, where and are the impact matrices of 4 × 4 dimensions. Based on the above multi-series local trend model, a multi-series damped local trend model consists of Equations (3) and (4) with a slope having a damping factor Φ: , where the damping factor is a 4 × 4 diagonal matrix. The absolute values of the diagonal Φ elements are constrained to be less than one to ensure a stationary trend (Athanasopoulos and de Silva 2012). Lastly, when the local trend is equal to zero, the transition equation (4) can be written as: , where is the impact matrix of 4 × 4 dimensions. Equations (3) and (7) are then referred to as the multi-series local level model. Model Estimation and Selection The unknown parameters in an M-STSM can be estimated with a maximum likelihood method. In this study, the R program (R Core Team 2016) is used. We first evaluate the log-likelihood using a routine from the package (Helske 2016). The approach combines the exact diffuse KFAS initialization method (Koopman and Durbin 2003) with a sequential processing approach (Koopman and Durbin 2000) to reduce the computational costs of high-dimensional multivariate modeling. The results along with the initial values of parameters are then fed into an optimizer to search for optimal parameters that maximize the log-likelihood. As a commonly observed issue
, where and are the impact matrices of 4 × 4 dimensions. Based on the above multi-series local trend model, a multi-series damped local trend model consists of Equations (3) and (4) with a slope having a damping factor Φ: , where the damping factor Φ is a 4 × 4 diagonal matrix. The absolute values of the diagonal elements are constrained to be less than one to ensure a stationary trend (Athanasopoulos and de Silva 2012). Lastly, when the local trend is equal to zero, the transition equation (4) can be written as: , where is the impact matrix of 4 × 4 dimensions. Equations (3) and (7) are then referred to as the multi-series local level model. Model Estimation and Selection The unknown parameters in an M-STSM can be estimated with a maximum likelihood method. In this study, the R program (R Core Team 2016) is used. We first evaluate the log-likelihood using a routine from the KFAS package (Helske 2016). The approach combines the exact diffuse initialization method (Koopman and Durbin 2003) with a sequential processing approach (Koopman and Durbin 2000) to reduce the computational costs of high-dimensional multivariate modeling. The results along with the initial values of parameters are then fed into an optimizer to search for optimal parameters that maximize the log-likelihood. As a commonly observed issue
Forecasting%20Seasonal%20Tourism%20Demand%20Using%20a%20Multi-Series%20Structural%20Time%20Series%20Method.docx