---
title: "VS0 - Introduction to pcds"
author: "Elvan Ceyhan"
date: '`r Sys.Date()` '
output:
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
bibliography: References.bib
vignette: >
  %\VignetteIndexEntry{VS0 - Introduction to pcds}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

<style>
.math {
  font-size: small;
}
</style>

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE,comment = "#>",fig.width=6, fig.height=4, fig.align = "center") 
options(rmarkdown.html_vignette.check_title = FALSE)
```

\newcommand{\X}{\mathcal{X}}
\newcommand{\Y}{\mathcal{Y}}
\newcommand{\NAS}{N_{AS}}
\newcommand{\NCS}{N_{CS}}
\newcommand{\NPE}{N_{PE}}
\newcommand{\y}{\mathsf{y}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\U}{\mathcal{U}}
\newcommand{\TY}{T(\Y_3)}
\newcommand{\mS}{\mathcal{S}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\argmin}{\text{arg min}}
\newcommand{\I}{\mathbf{I}}
\newcommand{\g}{\gamma}

First we load the `pcds` package:
```{r setup, message=FALSE, results='hide'}
library(pcds)
```

<!-- First we load the `pcds` package and the other packages: -->
<!-- ```{r setup, message=FALSE, results='hide'} -->
<!-- #library(pcds) -->
<!-- #library(tidyverse) -->
<!-- #library(plot3D) -->
<!-- #theme_set(theme_bw()) -->

<!-- library(miceadds) -->
<!-- miceadds::source.all("C:/Users/ezc0066/Documents/PCDRelated/pcds/R", grepstring="\\.R",  print.source=TRUE ) -->
<!-- ``` -->

# Introduction
`pcds` is an `R` package that stands for *proximity catch digraphs* (*PCDs*) and 
provides construction and visualization of the PCDs, 
and the spatial pattern tests (for inference on spatial interaction between classes or species) 
based on the two graph invariants of the PCDs. 
These invariants are the *domination number* and the *arc density*. 
The package provides a set of functions for the construction and visualization of three PCD families, 
namely 
*arc-slice PCDs* (*AS-PCDs*), 
*proportional-edge PCDs* (*PE-PCDs*), and 
*central similarity PCDs* (*CS-PCDs*),
and for spatial inference based on two of these PCD families,
namely PE- and CS-PCDs. 
Here, the spatial inference concerns testing class/species interaction for point pattern data,
usually in two dimensional space. 
The spatial interaction patterns of interest are segregation and association. 
*Segregation* is the pattern in which classes tend to repel each other 
in the sense that points tend to be clustered around points from the same class
(forming same-class clusters), 
while *association* is the pattern in which points from one class tend to cluster around points from another class
(forming mixed-class clusters).  
The one-dimensional versions of the PCD functions are also provided 
where AS-PCD is a special case of PE-PCDs or CS-PCDs. 
The one-dimensional versions are currently used for testing uniformity of points,
instead of interaction between classes (although it is possible to use them for this purpose as well). 
We only extend the PE-PCD construction and visualization to three dimensions in this package.

The vignette files for the `pcds` package are written as sections or chapters of a larger main vignette file
and are organized as "VSk_Title" where "k" is the section number 
and "Title" is the corresponding markdown file name,
starting with "VS0_Intro".
It is recommended to read the vignettes in this order for more efficient and informed use, 
however, they can be used in any order the users/readers prefer, 
but some tools or concepts may have been introduced in an earlier section (mostly with references).

The goal of these vignette files or sections is to facilitate graph abstraction of spatial point data and 
make it easier for users to adopt `pcds` by providing a comprehensive overview of the package's contents and 
detailed examples and illustration of certain functions. 
We begin with the 

- introduction and overview of PCDs,
- followed by worked out examples for an artificial 2D data, a real-life 2D data, and an artificial 1D data
with  discussions of how to work with `pcds`.

Then we, illustrate

- PCD construction in one Delaunay cell, which is an interval in 1D, a triangle in 2D, and a tetrahedron in 3D space,
- data generation from various spatial point patterns in 2D space,
- finding local extrema in Delaunay cells, and
- finally some auxiliary functions in Euclidean geometry.

The discussion covers the structure of function arguments, 
the required input data formats, and the various output formats. 
The subsequent sections provide visualization of the proximity regions, 
and the associated PCDs, and computation of domination number and arc-density of the PCDs, 
and the computation of the large-sample tests based on these invariants. 

# Package Contents

For ease of exposition,
we have grouped the package contents according their functionality and theme:

 - Utility functions
 - PCD functions
 - Pattern Generation Functions
 - S3 methods^[For example, 
 calling the generic function `summary(x)` on an object `x` with class `PCDs` actually dispatches the method `summary.PCDs` on `x`,
 which is equivalent to calling `summary.PCDs(x)`. 
 For a nice introduction, see _Advanced R_ by Hadley Wickham, available [online](http://adv-r.had.co.nz/OO-essentials.html).]
 - Datasets
 
Because all of these groups contain many functions, 
we organize them into subgroups by purpose. 
Below, we display each group of functions in a table with one column per subgroup. 

-----------------------------------------------------------------------------
Utility              PCDs              Pattern Generation    S3 Methods 
--------------- -------------------- -------------------- -------------------
AuxDomination      ArcSliceFunctions    PatternGen        ClassFunctions     

AuxGeometry      PropEdge*          \   \        

AuxTriangular    CentSim*         \    \     

AuxExtrema  \    \   \ 

-----------------------------------------------------------------------------
Table: Package Contents: Grouping of the Functions

PropEdge* contains PropEdge1D, PropEdge2D, and PropEdge3D functions, whereas CentSim* contains CentSim1D and CentSim2D functions only.

ClassFunctions contain functions like `summary`, `print.summary`, and `plot` of the following object classes: 
`Lines`,`TriLines`, `Lines3D`, `Planes`, `Patterns`, `Uniform`, `Extrema`, and `PCDs`. 
Among these objects, `Lines`, `TriLines`, `Lines3D`, and `Planes` facilitate visualization of the various geometric structures, 
proximity regions, and the corresponding digraphs, 
while `Patterns` is used for spatial point pattern generation (with `Uniform` being a special case), 
and `PCDs` pertains to the actual PCDs (the number of arcs, visualization of the digraph, and so on).

In all the `pcds` functions,
points are vectors, and data sets are either matrices or data frames.

# Proximity Catch Digraphs 
We illustrate PCDs in a two-class setting, 
extension to multi-class setting can be done in a pair-wise fashion or one-vs-rest
fashion for the classes.
For two classes, $\X$ and $\Y$, of points, 
let $\X$ be the class of interest (i.e. the *target class*)
and $\Y$ be the reference class (i.e. the *non-target class*)
and $\X_n$ and $\Y_m$ be samples of size $n$ and $m$ from classes $\X$ and $\Y$, respectively.
<!-- That is, we specify $\X$ as the *target class* and $\Y$ as the *non-target class*.  -->
The *proximity map* $N(\cdot): \Omega \rightarrow 2^{\Omega}$ associates with each point $x \in \X$, 
a *proximity region* $N(x) \subset \Omega$. 
Consider the data-random (or vertex-random) proximity catch digraph (PCD) $D=(V,A)$ 
with vertex set $V=\X$ and arc set $A$ defined by $(u,v) \in A \iff \{u,v\}\subset \X$ and $v \in N(u)$. 
The digraph $D$ depends on the (joint) distribution of $\X$ and on the map $N(\cdot)$. 
The adjective proximity --- for the digraph $D$ and for the map $N(\cdot)$ --- 
comes from thinking of the region $N(x)$ as representing those points in $\Omega$ "close" to $x$. 
The binary relation $u \sim v$, which is defined as $v \in N(u)$, is asymmetric, 
thus the adjacency of $u$ and $v$ is represented with *directed edges* or *arcs* which yield a digraph instead of a graph.
See @chartrand:2010 and @west:2001 for more on graphs and digraphs.
<!-- Let the data set $\X$ be composed of two non-empty sets,  -->
<!-- $\X$ and $\Y$, with sample sizes $n:=|\X|$ and $m:=|\Y|$, respectively.  -->
<!-- We investigate the PCD $D$, associated with $\X$ against $\Y$.  -->

<!-- Hence, in the definitions of our PCDs, the only difference is switching the roles of $\X$ and $\Y$.  -->

In the PCD approach the points correspond to observations from class $\X$ and
the proximity regions are defined to be (closed) regions (usually convex regions or simply triangles)
based on class $\X$ and $\Y$ points
and the proximity region for a class $\X$ point get larger as the distance of this class $\X$ point
from class $\Y$ points increases.

## Proximity Map Families
We briefly define three proximity map families.
Let $\Omega = \R^d$
and let $\Y_m=\left \{\y_1,\y_2,\ldots,\y_m \right\}$ be $m$ points in
general position in $\R^d$ and $T_i$ be the $i^{th}$ Delaunay cell
for $i=1,2,\ldots,J_m$, where $J_m$ is the number of Delaunay cells
based on $\Y_m$.
Let $\X_n$ be a set of iid random variables from distribution $F$ in
$\R^d$ with support $\mS(F) \subseteq \C_H(\Y_m)$
where $\C_H(\Y_m)$ stands for the convex hull of $\Y_m$.
For illustrative purposes, we focus on $\R^2$ where
a Delaunay tessellation is a *triangulation*, provided that no more
than three $\Y$ points are cocircular (i.e., lie on the same circle).
Furthermore, for simplicity,
let $\Y_3=\{\y_1,\y_2,\y_3\}$ be three non-collinear points
in $\R^2$ and $\TY=T(\y_1,\y_2,\y_3)$ be the triangle
with vertices $\Y_3$.
Let $\X_n$ be a set of iid random variables from $F$ with
support $\mS(F) \subseteq \TY$.
Let $\U(T\left(\Y_3 \right))$ be the uniform distribution on $T\left(\Y_3 \right)$.
If $F=\U(\TY)$, a composition of translation,
rotation, reflections, and scaling, denoted $\phi_b(\TY)$,
will take any given triangle $\TY$
to the standard basic triangle $T_b=T((0,0),(1,0),(c_1,c_2))$
with $0 < c_1 \le 1/2$, $c_2>0$,
and $(1-c_1)^2+c_2^2 \le 1$, preserving uniformity.
That is, if $X \sim \U(\TY)$ is transformed in the same manner to,
say $X'$, then we have $X' \sim \U(T_b)$.
In fact this will hold for data from any distribution $F$
up to scale.
Furthermore, $T_b$ can be transformed to the *standard equilateral triangle*
$T_e=T(A,B,C)$ with vertices $A=(0,0)$, $B=(1,0)$, and $C=(1/2,\sqrt{3}/2)$ by a transformation $\phi_e$
and $\phi_e(X') \sim \U(T_e)$.
That is uniform points in any triangle can be mapped to points uniformly distributed in the standard equilateral triangle
by a composition of $\phi_b(\TY)$ and $\phi_e$ (in the form $\phi_e \circ \phi_b(\TY)$).

## Arc-Slice Proximity Maps and Associated Proximity Regions {#sec:AS-PCD-construction}
We define the *arc-slice proximity region* with $M$-vertex regions for a point $x \in \TY$ as follows;
see also Figure \@ref(fig:NAS-example).
Using a center $M$ of $\TY$,
we partition $\TY$ into "vertex regions" $R_V(\y_1)$, $R_V(\y_2)$, and $R_V(\y_3)$.
If $M$ is in the interior of $\TY$,
we use line segments from the center $M$ of $\TY$ to the opposite edges
as extensions of the lines joining the vertices and the center $M$
to form the vertex regions.
For $x \in \TY \setminus \Y_3$, let $v(x) \in \Y_3$ be the
vertex in whose region $x$ falls, so $x \in R_V(v(x))$.
If $x$ falls on the boundary of two vertex regions,
we assign $v(x)$ arbitrarily to one of the adjacent regions.
The arc-slice proximity region is 
$\NAS(x):=\overline B(x,r(x)) \cap \TY$ where $\overline B(x,r(x))$ is the closed
ball centered at $x$ with radius $r(x):=d(x,v(x))$.
To make the dependence on $M$ explicit, we also use the notation$\NAS(\cdot,M)$.
A natural choice for the radius is $r(x):=\min_{\y \in \Y}d(x,\y)$
which implicitly uses the $CC$-vertex regions,
since $x \in R_{CC}(\y)$ iff $\y=\argmin_{u \in \Y}d(x,u)$.
<!-- To make this dependence explicit, we use $\NAS(\cdot,M_{CC})$. -->
See Figure \@ref(fig:NAS-example) for
$\NAS(x,M_{CC})$ with $x \in R_{CC}(\y_2)$.
See @ceyhan:comp-geo-2010 for more detail on AS proximity regions.

![(#fig:NAS-example) Illustration of the construction of arc-slice proximity region,
$\NAS(x,M_{CC})$ with an $x \in R_{CC}(\y_2)$.](NAS_definition.png){width=65%}

## Proportional-Edge Proximity Maps and Associated Proximity Regions {#sec:PE-PCD-construction}
<!-- For the expansion parameter $r \in [1,\infty]$, -->
We define the *proportional-edge proximity map* with expansion parameter $r \ge 1$ as follows;
see also Figure \@ref(fig:NPE-example).
Using a center $M$ of $\TY$,
we partition $\TY$ into "$M$-vertex regions" as in Section \@ref(sec:PE-PCD-construction).
Let $e(x)$ be the edge of $\TY$ opposite $v(x)$, the vertex whose region contains $x$,
$\ell(x)$ be the line parallel to $e(x)$ through $x$,
and $d(v(x),\ell(x))$ be the Euclidean distance from $v(x)$ to $\ell(x)$.
For $r \in [1,\infty)$, let $\ell_r(x)$ be the line parallel to $e(x)$
such that $d(v(x),\ell_r(x)) = rd(v(x),\ell(x))$ and $d(\ell(x),\ell_r(x)) < d(v(x),\ell_r(x))$.
Let $T_{PE}(x,r)$ be
the triangle similar to
and with the same orientation as $\TY$
having $v(x)$ as a vertex
and $\ell_r(x)$ as the opposite edge.
Then the *proportional-edge proximity region*
$\NPE(x,r)$ is defined to be $T_{PE}(x,r) \cap \TY$.
To make the dependence on $M$ explicit, we also use the notation$\NPE(x,r,M)$.
A natural choice for the center is the center of mass (CM) yielding the $CM$-vertex regions,
which have the same area (equaling one-third of the area of $\TY$).
Notice that $r \ge 1$ implies $x \in \NPE(x,r)$.
Note also that
$\lim_{r \rightarrow \infty} \NPE(x,r) = \TY$
for all $x \in \TY \setminus \Y_3$,
so we define $\NPE(x,\infty) = \TY$ for all such $x$.
For $x \in \Y_3$, we define $\NPE(x,r) = \{x\}$ for all $r \in [1,\infty]$.
See @ceyhan:dom-num-NPE-SPL, @ceyhan:arc-density-PE, and @ceyhan:test2014 for more detail.

![(#fig:NPE-example) Illustration of the construction of proportional-edge proximity region, $\NPE(x,r=2)$ (shaded region)
for an $x \in R_V(\y_1)$ where $d_1=d(v(x),\ell(v(x),x))$ and
$d_2=d(v(x),\ell_2(v(x),x))=2\,d(v(x),\ell(v(x),x))$.](NPE_definition.png){width=75%}

## Central Similarity Proximity Maps and Associated Proximity Regions {#sec:CS-PCD-construction}

<!-- For the expansion parameter $\tau \in (0,\infty]$, -->
We define the *central similarity proximity map* with expansion parameter $\tau>0$ as follows;
see also Figure \@ref(fig:NCS-example).
Let $e_j$ be the edge opposite vertex $\y_j$ for $j=1,2,3$,
and let "$M$-edge regions" $R_E(e_1)$, $R_E(e_2)$, $R_E(e_3)$
partition $\TY$ using line segments from the
center $M$ in the interior of $\TY$ to the vertices.
For $x \in (\TY)^o$, let $e(x)$ be the
edge in whose region $x$ falls; $x \in R_E(e(x))$.
If $x$ falls on the boundary of two edge regions we assign $e(x)$ arbitrarily.
For $\tau > 0$, 
the *central similarity proximity region*
$\NCS(x,\tau)$ is defined to be the triangle $T_{CS}(x,\tau) \cap \TY$ with the following properties:

 (i) For $\tau \in (0,1]$,
the triangle
$T_{CS}(x,\tau)$ has an edge $e_\tau(x)$ parallel to $e(x)$ such that
$d(x,e_\tau(x))=\tau\, d(x,e(x))$
and
$d(e_\tau(x),e(x)) \le d(x,e(x))$
and
for $\tau >1$,
$d(e_\tau(x),e(x)) < d(x,e_\tau(x))$
where $d(x,e(x))$ is the Euclidean distance from $x$ to $e(x)$,

 (ii) the triangle $T_{CS}(x,\tau)$ has the same orientation as and is similar to $\TY$,
 
 (iii) the point $x$ is at the center of mass of $T_{CS}(x,\tau)$.

Note that (i) implies that the expansion parameter is $\tau$,
(ii) implies "similarity", and
(iii) implies "central" in the name, (parameterized) *central similarity proximity map*.
To make the dependence on $M$ explicit, we also use the notation$\NCS(x,\tau,M)$.
A natural choice for the center is the center of mass (CM) yielding the $CM$-edge regions,
which have the same area (equaling one-third of the area of $\TY$).
Notice that $\tau>0$ implies that $x \in \NCS(x,\tau)$ and,
by construction, we have
$\NCS(x,\tau)\subseteq \TY$ for all $x \in \TY$.
For $x \in \partial(\TY)$ and $\tau \in (0,\infty]$, we define $\NCS(x,\tau)=\{x\}$.
For all $x \in \TY^o$ the edges
$e_\tau(x)$ and $e(x)$ are coincident iff $\tau=1$.
Note also that
$\lim_{\tau \rightarrow \infty} \NCS(x,\tau) = \TY$
for all $x \in (\TY)^o$,
so we define $\NCS(x,\infty) = \TY$ for all such $x$.
The central similarity proximity maps in @ceyhan:CS-JSM-2003 and @ceyhan:arc-density-CS
are $\NCS(\cdot,\tau)$ with $\tau=1$ and $\tau \in (0,1]$, respectively,
and in @ceyhan:test2014 with $\tau>1$.

![(#fig:NCS-example) Illustration of the
construction of central similarity proximity region, $\NCS(x,\tau=1/2)$ (shaded region)
for an $x \in R_E(e_3)$
where $h_2=d(x,e_3^\tau(x))=\frac{1}{2}\,d(x,e(x))$ and $h_1=d(x,e(x))$.](NCS_definition.png){width=75%}

## Delaunay Tessellation
The convex hull of the non-target class $C_H(\Y_m)$ can be partitioned 
into Delaunay cells through the *Delaunay tessellation* of $\Y_m \subset \mathbb{R}^d$. 
The Delaunay tessellation becomes a *triangulation* in $\mathbb R^2$
which partitions $C_H(\Y_m)$ into non-intersecting triangles. 
For $\Y$ points in general position, 
the triangles in the Delaunay triangulation satisfy the property 
that the circumcircle of a triangle contain no $\Y$ points 
except for the vertices of the triangle. 
In higher dimensions (i.e., $d >2)$, Delaunay cells are $d$-simplices (for example, a tetrahedron in $\mathbb{R}^3$).
Hence, the $C_H(\Y_m)$ is the union of a set of disjoint $d$-simplices $\{\mathfrak S_k\}_{k=1}^K$ 
where $K$ is the number of $d$-simplices, or Delaunay cells. 
Each $d$-simplex has $d+1$ non-co(hyper)planar vertices 
where none of the remaining points of $\Y_m$ are in the interior of the circumsphere of the 
simplex (except for the vertices of the simplex which are points from $\Y_m$). 
Hence, simplices of the Delaunay tessellations are more likely to be acute 
(simplices will not have substantially large inner angles). 
Note that Delaunay tessellation is the dual of the *Voronoi diagram* of the points $\Y_m$. 
A *Voronoi diagram* is a partitioning of $\mathbb{R}^d$ into convex polytopes 
such that the points inside each polytope is closer to the point associated 
with the polytope than any other point in $\Y$. 
Hence, a polytope $V(y)$ associated with a point $\y \in \Y_m$ is defined as 
$$V(y)=\left\{v \in \mathbb{R}^d: \lVert v-y \rVert \leq \lVert v-z \rVert \ \text{ for all } z \in \Y_m \setminus \{y\} \right\}.$$ 

Here, $\lVert\cdot\rVert$ stands for the usual Euclidean norm. 
Observe that the Voronoi diagram is unique for a fixed set of points $\Y_m$. 
A *Delaunay graph* is constructed by joining the pairs of points in $\Y_m$ 
whose boundaries of Voronoi polytopes have nonempty intersections.
The edges of the Delaunay graph constitute a partitioning of $C_H(\Y_m)$
providing the *Delaunay tessellation*. 
By the uniqueness of the Voronoi diagram, 
the Delaunay tessellation is also unique (except for cases where $d+1$ or more points lying on the same hypersphere). 
<!-- See Figure \@ref(fig:DTfig)  -->
Run the below code
for an illustration of Delaunay triangulation of 20 uniform $\Y$ points in the unit square
$(0,1) \times (0,1)$.
More detail on Voronoi diagrams and Delaunay tessellations can be found in @okabe:2000.

```{r DTfig, eval=FALSE, fig.cap = "Delaunay triangulation of 20 uniform $Y$ points in the unit square $(0,1)$-by-$(0,1)$."}
ny<-20; 

set.seed(1)
#Xp<-cbind(runif(nx),runif(nx))
Yp<-cbind(runif(ny),runif(ny))

#oldpar <- par(no.readonly = TRUE)
plotDeltri(Yp,Yp,xlab="",ylab="",main="Delaunay Triangulation of Y points")
```

## Graph Invariants: Arc Density and Domination Number
**Arc Density:** 
The *arc density* of a digraph $D=(V,A)$
of order $|V| = n$,
denoted $\rho(D)$,
is defined as
$$
\rho(D) = \frac{|A|}{n(n-1)}
$$
where $|\cdot|$ stands for set cardinality (@janson:2000).
Thus $\rho(D)$ represents the ratio of the number of arcs
in the digraph $D$ to the number of arcs in the complete symmetric
digraph of order $n$, which has $n(n-1)$ arcs.

If $X_1,X_2,\ldots,X_n \stackrel{iid}{\sim} F$, then
the relative density
of the associated data-random PCD,
denoted $\rho(\X_n;h,N)$, is a $U$-statistic,
$$\rho(\X_n;h,N) =
  \frac{1}{n(n-1)}
   \underset{i<j}{\sum\sum}h(X_i,X_j;N) $$
  
where
\begin{aligned}
h(X_i,X_j;N)&= \I \{(X_i,X_j) \in A\}+ \I \{(X_j,X_i) \in A\} \nonumber \\
       &= \I \{X_j \in N(X_i)\}+ \I \{X_i \in N(X_j)\}
\end{aligned}
with $\I\{\cdot\}$ being the indicator function.
We denote $h(X_i,X_j;N)$ as $h_{ij}$ for brevity of notation.
Since the digraph is asymmetric, $h_{ij}$ is defined as
the number of arcs in $D$ between vertices $X_i$ and $X_j$, in order to produce a symmetric kernel with finite variance (@lehmann:2004).

See @ceyhan:arc-density-PE, @ceyhan:arc-density-CS, and @ceyhan:test2014
for arc density of PE-PCDs and its use for spatial interaction for 2D data;
and @ceyhan:metrika-2012 and @ceyhan:revstat-2016 for arc density of PE-PCDs for 1D data
and its use for testing uniformity.

**Domination Number:** 
In a digraph $D=(V,A)$, a vertex $v \in V$ *dominates*
itself and all vertices of the form $\{u: (v,u) \in A\}$.
A *dominating set* $S_D$ for the digraph $D$ is a subset of $V$
such that each vertex $v \in V$ is dominated by a vertex in $S_D$.
A *minimum dominating set* $S^*_{D}$ is a dominating set of
minimum cardinality and the *domination number* $\g(D)$ is
defined as $\g(D):=|S^*_{D}|$ (see, e.g., @lee:1998).
<!-- where $|\cdot|$ denotes the set cardinality functional. -->
If a minimum dominating set is of size one, we call it a *dominating point*.
Note that for $|V|=n>0$, $1 \le \g(D) \le n$,
since $V$ itself is always a dominating set.
See @ceyhan:dom-num-NPE-SPL  and  @ceyhan:dom-num-NPE-Spat2011
for the domination number and its use for testing spatial interaction patterns in 2D data
and @ceyhan:stat-2020 for testing uniformity of 1D data.

The distribution of the domination number and arc density for AS-PCDs do not change for uniform data in a triangle $\TY$
when the data points are transformed to (uniform) points in the standard basic triangle $T_b$ (using $\phi_b(\TY)$)
but not when $\phi_e$ is applied to the uniform data in $T_b$.
So, one can focus on $T_b$ for computations and derivations regarding the AS-PCD invariants for uniform data.
On the other hand, the distribution of the domination number and arc density for PE- and CS-PCDs do not change for uniform data in a triangle $\TY$
when the data points are transformed to (uniform) points in the standard equilateral triangle $T_e$ (using $\phi_e \circ \phi_b(\TY)$).
That is, distribution of these graph invariants are geometry invariant for uniform data in triangles.
So, one can focus on $T_e$ for computations and derivations regarding the PE- and CS-PCD invariants for uniform data.

# References
