Table 6.

Deviance information criterion (DIC), Watanabe–Akaike information criterion (WAIC) and marginal predictive likelihood (MPL) of a subset of models of the FGG-GMRF model in Eq. (33)^a

Model specificationb	Likelihood	Time-varying effect	Edge length (m)	DIC	WAIC	MPL
M1.1.1.0	Poisson	RW1	–	5184.089	5623.433	− 2850.213
M1.2.1.0	NB	RW1	–	3281.386	3283.865	− 1778.259
M1.1.2.0	Poisson	RW2	–	5229.130	5616.038	− 2893.252
M1.2.2.0	NB	RW2	–	3380.390	3381.260	− 1941.200
M2.1.1.1	Poisson	RW1	1000	2657.428	2640.887	− 1582.563
M2.2.1.1	NB	RW1	1000	2676.950	2671.954	− 1583.277
M2.1.1.2	Poisson	RW1	1100	2663.998	2660.371	− 1579.446
M2.2.1.2	NB	RW1	1100	2676.937	2671.849	− 1582.548
M2.1.1.3	Poisson	RW1	1200	2663.953	2660.405	− 1580.599
M2.2.1.3	NB	RW1	1200	2680.719	2676.344	− 1582.903
M2.1.1.4	Poisson	RW1	1300	2658.124	2642.776	− 1583.341
M2.2.1.4	NB	RW1	1300	2681.375	2675.452	− 1582.484
M2.1.1.5	Poisson	RW1	1400	2663.895	2661.152	− 1580.897
M2.2.1.5	NB	RW1	1400	2680.761	2676.639	− 1582.581
M2.1.1.6	Poisson	RW1	1500	2664.065	2661.030	− 1580.177
M2.2.1.6	NB	RW1	1500	2680.761	2676.639	− 1582.581
M2.1.2.1	Poisson	RW2	1000	2658.819	2644.598	− 1597.562
M2.2.2.1	NB	RW2	1000	2669.423	2653.978	− 1599.671
M2.1.2.2	Poisson	RW2	1100	2658.947	2642.814	− 1599.163
M2.2.2.2	NB	RW2	1100	2658.947	2642.814	− 1599.163
M2.1.2.3	Poisson	RW2	1200	2665.287	2663.565	− 1594.933
M2.2.2.3	NB	RW2	1200	2681.804	2677.917	− 1597.695
M2.1.2.4	Poisson	RW2	1300	2659.083	2643.993	− 1598.382
M2.2.2.4	NB	RW2	1300	2683.923	2679.331	− 1596.700
M2.1.2.5	Poisson	RW2	1400	2665.050	2662.933	− 1595.102
M2.2.2.5	NB	RW2	1400	2683.923	2679.331	− 1596.700
M2.1.2.6	Poisson	RW2	1500	2664.968	2662.321	− 1595.516
M2.2.2.6	NB	RW2	1500	2683.821	2679.441	− 1596.753
M3.1.1.1	Poisson	RW1	1000	2660.567	2644.342	− 1586.289
M3.2.1.1	NB	RW1	1000	2667.208	2659.532	− 1588.466
M3.1.1.2	Poisson	RW1	1100	2658.277	2641.471	− 1585.928
M3.2.1.2	NB	RW1	1100	2667.213	2659.537	− 1588.420
M3.1.1.3	Poisson	RW1	1200	2658.277	2641.472	− 1585.928
M3.2.1.3	NB	RW1	1200	2667.213	2659.537	− 1588.419
M3.1.1.4	Poisson	RW1	1300	2660.667	2644.078	− 1586.228
M3.2.1.4	NB	RW1	1300	2667.213	2659.536	− 1588.410
M3.1.1.5	Poisson	RW1	1400	2660.510	2644.422	− 1586.429
M3.2.1.5	NB	RW1	1400	2667.191	2659.512	− 1588.405
M3.1.1.6	Poisson	RW1	1500	2660.598	2644.207	− 1586.461
M3.2.1.6	NB	RW1	1500	2667.191	2659.512	− 1588.400
M3.1.2.1	Poisson	RW2	1000	2661.441	2647.045	− 1600.555
M3.2.2.1	NB	RW2	1000	2668.788	2663.003	− 1602.876
M3.1.2.2	Poisson	RW2	1100	2659.836	2645.165	− 1599.883
M3.2.2.2	NB	RW2	1100	2659.836	2645.165	− 1599.883
M3.1.2.3	Poisson	RW2	1200	2659.544	2645.097	− 1599.448
M3.2.2.3	NB	RW2	1200	2668.788	2663.003	− 1602.876
M3.1.2.4	Poisson	RW2	1300	2661.572	2646.658	− 1599.695
M3.2.2.4	NB	RW2	1300	2668.788	2663.003	− 1602.871
M3.1.2.5	Poisson	RW2	1400	2661.619	2647.490	− 1600.489
M3.2.2.5	NB	RW2	1400	2668.803	2663.020	− 1602.851
M3.1.2.6	Poisson	RW2	1500	2661.708	2647.171	− 1600.532
M3.2.2.6	NB	RW2	1500	2668.803	2663.020	− 1602.847

The second number refers to the likelihood: 1: Poisson and 2: Negative Binomial, the third to the time varying effect of the coefficients, 1: random walk of order 1 (RW1) and 2: random walk of order 2 (RW2), the fourth digit defines the edge length, 0: edge length not relevant, 1: 1000 m, 2: 1,100 m, 3: 1,200 m, 4: 1,300 m, 5: 1,400 m, and 6: 1,500 m. Note that there is no edge length for the M1 models because they do not need meshing. M1 contains 4 sub-models, and M2 and M3 contain 24 sub-models each

^aThe FGG-GMRF models are specified as $\mathbb{E}\left[{\mathbf{y}}_{\mathit{t}},|,{\mathbf{E}}_{\mathit{t}},{\mathbf{\theta }}_{\mathbf{t}}\right]$

^bThe following notation applies. M denotes Model. M1 is the sub-model with covariates only, i.e., ${\mathit{\eta }}_t={\mathrm{\beta }}_0{1}_{(n_{\mathcal{A}}+n_p)}+{\sum }_{k=1}^4\left({\beta }_k,+,{\zeta }_{k,t}\right){\mathbf{z}}_{k,t}$, M2 the model with covariates plus area and cell level interaction, i.e., ${\mathit{\eta }}_t={\mathrm{\beta }}_0{1}_{(n_{\mathcal{A}}+n_p)}+{\sum }_{k=1}^K\left({\beta }_k,+,{\zeta }_{k,t}\right){\mathbf{z}}_{k,t}+{\mathbf{\delta }}_t+{\mathbf{\Phi }}_t$(Area interaction type IV with precision matrix ${\mathbf{Q}}_{\delta }={\tau }_{\delta }{\mathbf{R}}_{\omega }\otimes {\mathbf{R}}_{\varphi }),$. M3 the full model, i.e., ${\mathit{\eta }}_t={\mathrm{\beta }}_0{1}_{(n_{\mathcal{A}}+n_p)}+{\sum }_{k=1}^K\left({\beta }_k,+,{\zeta }_{k,t}\right){\mathbf{z}}_{k,t}+\ddot{\mathbf{\omega }}+\ddot{\mathit{\upsilon }}+{\varphi }_t{1}_{(n_{\mathcal{A}}+n_p)}+{\varsigma }_t{1}_{(n_{\mathcal{A}}+n_p)}+{\mathbf{\delta }}_t+{\mathbf{\Phi }}_t$