nncorr(X,
f = function(m1, m2) { m1 * m2 },
...,
use = "all.obs", method = c("pearson", "kendall", "spearman"))"ppp".f.cor.Three different values are computed: the unnormalised, normalised, and classical correlations. The unnormalised nearest neighbour correlation (Stoyan and Stoyan, 1994, section 14.7) is defined as $$\bar n_f = E[f(M, M^\ast)]$$ where $E[]$ denotes mean value, $M$ is the mark attached to a typical point of the point process, and $M^\ast$ is the mark attached to its nearest neighbour (i.e. the nearest other point of the point process). Here $f$ is any function $f(m_1,m_2)$ with two arguments which are possible marks of the pattern, and which returns a nonnegative real value. Common choices of $f$ are: for continuous real-valued marks, $$f(m_1,m_2) = m_1 m_2$$ for discrete marks (multitype point patterns), $$f(m_1,m_2) = 1(m_1 = m_2)$$ and for marks taking values in $[0,2\pi)$, $$f(m_1,m_2) = \sin(m_1 - m_2)$$ For example, in the second case, the unnormalised nearest neighbour correlation $\bar n_f$ equals the proportion of points in the pattern which have the same mark as their nearest neighbour.
Note that $\bar n_f$ is not a ``correlation'' in the usual statistical sense. It can take values greater than 1.
We can define a normalised nearest neighbour correlation
by
$$\bar m_f = \frac{E[f(M,M^\ast)]}{E[f(M,M')]}$$
where again $M$ is the
mark attached to a typical point, $M^\ast$ is the mark
attached to its nearest neighbour, and $M'$ is an independent
copy of $M$ with the same distribution.
This normalisation is also not a ``correlation''
in the usual statistical sense, but is normalised so that
the value 1 suggests ``lack of correlation'':
if the marks attached to the points of X are independent
and identically distributed, then
$\bar m_f = 1$.
The interpretation of values larger or smaller than 1 depends
on the choice of function $f$.
Finally if the marks of X are real numbers, we can also compute the
classical correlation, that is, the correlation coefficient
of the two random variables $M$ and $M^\ast$.
The classical correlation has a value between $-1$ and $1$.
Values close to $-1$ or $1$ indicate strong dependence between
the marks.
This function computes the unnormalised and normalised nearest neighbour correlations, and the classical correlation if appropriate.
The argument X must be a point pattern (object of class
"ppp") and must be a marked point pattern.
The argument f
must be a function, accepting two arguments m1
and m2 which are vectors of equal length containing mark
values (of the same type as the marks of X).
It must return a vector of numeric
values of the same length as m1 and m2.
The values must be non-negative.
The arguments use and method control
the calculation of the classical correlation using cor,
as explained in the help file for cor.
Other arguments may be passed to f through the ...
argument.
This algorithm assumes that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in X as X$window)
may have arbitrary shape.
Biases due to edge effects are
treated using the
data(finpines)
nncorr(finpines)
# heights of neighbouring trees are slightly negatively correlated
data(amacrine)
nncorr(amacrine, function(m1, m2) { m1 == m2})
# neighbouring cells are usually of different typeRun the code above in your browser using DataLab