Delineation of difference boundaries occurs through separation of some spatial locations from others. In BoundarySeer, spatial locations are categorized as boundary or not (for crisp boundaries) based on Boundary Likelihood Values (BLVs). For fuzzy boundaries, boundary membership is not an all or nothing thing.
As described in About wombling, a Boundary Element (BE) is a location with a "large" amount of change over space. The cutoff for a "large enough" BLV is somewhat arbitrary; most researchers declare locations with values in the upper 5th or 10th percentile to be BEs in crisp boundary delineation (Barbujani et al. 1989, Barbujani et al. 1990, Fortin and Drapeau 1995, Jacquez 1995). Within BoundarySeer, you can set BLV thresholds two ways, through a priori cutoffs, set in the wombling dialog, or using a BLV histogram. You may also set gradient angle thresholds for wombling on numeric raster and point data.
With numeric data, the threshold is given as a percentage, which tells BoundarySeer the number of BEs to select. For example, if you define the threshold as 10%, BoundarySeer selects those candidate BEs (cBEs) possessing the highest 10% BLVs.
The realized threshold may be slightly different from the stated threshold. BoundarySeer uses the percentage threshold to calculate the number of BEs, disregarding any fractional part in determining this number. For example, if your data set contains 85 cBEs and you select a 10% threshold, BoundarySeer will assign 8 locations to the set of BEs (giving a realized threshold of (8/85)x100% = 9.4%).
Furthermore, BoundarySeer will not distinguish among locations that have tied BLVs. That is, if in the above example the 8th highest BLV is also tied with the 9th and 10th highest values, BoundarySeer assigns all three locations to the set of BEs. In this case, the realized threshold is ( 10/85 ) x100% = 11.8%. You may find it useful to create several sets of BEs using different thresholds for comparison.
You may choose a threshold from the distribution of BLVs in the data. This method allows less arbitrary cutoffs, as you can place cutoffs in breaks in the distribution. For more information, see Defining thresholds using histograms.
Using thresholds to identify BEs has been criticized as subjective, in that, for a given threshold, a fixed number of BEs are always found, whether or not their rates of change are statistically unusual. Jacquez and Maruca (1998) have begun work on an alternative. Their approach involves a local and global statistic to determine (a) where statistically significant BEs are, and (b) whether the boundaries for the entire surface are statistically unusual or easily explained by chance. The local statistic, calculated for each pair of adjacent cBEs, is maximized when both standardized gradient magnitudes are large, and gradient angles are similar and perpendicular to the line connecting their locations. They proposed several null hypotheses, including complete spatial randomness and spatial autocorrelation without boundaries, and have begun power analyses for both crisp and fuzzy boundaries.
See also:
Subboundaries: gradient angle thresholds