Bayesian Analysis of Double Seasonal Autoregressive Models
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Bayesian Analysis of Double Seasonal Autoregressive Models Ayman A. Amin Menoufia University, Menoufia, Egypt Abstract In this paper we use the Gibbs sampling algorithm to present a Bayesian analysis to multiplicative double seasonal autoregressive (DSAR) models, considering both estimation and prediction problems. Assuming the model errors are normally distributed and using natural conjugate and g priors on the initial values and model parameters, we show that the conditional posterior distributions of the model parameters and variance are multivariate normal and inverse gamma respectively, and the conditional predictive distribution of the future observations is a multivariate normal. Using these closed-form conditional posterior and predictive distributions, we apply the Gibbs sampling to approximate empirically the marginal posterior and predictive distributions, enabling us easily to carry out multiple-step ahead predictions. The proposed Bayesian method is evaluated using simulation study and real-world time series dataset. AMS (2000) subject classification. Primary 37M10; Secondary 62F15. Keywords and phrases. Multiplicative seasonal autoregressive, Multiple seasonality, Posterior analysis, Predictive analysis, MCMC methods, Gibbs sampler, Internet traffic data
1 Introduction High frequency time series observed at small time units may be characterized by exhibiting multiple seasonal patterns. For example, hourly internet amount of traffic data can exhibit intraday and intraweek seasonal patterns. Other examples of high frequency time series containing multiple seasonal patterns include hourly access to computer web sites, hourly electricity load, hourly volumes of call center arrivals, and half-hourly demand for public transportation. The idea of modeling time series with multiple seasonalities can be traced back to 1971 when Thompson and Tiao (1971) showed that monthly disconnections of the telephone have double (annual and quarterly) seasonal patterns. Accordingly, seasonal autoregressive moving average (SARMA) models need to be modified and extended to fit and
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forecast time series with multiple seasonalities, and for more details about SARMA models see for example Box et al. (2016). The non-Bayesian analysis of double SARMA (DSARMA) models has been the subject of interest of many researchers, and these models are extensively studied and employed in modeling and forecasting double seasonal time series data, see for example Taylor et al. (2006), Taylor (2008a, b), Baek (2010), Cruz et al. (2011), Mohamed et al. (2011), Au et al. (2011), Cortez et al. (2012), and Kim (2013). Different approaches have been proposed in the literature for the Bayesian analysis of SARMA models to fit and forecast time series with single seasonality. Analytical approximation is one of these approaches, which simply approximates the posterior and predictive densities to be standard closedform distributions that are analytically tractable, see for example Broemeling and Shaarawy (1984, 1988), and Shaarawy and Ali (2003). Howev
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