\documentstyle{article}
\title{Complete Spatial Randomness and Quadrat Methods\\ 
{\large GRASS Tutorial on {\tt s.qcount}}}
\author{James Darrell McCauley}
\date{08 January 1993}
\newcommand{\RR}{\mbox{\protect\makebox[.15em][l]{I}R}}
\begin{document}
\bibliographystyle{plain}
\maketitle
Cressie~\cite{cressie91} defines the concept of {\em
complete spatial randomness\/} (csr) as synonymous
with a {\em homogeneous\/} Poisson process in
$\RR^d\/$ (here the concern is $d=2$). In layman's
terms, the definition states that events are equally
likely to occur anywhere within an area
$A\subset\RR^d$.

\begin{figure}[h]
\begin{center}
\input{csr.latex}
\centerline{(a)}
\input{reg.latex}
\centerline{(b)}
\input{cls.latex}
\centerline{(c)}
\end{center}
\caption{Realization of two-dimensional Poisson processes of 50
points on the unit square exhibiting (a)~complete
spatial randomness, (b)~regularity, and (c)~clustering.}
\label{fig:csr}
\end{figure}

\begin{figure}[h]
% load 'stat.inc
% plot [0:10] cchi(x) title "$\chi^2\/$ CDF", chi(x) title "$\chi^2\/$ PDF"
\begin{center}
\input{chisq.latex}
\end{center}
\caption{Probability Density Function and Cumulative Distribution
Function for a $\chi^2\/$ Random Variable.}
\label{fig:chisq}
\end{figure}

There are two types departure from a csr: regularity
and clustering.  Figure~\ref{fig:csr} shows
realizations of three processes for $N\left(A\right) =
50.$ Notice in figure~\ref{fig:csr}a that this
process may {\em appear\/} to be clustered.
This is because event-to-nearest-event distances ($W$) of
a homogeneous Poisson process can be modeled
as $\chi_2^2\/$ random variables. Recall the form of
the $\chi_2^2$ probability density function
(fig.~\ref{fig:chisq}). Most of the probability
is near zero. It follows that, given a suitable statistic,
csr can be tested using Pearsons's $\chi^2$ goodness-of-fit test.

Various indices and statistics
measure departure from csr, ie., the pattern.
Tables 8.3 and 8.6 in
Cressie's book~\cite{cressie91} summarizes six {\em
quadrat count\/} indices and 17 {\em nearest
neighbor\/} test statistics for quantifying
departure from csr.   The former,  also discussed
in Ripley~\cite{ripley81}, are implemented
in {\tt s.qcount} and described later.

{\tt s.qcount}
chooses {\em n\/} circular
regions of radius {\em r\/} such that they
are completely within the bounds of the current geographic
region and no two
regions overlap.
The regions are called random
{\em quadrats}.\footnote{Cressie~\protect\cite{cressie91} 
defined random quadrats this way, however Ripley~\protect\cite{ripley81}
used rectangular quadrats that potentially overlapped.}
The number of sites falling within each 
quadrat are counted and indices are calculated to estimate 
the departure of site locations from complete spatial 
randomness. This is illustrated in figure~\ref{fig:quads}.

\begin{figure}[h]
\begin{center}
\input pinequads.latex
\end{center}
\caption{Randomly placed quadrats ($n=100$) with 584 sample points. This figure
was produced using the output of {\tt s.qcount} and {\tt g.gnuplot}.}
\label{fig:quads}
\end{figure}

The six indices and their realizations for the sampling
shown in figure~\ref{fig:quads} are shown in Table~\ref{tbl:indices}.
Cressie~\cite{cressie91} gives a short summary of these on pages~590
and 591 and Ripley~\cite{ripley81} discusses them on pages~102--106.
The original reference to each index is given in Table~\ref{tbl:indices}
in case you want to read more about them.

\begin{table}[h]
\begin{center}
\begin{minipage}{.8\textwidth}
\caption{Indices for Quadrat Count Data.  Adapted from
Cressie~\protect\cite{cressie91}, this table shows the
statistics computed for the quadrats in figure~\protect\ref{fig:quads}}
\label{tbl:indices}
\begin{center}
\begin{tabular}{lccl}\hline
Index & Estimator\protect\footnote{$X_i$ the
number of sites in the $i$th quadrat, $\bar{X}\/$ is the mean of the
quadrat counts, and $S^2\/$ is the sample variance.}
 & Realization & \multicolumn{1}{c}{Reference}\\ \hline
$I$ & $\frac{S^2}{\bar{X}}$ & 2.128 & Fisher~\cite{fisher22} \\ \\
$ICS$ & $\frac{S^2}{\bar{X}}-1$ & 1.128& David and Moore~\cite{david54}\\ \\
$ICF$ & $\frac{\bar{X}^2}{S^2-\bar{X}}$ & 1.383& Douglas~\cite{douglas75}\\ \\
$\stackrel{*}{X}$ & $\bar{X}-\frac{S^2}{\bar{X}}-1$
				& 2.688& Lloyd~\cite{lloyd67}\\ \\
$IP$ & $\frac{\stackrel{*}{X}}{\bar{X}}$ 
				& 1.723 & Lloyd~\cite{lloyd67}\\ \\
$I_\delta$ & 
$\frac{n\sum_{i=1}^n X_i\left(X_i-1\right)}{n\bar{X} \left(n\bar{X}-1\right)}$ 
%$\frac{n\bar{X} \cdot IP}{n\bar{X}-1}$ 
& 1.720 & Morisita~\cite{morisita59}\\ \hline
\end{tabular}
\end{center}
\end{minipage}
\end{center}
\end{table}

\clearpage
\bibliography{quadrat}
\end{document}
