Discrete-time autoregressive model for unequally spaced time-series observations

¹ Departmento de Matemáticas, Facultad de Ciencia, Universidad de Santiago de Chile, Av. Libertador Bernardo O’Higgins 3663. Estacion Central, Santiago, Chile
e-mail: susana@mat.puc.cl
² Departmento de Estadística, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 7 820436 Macul, Santiago, Chile
³ Millennium Institute of Astrophysics, Santiago, Chile
⁴ Max-Planck-Institut für Astronomie, Heidelberg, Germany
⁵ Faculty of Engineering and Sciences, Universidad Adolfo Ibañez, Diagonal Las Torres 2700, Peñalolén, Santiago, Chile

Received: 27 March 2019
Accepted: 14 June 2019

Abstract

Most time-series models assume that the data come from observations that are equally spaced in time. However, this assumption does not hold in many diverse scientific fields, such as astronomy, finance, and climatology, among others. There are some techniques that fit unequally spaced time series, such as the continuous-time autoregressive moving average (CARMA) processes. These models are defined as the solution of a stochastic differential equation. It is not uncommon in astronomical time series, that the time gaps between observations are large. Therefore, an alternative suitable approach to modeling astronomical time series with large gaps between observations should be based on the solution of a difference equation of a discrete process. In this work we propose a novel model to fit irregular time series called the complex irregular autoregressive (CIAR) model that is represented directly as a discrete-time process. We show that the model is weakly stationary and that it can be represented as a state-space system, allowing efficient maximum likelihood estimation based on the Kalman recursions. Furthermore, we show via Monte Carlo simulations that the finite sample performance of the parameter estimation is accurate. The proposed methodology is applied to light curves from periodic variable stars, illustrating how the model can be implemented to detect poor adjustment of the harmonic model. This can occur when the period has not been accurately estimated or when the variable stars are multiperiodic. Last, we show how the CIAR model, through its state space representation, allows unobserved measurements to be forecast.