Theoretically, the point interaction \(t(y)\) at a point \(y\) is calculated
as the proportion of available area in a circular region \(Y\) or radius \(r\) centred
at \(y\) that overlaps with circles of radius \(r\) centred at other presence locations
(Baddeley & Turner, 2005).
This function discretises the study region at the same spatial resolution
as availability by defining the matrix occupied, a fine grid of locations
spanning the study region initialised to zero. The values of occupied within a
distance of r of each presence location are then augmented by 1, such that
occupied then contains the total number of presence locations with which each
grid location interacts. To prevent unavailable areas from being included in the
calculation of point interactions, the values of occupied at grid locations for which
availability = 0 are set to zero.
\(t(y)\) is then estimated as the proportion of available grid locations within \(Y\)
that overlap circular regions around other presence locations.
The availability matrix is particularly useful for regions that have inaccessible areas
(e.g. due to the presence of ocean or urban areas).
Finer resolutions of the availability matrix will yield more precise estimates but
at a cost of greater computation time.