Spatial and Spatio-Temporal Models using 'INLA'.

INLAspacetime

This is a R package to implement certain spatial and spatio-temporal models, including some of the spatio-temporal models proposed here. It uses the cgeneric interface in the INLA package, to implement models by writing C code to build the precision matrix compiling it so that INLA can use it internally.

We have implemented

1. some of the models presented in A diffusion-based spatio-temporal extension of Gaussian Matérn fields (2024). Finn Lindgren, Haakon Bakka, David Bolin, Elias Krainski and Håvard Rue. SORT 48 (1) January-June 2024, 3-66. (https://www.idescat.cat/sort/sort481/48.1.1.Lindgren-etal.pdf)

2. the barrier (and transparent barriers) model proposed in https://doi.org/10.1016/j.spasta.2019.01.002

Vignettes

Please check here

Installation

The ‘INLA’ package is a suggested one, but you will need it for actually fitting a model. You can install it with

install.packages("INLA",repos=c(getOption("repos"),INLA="https://inla.r-inla-download.org/R/testing"), dep=TRUE) 

You can install the current CRAN version of INLAspacetime:

install.packages("INLAspacetime")

You can install the latest version of INLAspacetime from GitHub with

## install.packages("remotes")
remotes::install_github("eliaskrainski/INLAspacetime",  build_vignettes=TRUE)

A spacetime example

Simulate some fake data.

set.seed(1)
n <- 5
dataf <- data.frame(
    s1   = runif(n, -1, 1),
    s2   = runif(n, -1, 1),
    time = runif(n, 1, 4),
    y    = rnorm(n, 0, 1))
str(dataf)
#> 'data.frame':    5 obs. of  4 variables:
#>  $ s1  : num  -0.469 -0.256 0.146 0.816 -0.597
#>  $ s2  : num  0.797 0.889 0.322 0.258 -0.876
#>  $ time: num  1.62 1.53 3.06 2.15 3.31
#>  $ y   : num  -0.00577 2.40465 0.76359 -0.79901 -1.14766

Loading packages:

library(fmesher)
library(INLA)
library(INLAspacetime)
#> see more on https://eliaskrainski.github.io/INLAspacetime

Define spatial and temporal discretization meshes

smesh <- fm_mesh_2d(
  loc = cbind(0,0), 
  max.edge = 5, 
  offset = 2)
tmesh <- fm_mesh_1d(
  loc = 0:5)

Define the spacetime model object to be used

stmodel <- stModel.define(
    smesh = smesh, ## spatial mesh
    tmesh = tmesh, ## temporal mesh
    model = '121', ## model, see the paper
    control.priors = list(
        prs = c(1, 0.1), ## P(spatial range < 1) = 0.1
        prt = c(5, 0), ## temporal range fixed to 5
        psigma = c(1, 0.1) ## P(sigma > 1) = 0.1
        )
    )
#> Warning in stModel.define(smesh = smesh, tmesh = tmesh, model = "121",
#> control.priors = list(prs = c(1, : Setting 'useINLAprecomp = FALSE' to use new
#> code.

Fit the model

Define a projector matrix from the spatial and temporal meshes to the data

Aproj <- inla.spde.make.A(
    mesh = smesh,
    loc = cbind(dataf$s1, dataf$s2),
    group = dataf$time,
    group.mesh = tmesh
)

Create a ‘fake’ column to be used as index. in the f() term

dataf$st <- NA

Setting the likelihood precision (as fixed)

ctrl.lik <- list(
  hyper = list(
    prec = list(
      initial = 10, 
      fixed = TRUE)    
  )
)

Combine a ‘fake’ index column with A.local

fmodel <- y ~ f(st, model = stmodel, A.local = Aproj)

Call the main INLA function:

fit <- inla(
    formula = fmodel,
    data = dataf,
    control.family = ctrl.lik)

Posterior marginal summaries for fixed effect and the model parameters that were not fixed.

fit$summary.fixed
#>                 mean      sd 0.025quant  0.5quant 0.975quant      mode
#> (Intercept) 0.693389 4.03265  -6.962331 0.5227188   9.417425 0.5550712
#>                      kld
#> (Intercept) 7.398472e-05
fit$summary.hyperpar
#>                   mean        sd 0.025quant 0.5quant 0.975quant     mode
#> Theta1 for st 1.199222 0.4918533  0.3653818 1.161539   2.277396 0.974993
#> Theta2 for st 1.435517 0.1710676  1.1031120 1.434032   1.776667 1.427752

Using the inlabru

library(inlabru)

Setting the observation (likelihood) model object

data_model <- bru_obs(
  formula = y ~ ., 
  family = "gaussian",
  control.family = ctrl.lik, 
  data = dataf)

Define the data model: the linear predictor terms

linpred <- ~ 1 +
    field(list(space = cbind(s1, s2), 
               time = time),
          model = stmodel)

Fitting

result <- bru(
  components = linpred,
  data_model)

Summary of the model parameters

result$summary.fixed
#>                mean       sd 0.025quant 0.5quant 0.975quant      mode
#> Intercept 0.6690302 3.970182  -6.887199 0.509471   9.214066 0.5379221
#>                    kld
#> Intercept 5.683968e-05
result$summary.hyperpar
#>                      mean        sd 0.025quant 0.5quant 0.975quant      mode
#> Theta1 for field 1.190438 0.4868951  0.3623876 1.153809   2.256071 0.9726162
#> Theta2 for field 1.435268 0.1709839  1.1033563 1.433674   1.776580 1.4269195