# dbmss Package Reference

### Distance-Based Measures of Spatial Structures

Source:`vignettes/articles/reference.Rmd`

`reference.Rmd`

The *dbmss* package allows simple computation of spatial
statistic functions of distance to characterize the spatial structures
of mapped objects, including classical ones (Ripley’s K and others) and
more recent ones used by spatial economists (Duranton and Overman’s
\(K_d\), Marcon and Puech’s \(M\)). It relies on *spatstat* for
some core calculation.

This vignette contains its reference guide.

Characterizing spatial distributions of objects concerns numerous researchers in various fields. Amongst many questions, ecologists have been addressing the spatial attraction between species (Duncan 1991) or the non-independence of the location of dead trees in a forest (Haase et al. 1997). Analyzing the spatial distribution of plants, economists may be concerned by the location of new entrants (Duranton and Overman 2008) or by the location of shops according to the types of good sold (Picone, Ridley, and Zandbergen 2009). In epidemiology, researchers want to identify the spatial distribution of sick individuals in comparison to the population (Diggle and Chetwynd 1991). In these fields of research, the point process theory undoubtedly helps dealing with these questions. Exploratory statistics of point patterns widely rely on Ripley’s seminal work (Ripley 1976, 1977), namely the \(K\) function. A review of similar methods has been made by Marcon and Puech (2017) who called them distance-based measures of spatial concentration. We will write spatial structures here since both dispersion and concentration can be characterized. They are considered as novel and promising tools in spatial economics (Combes, Mayer, and Thisse 2008). The traditional approach to detect localization, i.e. the degree of dissimilarity between the geographical distribution of an industry and that of a reference (Hoover 1936), relies on discrete space (a country is divided in regions for example) and measures of inequality between zones, such as the classical Gini (1912) index or the more advanced Ellison and Glaeser (1997) index. This approach suffers several limitations, mainly the Modifiable Areal Unit Problem (MAUP): results depend on the way zones are delimited and on the scale of observation (Openshaw and Taylor 1979). Distance-based methods have the advantage to consider space as continuous, i.e. without any zoning, allowing detecting spatial structures at all scales simultaneously and solve MAUP issues.

These methods estimate the value of a function of distance to each
point calculated on a planar point pattern, typically objects on a map.
They all consist in counting *neighbors* (up to or exactly at the
chosen distance) around each *reference point* and transform
their number into a meaningful statistic. There are basically three
possible approaches: just count neighbors, count neighbors per surface
area or calculate the proportion of neighbors of interest among all
neighbors. They define the three families of functions: absolute (how
many neighbors are there?), topographic (how many neighbors per unit of
area?) and relative (what is the ratio of neighbors of interest?). The
function values are not the main motivation. The purpose is rather to
test the point pattern against the null hypothesis that it is a
realization of a known point process which does not integrate a property
of interest. The basic purpose of Ripley’s \(K\) is to test the observed point pattern
against complete spatial randomness (CSR), i.e. a homogeneous Poisson
process, to detect dependence between point locations (the null
hypothesis supposes independent points) assuming homogeneity (i.e. the
probability to find a point is the same everywhere). Ripley-like
functions, available in the *dbmss* package, can be classified in
three families:

- Topographic measures such as \(K\) take space as their reference. They have been widely used in ecology (Fortin and Dale 2005). They have been built from the point process theory and have a strong mathematical background.
- Relative measures such as \(M\) (Marcon and Puech 2010) compare the structure of a point type to that of another type (they can be considered as cases and controls). They have been developed in economics, where comparing the distribution of a sector of activity to that of the whole economic activity is a classical approach (Combes, Mayer, and Thisse 2008), but introduced only recently in ecology (Marcon, Puech, and Traissac 2012).
- Absolute functions such as \(K_d\) (Duranton and Overman 2005) have no reference at all but their value can be compared to the appropriate null hypothesis to test it.

Relative and absolute functions have been built from descriptive
statistics of point patterns, not related to the underlying point
processes, so they are seen as heuristic and ignored by the statistical
literature (Illian et al. 2008).
Topographic functions are implemented in the *spatstat* package
[Baddeley and Turner (2005)} but absolute
and relative functions are missing. We fill this gap by proposing the
*dbmss* package. It makes the computation of the whole set of
distance-based methods simple for empirical researchers by introducing
measures that are not available elsewhere and wrapping some topographic
measures available in *spatstat* so that all can be used the same
way.

Estimated values of the functions must be tested against a null
hypothesis. The usual empirical way to characterize a spatial structure
consists in computing the appropriate function and comparing it to the
quantiles of a large number of simulations of the null hypothesis to
reject (Kenkel 1988). We propose extended
possibilities to evaluate confidence envelopes, including *global
envelopes* (Duranton and Overman
2005), a goodness-of-fit test (Diggle
2013) and an analytic test (Lang and
Marcon 2013; Marcon, Traissac, and Lang 2013).

Definitions of all functions and formulas for their estimation can be found in Marcon and Puech (2017) and are not repeated here, but they are summarized in the statistical background section. Their implementation is presented in the package content section.

## Rationale and statistical background

We consider a map of points which often represents establishments in economic geography or trees in vegetation science. These points have two marks: a type (an industrial sector, a species…) and a weight (a number of employees, a basal area…). We want to apply to this point pattern a variety of exploratory statistics which are functions of distance between points, able to test the null hypothesis of independence between point locations. These functions are either topographic, absolute or relative. They can be interpreted as the ratio between the observed number of neighbors and the expected number of neighbors if points where located independently from each other. If reference and neighbor points are of the same type, the functions are univariate, to study concentration or dispersion. They are bivariate if the types differ, to address the colocation of types. We detail this approach below.

### Topographic, homogeneous functions

Topographic, homogeneous functions are Ripley’s \(K\) and its derivative \(g\). Their null hypothesis is a Poisson homogeneous process: rejecting it means that the process underlying the observed pattern is either not homogeneous or not independent. These functions are applied when homogeneity is assumed so independence only is tested by comparing the observed values of the function to their confidence envelope under CSR. Bivariate functions are tested against the null hypothesis of random labeling (points locations are kept unchanged but marks are redistributed randomly) or population independence (the reference point type is kept unchanged, the neighbor point type is shifted) following Goreaud and Pélissier (2003). The random labeling hypothesis considers that points preexist and their marks are the result of a process to test (e.g. are dead trees independently distributed in a forest?). The population independence one considers that points belong to two different populations with their own spatial structure and wants to test whether they are independent from each other.

Edge effect correction is compulsory to compute topographic
functions: points located close to boundaries have less neighbors
because of the lack of knowledge outside the observation window. The
*spatstat* package provides corrections following Ripley (1988), we use them.

### Topographic, inhomogeneous functions

\(K_{inhom}\) (Baddeley, Møller, and Waagepetersen 2000) is
the generalization of \(K\) to
inhomogeneous processes: it tests independence of points assuming the
intensity of the process is known. Empirically, it generally has to be
estimated from the data after assumptions on the way to do it relying on
the theoretical knowledge of the process. The null hypothesis
(*random position*) is that the pattern comes from an
inhomogeneous Poisson process of this intensity, which can be simulated.
Applying \(K_{inhom}\) to a single
point type allows using the *random location* null hypothesis,
following Duranton and Overman (2005):
observed points (with their marks) are shuffled among observed locations
to test for independence. Bivariate \(K_{inhom}\) null hypotheses may be random
labeling or population independence as defined by Marcon and Puech (2010): reference points are
kept unchanged, other points are redistributed across observed
locations.

\(K_{mm}\) (Penttinen 2006; Penttinen, Stoyan, and Henttonen
1992) generalizes \(K\) to
weighted points (weights are continuous marks of the points). Its null
hypothesis in *dbmss* is random location. Penttinen, Stoyan, and Henttonen (1992) inferred
the point process from the point pattern, and used the inferred process
to simulate the null hypothesis patterns. This requires advanced spatial
statistics techniques and knowledge about the process that is generally
not available. The random location hypothesis is a way to draw null
patterns simply, but ignoring the stochasticity of the point
process.

The \(D\) (Diggle and Chetwynd 1991) function compares the \(K\) function of points of interest (cases) to that of other points (controls). Its null hypothesis is random labeling.

### Absolute functions

In their seminal paper, Duranton and Overman (Duranton and Overman 2005) study the distribution of industrial establishments in Great Britain. Every establishment, represented by a point, is characterized by its position (geographic coordinates), its sector of activity (point type) and its number of employees (point weight). The \(K_d\) function (Duranton and Overman 2005) is the probability density to find a neighbor a given distance apart from a point of interest in a finite point process. The \(K^{emp}\) function integrates the weights of points: it is the density probability to find an employee \(r\) apart from an employee of interest.

\(K_d\) and \(K^{emp}\) are absolute measures since their value is not normalized by the measure of space or any other reference: for a binomial process, \(K_d\) increases proportionally to \(r\) if the window is large enough to ignore edge effects (the probability density is proportional to the perimeter of the circle of radius \(r\), Bonneu and Thomas-Agnan 2015), then edge effects make it decrease to 0 when \(r\) becomes larger than the window’s size: it is a bell-shaped curve. \(K_d\) values are not interpreted but compared to the confidence envelope of the null hypothesis, which is random location. Bivariate functions null hypotheses are random labeling, following Duranton and Overman (2005), i.e. point types are redistributed across locations while weights are kept unchanged, or population independence (as for \(K_{inhom}\)). It is not corrected for edge effects. \(K_d\) was designed to characterize the spatial structure of an economic sector, comparing it to the distribution of the whole activity. From this point of view, it has been considered as a relative function Marcon and Puech (2010). We prefer to be more accurate and distinguish it from strict relative functions which directly calculate a ratio or a difference between the number of points of the type of interest and the total number of points. What makes it relative is only its null hypothesis: changing it for random location (that of univariate \(K_{inhom}\)) would make univariate \(K_d\) behave as a topographic function (testing independence of the distribution supposing its intensity is that of the whole activity).

\(K_d\) is a leading tool in spatial economics. A great number of its applications can be found in the literature that confirms the recent interest for distance-based methods in spatial economics. A recent major study can be found in Ellison, Glaeser, and Kerr (2010).

### Relative functions

The univariate and bivariate \(M\)
function (Marcon and Puech 2010) are the
ratio of neighbors of interest up to distance \(r\) normalized by its value over the whole
domain. Their null hypotheses are the same as \(K_d\)’s. They do not suffer edge effects.
Marcon and Puech (2010) show that the
\(M\) function respect most of the
axioms generally accepted as the *good properties* to evaluate
geographic concentration in spatial economics (Combes and Overman 2004; Duranton and Overman
2005).

\(m\) (Lang, Marcon, and Puech 2020) is the density function measuring the same ratio as \(M\), at distance \(r\).

### Unification

Empirically, all estimators can be seen as variations in a unique framework: neighbors of each reference point are counted, their number is averaged and divided by a reference measure. Last, this average local result is divided by its reference value, calculated over the whole point pattern instead of around each point.

Choosing reference and neighbor point types allows defining univariate or bivariate functions, counting neighbors up to or at a distance defines cumulative or density functions, taking an area or a number of points as the reference measure defines topographic or relative functions. These steps are detailed for two functions to clarify them: we focus on Ripley’s \(g\) and Marcon and Puech’s \(M\) bivariate function. See Marcon and Puech (2017) for a full review.

Reference points are denoted \(x_i\), neighbor points are \(x_j\). For density functions such as \(g\), neighbors of \(x_i\) are counted at a chosen distance \(r\):

\[n\left(x_i,r\right){\rm =}\sum_{j{\rm ,\ }i\ne j}{k\left(\left\|x_i{\rm -}x_j\right\|,r\right)c\left(i,j\right)}\]

\(k(\left\|x_i{\rm -}x_j\right\|,r)\) is a kernel estimator, necessary to evaluate the number of neighbors at distance \(r\), and \(c(i,j)\) is an edge-effect correction (points located close to boundaries have less neighbors because of the lack of knowledge outside the observation window).

To compute the bivariate \(M\) function, reference points are of a particular type in a marked point pattern: \(x_i \in \mathcal{R}\), where \(\mathcal{R}\) is the set of points of the reference type. Neighbors of the chosen type are denoted \(x_j \in \mathcal{N}\). In cumulative functions such as \(M\), neighbors are counted up to \(r\):

\[n\left(x_i,r\right){\rm =}\sum_{x_j \in \mathcal{N}, i \ne j}{{\mathbf 1}\left(\left\|x_i{\rm -}x_j\right\|\le r\right)w\left(x_j\right)}\]

Points can be weighted, \(w(x_j)\) is the neighbor’s weight.

The number of neighbors is averaged then. \(n\) is the number of reference points:

\[\bar{n}\left(r\right)=\frac{1}{n}\sum_i{n\left(x_i,r\right)}\]

The average number of neighbors is compared to a reference measure. It may be a measure of space (the perimeter of the circle of radius \(r\) for \(g\)), defining topographic functions:

\[z\left(r\right)=2 \pi r\]

It may also be the average number of neighbors of all types in a relative function such as \(M\):

\[z\left(r\right)=\frac{1}{n}\sum_{j,i\ne j}{{\mathbf 1}\left(\left\|x_i{\rm -}x_j\right\|\le r\right)w\left(x_j\right)}\]

Finally, \(\frac{\bar{n}(r)}{z(r)}\) is compared to the same ratio computed on the whole window. For \(g\):

\[\frac{\bar{n}_0}{z_0}{\rm =}\frac{n-1}{A}\]

\(A\) is the area of the window, \(\bar{n}_0\) and \(z_0\) are the limit values of \(\bar{n}(r)\) and \(z(r)\) when \(r\) gets larger than the window’s size. For \(M\), it comes:

\[\frac{{\bar{n}}_0}{z_0}=\sum_i{\frac{W_\mathcal{N}}{W-w\left(x_i\right)}}\]

\(W_\mathcal{N}\) is the total weight of neighbor points, \(W\) that of all points. Finally, despite the functions are quite different (density vs. cumulative, topographic vs. relative, univariate vs. bivariate), both estimators can be written as \({\frac{{\bar{n}}}{z}}/{\frac{{\bar{n}}_0}{z_0}}\). Their value (except for absolute functions) can be interpreted as a location quotient: \(g(r)=2\) or \(M(r)=2\) means than twice more neighbors are observed at (or up to) distance \(r\) than expected on average, i.e. ignoring the point locations in the window. The appropriate function will be chosen among the toolbox according to the question raised.

## Package content

The *dbmss* package contains a full (within the limits of the
literature reviewed in section 2) set of functions to characterize the
spatial structure of a point pattern, including tools to compute the
confidence interval of the counterfactual. It allows addressing big
datasets thanks to C++ code used to calculate distances between pairs of
points (using *Rcpp* and
*RcppParallel* infrastructure: Eddelbuettel and François 2011;
Allaire et al. 2023). Computational requirements actually are an
issue starting from say 10,000 points (see
Ellison, Glaeser, and Kerr 2010 for instance). Memory requirement
is \(O(n)\), i.e. proportional to the
number of points to store their location and type. We use loops to
calculate distances and increment summary statistics rather than store a
distance matrix which is \(O(n^2)\),
following Scholl and Brenner (2015).
Computation time is \(O(n^2)\) because
\(n(n-1)/2\) pair distances must be
calculated.

We consider planar points patterns (sets of points in a 2-dimensional
space) with marks of a special kind: each point comes with a continuous
mark (its weight) and a discrete one (its type). We call this special
type of point pattern *weighted, marked, planar point patterns*
and define objects of class `wmppp`

, which derives from
`ppp`

class as defined in *spatstat*. Marks are a
dataframe with two columns, `PointWeight`

containing the
weights of points, and `PointTypes`

containing the types, as
factors.

A `wmppp`

object can be created by the
`wmppp()`

function which accepts a dataframe as argument, or
converted from a `ppp`

object by `as.wmppp()`

.
Starting from a csv file containing point coordinates, their type and
their weight in four columns, a `wmppp`

object can be created
by just reading the file with `read.csv()`

and applying
`wmppp()`

to the result. Options are available to specify the
observation window or guess it from the point coordinates and set
default weights or types to points when they are not in the dataframe,
see the package help for details. The simplest code to create a
`wmppp`

with 100 points is as follows. It draws point
coordinates between 0 and 1, and creates a `wmppp`

with a
default window, all points are of the same type named *All* and
their weight is 1.

`## Loading required package: Rcpp`

`## Loading required package: spatstat.explore`

`## Loading required package: spatstat.data`

`## Loading required package: spatstat.geom`

`## spatstat.geom 3.2-7`

`## Loading required package: spatstat.random`

`## spatstat.random 3.2-2`

`## Loading required package: nlme`

`## spatstat.explore 3.2-5`

```
Pattern <- wmppp(data.frame(X = runif(100), Y = runif(100)))
summary(Pattern)
```

```
## Marked planar point pattern: 100 points
## Average intensity 102.9038 points per square unit
##
## Coordinates are given to 8 decimal places
##
## Mark variables: PointWeight, PointType
## Summary:
## PointWeight PointType
## Min. :1 All:100
## 1st Qu.:1
## Median :1
## Mean :1
## 3rd Qu.:1
## Max. :1
##
## Window: rectangle = [0.004359, 0.9922431] x [0.0029645, 0.9866648] units
## (0.9879 x 0.9837 units)
## Window area = 0.971782 square units
```

### Distance-based functions

All functions are named `Xhat`

where `X`

is the
name of the function:

- Ripley’s \(g\) and \(K\), and \(K\)’s normalization, Besag’s \(L\) (1977);
- Penttinen’s \(K_{mm}\) and \(L_{mm}\);
- Diggle and Chetwynd’s \(D\);
- Baddeley et al.’s \(K_{inhom}\) and its derivative \(g_{inhom}\);
- Marcon and Puech’s \(M\);
- Duranton and Overman’s \(K_d\) (including its weighted version \(K^{emp}\)).

The suffix `hat`

has been used to avoid confusion with
other functions in R: `D`

exists in the *stats*
package for example.

Arguments are:

- A weighted, marked planar point pattern (a
`wmppp`

class object). The window can be a polygon or a binary image, as in*spatstat*. - A vector of distances.
- Optionally a reference and a neighbor point type to calculate bivariate functions, or equivalently the types of cases and controls for the \(D\) function.
- Some optional arguments, specific to some functions.

Topographic functions require edge-effect corrections, provided by
*spatstat*: the `best`

correction is systematically
used. Relative functions ignore the window. Technical details are
provided in help files.

These functions return an `fv`

object, as defined in
*spatstat*, which can be plotted.

### Confidence envelopes

The classical confidence intervals, calculated by Monte-Carlo
simulations (Kenkel 1988) are obtained by
the `XEnvelope`

functions, where `X`

is the
function’s name. Arguments are the number of simulations to run, the
risk level, those of the function and the null hypothesis to simulate.
These functions return a `dbmssEnvelope`

object which can be
plotted.

Null hypotheses have been discussed by Goreaud and Pélissier (2003) for topographic functions such as \(K\) and by Marcon and Puech (2010) for relative functions. The null hypothesis for univariate functions is random position (points are drawn in a Poisson process for topographic functions) or random location (points are redistributed across actual locations for relative functions). Bivariate functions support random labeling and population independence null hypotheses. The possible values of arguments are detailed in the help file of each function.

Building a confidence envelope this way is problematic because the
test is repeated at each distance. The underestimation of the risk has
been discussed by Loosmore and Ford
(2006). Duranton and Overman (2005)
proposed a global envelope computed by the repeated elimination of
simulations reaching an extreme value at any distance until the desired
level is reached. The argument `Global = TRUE`

is used to
obtain it instead of the local one.

### Examples

We illustrate the main features of the package by two examples. The
first one comes from the economic literature (Bonneu 2007) ^{1}

```
load("CSBIGS.Rdata")
Category <- cut(
Emergencies$M,
quantile(Emergencies$M, probs = c(0, 0.9, 1)),
labels = c("Other", "Biggest"),
include.lowest = TRUE
)
X <- wmppp(
data.frame(X = Emergencies$X, Y = Emergencies$Y, PointType = Category),
win=Region
)
X$window$units <- c("meter","meters")
X2 <- split(X)
marks(X2$Other) <- rep(1, X2$Other$n)
marks(X2$Biggest) <- rep(1, X2$Biggest$n)
par(mfrow = c(1, 2), mar = c(0, 0, 0, 0))
plot(X2$Other, main="", maxsize = 1, legend = FALSE)
text(514300, 1826800, "a")
plot(X2$Biggest, main = "", maxsize = 1, legend = FALSE)
text(514300, 1826800, "b")
```

The map of emergencies in the urban area of Toulouse, France, during year 2004 (about 33 km from south to north) shows (a) 20,820 emergencies recorded (many points are confused at the figure scale) and (b) the locations of the 10 percent most serious ones. The workload associated to each emergency (the number of men \(\times\) hours it required) is known. The original study tested the dependence between workload and location of emergencies: it did not exclude the null hypothesis of random labeling. We have a complementary approach here: we consider the 10 percent more serious emergencies, i.e. those which caused the highest workload. \(K_d\) may detect concentration (or dispersion) if, at a distance \(r\) from a serious emergency, the probability to find another serious emergency is greater (or lower) than that of finding an emergency whatever its workload:

The `Emergencies`

data frame contains point coordinates
(in meters) in columns `X`

and `Y`

and workload in
column `M`

. The second line of the code creates a vector
containing a factor describing the workload to separate its 10% higher
values. A `wmppp`

object is created then, containing the
points and their mark. The `KdEnvelope`

function is run from
0 to 10km by steps of 100m for the most serious emergencies. The figure
below shows that the 10% most serious emergencies are more dispersed
than the distribution of all emergencies at all distances up to
10km.

```
KdE <- KdEnvelope(
X,
r = seq(0, 10000, 100),
NumberOfSimulations = 1000,
ReferenceType = "Biggest",
Global = TRUE
)
autoplot(KdE, main = "")
```

The solid, black curve is \(K_d\). The dotted red curve is the average simulated value and the shaded area is the confidence envelope under the null hypothesis of random location. The risk level is 5%, 1000 simulations have been run. distances are in meters.

This opens the way to discuss on the optimal location of fire stations.

The second example uses the `paracou16`

point pattern
provided in the package. It represents the distribution of trees in a
4.1-ha tropical forest plot in the Paracou field station in French
Guiana (Gourlet-Fleury, Guehl, and Laroussinie
2004). It contains 2426 trees, whose species is either *Qualea
rosea*, *Vouacapoua americana* or Other (one of more than 300
species). Weights are basal areas (the area of the stems virtually cut
1.3 meter above ground), measured in square centimeters.

```
autoplot(
paracou16,
labelSize = expression("Basal area (" ~cm^2~ ")"),
labelColor = "Species"
)
```

On the map, circles are centered on trees in the forest plot (the containing rectangle is 200m wide by 250m long). Circle sizes are proportional to the basal areas of trees.

The question to test is dependence between the distributions of the two species of interest. Bivariate \(M(r)\) is calculated for \(r\) between 0 and 30 meters. 1000 simulations are run to build the global confidence envelope.

```
Envelope <- MEnvelope(
paracou16,
r = seq(0, 30, 2),
NumberOfSimulations = 1000,
Alpha = 0.05,
ReferenceType = "V. Americana",
NeighborType = "Q. Rosea",
SimulationType = "RandomLabeling",
Global = TRUE
)
autoplot(Envelope, main = "", ylim = c(0, 20))
```

\(M(r)\) values of *Qualea
rosea* around *Vouacapoua Americana* are plotted. The solid,
black curve is \(M\). The dotted red
curve is the average simulated value. The shaded area is the confidence
envelope. \(M=1\) is expected if points
are independently distributed. The risk level is 5%, 1000 simulations
have been run. Distances are in meters.

The calculated function is \(M\),
showing the repulsion between *V. Americana* and *Q.
rosea* up to 30 m. Significance is unclear, since the observed
values of the function are very close to the lower bound of the
envelope. The complete study, with a larger dataset allowing significant
results, can be found in Marcon, Puech, and
Traissac (2012).

### Goodness-of-fit test

A Goodness-of-fit test for \(K\) has
been proposed by Diggle (2013), applied to
\(K\) by Loosmore and Ford (2006) and to \(M\) by Marcon,
Puech, and Traissac (2012). It calculates the distance between
the actual values of the function and its average value obtained in
simulations of the null hypothesis. The same distance is calculated for
each simulated point pattern, and the returned \(p\)-value of the test if the ratio of
simulations whose distance is larger than that of the real point
pattern. The test is performed by the `GoFtest`

function
whose argument is the envelope previously calculated (actually, the
function uses the simulation values).

Applied to the example of Paracou trees, the \(p\)-value is:

`GoFtest(Envelope)`

`## [1] 0.267`

### Ktest

The `Ktest`

has been developed by Lang and Marcon (Lang and Marcon 2013; Marcon, Traissac, and Lang
2013). It does not rely on simulations and returns the \(p\)-value to erroneously reject complete
spatial randomness (CSR) given the values of \(K\). It relies on the exact variance of
\(K\) calculated with edge-effect
corrections. It only works in a rectangular window.

The following example tests a 1.5-ha subset of `paracou16`

(100m by 150m, origin at the southwestern corner).

```
data("paracou16")
RectWindow <- owin(c(300, 400), c(0, 150))
X <- paracou16[RectWindow]
(pKtest <- Ktest(X, seq(5, 50, 5)))
```

`## [1] 0.002682576`

It rejects CSR (p=0.0027).

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