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The approach uses quasi-random sampling (Grafston & Tille, 2013, Foster et al., 2018) to generate a quadrature scheme for numerical approximation of a Poisson point process model (Berman & Turner 1992; Warton & Shepard 2010). Quasi-random sampling quadrature are a form of spatially-balanced survey design or point stratification that aims to reduce the frequency of placing samples close to each other (relative to pseudo-random or grid designs). A quasi-random quadrature design improves efficiency of background point sampling (and subsequent modelling) by reducing the amount of spatial auto-correlation between data implying that each sample is providing as much unique information as possible (Grafston & Tille, 2013, Foster et al., 2018) and thus reducing low errors for geostatistical prediction (Diggle & Ribeiro, 2007). Because the quasi-random design is not on a regular grid we use Dirichlet tessellation to generate polygons for each point in the quadrature scheme. Areal weights are then derived from these polygons.

We note that this approach is different to a targeted background scheme for generating pseudo-absences. If the users intent is to reduce sighting biases via a targeted background scheme, we recommend that bias is accounted for via covariates (distance to roads) or an offset (a known amount of effort) (e.g Warton et al., 2013; Renner et al, 2015).

The ppmData package can be installed using devtools or remotes R packages.

install.packages('devtools')
devtools::install_github('skiptoniam/ppmData')

Here is an example using a dataset from the ppmData package. We setup a quadrature scheme for the species Tasmaphena sinclairi located within Tasmania, Australia.

library(ppmData)
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")
npoints <- 1000
ppmdata1 <- ppmData(npoints = npoints, presences=presences,
                    window = covariates[[1]], covariates=covariates)

Here we plot the quadrature scheme. The green points represent the known locations of Tasmaphena sinclairi. The black points represent the quadrature locations. Quasi-random quadrature where the integration points are generated using a quasi-random areal sample.

plot(ppmdata1)

ppp <- ppmdata1$ppmData
form  <- presence/weights ~ 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 predict the relative intensity across the window, this will give us something similar to the expected count of species presences per-cell.

pred <- predict(covariates,
                ft.ppm,
                type="response")
plot(pred*prod(res(pred))) # scale response by area of cell.
points(presences[!duplicated(presences),1:2],pch=16,cex=0.75,col=gray(0.2,0.75))

Using ppmData to generate a quasi-random scheme

system.time(ppmdata2 <- ppmData(npoints = 100000,
                                presences=presences,
                                window = covariates[[1]],
                                covariates=covariates))

## There were 119 duplicated points unique to X, Y & SpeciesID, they have been removed.

## Developing a quadrature scheme for a single species dataset.

## There are 45 presence observations for this point process

## There are 100000 background quadrature (integration) points

## There are a total of 100045 sites in the model.matrix

##    user  system elapsed 
##   4.355   0.024   4.382

Using a spatstat to generate a quasi-random scheme for the snails dataset.

suppressPackageStartupMessages(library(spatstat))
e <- ext(covariates[[1]])
W <- owin(e[1:2],e[3:4])
snails_ppp <- ppp(presences[,1],presences[,2],W)
D <- rQuasi(100000,W)
system.time(Q <- quadscheme(snails_ppp,D,method="dirichlet", exact=FALSE))

##    user  system elapsed 
## 258.793   0.507 259.491

Please note that the ppmData project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.

Fork the ppmData repository

Clone your fork using the command:

git clone https://github.com/<username>/ppmData.git

Contribute to your forked repository.

Create a pull request.

If your code passes the necessary checks and is documented, your changes and/or additions will be merged in the main ppmData repository.

If you notice an issue with this repository, please report it using Github Issues. When reporting an implementation bug, include a small example that helps to reproduce the error. The issue will be addressed as quickly as possible.

If you have questions or need additional support, please open a Github Issues or send a direct email to [email protected].

Diggle, P. J., P. J. Ribeiro, Model-based Geostatistics. Springer Series in Statistics. Springer, 2007.

Foster, S.D., Monk, J., Lawrence, E., Hayes, K.R., Hosack, G.R. and Przeslawski, R., 2018. Statistical considerations for monitoring and sampling. Field manuals for marine sampling to monitor Australian waters, pp.23-41.

Grafstrom, Anton, and Yves Tille. Doubly balanced spatial sampling with spreading and restitution of auxiliary totals. Environmetrics 24.2 (2013): 120-131.

Renner, I.W., Elith, J., Baddeley, A., Fithian, W., Hastie, T., Phillips, S.J., Popovic, G. and Warton, D.I., 2015. Point process models for presence only analysis. Methods in Ecology and Evolution, 6(4), p.366-379.

Warton, D. I., and L. C. Shepherd. Poisson point process models solve the pseudo-absence problem for presence-only data in ecology. The Annals of Applied Statistics 4.3 (2010): 1383-1402.

Warton, D.I., Renner, I.W. and Ramp, D., 2013. Model-based control of observer bias for the analysis of presence-only data in ecology. PloS one, 8(11), p.e79168.