Bayesian Inference of Triple Seasonal Autoregressive Models

Recently, autoregressive (AR) time-series models have been extended to model time-series with double seasonality. However, in some real applications, high frequency time-series can exhibit triple seasonal patterns. Therefore, in this paper we aim to extend the AR models to fit time-series with three seasonality layers, and accordingly we introduce the Bayesian inference for triple seasonal autoregressive (TSAR) models. In this Bayesian inference, we first assume the normal distribution for the TSAR model errors and employ different priors on the TSAR model parameters, including normal-gamma, g and Jeffreys' priors. Based on the normally distributed errors and employed model parameters' priors, we derive the marginal posterior distributions of different TSAR model parameters in closed forms. Particularly, we show that the marginal posterior of the TSAR model coefficients vector to be a multivariate t distribution and the marginal posterior of the TSAR model precision to be a gamma distribution. We conduct an extensive simulation study aiming to evaluate the efficiency of our proposed Bayesian inference, and also we apply our work to real hourly time-series on electricity load in some European countries.