Within BoundarySeer, statistics can be evaluated under a null hypothesis that includes some spatial pattern, such as spatial autocorrelation. Many spatial statisticians consider such a null hypothesis to be more tenable than complete spatial randomness (Fortin and Jacquez 2000). BoundarySeer accounts for spatial autocorrelation (or other spatial or nonspatial patterns) by restricting the randomizations during the Monte Carlo process, so that each observation is more likely to be 'sampled' at some locations and less likely at others.
To restrict randomizations, BoundarySeer uses a matrix of probabilities called a generator matrix. For a data set with N sample locations (and therefore N sets of observations), the generator matrix G is an N X N matrix. The matrix elements, gij, give the relative probability of assigning observation vector i to location j, given that all locations are available for assignment. The observation vector is the list of the values of each variable at a particular location. During the process of randomization, observations are chosen at random and assigned to locations, and as these locations then become unavailable, the relative probabilities are transformed into actual probabilities that allow further assignments to be made.
Here is a summary of the process of how BoundarySeer uses a generator matrix to randomize data (assuming the matrix has already been calculated):
Select an observation vector at random from those available.
Calculate the actual assignment probabilities from elements of the generator matrix.
Select a location at random, according to probabilities calculated in step 2.
Make the assignment and adjust the generator matrix accordingly by removing the row and column corresponding the observation vector and location (respectively) that have just been assigned.
Repeat steps 1-4 until all observation vectors have been assigned.
See also: