# Part 3: Spatial analysis of geotagged data

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## Part 3: Spatial analysis of geotagged data

See the other parts in this series of blog

posts.

In parts 1 and 2 we extracted spatial coordinates from our photos and

then made an interactive web map that included data associate with those

photos. Here I describe how we can build a spatial statistical model to

interpolate to unmeasured locations.

Here we will build an interactive map showing model results from an interpolation of oyster presence/absence.

First up we need to load some packages:

```
library(readr)
library(INLA)
library(sp)
library(raster)
library(rgeos)
```

The `INLA`

package is crucial here. We will be

using `INLA`

to perform Bayesian Kriging. In non-technical terms,

Bayesian kriging lets us fill in the gaps between observations of oyster

counts by interpolating from nearby points.

### Getting the data in the right form

Before we begin, we need to load in the data, and transform it into a

spatial data frame. In particular, it is important to transform the

lon-lats into UTM coordinates. UTM coordinates allow us to measure

distances between sample sites in metres:

```
dat <- read_csv('Oysters_merged.csv')
n <- nrow(dat)
utmproj <- "+proj=utm +zone=10 +north +datum=WGS84 +units=m +no_defs +ellps=WGS84 +towgs84=0,0,0"
spdat <- dat
coordinates(spdat) <- ~ GPSLongitude + GPSLatitude
proj4string(spdat) <- "+init=epsg:4326"
spdf <- spTransform(spdat, CRS(utmproj))
plot(spdf)
```

You should get a plot like that pictured just of the sample sites.

### Creating an INLA “Mesh”

Now we can start building the specialised data-structures we need for

the `INLA`

package. First up, we have to build what’s called a mesh.

This is a Delaunay

triangulation

built around our data points. It accounts for the non-regular spatial

structuring of the sampling (I wandered around on the rocky shore

throwing the quadrat at intervals).

Below we provide two parameters to `max.edge`

, this will enable us to

buffer the edges to obtain a slightly large spatial domain.

```
max.edge.length <- c(25, 40)
loc1 <- as.matrix(coordinates(spdf))
mesh <- inla.mesh.2d(loc=loc1, max.edge = max.edge.length, offset=1, cutoff = 5)
plot(mesh)
plot(spdf, add = T, col = 'red')
```

You can play with the edge length, offset and cutoff parameters to vary

how the triangulations turns out. You can get more guidance at the

`INLA`

page and in this

tutorial.

An important step is to check that our maximum edge lengths are less

than the estimated range (otherwise it is pointless including a spatial

effect!). We will check the range size below, once we have fitted the

model.

You might like to think of a-prior reasons for modelling a certain

range, for instance, what is a reasonable spatial scale for ‘clumping’

of oyster patches?

### Build INLA stack

Now we need to implement a few more steps to build up an appropriate

‘stack’ of data for `INLA`

. Note that I have transformed oyster count to

presence – absence. We will build a binomial model and just predict

whether any given quadrat had oysters or not.

```
A.data <- inla.spde.make.A(mesh, loc1)
spde <- inla.spde2.matern(mesh, alpha = 2)
spdf$presabs <- ifelse(spdf$oysters_live>0, 1, 0)
stk <- inla.stack(data=list(y=spdf$presabs), A=list(A.data, 1),
effects=list(s=1:mesh$n, intercept=rep(1, n)),
remove.unused = FALSE, tag = "est")
```

Note that here we choose `alpha = 2`

. `alpha`

is a smoothness parameter

that must be 0 <= `alpha`

<= 2, higher values will give smoother

spatial interpolation. Also note that we have to specify ‘data’ for the

intercept (just a vector of `1`

).

The `tag = "est"`

argument just gives a name tag to the fitted model. In

other applications you might want to join the estimation stack with a

prediction stack (just build two stacks, one where the response is `NA`

and then join them by applying `inla.stack`

to both).

### Model

Now, after that considerable amount of prepartion, we get to fitting the

model. We use standard **R** syntax for the formula, with the addition

of the `f()`

term to specify a function.

```
formula <- y ~ 0 + intercept + f(s, model=spde)
mod <- inla(formula, data=inla.stack.data(stk),
control.predictor=list(A = inla.stack.A(stk)),
family = 'binomial')
```

We can extract summary parameters using `summary`

. Here we are

interested in the random effects, it is good practice to call

`inla.hyperpar`

before we inspect random effects:

```
hyper <- inla.hyperpar(mod)
summary(hyper)
```

### Checking the range parameter

We should also check that our triangle edge length is less than the

estimated spatial range. We can do that by making a call to the model

object, asking for the estimated parameters for the random field

```
rf <- inla.spde.result(inla = mod, name = "s",
spde = spde, do.transf = TRUE)
plot(rf$marginals.range[[1]], type = "l",
xlab = "Range (m)", ylab = "Density",
xlim = c(0, 1000))
abline(v = max.edge.length[2], col = 'red')
```

The red line shows our edge length parameter, which is well less than

the model estimate for the spatial range.

### Model predictions

Now we have a fitted model, we can extracted predicted probabilities of

oyster presence and map them. First transform the predictions from the

logit scale back to probabilities:

```
ypred <- exp(mod$summary.random$s$mean)/(1 + exp(mod$summary.random$s$mean))
```

Then we can turn our predictions into a spatial points, that can be be

mapped onto a raster for plotting:

```
extent <- extent(spdf)
xdims <- 100; ydims <- 200
xlim <- c(extent[1], extent[2]); ylim = c(extent[3], extent[4]);
proj <- inla.mesh.projector(mesh, xlim = xlim, ylim = ylim, dims=c(xdims, ydims))
field.proj <- inla.mesh.project(proj, ypred)
datpred <- data.frame(x = rep(proj$x, ydims), y = rep(proj$y, each = xdims), pred = as.numeric(field.proj))
coordinates(datpred) <- ~x + y
proj4string(datpred) <- utmproj
dat3 <- spTransform(datpred, crs("+init=epsg:4326"))
r <- raster(extent(dat3), ncols = xdims, nrows= ydims, crs = crs(spdat))
icell <- cellFromXY(r, dat3)
r[icell] <- as.numeric(dat3$pred)
```

And finally the plot:

```
cols <- RColorBrewer::brewer.pal(9, "Purples")
plot(r, col = cols)
points(spdf)
plot(spdat, add = TRUE)
```

So it looks like oysters are more common in the southern portion of the

survey area. But note that our predictions extent a fair way from the

points. We might want to redo these with a more constrained spatial

region.

Finally, we could map this onto a satellite image to get a better view

of what is going on. I am just using the same `leaflet`

code as

before:

```
library(leaflet)
brks <- seq(0,1, by = 0.1)
ncol <- length(brks)
oystercols <- RColorBrewer::brewer.pal(min(ncol, 9), 'Reds')
pal <- colorBin(oystercols, dat$oysters_live, bins = brks)
x1 <- -124.5997
y1 <- 49.52848
leaflet() %>%
addProviderTiles("Esri.WorldImagery") %>%
setView(lng = x1, lat = y1, zoom = 16) %>%
addRasterImage(r, opacity = 0.8, color = pal) %>%
addCircleMarkers(lng = dat$GPSLongitude, lat = spdat$GPSLatitude,
radius = 4) %>%
addLegend("topright", pal = pal,
values = brks,
title = "Chance of live oysters (Crassostrea gigas)",
opacity = 1)
```

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