Table 2. Summary of the Bayesian hierarchical model.

Stages	Description
First stage	– The observational model, $y|x\sim \pi (y|x)$
	– y is the observations
Second stage	– Components in Equation (9), model as Gaussian Markov random fields (GMRF) with precision matrix Q, i.e. inverse covariance matrix, $x|\theta \sim \pi (x|\theta )$
	– $x=(\alpha ,{\xi }_i,{\phi }_t,{\gamma }_t,{\delta }_{it}{)}^{\prime }$
Third stage	– Hyperparameters, $\theta \sim \pi (\theta )$
	– $\theta =({\sigma }_s^2,{\lambda }_s,{\sigma }_{\phi }^2,{\sigma }_{\gamma }^2,\rho ,{\sigma }_{\delta }^2{)}^{\prime }$