How to use ppmData to fit a Inhomogenous Poisson Point Process.

Load the required libraries

library(ppmData)
#> Loading required package: terra
#> terra 1.9.50

Setting up a ppm quadrature.

Basically you need a set of environmental or spatial rasters which will be used as covariates in the inhomogeneous point process models to estimate the intensity of the point pattern. In this case we are using a snail species from Tasmania, Australia. We aim to use ppmData to set up a quadrature scheme which can be used in any glm style model to fit a weighted Poisson GLM. For example, users could use mgcv or glmnet for extra flexibility of regression coefficients or regularization of covariates, respectively.

ppmData comes with a set of species presences and raster data to run examples in the package and vignettes. There is a marked point pattern for the recorded locations of a group of terrestrial snails observed in Tasmania, Australia. The species identification represent a mark at each location. There is also a set of covariates that include some commonly used environmental gradients and a few observer layers (such as distance from main roads).

For this example we will just use a single species, and treat it a s regular point pattern.

path <- system.file("extdata", package = "ppmData")
lst <- list.files(path=path,pattern='*.tif',full.names = TRUE)
covariates <- rast(lst)
presences <- subset(snails,SpeciesID %in% "Tasmaphena sinclairi")
presences <- presences[!duplicated(presences),]

We can plot the recorded presences of Tasmaphena sinclairi.

plot(covariates[[1]], legend=FALSE,axes=FALSE)
points(presences,pch=16,cex=0.75,col=gray(0.2,0.75))

Recorded presences of Tasmaphena sinclairi in Tasmania, Australia.

To fit a inhomogeneous Poisson point process for Tasmaphena sinclairi we need to set up a quadrature scheme. We can do this using the ppmData function. We need to pass a set of presences, a raster which represents a window or region in which the presences are observed and the model is to be fitted. We also need to set of covariates which we’ll use to describe the distribution of the observed presences.

window <- covariates[[1]]
ppmdata <- ppmData(presences=presences,
                   window = window,
                   covariates = covariates)
#> Developing a quadrature scheme for a single species dataset.
#> There are 45 presence observations for this point process
#> There are 1024 background quadrature (integration) points
#> There are a total of 1069 sites in the model.matrix

We can now plot the quadrature scheme and see what this looks like within our window.

plot(ppmdata)

Quadrature scheme for Tasmaphena sinclairi.

In the example above, we have not provided the number of quadrature (background) points for modelling and really need to do so if we want to generate a reasonable model for point process, there is actually a trade off between the number of quadrature points and the computation time required to run a ppm. As the number of quadrature points (\(Q\)) approaches infinity (\(Q_{n\to\infty}\)) the integrate estimated using the quadrature should become more and more accurate, at the cost of greater computation time.

We can increase the number of quadrature points using the npoints parameter in ppmData. Let’s bump it up to say 10000, I would typically choose a few more, say 50000 or 100000, but the best way to assess this is to compare the log-likelihoods of model fits using different numbers of quadrature [e.g. Warton & Shepard 2010].

ppmdata2 <- ppmData(presences=presences,
                    window = window,
                    covariates = covariates,
                    npoints = 10000,
                    na.rm=TRUE)
#> Developing a quadrature scheme for a single species dataset.
#> There are 45 presence observations for this point process
#> There are 10000 background quadrature (integration) points
#> There are a total of 10045 sites in the model.matrix

If we look at this quadrature scheme we can see that there are more background points.

plot(ppmdata2)

Quadrature scheme for Tasmaphena sinclairi with 10000 quasirandom quadrature points.

We can now think about fitting a ppm using glm from the R stats package [@r_core_team_r_2013]. We just need to extract the data.frame from the ppmData object and to specify a formula for the model. The only extra step is making the formula response equal to presence/weights (which gives us \(z\) in the @berman_approximating_1992 approximation). The right side of the formula will represent the covariates and their functional forms. In this example we specify independent 2\(^{nd}\) degree polynomials for each covariate, except for the spatial coordinates X & Y which we include in the model as interacting 2\(^{nd}\) degree polynomials. Once this is done, we define the weights in the glm function call and we now can fit an IPPM using ppmData and glm.

ppp <- ppmdata2$ppmData
form  <- presence/weights ~ poly(X,Y, degree = 2) + 
                            poly(max_temp_hottest_month, degree = 2) + 
                            poly(annual_mean_precip, degree = 2) + 
                            poly(annual_mean_temp, degree = 2) + 
                            poly(distance_from_main_roads, degree = 2)

ft.ppm <- glm(formula = form, data = ppp,
              weights = as.numeric(ppp$weights),
              family = poisson())

One we have fitted the model, we might want to do predict the relative intensity across the window, this will give us something similar to the expected count of presences per cell or area.

One thing to note, is that if we include longitude and latitude in the models (X & Y), for spatial prediction we need to include rasters which represent those values. Below is a function: xyFromWindow, which will return the longitude and latitude of each cell center in the a spatial raster format. We can append that to the multi-layered raster object and use it for prediction. We can use the terra::predict function here to predict the ppm to the raster layer data. This might not work for all model types (e.g glmnet).

## A function for return longitude and latitude as rasters.
xyFromWindow <- function(window, coord.name = c("X","Y"), mask.na=FALSE){

  grid.locations <- terra::xyFromCell(window,1:ncell(window))
  X <- Y <- window
  X[] <- grid.locations[,1]
  Y[] <- grid.locations[,2]
  names(X) <- coord.name[1]
  names(Y) <- coord.name[2]
  st.lonlat <- c(X,Y)
  if(mask.na){
    st.lonlat <- mask(st.lonlat,window)
  }
  return(st.lonlat)
}

xy <- xyFromWindow(window)
covariates2 <- c(xy,covariates)

## Here we can predict using the terra prediction methods.
pred <- predict(covariates2,
                ft.ppm,
                type="response")

plot(pred, col=(hcl.colors(11,palette= "Plasma")),
     main="Predicted intensity",
     plg=list(title=expression(paste(lambda[i])),x=142,y=-43),axes=FALSE)

Now we have a prediction for the fitted ppm model. By predicting the model on the “response” scale, we are getting the predicted intensity. However, glm assumes the weights for each observation for prediction equals one. So we need to re-scale the prediction to the cell area size. This will give us the expected count of presences per cell.

# scale prediction by area of cell
pred2 <- pred*prod(res(pred))

plot(pred2, col=(hcl.colors(11,palette= "Plasma")),
     main="Predicted intensity per cell",
     plg=list(title=expression(paste(lambda[i])),x=142,y=-43),axes=FALSE)

Using spatstat functions

We can port the ppmData objects to spatstat quite easily using the as.spatstat function call. Once we have done this, then it easy to use spatstat to do analyses and model testing if so desired.

library(spatstat)
#> Loading required package: spatstat.data
#> Loading required package: spatstat.univar
#> spatstat.univar 3.2-0
#> Loading required package: spatstat.geom
#> spatstat.geom 3.8-3
#> 
#> Attaching package: 'spatstat.geom'
#> The following objects are masked from 'package:terra':
#> 
#>     area, delaunay, is.empty, rescale, rotate, shift, where.max,
#>     where.min
#> Loading required package: spatstat.random
#> spatstat.random 3.5-2
#> Loading required package: spatstat.explore
#> Loading required package: nlme
#> spatstat.explore 3.8-3
#> Loading required package: spatstat.model
#> Loading required package: rpart
#> spatstat.model 3.7-2
#> Loading required package: spatstat.linnet
#> spatstat.linnet 3.5-4
#> 
#> spatstat 3.6-3 
#> For an introduction to spatstat, type 'beginner'
ss <- as.spatstat(ppmdata2)
Q <- ss$Q
ss.covars <- ss$covariates
ss.form <- Q ~ poly(max_temp_hottest_month, degree = 2) + 
               poly(annual_mean_precip, degree = 2) + 
               poly(annual_mean_temp, degree = 2) + 
               poly(distance_from_main_roads, degree = 2)
ss.fit <- ppm(ss.form,data=ss.covars,Poisson())
ss.pred <- predict(ss.fit,
                   covariates = ss.covars,
                   type="intensity",
                   ngrid=720)
plot(ss.pred, main="Predicted intensity")

There are also a bunch of other functions which can covert ppmData objects to say a ppp or extract a spatstat owin from a ppmData object.

library(spatstat)
snail <- as.ppp(ppmdata2)
ow <- as.owin(ppmdata2)
par(mfrow=c(1,2))
plot(snail,main=expression(paste(italic('Tasmaphena sinclairi'), " point pattern")))
plot(ow,"spatstat owin")

- Setting up a ppm quadrature.
- Using spatstat functions