Covariance Structure

This vignette details the different covariance structures available in clustTMB.

Random Effect Covariance Matrices

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

Spatial GMRF

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) κ, 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) τ, which is a function of κ and the marginal spatial variance σ2:

$$\tau = \frac{1}{2\sqrt{\pi}\kappa\sigma}.$$ 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$.

ωi ∼ GMRF(Q[κ, τ])

Spatial Example

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: INLA

Coordinates 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 predictions

Models 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 
##  0.1780772  0.5708861  0.1653883  2.0157791  4.3160894  5.4259846  6.7095834 
##      betad      betad      betad      betad      betad      betad      betad 
##  1.0064908  3.6030478  5.2113126  6.2155341  0.1259825  3.1475086  4.2016944 
##      betad      betad      betad      betad      betad      theta      theta 
##  5.2523449 -1.4361475  3.1133006  4.2118630  5.1996620 -1.2100914 -2.9055241 
##      theta      theta      theta      theta      theta      theta      theta 
## -1.2794655 -1.2502142 -2.5623894 -3.1154839 -2.2459339 -2.3607637 -1.8075093 
##      theta      theta      theta      theta      theta      theta      theta 
## -4.0486922 -2.6845125 -3.0661888 -2.4648513 -3.3381053 -2.7804485 -2.6686140 
##  ln_kappag 
## -5.9132466
# Minimum negative log likelihood
mod$opt$objective
## [1] 2318.889

Gating Network Examples

When random effects, 𝕦, are specified in the gating network, the probability of cluster membership πi, g for observation i is fit using multinomial regression:

$$ \begin{align} \mathbb{\eta}_{,g} &= X\mathbb{\beta}_{,g} + \mathbb{u}_{,g} \\ \mathbb{\pi}_{,g} &= \frac{ exp(\mathbb{\eta}_{,g})}{\sum^{G}_{g=1}exp(\mathbb{\eta}_{,g})} \end{align} $$