aij = area (m2) of patch ij.
pij = perimeter (m) of patch ij.
ni = number of patches in the landscape of patch type (class) i.
Description PAFRAC equals 2 divided by the slope of regression line obtained by regressing the logarithm of patch area (m2) against the logarithm of patch perimeter (m). That is, 2 divided by the coefficient b1 derived from a least squares regression fit to the following equation: ln(area) = b0 + b1ln(perim). Note, PAFRAC excludes any background patches.
Units None
Range 1 ≦ PAFRAC ≦ 2 A fractal dimension greater than 1 for a 2-dimensional landscape mosaic indicates a departure from a Euclidean geometry (i.e., an increase in patch shape complexity). PAFRAC approaches 1 for shapes with very simple perimeters such as squares, and approaches 2 for shapes with highly convoluted, plane-filling perimeters. PAFRAC employs regression techniques and is subject to small sample problems. Specifically, PAFRAC may greatly exceed the theoretical range in values when the number of patches is small (e.g., < 10), and its use should be avoided in such cases. In addition, PAFRAC requires patches to vary in size. Thus, PAFRAC is undefined and reported as "N/A" in the "basename".class file if all patches are the same size or there is < 10 patches.
Comments Perimeter-area fractal dimension is appealing because it reflects shape complexity across a range of spatial scales (patch sizes). However, like its patch-level counterpart (FRACT), perimeter-area fractal dimension is only meaningful if the log-log relationship between perimeter and area is linear over the full range of patch sizes. If it is not (and this must be determined separately), then fractal dimension should be computed separately for the range of patch sizes over which it is constant. Note, because this index employs regression analysis, it is subject to spurious results when sample sizes are small. In landscapes with only a few patches, it is not unusual to get values that greatly exceed the theoretical limits of this index. Thus, this index is probably most useful if sample sizes are large (e.g., n ≥ 20), although FRAGSTATS computes the index for moderate sample sizes as well (i.e., n ≥ 10). In addition, it is important to realize that the perimeter-area fractal dimension computed in FRAGSTATS is based on the regression of log area on log perimeter; that is, ln(area) = b0 + b1ln(perim). It is equally valid to compute fractal dimension by regressing log perimeter on log area; that is, ln(perim) = b0 + b1ln(area), in which case the fractal dimension (D) is equal to 2 times the slope (b1). These two approaches give slightly different answers and it is not clear that one is superior to the other. Both approaches are used in practice, so it behooves you to note the manner by which fractal dimension is computed when comparing among studies.