# 11  Point Pattern Analysis

Point pattern analysis is concerned with describing patterns of points over space and making inference about the process that could have generated an observed pattern. The main focus here lies on the information carried in the locations of the points, and typically these locations are not controlled by sampling but a result of a process we are interested in, such as animal sightings, accidents, disease cases, or tree locations. This is opposed to geostatistical processes (Chapter 12) where we have values of some phenomenon everywhere but observations limited to a set of locations that we can control, at least in principle. Hence, in geostatistical problems the prime interest is not in the observation locations but in estimating the value of the observed phenomenon at unobserved locations. Point pattern analysis typically assumes that for an observed area, all points are available, meaning that locations without a point are not unobserved as in a geostatistical process, but are observed and contain no point. In terms of random processes, in point processes locations are random variables, where, in geostatistical processes the measured variable is a random field with locations fixed.

This chapter is confined to describing the very basics of point pattern analysis, using package spatstat , and related packages by the same authors. The spatstat book of Baddeley, Rubak, and Turner (2015) gives a comprehensive introduction to point pattern theory and the use of the spatstat package family, which we will not try to copy. Inclusion of particular topics in this chapter should not be seen as an expression that these are more relevant than others. In particular, this chapter tries to illustrate interfaces existing between spatstat and the more spatial data science oriented packages sf and stars. A further book that introduces point patterns analysis is Stoyan et al. (2017). R package stpp for analysing spatiotemporal point processes is discussed in Gabriel, Rowlingson, and Diggle (2013).

Important concepts of point patterns analysis are the distinction between a point pattern and a point process: the latter is the stochastic process that, when sampled, generates a point pattern. A dataset is always a point pattern, and inference involves figuring out the properties of a process that could have generated a pattern like the one we observed. Properties of a spatial point process include

• first order properties: the intensity function measures the number of points per area unit; this function is spatially varying for a inhomogeneous point process
• second order properties: given a constant or varying intensity function, describe whether points are distributed independently from one another, tend to attract each other (clustering), or repulse each other (more regularly distributed than under complete spatial randomness)

## 11.1 Observation window

Point patterns have an observation window. Consider the points generated randomly by

library(sf)
# Linking to GEOS 3.11.1, GDAL 3.6.2, PROJ 9.1.1; sf_use_s2() is TRUE
n <- 30
set.seed(13531) # remove this to create another random sequence
xy <- data.frame(x = runif(n), y = runif(n)) |>
st_as_sf(coords = c("x", "y"))

then these points are (by construction) uniformly distributed, or completely spatially random, over the domain $$[0,1] \times [0,1]$$. For a larger domain, they are not uniform, for the two square windows w1 and w2 created by

w1 <- st_bbox(c(xmin = 0, ymin = 0, xmax = 1, ymax = 1)) |>
st_as_sfc()
w2 <- st_sfc(st_point(c(1, 0.5))) |> st_buffer(1.2)

this is shown in Figure 11.1.

Code
par(mfrow = c(1, 2), mar = c(2.1, 2.1, 0.1, 0.5), xaxs = "i", yaxs = "i")
plot(w1, axes = TRUE, col = 'grey')
plot(w2, axes = TRUE, col = 'grey')
plot(xy, add = TRUE, cex = .5) Figure 11.1: Depending on the observation window (grey), the same point pattern can appear completely spatially random (left), or clustered (right)

Point patterns in spatstat are objects of class ppp that contain points and an observation window (an object of class owin). We can create a ppp from points by

library(spatstat) |> suppressPackageStartupMessages()
as.ppp(xy)
# Planar point pattern: 30 points
# window: rectangle = [0.009, 0.999] x [0.103, 0.996] units

where we see that the bounding box of the points is used as observation window when no window is specified. If we add a polygonal geometry as the first feature of the dataset, then this is used as observation window:

(pp1 <- c(w1, st_geometry(xy)) |> as.ppp())
# Planar point pattern: 30 points
# window: polygonal boundary
# enclosing rectangle: [0, 1] x [0, 1] units
c1 <- st_buffer(st_centroid(w2), 1.2)
(pp2 <- c(c1, st_geometry(xy)) |> as.ppp())
# Planar point pattern: 30 points
# window: polygonal boundary
# enclosing rectangle: [-0.2, 2.2] x [-0.7, 1.7] units

To test for homogeneity, one could carry out a quadrat count, using an appropriate quadrat layout (a 3 $$\times$$ 3 layout is shown in Figure 11.2)

par(mfrow = c(1, 2), mar = rep(0, 4))
plot(q1, main = "")
plot(q2, main = "")
plot(xy, add = TRUE) Figure 11.2: 3 $$\times$$ 3 quadrat counts for the two point patterns

and carry out a $$\chi^2$$ test on these counts:

quadrat.test(pp1, nx=3, ny=3)
# Warning: Some expected counts are small; chi^2 approximation may be
# inaccurate
#
#   Chi-squared test of CSR using quadrat counts
#
# data:  pp1
# X2 = 8, df = 8, p-value = 0.9
# alternative hypothesis: two.sided
#
# Quadrats: 9 tiles (irregular windows)
# Warning: Some expected counts are small; chi^2 approximation may be
# inaccurate
#
#   Chi-squared test of CSR using quadrat counts
#
# data:  pp2
# X2 = 43, df = 8, p-value = 2e-06
# alternative hypothesis: two.sided
#
# Quadrats: 9 tiles (irregular windows)

which indicates that for the second case we have an indication that this is not a CSR (completely spatially random) pattern. As indicated by the warning, we should take the p-values with a large grain of salt because we have too small expected counts.

Kernel densities can be computed using density, where kernel shape and bandwidth can be controlled. Here, cross-validation is used by function bw.diggle to specify the bandwidth parameter sigma; plots are shown in Figure 11.3.

den1 <- density(pp1, sigma = bw.diggle)
den2 <- density(pp2, sigma = bw.diggle)
Code
par(mfrow = c(1, 2), mar = c(0,0,1.1,2))
plot(den1)
plot(den2)
plot(pp1, add=TRUE)

The density maps created this way are obviously raster images, and we can convert them into stars object by

library(stars)
s1 <- st_as_stars(den1)
(s2 <- st_as_stars(den2))
# stars object with 2 dimensions and 1 attribute
# attribute(s):
#        Min.  1st Qu. Median Mean 3rd Qu. Max. NA's
# v  1.03e-14 0.000153  0.304 6.77    13.1 42.7 3492
# dimension(s):
#   from  to offset  delta x/y
# x    1 128   -0.2 0.0187 [x]
# y    1 128   -0.7 0.0188 [y]

and we can verify that the area under the density surface is similar to the sample size (30), by

s1$a <- st_area(s1) |> suppressMessages() s2$a <- st_area(s2) |> suppressMessages()
with(s1, sum(v * a, na.rm = TRUE))
#  29
with(s2, sum(v * a, na.rm = TRUE))
#  30.7

More exciting applications involve modelling the density surface as a function of external variables. Suppose we want to model the density of pp2 as a Poisson point process (meaning that points do not interact with each other), where the intensity is a function of distance to the centre of the “cluster”, and these distance are available in a stars object:

pt <- st_sfc(st_point(c(0.5, 0.5)))
st_as_sf(s2, as_points = TRUE, na.rm = FALSE) |>
st_distance(pt) -> s2$dist we can then model the densities using ppm, where the name of the point pattern object is used on the left-hand side of the formula: (m <- ppm(pp2 ~ dist, data = list(dist = as.im(s2["dist"])))) # Nonstationary Poisson process # Fitted to point pattern dataset 'pp2' # # Log intensity: ~dist # # Fitted trend coefficients: # (Intercept) dist # 4.54 -4.25 # # Estimate S.E. CI95.lo CI95.hi Ztest Zval # (Intercept) 4.54 0.341 3.87 5.21 *** 13.32 # dist -4.25 0.701 -5.62 -2.88 *** -6.06 The returned object is of class ppm, and can be plotted: Figure 11.4 shows the predicted surface. The prediction standard error can also be plotted. Code plot(m, se = FALSE) Figure 11.4: Predicted densities of a ppm model The model also has a predict method, which returns an im object that can be converted into a stars object by predict(m, covariates = list(dist = as.im(s2["dist"]))) |> st_as_stars() # stars object with 2 dimensions and 1 attribute # attribute(s): # Min. 1st Qu. Median Mean 3rd Qu. Max. NA's # v 0.0694 0.527 2.12 6.62 7.3 89.9 3492 # dimension(s): # from to offset delta x/y # x 1 128 -0.2 0.0187 [x] # y 1 128 -0.7 0.0187 [y] ## 11.2 Coordinate reference systems All routines in spatstat are layed out for two-dimensional data with Cartesian coordinates. If we try to convert an object with ellipsoidal coordinates, we get an error: system.file("gpkg/nc.gpkg", package = "sf") |> read_sf() |> st_geometry() |> st_centroid() |> as.ppp() # Error: Only projected coordinates may be converted to spatstat # class objects When converting to a spatstat data structure we loose the coordinate reference system we started with. It can be set back to sf or stars objects by using st_set_crs. ## 11.3 Marked point patterns, points on linear networks A few more data types can be converted to and from spatstat. Marked point patterns are point patterns that have a “mark”, which is either a categorical label or a numeric label for each point. A dataset available in spatstat with marks is the longleaf pines dataset, containing diameter at breast height as a numeric mark: longleaf # Marked planar point pattern: 584 points # marks are numeric, of storage type 'double' # window: rectangle = [0, 200] x [0, 200] metres ll <- st_as_sf(longleaf) print(ll, n = 3) # Simple feature collection with 585 features and 2 fields # Geometry type: GEOMETRY # Dimension: XY # Bounding box: xmin: 0 ymin: 0 xmax: 200 ymax: 200 # CRS: NA # First 3 features: # spatstat.geom..marks.x. label geom # NA NA window POLYGON ((0 0, 200 0, 200 2... # 1 32.9 point POINT (200 8.8) # 2 53.5 point POINT (199 10) Values can be converted back to ppp with as.ppp(ll) # Warning in as.ppp.sf(ll): only first attribute column is used for # marks # Marked planar point pattern: 584 points # marks are numeric, of storage type 'double' # window: polygonal boundary # enclosing rectangle: [0, 200] x [0, 200] units Line segments, in spatstat objects of class psp can be converted back and forth to simple feature with LINESTRING geometries following a POLYGON feature with the observation window, as in print(st_as_sf(copper$SouthLines), n = 5)
# Simple feature collection with 91 features and 1 field
# Geometry type: GEOMETRY
# Dimension:     XY
# Bounding box:  xmin: -0.335 ymin: 0.19 xmax: 35 ymax: 158
# CRS:           NA
# First 5 features:
#     label                           geom
# 1  window POLYGON ((-0.335 0.19, 35 0...
# 2 segment LINESTRING (3.36 0.19, 10.4...
# 3 segment LINESTRING (12.5 0.263, 11....
# 4 segment LINESTRING (11.2 0.197, -0....
# 5 segment LINESTRING (6.35 12.8, 16.5...

Finally, point patterns on linear networks, in spatstat represented by lpp objects, can be converted to sf by

print(st_as_sf(chicago), n = 5)
# Simple feature collection with 620 features and 4 fields
# Geometry type: GEOMETRY
# Dimension:     XY
# Bounding box:  xmin: 0.389 ymin: 153 xmax: 1280 ymax: 1280
# CRS:           NA
# First 5 features:
#     label seg tp marks                           geom
# 1  window  NA NA  <NA> POLYGON ((0.389 153, 1282 1...
# 2 segment  NA NA  <NA> LINESTRING (0.389 1254, 110...
# 3 segment  NA NA  <NA> LINESTRING (110 1252, 111 1...
# 4 segment  NA NA  <NA> LINESTRING (110 1252, 198 1...
# 5 segment  NA NA  <NA> LINESTRING (198 1277, 198 1...

where we only see the first five features; the points are also in this object, as variable label indicates

table(st_as_sf(chicago)\$label)
#
#   point segment  window
#     116     503       1

Potential information about network structure, how LINESTRING geometries are connected, is not present in the sf object. Package sfnetworks would be a candidate package to hold such information in R or to pass on network data imported from OpenStreetMaps to spatstat.

## 11.4 Spatial sampling and simulating a point process

Package sf contains an st_sample method that samples points from MULTIPOINT, linear or polygonal geometries, using different spatial sampling strategies. It natively supports strategies “random”, “hexagonal”, “Fibonacci” (Section 11.5), and “regular”, where “regular” refers to sampling on a square regular grid and “hexagonal” essentially gives a triangular grid. For type “random”, it can return exactly the number of requested points, for other types this is approximate.

st_sample also interfaces point process simulation functions of spatstat, when other values for sampling type are chosen. For instance the spatstat function rThomas is invoked when setting type = Thomas (Figure 11.5):

kappa <- 30 / st_area(w2) # intensity
th <- st_sample(w2, kappa = kappa, mu = 3, scale = 0.05,
type = "Thomas")
nrow(th)
#  90
Code
par(mar = rep(0, 4))
plot(w2)
plot(th, add = TRUE) Figure 11.5: Thomas process with mu = 3 and scale = 0.05

The help function obtained by ?rThomas details the meaning of the parameters kappa, mu, and scale. Simulating point processes means that the intensity is given, not the sample size. The sample size within the observation window obtained this way is a random variable.

## 11.5 Simulating points on the sphere

Another spatial random sampling type supported by sf natively (in st_sample) is simulation of random points on the sphere. An example of this is shown in Figure 11.6, where points were constrained to those in oceans. Points approximately regularly distributed over a sphere are obtained by st_sample with type = "Fibonacci" .

Code
par(mar = rep(0, 4), mfrow = c(1, 2))
library(s2)
g <- as_s2_geography(TRUE) # Earth
co <- s2_data_countries()
oc <- s2_difference(g, s2_union_agg(co)) # oceans
b <- s2_buffer_cells(as_s2_geography("POINT(-30 -10)"), 9800000) # visible half
i <- s2_intersection(b, oc) # visible ocean
co <- s2_intersection(b, co)
ortho = st_crs("+proj=ortho +lat_0=-10 +lon_0=-30")
# background:
st_transform(st_as_sfc(i), ortho) |> plot(col = 'lightblue')
st_transform(st_as_sfc(co), ortho) |> plot(col = NA, add = TRUE, border = 'grey')
# sampling randomly from globe:
sf_use_s2(FALSE) |> suppressMessages()
st_as_stars() |> st_bbox() |> st_as_sfc() |> st_sample(1000, exact = FALSE) |>
suppressMessages() -> pts
sf_use_s2(TRUE) |> suppressMessages()
pts |> s2_intersection(i) |> st_as_sfc() -> pts
st_transform(pts, ortho) |> plot(add = TRUE, pch = 3, cex = .5)
# right: background:
st_transform(st_as_sfc(i), ortho) |> plot(col = 'lightblue')
st_transform(st_as_sfc(co), ortho) |> plot(col = NA, add = TRUE, border = 'grey')
# Fibonacci:
sf_use_s2(FALSE) |> suppressMessages()
st_as_stars() |> st_bbox() |> st_as_sfc() |>
st_sample(1000, type = "Fibonacci", exact = FALSE) |> suppressMessages() -> pts
sf_use_s2(TRUE) |> suppressMessages()
pts |> s2_intersection(i) |> st_as_sfc() -> pts
st_transform(pts, ortho) |> plot(add = TRUE, pch = 3, cex = .5) Figure 11.6: Points sampled on the globe over the oceans: randomly (left) and approximately regular (Fibonacci; right), shown in an orthographic projection

## 11.6 Exercises

1. After loading spatstat, recreate the plot obtained by plot(longleaf) by using ggplot2 and geom_sf(), and by sf::plot().
2. Convert the sample locations of the NO$$_2$$ data used in Chapter 12 to a ppp object, with a proper window.
3. Compute and plot the density of the NO$$_2$$ dataset, import the density as a stars object, and compute the volume under the surface.