To get smoother transitions between regions with a dense network of station locations and regions with only a few observation stations, control parameters in the mesh functions can be set. Depending on the specific choice of these mesh parameters, the desired mesh config- uration can be created.
There are two main parameters influencing the mesh configuration. The cutoff value specifies the minimum distance between two points. Points that are closer to each other than the value chosen for cutoff are merged to a single location in the mesh construction. The refineparameter specifies whether the mesh construction should be refined by setting it either to TRUE or FALSE. Alternatively, the refine argument allows to specify constraints for the refinement, like the maximum length of the edges and the minimum interior angle of the triangles. By choosing different values for these two parameters different mesh types can be obtained: Meshes with very small or very large triangles, meshes with smooth transitions between the triangle sizes, or meshes with an abrupt change of the triangle sizes. A small empirical case study showed that the mesh configuration has a large influence on the predic- tive quality of the estimated MEMOS model. Figure 5.2 shows examples of four different mesh configurations obtained by choosing different values for cutoff and refine for the station locations in Germany. All four meshes in Figure 5.2 were obtained by specifying the argument loc in inla.mesh.create() as the 518 distinct observation locations of the
full data set containing the dates from February 2010 to April 2011. cutoff=0.8, refine=list(max.edge=0.4) Longitude Latitude 6 8 10 12 14 48 50 52 54 56 cutoff=0.6, refine=TRUE Longitude Latitude 6 8 10 12 14 16 48 50 52 54 56 cutoff=0.3, refine=list(max.edge=0.9) Longitude Latitude 6 8 10 12 14 16 48 50 52 54 56
cutoff=0, refine=list(max.edge=5, min.angle=0.1)
Longitude Latitude 6 8 10 12 14 16 48 50 52 54 56
Figure 5.2: Mesh configurations with different parameters
The mesh in the top left panel of Figure 5.2 has a high cutoff value and allows for short edges only, resulting in a high number of small triangles. The call to obtain this mesh is
inla.mesh.create(loc=locations, cutoff=0.8, refine=list(max.edge=0.4)).
The mesh in the top right panel has a high cutoff value as well, but no constraints on the edge length. Therefore the mesh has less and larger triangles. While the transition between the triangle sizes is very smooth in the top left mesh, it is less smooth in the top right mesh. Here the triangles covering the borders of Germany are larger than in the interior. However, the transition is still smoother than in the bottom right mesh. This mesh is obtained via the call
inla.mesh.create(loc=locations, cutoff=0.5, refine=TRUE).
The bottom left mesh has a small cutoff value, yet the constraint on the edge length is not as tough as in the top left mesh. This results in somewhat larger triangles than in the top left mesh, while the transition between the triangle sizes in the top left mesh is still quite smooth. This mesh is produced via the call
inla.mesh.create(loc=locations, cutoff=0.3, refine=list(max.edge=0.9)).
The bottom right mesh has a cutoff value of zero and very soft constraints on the edge length. This choice of parameters results in a crude triangulation with a very rough transition between large and smaller triangles. The call for this mesh is
inla.mesh.create(loc=locations, cutoff=0, refine=list(max.edge=5, min.angle=0.1)).
It should be noted, that the arguments in this call are the same as for the mesh in Figure 5.1. The only difference is that the mesh in Figure 5.1 is constructed from the locations present in the combined data set containing the 25 training days September 8, 2010 to October 2, 2010 as well as the data for the prediction day October 3, 2010, as already described in Section 5.6. In contrast, the meshes in Figure 5.2 are constructed from the observation locations present in the complete data set containing dates from February 2010 to April 2011.
From the empirical inspection of different mesh types, a rule of thumb for the application within the MEMOS model was derived.
A mesh where the triangles and vertices reflect the structure of the observation stations yields better predictive performance. Where the density of stations is higher, small triangles should be employed, and in regions with very low density only a few and large triangles should be placed. Using an overall smooth triangulation with small triangles everywhere, regardless of the real density of stations, yields a deterioration of the predictive performance. A further experience is that the model yields better predictive performance when the number of vertices resembles the number of observation stations. The predictive performance deteriorates when the number of vertices becomes much larger than the number of observations available, such as in the top left panel of Figure 5.2. A mesh reflecting the structure of the observation stations, such as the one in the bottom right panel of Figure 5.2, yields the best results concerning predictive quality. A possible explanation is that the vertices of the mesh are the discrete points where the SPDE solution is approximated. If there are more locations where an approximation is computed than observations, this might have a negative effect on the quality of results. Especially in the case of a too large number of vertices not having any observations in their vicinity, the accuracy of the approximated SPDE solution at those vertices might deteriorate.