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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.

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  • $\begingroup$ These are interesting academic points, but it is difficult to see any practical application. Once you have determined--using, say, the join count statistic--that the process is not CSR (and few spatial processes are), there is no point to using additional statistics to re-prove the same thing. The value of these statistics lies in their descriptive power and in what they might suggest about the causes of the patterns. One more test of a (meaningless) null hypothesis would be of little help. $\endgroup$ – whuber Nov 13 '13 at 16:00
  • $\begingroup$ whuber, to be polite, you are making a lot of assumptions about why I am wanting to do this, are you not? If you understood the standardized join-count that I mention, then you would know that its primary use is to measure dispersion while conditioning on the proportion of cells that take on any such value. Without standardization, the join-count is a measure of density and dispersion, and so the nonstandardized version is confounded. I suggest that people stick to what is being asked. I asked for specific guidance. I know why it is important. $\endgroup$ – wvguy8258 Nov 13 '13 at 19:34
  • $\begingroup$ When I spoke of probability, it is the same as stating "distance from the mean shown in standard deviation units" or some such thing. That is what the scaling is. Asking about probability does not imply hypothesis testing. And as was stated, most landscape patterns are far from the expectation given a random process, and so any null hypothesis is surely rejected. $\endgroup$ – wvguy8258 Nov 13 '13 at 19:38
  • $\begingroup$ Obviously it is not the case that "any guidance is appreciated." I am not challenging your knowledge and would like to suggest you not challenge mine, because that only makes you seem churlish. I was responding to what is written in your question, which appears to be framed from the point of view of computing null distributions of "random allocation processes." If you believe I have misinterpreted your question, then take that as evidence that others may similarly misinterpret it and please make an effort to clarify what you are looking for. $\endgroup$ – whuber Nov 13 '13 at 19:48
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    $\begingroup$ Sorry.I have just had semi-bad experiences here multiple times where someone does not address your question,but instead tells you it is not a good question without fully understanding why you are asking it.If I explained the entire background,would anyone want to read something even longer?I am wanting to compute null distributions from random allocation processes.So, you got that correct, and I see no reason to edit. I did not mention wanting to test null hypotheses.There are reasons other than that to want to know something of the null distribution. $\endgroup$ – wvguy8258 Nov 14 '13 at 22:59

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