Adaptive spatio-temporal intensity by partitioning algorithm

First example: Log-Gaussian spatio-temporal point pattern

Generating an observation

First, we want to check the intensity using an artificial point pattern. Therefore, we simulate a spatio-temporal point pattern (in this case, we fix the number of points prior to simulation).

# Number of points coming from a Poisson distribution with mean 1000
N <- rpois(1, 1000)

Then we set a covariance model for the underlying random field; in our case, we chose a non-separable Geneiting covariance function with its parameters. We set the additional parameters for the random field simulation as the mesh, variance and scales.

logGaussianPP <- rlgcp(npoints = N, nx = 128, ny = 128, nt = 64,
                       separable = F, model = "gneiting", scale = c(0.5, 0.8),
                       param = c(1, 1, .1, 0.1, 1, 2), var.grf = 2, mean.grf = 1)

We want to plot the point pattern generated above in 2D, so we employ the spatstat package where we see our point pattern as a marked spatial point pattern.

# Spatstat format
XX <- ppp(x = logGaussianPP$xyt[, 1], y = logGaussianPP$xyt[, 2], 
          marks = logGaussianPP$xyt[, 3], window = owin())

# Set the color scheme
colmap <- colourmap(rainbow(512), range = range(marks(XX)))
sy <- symbolmap(pch = 21, bg = colmap, range = range(marks(XX)))

# Plotting
plot(XX, symap = sy, 'log-Gaussian Cox point pattern')

Estimating global bandwidths

Before using the adaptive estimator, we should set the global bandwidths, i.e., the bandwidths for estimating the pilot intensities. These pilot intensities will assign a particular bandwidth to each data point.

# Global bandwidths
bwS0 <- OS(XX) # The spatial bandwidth will be the oversmoothing version
bwt0 <- bw.SJ(XX$marks) # We employ Sheather & Jones's bandwidth for time 

# Spatial and temporal bandwidths based on pilot estimations
bwS <- bw.abram(unmark(XX), h0 = bwS0)
bwt <- bw.abram.temp(t = XX$marks, h0 = bwt0)

Intensity estimates

First we use a fixed-bandwidth kernel estimate for the intensity using a function provided by the sparr package.

# Fixed bandwidth estimate (non-separable)
classic.dens <- spattemp.density(pp = unmark(XX), tt = XX$marks,
                                 sres = 128, tres = 64,
                                 lambda = bwt0, h = bwS0,
                                 sedge = "uniform")

Now we compute the adaptive intensity by using our partitioning algorithm. For doing that, we have to set first a fixed number of groups for the spatial and the temporal bandwidths; it could be done manually or let the function decide based on a rule-of-thumb. In addition, we set a temporal resolution of 64 pixels (the default is 128 for space and for time). We can evaluate the intensity at the data points (at = “points”), or obtain snapshots (at = “bins”, the default).

# In this case we use 8 groups for space and 4 for time
adapt.dens <- dens.par(X = XX, 
                       dimt = 64,
                       bw.xy = bwS, bw.t = bwt, 
                       ngroups.xy = 8, ngroups.t = 4,
                       at = "bins")

We plot some snapshots of the estimates that we have computed.

# We select some fixed times for visualisation
I <- c(13, 17, 21, 31, 50)

# We subset the lists
CN <- as.imlist(classic.dens$z[I])
AN <- as.imlist(adapt.dens[I])

# We generate the plots
plot.imlist(CN, ncols = 5, main = 'Classic fixed-bandwidth estimate')
plot.imlist(AN, ncols = 5, main = 'Adaptive non-separable estimate')

Second example: Amazonia Data

Loading and visualising data

We want to estimate the intensity of the Amazonia fires.

# Loading dataset
data("amazon")

Now, given that the number of points is huge, for visualisation we select a sample of 5000.

# Extract a sample of 5000 data points
AmazonReduced <- amazon[sample.int(amazon$n, 5000)]

# Set the color scheme
colmap <- colourmap(rainbow(512), range = range(marks(AmazonReduced)))
sy <- symbolmap(pch = 21, bg = colmap, range = range(marks(AmazonReduced)))

# Plotting
plot(AmazonReduced, symap = sy, 'Sample of Amazonia fires')

Separability test

Before estimating the intensity, it is very convenient to know whether separability holds. When separability holds, estimating the spatio-temporal intensity becomes easier and faster. So we perform a separability test to have an approximate answer to that question. The test is performed in a Monte Carlo fashion; therefore, we have to fix a number of Monte Carlo samples prior to the test, say 1500.

separability.test(X = amazon, nperm = 1500)

Given the test result, we should not assume separability.

Estimating global bandwidths

We select the bandwidths for estimating the pilot intensities.

# Global bandwidths
bwS0 <- OS(amazon) # The spatial bandwidth will be the oversmoothing version
bwt0 <- bw.nrd(amazon$marks) # We employ Scott bandwidth for time 

# Spatial and temporal bandwidths based on pilot estimations
bwS <- bw.abram(amazon, h0 = bwS0)
bwt <- bw.abram.temp(t = amazon$marks, h0 = bwt0)

Intensity estimates

We use a fixed-bandwidth kernel estimate for the intensity

# Fixed bandwidth estimate (non-separable)
# This could be time consuming
classic.dens <- spattemp.density(pp = unmark(amazon), h = bwS0, tt = amazon$marks,
                                 sres = 64, tres = 64,
                                 lambda = bwt0,
                                 sedge = "uniform")

Now we compute the adaptive intensity by using our partitioning algorithm.

# In this case we let the algorithm to choose the numbers of groups
# It could be time consuming
adapt.dens <- dens.par(X = amazon, 
                       dimyx = 64, dimt = 64,
                       bw.xy = bwS, bw.t = bwt, 
                       at = "bins")

We plot some snapshots of the estimates that we have computed.

# We select some fixed times for visualisation
I <- c(12, 18, 23, 31, 55)

# We subset the lists
CN <- as.imlist(classic.dens$z[I])
AN <- as.imlist(adapt.dens[I])

# We generate the plots
plot.imlist(CN, ncols = 5, main = 'Classic fixed-bandwidth estimate')
plot.imlist(AN, ncols = 5, main = 'Adaptive non-separable estimate')

Dynamic intensity estimate

animation::saveVideo(
  for(i in 1:length(adapt.dens)){
    plot(adapt.dens[[i]], main = paste("Time",i))
  },
  video.name="amazon.mp4", other.opts = '-b:v 1M -pix_fmt yuv420p', 
  ani.width = 640, ani.height = 640, interval = 1 / 12)