How to interpret the spatial random field

[Kurt McLaren]

I'm modelling species spatio-temporal density in a montane forest. First, I identified the most important topographic parameter using a training and validation dataset.

Usually for my models, I have to specify the practical range and the marginal standard deviation to reduce the probability that the spatial effect explains the data better than the covariates as follows: 

prior.median.sd = 0.0001; prior.median.range = 2

spde = inla.spde2.pcmatern(mesh, prior.range = c(prior.median.range, .5), prior.sigma = c(prior.median.sd, .5))

I use the following formula after setting up a spatial and temporal mesh:

U <- 0.001

hyper.prec <- list(theta = list(prior="pc.prec", param=c(U, 0.05)))

formula <- Den ~ -1 + Intercept +  f(TPI, model = "rw1", hyper = hyper.prec)  + f(i, model=spde, group = i.group, control.group = list (model = "ar1"))

################################

where TPI is topographic position index. 

My result is below. As you can see, Rho is not significant (go to the bottom below the first output). I assume that this means that my spatial random effect changes over time? 

Also, because of the marginal standard deviation that I use, I usually do not get the spatial variance as being important (again at the bottom below the first output). I have included a pdf of my spatial random field. Given that the variance is not significant, how should I interpret the random field?  Do the areas of higher or lower spatial autocorrelation mean anything, and can this be construed as density changing at various locations resulting in some areas having much similar/different densities than others?

Additional information: I use a relatively small number of clumped plots. I use the center of each plot and a polygon of the forest area to build my mesh. 

Thanks,

Kurt

 Call:

   c("inla(formula = formula, family = family, contrasts = 

   inla.setOption(scale.model.default = TRUE), ", " data = inla.stack.data(stack.est, spde 

   = spde), verbose = TRUE, ", " control.compute = list(cpo = TRUE, dic = TRUE, config = 

   TRUE), ", " control.predictor = list(A = inla.stack.A(stack.est), compute = TRUE), ", " 

   keep = FALSE)") 

Time used:

    Pre = 8.42, Running = 125, Post = 2.52, Total = 136 

Fixed effects:

           mean    sd 0.025quant 0.5quant 0.975quant  mode kld

Intercept 0.513 0.006      0.501    0.513      0.526 0.513   0

Random effects:

  Name    Model

    TPI RW1 model

   i SPDE2 model

Model hyperparameters:

                                            mean      sd 0.025quant 0.5quant 0.975quant     mode

Precision for the Gaussian observations   93.209   9.330     76.206    92.75    112.880   91.841

Precision for TPI                       1696.909 315.876   1166.916  1665.80   2401.157 1604.169

Range for i                                4.292   8.345      0.407     2.06     21.978    0.865

Stdev for i                                0.000   0.000      0.000     0.00      0.000    0.000

GroupRho for i                             0.051   0.737     -0.997     0.15      0.989   -1.000

Expected number of effective parameters(stdev): 5.33(0.519)

Number of equivalent replicates : 52.58 

Deviance Information Criterion (DIC) ...............: -465.51

Deviance Information Criterion (DIC, saturated) ....: 288.30

Effective number of parameters .....................: 6.41

Marginal log-Likelihood:  71.87 

CPO and PIT are computed

Posterior marginals for the linear predictor and

 the fitted values are computed

*********************************************************************************

                               mean                   sd               0.025quant     0.5quant   0.975quant

obs.var             1.083548e-02 1.078099e-03  8.872058e-03 1.078004e-02 1.310143e-02

tpi.var                6.094432e-04 1.118437e-04  4.169475e-04 6.001132e-04 8.547283e-04

spde.var.nom     3.582072e-08 8.279055e-08 -7.685764e-10 6.868704e-09 2.237234e-07

spde.range.nom 4.202139e+00 6.985897e+00  3.832717e-01 2.028437e+00 2.162595e+01

AR.rho                  5.106370e-02 7.368315e-01 -9.968896e-01 1.502972e-01 9.886810e-01

[Kurt McLaren]

Sorry, I wrote this in haste, please forgive my manners. I would appreciate any help anyone can offer. Thank you in advance.