1 About

Warning: This topic is no longer maintained, please see Barrier Model applied to the Archipelago Data instead.

This topic covers how to visualize spatial models near a coast. The method for visualizing spatial models can be used for other models. The knowledge we gain on the spatial models we are looking at is useful when using these models.

1.1 Initialisation and dependency

We load the libraries and functions we need. You may need to install these libraries (Installation and general troubleshooting). Feel free to save the web location where the data is as an R-file on your computer. We also set random seeds to be used later.

library(INLA); library(fields)
library(rgeos)
library(viridisLite)

set.seed(2016)
set.inla.seed = 2016

1.2 Data

1.2.1 Download data

An internet connection is required, unless you have already downloaded the repository.

dir.create("data/")
download.file(url = "https://haakonbakkagit.github.io/data/WebSiteData-Archipelago.RData", destfile = "data/WebSiteData-Archipelago.RData")

1.2.2 Load data

## Load data
load(file = "data/WebSiteData-Archipelago.RData")
# - if you have saved the file locally

## What is loaded
# - poly.water is our study area
# - df is our dataframe to be analysed
# - dat is the orginial dataframe
str(poly.water, 1)
## Formal class 'SpatialPolygons' [package "sp"] with 4 slots

1.2.3 Data citations

For a description of the data see (Kallasvuo, Vanhatalo, and Veneranta 2016). Data collection was funded by VELMU and Natural Resources Institute Finland (Luke).

2 Generic model setup

We choose one of the four fish species to model, the smelt

df$y = df$y.smelt

To keep track of our models, we define a list containing all the models.

M = list()
# - the list of all models
M[[1]] = list()
# - the first model
M[[1]]$shortname = "stationary-no-cov"
# - a short description, the name of the model
M[[2]] = list()
M[[2]]$shortname = "stationary-all-cov"
M[[3]] = list()
M[[3]]$shortname = "barrier-no-cov"
M[[4]] = list()
M[[4]]$shortname = "barrier-all-cov"

2.1 The mesh: Dependency BTopic104

The following code is from the dependency. The mesh we are using was called mesh4 in BTopic104. You probably want to read this dependency topic.

We had to adjust max.edge. When running with max.edge=0.95 we got ranges near 3. As we mentioned in BTopic104, the max.edge must be at most ca range/5.

max.edge = 0.6
bound.outer = 4.6
mesh = inla.mesh.2d(boundary = poly.water,
                     loc=cbind(df$locx, df$locy),
                    max.edge = c(1,5)*max.edge,
                    cutoff = 0.06,
                    offset = c(max.edge, bound.outer))
plot(mesh, main="Our mesh", lwd=0.5); points(df$locx, df$locy, col="red")

mesh$n
## [1] 6539

2.2 The stack

To connect the observations \(y_i\) (at locations \(s_i\)) to the mesh nodes s, we have the so-called A-matrix.

A.i.s = inla.spde.make.A(mesh, loc=cbind(df$locx, df$locy))

Next we set up the stack, needed for all spatial models in INLA.

 stk <- inla.stack(data=list(y=df$y, e=df$exposure), 
                    effects=list(s=1:mesh$n,
                                 data.frame(m=1, df[ ,5:11]), 
                                 # - m is the intercept
                                 iidx=1:nrow(df)),
                   A=list(A.i.s, 1, 1),
                    remove.unused = FALSE, tag='est')

3 Stationary models

To set up the stationary spatial model, we first define the spatial Model Component.

spde = inla.spde2.pcmatern(mesh, prior.range = c(6, .5), prior.sigma = c(3, 0.01))
# - We put the prior median at approximately 0.5*diff(range(df$locy))
# - - this is roughly the extent of our study area
# - The prior probability of marginal standard deviation 3 or more is 0.01.

Then we define the two formulas.

hyper.iid = list(prec = list(prior = 'pc.prec', param = c(3, 0.01))) 
# - the param the same as prior.sigma above, with the same interpretation
M[[1]]$formula = y~ -1+m + f(s, model=spde) + f(iidx, model="iid", hyper=hyper.iid)
# - no covariates (except intercept m)
M[[2]]$formula = as.formula(paste( "y ~ -1 + ",paste(colnames(df)[5:11], collapse = " + ")))
M[[2]]$formula = update(M[[2]]$formula, .~. +m + f(s, model=spde) + f(iidx, model="iid", hyper=hyper.iid))

Let us check that our objects are correct.

print(M[[1]])
## $shortname
## [1] "stationary-no-cov"
## 
## $formula
## y ~ -1 + m + f(s, model = spde) + f(iidx, model = "iid", hyper = hyper.iid)
print(M[[2]])
## $shortname
## [1] "stationary-all-cov"
## 
## $formula
## y ~ dptLUKE + dptavg15km + dist30m + joetdsumsq + lined15km + 
##     swmlog10 + temjul15 + m + f(s, model = spde) + f(iidx, model = "iid", 
##     hyper = hyper.iid) - 1

We will not run the INLA calls until we have set up all the models.

4 Barrier models

First we divide up the mesh accoring to our study area polygon.

tl = length(mesh$graph$tv[,1])
# - the number of triangles in the mesh
posTri = matrix(0, tl, 2)
for (t in 1:tl){
  temp = mesh$loc[mesh$graph$tv[t, ], ]
  posTri[t,] = colMeans(temp)[c(1,2)] 
}
posTri = SpatialPoints(posTri)

# - compute the triangle positions
normal = over(poly.water, posTri, returnList=T)
# - checking which mesh triangles are inside the normal area
normal = unlist(normal)
barrier.triangles = setdiff(1:tl, normal)

poly.barrier = inla.barrier.polygon(mesh, barrier.triangles)
barrier.model = inla.barrier.pcmatern(mesh, barrier.triangles = barrier.triangles, prior.range = c(6, .5), prior.sigma = c(3, 0.01))
# - this creates the INLA object for the model
M[[3]]$formula = y~ -1+m + f(s, model=barrier.model) + f(iidx, model="iid", hyper=hyper.iid)
# - no covariates (except intercept m)
M[[4]]$formula = as.formula(paste( "y ~ -1 + ",paste(colnames(df)[5:11], collapse = " + ")))
M[[4]]$formula = update(M[[4]]$formula, .~. +m + f(s, model=barrier.model) + f(iidx, model="iid", hyper=hyper.iid))

Then we check

print(M[[3]])
## $shortname
## [1] "barrier-no-cov"
## 
## $formula
## y ~ -1 + m + f(s, model = barrier.model) + f(iidx, model = "iid", 
##     hyper = hyper.iid)
print(M[[4]])
## $shortname
## [1] "barrier-all-cov"
## 
## $formula
## y ~ dptLUKE + dptavg15km + dist30m + joetdsumsq + lined15km + 
##     swmlog10 + temjul15 + m + f(s, model = barrier.model) + f(iidx, 
##     model = "iid", hyper = hyper.iid) - 1

4.1 Plot data

Now that we have a polygon for representing the land area (physical barrier), we plot the data locations.

par(mar=rep(.1, 4))
plot(df$locx, y=df$locy, pch=20, asp=1)
plot(poly.barrier, add=T, border="black", col="grey")

5 Running all the models

Set up the initial values.

## Initial values
# - speeds up computations
# - improves accuracy of computations
# - set these to NULL the first time you run the model
M[[1]]$init = c(2.509,1.199,-0.574)
M[[2]]$init = c(1.162,0.313,-0.627)
M[[3]]$init = c(0.833,2.244,-0.471)
M[[4]]$init = c(0.044,1.274,-0.596)

Next, we run the inference for the our models, which can take some time.

for (i in 1:length(M)){
    print(paste("Running:  ", M[[i]]$shortname))
    M[[i]]$res = inla(M[[i]]$formula,
                      data=inla.stack.data(stk),
                      control.predictor=list(A=inla.stack.A(stk)),
                      family="poisson", E = e,
                      control.inla= list(int.strategy = "eb"),
                      control.mode=list(restart=T, theta=M[[i]]$init))  
}
## [1] "Running:   stationary-no-cov"
## [1] "Running:   stationary-all-cov"
## [1] "Running:   barrier-no-cov"
## [1] "Running:   barrier-all-cov"
# - time: < 20 min

The initial values that we set M[[i]]$init:

for (i in 1:length(M)){
  print(paste(round(M[[i]]$res$internal.summary.hyperpar$mode, 3), collapse = ','))
}
## [1] "2.505,1.199,-0.574"
## [1] "1.144,0.303,-0.629"
## [1] "0.794,2.362,-0.455"
## [1] "0.006,1.347,-0.562"

5.1 Summaries

You can run these to see the model output.

#summary(M[[1]]$res)
#summary(M[[2]]$res)
#summary(M[[3]]$res)
#summary(M[[4]]$res)

5.2 Logfiles

To understand how well the computations of the posterior worked, we look at the logfiles.

#M[[i]]$res$logfile

Discussing INLA inference in detail is out of scope of this topic. What we conclude from this is that the posteriors for the models without covariates are nicely behaved, and INLA is very successful. For the models with covariates, the posterior is a bit flat, and INLA is having a bit of difficulty. However, it is still sufficiently good.

6 Plot posterior spatial fields

6.1 Define the plotting function BTopic103

local.plot.field = function(field, mesh, xlim, ylim, ...){
  stopifnot(length(field) == mesh$n)
  # - error when using the wrong mesh
  if (missing(xlim)) xlim = poly.water@bbox[1, ] 
  if (missing(ylim)) ylim = poly.water@bbox[2, ]
  # - choose plotting region to be the same as the study area polygon
  proj = inla.mesh.projector(mesh, xlim = xlim, 
                             ylim = ylim, dims=c(300, 300))
  # - Can project from the mesh onto a 300x300 grid 
  #   for plots
  field.proj = inla.mesh.project(proj, field)
  # - Do the projection
  image.plot(list(x = proj$x, y=proj$y, z = field.proj), 
             xlim = xlim, ylim = ylim, col = plasma(17), ...)  
}

6.2 Plot the models without covariates

for (i in c(1,3)) {
  field = M[[i]]$res$summary.random$s$mean + M[[i]]$res$summary.fixed['m', 'mean']
  local.plot.field(field, mesh, main=paste(M[[i]]$shortname), zlim=c(-10.5, 1))
  plot(poly.barrier, add=T, col="grey")
  points(df$locx, df$locy)
}

6.3 Plot the models with covariates

for (i in c(2,4)) {
  field = M[[i]]$res$summary.random$s$mean + M[[i]]$res$summary.fixed['m', 'mean']
  local.plot.field(field, mesh, main=paste(M[[i]]$shortname), zlim=c(-9, -4.5))
  plot(poly.barrier, add=T, col="grey")
  points(df$locx, df$locy)
}

7 Compare fixed effects

Do a nice ggplot of the fixed effects between the spatial models with covariates.

## Set up dataframe with relevant results
res = M[[2]]$res$summary.fixed[ ,c(4,3,5)]
# - all covar, some quantiles
for(i in c(4)) res = rbind(res, M[[i]]$res$summary.fixed[ ,c(4,3,5)])

colnames(res) = c("E", "L", "U")
rownames(res)=NULL
n.covar = nrow(M[[2]]$res$summary.fixed)
res$model = factor(rep(c("MS", "MB"), each=n.covar, 
                   levels = c("MS", "MB")))
res$covar = factor(rep(rownames(M[[2]]$res$summary.fixed), 2))

if(require(ggplot2)) {
  ggplot(res, aes(x = model, y = E)) +
    facet_wrap(~covar, scales = "free_y") +
    geom_point(size = 3) +
    geom_errorbar(aes(ymax = U, ymin = L)) +
    xlab(NULL) + ylab(NULL) 
}

8 Comments

This analysis is done in more detail in the paper (Bakka et al. 2016).

References

Bakka, H., J. Vanhatalo, J. Illian, D. Simpson, and H. Rue. 2016. “Accounting for Physical Barriers in Species Distribution Modeling with Non-Stationary Spatial Random Effects.” arXiv preprint arXiv:1608.03787. Norwegian University of Science; Technology, Trondheim, Norway.
Kallasvuo, Meri, Jarno Vanhatalo, and Lari Veneranta. 2016. “Modeling the Spatial Distribution of Larval Fish Abundance Provides Essential Information for Management.” Canadian Journal of Fisheries and Aquatic Sciences, no. ja.