Hello, I am looking for suggestions as to how solve what follows.
Let's assume I have some Thiessen polygons built around some locations (crosses in the attached image). Some points (red dots in the attached image) lie within the polygons. For each polygon I want to calculate the expected number of points, against which compare the observed number of points, and to calculate the significance (i.e., p-value) of the observed number of points under the Null Hypothesis of a random distribution of points within the study area as a whole.
We actually know (a) the area of each polygon, (b) the total area of the study plot (=sum of the polygons' area), (c) the total number of points, and (d) the number of points falling inside each polygon.
I guess one should use the binomial distribution, but I seem I can't figure out how to exact implement the calculation. Where I am stuck is in working out the value of p, which in the case of polygons of equal size (i.e., quadrats) should be the reciprocal of the number of quadrats into which the study area is divided. But I do not know how to calculate p in case of polygons of unequal size.
In case of quadrats (equal sized polygons), we should use a binomial distribution with:
p=1/x (where x is the number of quadrats)
n=number of events in the pattern (i.e., total number of points)
k=number of events in a quadrat.
In the specific case I described, would p be equal to the fraction of each polygon's area relative to the whole study plot (i.e., p=polygon area/sum of all the polygons area)? Thanks for any suggestion. Best Gm
Dear Gianmarco,
if I understood the problem correctly, this should be the solution. If the points are randomly scattered across the study area, each polygon should receive a fraction of points proportional to the relative area of the polygon (the proportion of the total area occupied by particular polygon). For example, if you have 100 points and three polygons occupying 0.5, 0.4 and 0.1 of the total area (the sum of all three polygon areas), then you would expect to have 0.5*100=50 points in polygon 1, 0.4*100=40 points in polygon 2 and 0.1*100=10 points in polygon 3. When you calculate the expected number of points for each polygon, then you can use one-sample chi square test to test for randomness. For each polygon you calculate the chi square, sum the individual polygon chi squares and this is the total chi square statistic. You should then compare the value of this statistic to a table of values with degrees of freedom equal to the number of polygons minus one to see if it is above or below the conventional 0.05 significance level.
Best wishes,
Marko
Dear Marko,
thanks indeed for your reply and for the details provided. Waiting for a reply here, I also performed some further search on my own, and I have found that O'Sallivan-Unwin, "Geographic Information Analysis", 2010, p. 104 indicate:
...the probability p in the quadrat counting case is given by the size of each quadrat relative to the size of the study region.....This gives us the final expression for the probability distribution of the quadrat counts for a point pattern generated by Independent Random Process....which is simply a binomial distribution with p=1/x [in case of quadrats of equal size].....
Therefore, I believe that another way to test the significance of the observed counts is using binomial distribution. In R it is:
dbinom(by polygon n points, total n of points, %polygonarea).
For each polygon, the probability of observing a number of points from 1 to the actual abserved count, is:
pbinom(by polygon n points, total n of points, %polygonarea).
I have implemented a couple of functions based on the above in my GmAMisc R package, whose description may be found here on RG (Deleted research item The research item mentioned here has been deleted
, or on GitHub (https://github.com/gianmarcoalberti/GmAMisc).Thank you again for your input; I hope this discussion proves useful to others jumping in here in the future.
Best
Gm
Dear Gianmarco,
you are welcome, the chi square approach I suggested is an approximation, and it seems to me that you are looking for an exact method for each individual polygon.
It has just occurred to me that the Poisson distribution approach can be used as well. For example, you can calculate the overall point density by dividing the total number of points with the total area (entire study area), this would give you a lambda parameter of the Poisson distribution. Then you could use the Poisson distribution to calculate the expected number of points in each polygon given its area and to calculate the 95% confidence intervals for this estimate. If the actual (empirical) number of points falls out of the 95% CI then the number of points received by that particular polygon is not generated by a random process.
Best,
Marko
You may want to have a look at this classic paper on the distribution of flying bombs over London in WW2; it uses a Poisson analysis to compare the actual landings on a grid with a theoretically even distribution of counts
R. D. Clarke (1946) An application of the Poisson distribution
Journal of the Institute of Actuaries , Volume 72, Issue p. 481 https://doi.org/10.1017/S0020268100035435
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