This vignette details the different covariance structures available in clustTMB.
| Covariance | Notation | No..of.Parameters | Data.requirements |
|---|---|---|---|
| Spatial GMRF | gmrf | 2 | spatial coordinates |
| AR(1) | ar1 | 2 | unit spaced levels |
| Rank Reduction | rr(random = H) | JH - (H(H-1))/2 | |
| Spatial Rank Reduction | rr(spatial = H) | 1 + JH - (H(H-1))/2 | spatial coordinates |
clustTMB fits spatial random effects using a Gaussian Markov Random Field (GMRF). The precision matrix, \(Q\), of the GMRF is the inverse of a Matern covariance function and takes two parameters: 1) \(\kappa\), which is the spatial decay parameter and a scaled function of the spatial range, \(\phi = \sqrt{8}/\kappa\), the distance at which two locations are considered independent; and 2) \(\tau\), which is a function of \(\kappa\) and the marginal spatial variance \(\sigma^{2}\):
\[\tau = \frac{1}{2\sqrt{\pi}\kappa\sigma}.\] The precision matrix is approximated following the SPDE-FEM approach [@Lindgren2011], where a constrained Delaunay triangulation network is used to discretize the spatial extent in order to determine a GMRF for a set of irregularly spaced locations, i$.
\[\omega_{i} \sim GMRF(Q[\kappa, \tau])\]
Prior to fitting a spatial cluster model with clustTMB, users need to set up the constrained Delaunay Triangulation network using the R package, fmesher. This package provides a CRAN distributed collection of mesh functions developed for the package, R-INLA. For guidance on setting up an appropriate mesh, see Triangulation details and examples and Tools for mesh assessment from
In this example, the following mesh specifications were used:
loc <- meuse[, 1:2]
Bnd <- fmesher::fm_nonconvex_hull(as.matrix(loc), convex = 200)
meuse.mesh <- fmesher::fm_mesh_2d(as.matrix(loc),
max.edge = c(300, 1000),
boundary = Bnd
)## Loading required namespace: INLACoordinates are converted to a spatial point dataframe and read into the clustTMB model, along with the mesh, using the spatial.list argument. The gating formula is specified using the gmrf() command:
Loc <- sf::st_as_sf(loc, coords = c("x", "y"))
mod <- clustTMB(
response = meuse[, 3:6],
family = lognormal(link = "identity"),
gatingformula = ~ gmrf(0 + 1 | loc),
G = 4, covariance.structure = "VVV",
spatial.list = list(loc = Loc, mesh = meuse.mesh)
)## intercept removed from gatingformula
## when random effects specified## spatial projection is turned off. Need to provide locations in projection.list$grid.df for spatial predictionsModels are optimized with nlminb(), model results can be viewed with nlminb commands:
# Estimated fixed parameters
mod$opt$par## betag betag betag betad betad betad betad
## 0.1810561 0.5594793 0.1898442 2.0157745 4.3160880 5.4259819 6.7095831
## betad betad betad betad betad betad betad
## 1.0164082 3.6119034 5.2215817 6.2274867 0.1353846 3.1482198 4.2137115
## betad betad betad betad betad theta theta
## 5.2614025 -1.4361504 3.1132966 4.2118588 5.1996568 -1.2100810 -2.9055386
## theta theta theta theta theta theta theta
## -1.2794746 -1.2502187 -2.5718215 -3.1310896 -2.2406099 -2.3780380 -1.8212760
## theta theta theta theta theta theta theta
## -4.0603269 -2.6424666 -3.0432260 -2.4648411 -3.3381004 -2.7804404 -2.6686130
## ln_kappag
## -5.9346215 # Minimum negative log likelihood
mod$opt$objective## [1] 2318.922When random effects, \(\mathbb{u}\), are specified in the gating network, the probability of cluster membership \(\pi_{i,g}\) for observation \(i\) is fit using multinomial regression:
\[ \begin{align} \mathbb{\eta}_{,g} &= X\mathbb{\beta}_{,g} + \mathbb{u}_{,g} \\ \mathbb{\pi}_{,g} &= \frac{ exp(\mathbb{\eta}_{,g})}{\sum^{G}_{g=1}exp(\mathbb{\eta}_{,g})} \end{align} \]