(C115) Clumpiness Index

g_ii =    number of like adjacencies (joins) between pixels of patch type (class) i based on the double-count method.

g_ik =    number of adjacencies (joins) between pixels of patch types (classes) i and k based on the double-count method.

min-e_i =          minimum perimeter (in number of cell surfaces) of patch type (class) i for a maximally clumped class.

P_i =     proportion of the landscape occupied by patch type (class) i.

Description

CLUMPY equals the proportional deviation of the proportion of like adjacencies involving the corresponding class from that expected under a spatially random distribution. If the proportion of like adjacencies (G_i) is less than the proportion of the landscape comprised of the focal class (P_i) and P_i < 0.5, then CLUMPY equals G_i minus P_i, divided by P_i; else, CLUMPY equals G_i minus P_i, divided by 1 minus P_i. Note, it can be shown that G_i equals 1 when the patch type is maximally clumped, but this requires adjustment for the perimeter of the class. If a_i is the area of class i (in terms of number of cells) and n is the side of a largest integer square smaller than a_i, and m = a_i - n², then the minimum perimeter of class i (i.e., when it is maximally clumped), min-e_i, will take one of the three forms (Milne 1991, Bogaert et al. 2000):

min-e_i = 4n, when m = 0, or

min-e_i = 4n + 2, when n² < a_i ≤ n(1+n), or

min-e_i = 4n + 4, when a_i > n(1+n).

Note, g_ii in the numerator includes only internal like adjacencies; like adjacencies involving cells in the border are not included. The sum of g_ik in the denominator includes all adjacencies involving the focal class, including adjacencies involving background and all adjacencies involving the landscape boundary, regardless of whether a border is present or not. Cell adjacencies are tallied using the double-count method in which pixel order is preserved. Note, P_i is based on the total landscape area (A) including any internal background present.

Units

None

Range

-1 ≤ CLUMPY ≤ 1

Given any P_i , CLUMPY equals -1 when the focal patch type is maximally disaggregated; CLUMPY equals 0 when the focal patch type is distributed randomly, and approaches 1 when the patch type is maximally aggregated. Note, CLUMPY is undefined and reported as N/A in the output files when the class consists either of a single cell, comprises all but 1 cell, or comprises the entire landscape, because it is impossible to distinguish between clumped, random and dispersed distributions in these cases.

Comments

Clumpiness index is calculated from the adjacency matrix, which shows the frequency with which different pairs of patch types (including like adjacencies between the same patch type) appear side-by-side on the map. Clumpiness is scaled to account for the fact that the proportion of like adjacencies (G_i) will equal P_i for a completely random distribution (see previous discussion). The formula is contingent upon G_i and P_i because the minimum value of G_i has two forms which depend on P_i. Specifically, when P_i ≤ 0.5, G_i = 0 when the class is maximally disaggregated (i.e., subdivided into one cell patches) and is 1 when the class is maximally clumped. However, when P_i ≥ 0.5, G_i = 2P_i -1 when the class is maximally disaggregated and approaches 1 when the class is maximally clumped.