Nearest-Neighbour Correlation of Marked Point Pattern

Description

Computes the nearest-neighbour correlation of a marked point pattern.

Usage

     nncorr(X,
            f = function(m1, m2) { m1 * m2 },
            ...,
            use = "all.obs", method = c("pearson", "kendall", "spearman"))

Arguments

Details

The nearest neighbour correlation nbar of a marked point process X is a number measuring the dependence between the mark of a typical point and the mark of its nearest neighbour.

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

nbar[f] = E[f(M, M*)]

where E[] denotes mean value, M is the mark attached to a typical point of the point process, and M* is the mark attached to its nearest neighbour (i.e. the nearest other point of the point process).

Here f is any function f(m1,m2) 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(m1,m2)= m1 * m2

for discrete marks (multitype point patterns),

f(m1,m2)= (m1 == m2)

and for marks taking values in [0,2 * pi),

f(m1,m2) = sin(m1-m2).

For example, in the second case, the unnormalised nearest neighbour correlation nbar[f] equals the proportion of points in the pattern which have the same mark as their nearest neighbour.

Note that nbar[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

mbar[f] = E[f(M,M*)]/E[f(M,M')]

where again M is the mark attached to a typical point, M* 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 mbar[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*. 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 ‘border method’ edge correction.

Value

Labelled vector of length 2 or 3 containing the unnormalised and normalised nearest neighbour correlations, and the classical correlation if appropriate.

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

References

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

Examples

  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 type