gii = number of like adjacencies (joins) between pixels of patch type (class) i based on the doublecount method. gik = number of adjacencies (joins) between pixels of patch types (classes) i and k based on the doublecount method. Pi = 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 (Gi) is greater than or equal to the proportion of the landscape comprised of the focal class (Pi), then CLUMPY equals Gi minus Pi, divided by 1 minus Pi. Likewise, if Gi < Pi, and Pi ≥ 0.5, then CLUMPY equals Gi minus Pi, divided by 1 minus Pi. However, if Gi < Pi, and Pi < 0.5, then CLUMPY equals Pi minus Gi, divided by negative Pi. Note, all background edge segments are included in the sum of all adjacencies involving the focal class, including landscape boundary segments if a border is not provided. Cell adjacencies are tallied using the doublecount method in which pixel order is preserved, at least for all internal adjacencies (i.e., involving cells on the inside of the landscape). If a landscape border is present, adjacencies on the landscape boundary are counted only once, as are all adjacencies with background. Note, Pi is based on the total landscape area (A) including any internal background present. 
Units  Percent 
Range  1 ≦ CLUMPY ≦ 1 Given any Pi , 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 equals 1 only when the landscape consists of a single patch and includes a border comprised of the focal class. 
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 sidebyside on the map. Clumpiness is scaled to account for the fact that the proportion of like adjacencies (Gi) will equal Pi for a completely random distribution (see previous discussion). The formula is contingent upon Gi and Pi because the minimum value of Gi has two forms which depend on Pi. Specifically, when Pi ≦ 0.5, Gi = 0 when the class is maximally disaggregated (i.e., subdivided into one cell patches) and approaches 1 when the class is maximally clumped. However, when Pi ≥ 0.5, Gi = 2Pi  1 when the class is maximally disaggregated and approaches 1 when the class is maximally clumped. Note, when Gi > Pi, the formula given above assumes a maximum value of Gi = 1 (i.e., maximum clumping). This is not strictly true. In fact, the maximum value of Gi asymptotically approaches 1 as Pi increases to 1. At very small Pi, the maximum value of Gi is somewhat less. However, the bias is only nontrivial when the focal class consists of only a few cells. As the number of cells increases, the bias rapidly decreases and becomes trivial. Hence, when Gi > Pi CLUMPY is slightly biased low. That is, the computed degree of clumping is slightly less than the actual degree of clumping, but again, the difference is trivial under most conditions. This approach of assuming that a maximum value of Gi = 1 is necessary because it is impossible to calculate the true maximum value of Gi, taking into account potential like adjacencies of perimeter cell surfaces of the focal class when maximally clumped into a single compact patch. Recall that FRAGSTATS allows for the existence of a landscape border, which may consist of cells of the same class as the neighboring patches inside the landscape proper (a situation virtually guaranteed to occur in a moving window analysis). Unfortunately, there is no way to calculate the expected number of perimeter cell surfaces adjacent to the landscape boundary given any Pi  this depends on the exact configuration and positioning of the focal class when maximally clumped. Note, the maximum like adjacencies computed for the aggregation index does not include perimeter cell surfaces. Thus, calculating maximum Gi based on this approach will always underestimate the true value. The use of 1 as the maximum Gi guarantees a theoretical maximum (upper limit) value of 1 for CLUMPY. 