conditions, such as when time measurement errors are negligible.

A key contribution is the structured formulation of the time series model itself. It is expressed as the product of a signal component and a time component. The signal component can often be further split into a deterministic function defining the time-dependent mean (e.g., a constant or sinusoid) and a stochastic model describing the distribution of the true signal around that mean at any given time (e.g., a Gaussian). The time component models the intrinsic probability distribution of event occurrence times, which for light curves is typically a uniform distribution. This structure cleanly separates measurement noise from the intrinsic stochasticity of the physical process being studied. The paper also incorporates fully stochastic models like the Ornstein-Uhlenbeck process, where such a separation is not possible.

For model comparison and selection, the author advocates for the use of the “cross-validation likelihood” over the more traditional Bayesian evidence. While the evidence averages the likelihood over the prior parameter distribution, the cross-validation likelihood averages it over the posterior distribution from different partitions of the data. This is argued to be more robust to changes in the often subjective prior distributions, a crucial consideration in exploratory data analysis. Both quantities are estimated using Monte Carlo methods—sampling from the prior for the evidence and from the posterior for the cross-validation likelihood.

The method is demonstrated through application. First, it is validated on simulated data. Then, it is applied to a re-analysis of the light curves of 11 ultra-cool dwarf stars. The goal is to assess claims of variability and periodicity from a previous study. The Bayesian framework allows for a direct comparison of a null hypothesis (a constant signal with only measurement noise) against various variable models, including a constant mean with intrinsic Gaussian scatter, a sinusoidal model with scatter, and a pure Ornstein-Uhlenbeck process. The results confirm significant variability in 10 out of 11 objects. For one object, two plausible periodicities are identified. Interestingly, another object is best described not by a deterministic periodic model but by the Ornstein-Uhlenbeck process, suggesting its brightness fluctuations are characterized by a specific type of stochastic memory rather than a clear periodic signal. The paper concludes by emphasizing the power and generality of the Bayesian approach for time series analysis, despite its computational demands, positioning it as a rigorous alternative to more restrictive frequentist or ad-hoc methods.