Use of custom distribution to increase MCMC efficiency
The dbinomLocal_normal() distribution incorporates three features to increase computational efficiency.
1. Local detector evaluation
This step restricts calculations of the detection probabilities (here using a halfnormal function) to detectors (traps) within a radius where detections are realistically possible (Cyril Milleret et al. 2019). We use the function getLocalObjects to identify the set of detectors that are within a certain distance \(d_{max}\) of each habitat cell center (blue points in the plot below). These reduced sets of detectors are stored in the localIndices matrix and are later used in the local evaluation of the detection model to speed up calculations. The value of \(d_{max}\) should be as small as possible in order to reduce computation but always large enough so that for any particular individual, the set of local traps associated with the coordinates of the activity center s include all detectors at which that individual was detected. The \(d_{max}\) value will therefore affect the number of columns in localIndices Here, we use \(d_{max}=38\). with the coordinates of the activity center s … We also aggregated the habitat matrix to obtain larger habitat cells (lower resolution) and obtain objects with smaller dimensions. This reduces the number of habitat cells for which we have to identify the set of detectors that are within \(d_{max}\) of the cell center. The goal is to create the object localIndices of the smallest dimension possible, that balances the cost of looking up relevant grid cells and reducing calculations for each grid cell.
Here, we resize the habitat matrix by a factor of 24, which corresponds to the resizeFactor argument. This means that 24x24 cells are aggregated into a single cell. The resizeFactor value will affect how many rows localIndices will be composed of.
set.seed(2)
DetectorIndex <- getLocalObjects(habitatMask = data$habitat.mx,
coords = data$detector.xy,
dmax = 38,
resizeFactor = 24)
constants$y.maxDet <- dim(DetectorIndex$habitatGrid)[1]
constants$x.maxDet <- dim(DetectorIndex$habitatGrid)[2]
constants$ResizeFactor <- DetectorIndex$resizeFactor
constants$n.cells <- dim(DetectorIndex$localIndices)[1]
constants$maxNBDets <- DetectorIndex$numLocalIndicesMax
data$detectorIndex <- DetectorIndex$localIndices
data$nDetectors <- DetectorIndex$numLocalIndices
data$habitatIDDet <- DetectorIndex$habitatGrid2. Sparse representation of the observation matrix
We re-express y as a sparse representation of the detection matrix to reduce its size. In this representation, we turn the detection matrix y into three objects:
detIndices: where each row (corresponding to one individual) contains the identification numbers of detectors at which that individual was detected.y: a second matrix of identical dimension, containing the number of detections of a given individual at each detector. This second matrix is necessary for modelling non-binary detections (e.g. \(binomial\) observation models)detNums: a vector containing the number of detectors at which each individual was detected.
ySparse <- getSparseY(x = my.jags.input$y)
data$y <- ySparse$y[,,1]
data$yDets <- ySparse$detIndices[,,1]
data$nbDetections <- ySparse$detNums[,1]
constants$nMaxDetectors <- ySparse$maxDetNums3. Skip unnecessary calculations
The function dbinomLocal_normal() takes the logical argument indicator that specifies whether the individual i is available (\(z_i\) = 1) for detection or not (\(z_i\) = 0). When \(z_i\) = 0, calculations of \(p_{ij}\) are not performed and therefore increases MCMC efficiency.
