Extending time series models for irregular observational gaps with a moving average structure for astronomical sequences

Abstract

In this study we introduce a novel moving-average model for analyzing stationary time series observed irregularly in time. The process is strictly stationary and ergodic under normality and weakly stationary when normality is not assumed. Maximum likelihood estimation can be efficiently carried out through a Kalman algorithm obtained from the state-space representation of the model. The Kalman algorithm has order O(n) (where n is the number of observations in the sequence), from which it is possible to efficiently generate parameter estimators, linear predictors and their mean squared errors. Two procedures were developed for assessing parameter estimation errors: one based on the Hessian of the likelihood function and another one based on the bootstrap method. The behavior of these estimators was assessed through Monte Carlo experiments. Both methods give accurate estimation performance, even with relatively small number of observations. Moreover, it is shown that for non-Gaussian data, specifically for the Student’s-t and Generalized error distributions, the parameters of the model can be estimated precisely by maximum likelihood. The proposed model is compared to the continuous autoregressive moving average models (CARMA), showing better performance when the moving average parameter is negative or close to one. We illustrate the implementation of the proposed model with light curves of variable stars from the OGLE and HIPPARCOS surveys and stochastic objects from ZTF. The results suggest that the iMA model is a suitable alternative for modeling astronomical light curves, particularly when they have negative autocorrelation.