R Tutorial : Further testing (Spatial Statistics in R)

Key Takeaways

one problem with the quadrat test is that you have to choose a set of sub regions two few and you don't have many numbers for the statistical test to have much power too many and your regions don't have many points in them and you lose power too there are alternative tests that don't depend on arbitrary subdivisions instead relying on estimating properties of the spatial point process from the events one such property is the nearest neighbor distribution look at each event in your pattern and find the distance to the nearest event so here as nearest neighbor is B B's nearest neighbor is a and season areas neighbor is B that gives us three distances do this for every event plot a histogram to give an estimate of the probability density function of the nearest neighbor distribution a corresponding cumulative distribution function the probability of finding a nearest neighbor within a distance D is named G for a completely spatially random process the theoretical form of G can be worked out exactly so for a process with lambda events per unit area it looks like this spats that uses the GS function to estimate G given a PPP object and an optional distance vector the lines in this plot include two Corrections for edge effects because events near the edge of the window have less area in which the nearest neighbor might have been and this will bias the estimator so G s has a correction argument to choose these edge correction algorithms another useful function of spatial point processes is the ripley's reduced second moment measure better known as the K function K is the number of expected events to be found at a given distance from an event scaled by the intensity to estimate it for some given distance D visit each event in turn and count the number of other events in a circle of radius D take the average that's K of D do that for a number of values of D and you can plot the function for a completely spatially random process this gives K of equals pi times D squared the area of a circle of radius D but how much variation can be expected from an estimate K so that it can be used as a test for complete spatial randomness well first create 99 completely spatially random point patterns in your window and compute the 99 estimates of K you can plot them now compute K for the data and plot it over the simulation estimates is it bigger than the simulations at any point that's an indication of clustering at that scale since the data at that point outranks all 99 simulations you can say it rejects the null hypothesis at a p-value of 0.01 using 99 simulations just makes the division easy since you divided by 99 plus 1 for the data 100 to get the p-value this sort of test where you generate replicates from your hypothesis compute test statistics or functions and compared with the statistic from the data is called a Monte Carlo test it's commonly used in spatial statistics because working out the theoretical distribution of things can be somewhat tricky in the next few exercises you'll explore the nearest neighbor distribution and do some Monte Carlo testing