Let's say I have a grid of cells from a satellite image that take on the value of 0 or 1, with 1 being forest.
There are many spatial pattern metrics that measure things about this pattern such as some dimension of forest fragmentation.
A common one is the join-count statistic which is simple enough that its expectation and variance are calculable for a random allocation process. This allows one to estimate the probability of a join-count with an equal or greater value than that observed, given a random process. One only needs to know the relationship among grid cells (are they spatially connected/neighbors or not) and the probability of any cells taking on a specific value (e.g. the probability that a cell is forest and probability that it is not).
More complex spatial patterns do not typically have a standardized version. For instance, the perimeter of all forest patches can be divided by the surface area of all patches to calculate a perimeter-area ratio. The farther from one large circle (the minimum) then the more fragmented.
It would be easy to calculate the probability of any given spatial pattern (they would all be the same, I believe, if they contain the same number of forested/non-forested cells), however I am interested in knowing the expected perimeter-area ratio and variance so that measures such as this can be standardized.
Are there any texts someone can suggest?
I have read quite extensively within landscape ecology and related fields, but I do not believe this has been addressed, so I need a math text or even area of study to research of which I am not currently aware.
A simple approach would be Monte Carlo simulation, but this would be very computationally intensive for even moderately large collections of grid cells. Further, the patterns observed are typically quite rare in comparison to a random process and so you could allow the simulation to run quite a while and I believe never fill in the tails of the distribution created by it so as to get a good handle on the probability of observing a pattern with the same value or greater.
Any guidance related to obtaining my objective is appreciated.
The primary motivation of this is not to test "straw man" null hypotheses of random allocation but instead to be able to compare the patterns from two areas while removing some of the confounding due to differences in the number of cells allocated to each category.