```
library(sf)
# Linking to GEOS 3.10.2, GDAL 3.4.3, PROJ 8.2.1; sf_use_s2() is TRUE
<- 30
n set.seed(13531) # remove this to create another random sequence
<- data.frame(x = runif(n), y = runif(n)) |>
xy st_as_sf(coords = c("x", "y"))
```

# 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`

(Baddeley, Turner, and Rubak 2022), 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`

(Gabriel et al. 2022) 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 data set 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 or intensity function, which measures the number of points per area unit; this function is spatially varying for a
*inhomogeneous*point process - second order properties, e.g. pairwise interactions: given a constant or varying intensity function, are points distributed independently
*from one another*, or do they tend to attract each other (clustering) or repulse each other (appear regularly distributed, compared to complete spatial randomness)

## 11.1 Observation window

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

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

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

this is shown in Figure 11.1 .

```
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(xy, add = TRUE)
plot(w2, axes = TRUE, col = 'grey')
plot(xy, add = TRUE, cex = .5)
```

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:

```
<- c(w1, st_geometry(xy)) |> as.ppp())
(pp1 # Planar point pattern: 30 points
# window: polygonal boundary
# enclosing rectangle: [0, 1] x [0, 1] units
<- st_buffer(st_centroid(w2), 1.2)
c1 <- c(c1, st_geometry(xy)) |> as.ppp())
(pp2 # 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 x 3 layout is shown in Figure 11.2)

```
par(mfrow = c(1, 2), mar = rep(0, 4))
<- quadratcount(pp1, nx=3, ny=3)
q1 <- quadratcount(pp2, nx=3, ny=3)
q2 plot(q1, main = "")
plot(xy, add = TRUE)
plot(q2, main = "")
plot(xy, add = TRUE)
```

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)
quadrat.test(pp2, 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: 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 .

```
<- density(pp1, sigma = bw.diggle)
den1 <- density(pp2, sigma = bw.diggle) den2
```

```
par(mfrow = c(1, 2), mar = c(0,0,1.1,2))
plot(den1)
plot(pp1, add=TRUE)
plot(den2)
plot(pp1, add=TRUE)
```

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

```
library(stars)
# Loading required package: abind
<- st_as_stars(den1)
s1 <- st_as_stars(den2))
(s2 # 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 0.01875 [x]
# y 1 128 -0.7 0.01875 [y]
```

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

```
$a <- st_area(s1) |> suppressMessages()
s1$a <- st_area(s2) |> suppressMessages()
s2with(s1, sum(v * a, na.rm = TRUE))
# [1] 29
with(s2, sum(v * a, na.rm = TRUE))
# [1] 30.7
```

More exciting applications involve e.g. 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 center of the “cluster”, and these distance are available in a `stars`

object:

```
<- st_sfc(st_point(c(0.5, 0.5)))
pt 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 as the left-hand-side of the `formula`

:

```
<- ppm(pp2 ~ dist, data = list(dist = as.im(s2["dist"]))))
(m # Nonstationary Poisson process
#
# 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.

`plot(m, se = FALSE)`

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 0.01875 [x]
# y 1 128 -0.7 0.01875 [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
```

Also, when converting to a `spatstat`

data structure (e.g. to a `ppp`

, create a density image, convert back to `stars`

) we loose the coordinate reference system we started with. It can be set back e.g. 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 numeric mark:

```
longleaf# Marked planar point pattern: 584 points
# marks are numeric, of storage type 'double'
# window: rectangle = [0, 200] x [0, 200] metres
<- st_as_sf(longleaf)
ll 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*, as of which `LINESTRING`

is connected to others, is not present in the `sf`

object. Package `sfnetworks`

(van der Meer et al. 2022) would be a candidate package to hold such information, or e.g. 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):

```
<- 30 / st_area(w2) # intensity
kappa <- st_sample(w2, kappa = kappa, mu = 3, scale = 0.05,
th type = "Thomas")
nrow(th)
# [1] 82
```

## Code

```
par(mar = rep(0, 4))
plot(w2)
plot(th, add = TRUE)
```

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"`

(González 2010).

## Code

```
par(mar = rep(0, 4), mfrow = c(1, 2))
library(s2)
<- as_s2_geography(TRUE) # Earth
g <- s2_data_countries()
co <- s2_difference(g, s2_union_agg(co)) # oceans
oc <- s2_buffer_cells(as_s2_geography("POINT(-30 -10)"), 9800000) # visible half
b <- s2_intersection(b, oc) # visible ocean
i <- s2_intersection(b, co)
co = st_crs("+proj=ortho +lat_0=-10 +lon_0=-30")
ortho # 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()
|> s2_intersection(i) |> st_as_sfc() -> pts
pts # add:
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()
|> s2_intersection(i) |> st_as_sfc() -> pts
pts st_transform(pts, ortho) |> plot(add = TRUE, pch = 3, cex = .5)
```

## 11.6 Exercises

- After loading
`spatstat`

, recreate the plot obtained by`plot(longleaf)`

by using`ggplot2`

and`geom_sf()`

, and by`sf::plot()`

. - Convert the sample locations of the NO\(_2\) data used in Chapter 12 to a
`ppp`

object, with a proper window. - Compute and plot the density of the NO\(_2\) dataset, import the density as a
`stars`

object and compute the volume under the surface.