Covariance Structure • clustTMB

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) $\kappa$ , 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$.

$\omega_{i} \sim GMRF(Q[\kappa, \tau])$

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 
##  0.1778517  0.5710441  0.1653750  2.0157772  4.3160911  5.4259797  6.7095829 
##      betad      betad      betad      betad      betad      betad      betad 
##  1.0064936  3.6030483  5.2113125  6.2155353  0.1259828  3.1475076  4.2016919 
##      betad      betad      betad      betad      betad      theta      theta 
##  5.2523416 -1.4361461  3.1133028  4.2118636  5.1996631 -1.2100901 -2.9055603 
##      theta      theta      theta      theta      theta      theta      theta 
## -1.2794783 -1.2502294 -2.5624050 -3.1154962 -2.2459395 -2.3607764 -1.8075186 
##      theta      theta      theta      theta      theta      theta      theta 
## -4.0486716 -2.6845164 -3.0661892 -2.4648441 -3.3381097 -2.7804240 -2.6686100 
##  ln_kappag 
## -5.9132725

# Minimum negative log likelihood
mod$opt$objective

## [1] 2318.892

Gating Network Examples

When 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:

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