Formulation of the covariance models:
Gaussian Markov Random Field (MRF) and conditional autoregressive models are essentially the same thing, apart from details of specification.
adjacency and AR1 random effects can be seen as specific MRFs.
The common idea is the Markov-like property that the distribution of each element \(b_i\) of the random-effect b, given values of a few specific elements (the “neighbours” of \(i\)), is independent of other elements (i.e., of non-neighbours). The non-zero non-diagonal elements of a precision matrix characterize the neighbours.
Given the inferred vector b of values of the MRF on the lattice, the interpolation of the MRF in any focal point is of the form Ab where each row of A weights the values of b according to the position of the focal point relative to the vertices of the lattice. Following the original publications,
* for grids given by model=<inla.spde2 object>, the non-zero weights are the barycentric coordinates of the focal point in the enclosing triangle from the mesh triangulation (points from outside the mesh would have zero weights, so the predicted effect Ab=0);
* for regular grids (NULL model), the weights are computed as <Wendland function>(<scaled Euclidean distances between focal point and vertices>).
The IMRF model defines both a lattice in space, the precision matrix for a Gaussian MRF over this lattice, and the A matrix of weights. The full specification of the MRF on irregular lattices is complex. The \(\kappa\) (kappa) parameter considered by spaMM is the \(\kappa\) scale parameter considered by Lindgren et al and comparable to the \(\rho\) scale factor of the Matérn model. The \(\alpha\) argument of the INLA::inla.spde2.matern controls the smoothness of the approximated Matern model, as \(\alpha=\nu + d/2\)) where \(d\) is the dimension of the space.
Correlation models created by INLA::inla.spde2.pcmatern are handled so as to give the same correlation values as when INLA::inla.spde2.matern is used with the same mesh and alpha argument (thus, the extra functionalities of “pc”matern are ignored).
Not all options of the INLA functions may be compatible or meaningful when used with spaMM (only the effects of alpha and cutoff have been checked).
Normalization:
For the MRFs on default regular grids (missing model argument), the precision matrix is defined (up to a variance parameter) as M'M where the diagonal elements \(m_{ii}\) of M are 4+\(\kappa^2\) and the \(m_{ij}\) for the four nearest neighbours are -1 (note that M'M involves more than these four neighbours). The precision matrix defined in this way is the inverse of an heteroscedastic covariance matrix C, but (following Nychka et al.) by default a normalization is applied so that the random effect in each data position is homoscedastic (the precision matrix for the latent effect in grid positions is not modified, but it is the A matrix of weights which is is modified). As for other random effects, the variance is further controlled by a multiplicative factor \(\lambda\).
Without normalization, the covariance matrix of the random effect in data locations is \(\lambda\)ALL'A' (A being the above-described weight matrix, and L is a “square root” of C), and AL is the original “design matrix” of the random effect. \(\lambda\) may then be quite different from the marginal variance of the random effect, and is difficult to describe in a simple way.
For normalization, A is modified as WA where W is a diagonal matrix such that WAL is a correlation matrix (WALL'A'W' has unit diagonal); then, \(\lambda\) is the marginal variance of the random effect.
For irregular grids specified using the model argument, the precision matrix described by this object is also the inverse of an heteroscedastic covariance matrix, but here (again following original publicatiosn such as Lindgren at al. 2011) the normalization is not applied by default (and was not even an option before version 4.3.23). But for ease of presentation and interpretation, if for no other reason, the normalized model may be preferable.
Details for rectangular grids:
By default (meaning in particular that model is not used to specify a lattice defined by the INLA procedures), the IMRF lattice is rectangular (currently the only option) and is made of a core lattice, to which margins of margin steps are added on each side. The core lattice is defined as follows: in each of the two spatial dimensions, the range of axial coordinates is determined. The largest range is divided in nd-1 steps, determining nd values and step length \(L\). The other range is divided in steps of the same length \(L\). If it extends over (say) \(2.5 L\), a grid of 2 steps and 3 values is defined, and by default centered on the range (the extreme points therefore typically extend slightly beyond the grid, within the first of the additional steps defined by the margin; if not centered, the grid start from the lower coordinate of the range).
multIMRF implements multilevel IMRFs. It defines a sequence of IMRFs, with progressively finer lattices, a common \(\kappa\) value hy_kap for these IMRFs, and a single variance parameter hy_lam that determines \(\lambda\) values decreasing by a factor of 4 for successive IMRF terms. By default, each component IMRF is normalized independently as described above (as in Nychka et al. 2019), and hy_lam is the sum of the variances of these terms (e.g., if there are three levels and hy_lam=1, the successive variances are (1,1/4,1/16)/(21/16) ). The nd of the first IMRF is set to the coarse value, and its lattice is defined accordingly. If coarse=4 and margin=5, a grid of 14 coordinates is therefore defined over the largest range. In the second IMRF, the grid spacing is halved, so that new steps are defined halfway between the previous ones (yielding a grid of 27 step in the widest range). The third IMRF proceeds from the second in the same way, and so on.
To control initial or fixed values of multIMRF \(\kappa\) and variance parameters, which are hyper-parameter controlling several IMRF terms, the hyper syntax shown in the Examples should be used. hyper is a nested list whose possible elements are named "1", "2", ... referring to successive multIMRF terms in the input formula, not to successive random effect in the expanded formula with distinct IMRF terms (see Examples). But the different IMRF terms should be counted as distinct random effects when controlling other parameters (e.g., for fixing the variances of other random effects).