ei = total length of edge (or perimeter) of class i in terms of number of cell surfaces; includes all landscape boundary and background edge segments involving class i.min ei = minimum total length of edge (or perimeter) of class i in terms of number of cell surfaces (see below).max ei = maximum total length of edge (or perimeter) of class i in terms of number of cell surfaces (see below). Description nLSI equals the total length of edge (or perimeter) involving the corresponding class, given in number of cell surfaces, minus the minimum length of class edge (or perimeter) possible for a maximally aggregated class, also given in number of cell surfaces, which is achieved when the class is maximally clumped into a single, compact patch, divided by the maximum minus the minimum length of class edge. If ai is the area of class i (in terms of number of cells)[note, this is equivalent to the sum of patch areas across all patches of class i] and n is the side of the largest integer square smaller than ai (denoted ) and m = ai - n2, then the minimum edge or perimeter of class i, min-ei, will take one of the three forms (Milne 1991, Bogaert et al. 2000): min-ei = 4n, when m = 0, or min-ei = 4n + 2, when n2 < ai ≦ n(1+n), or min-ei = 4n + 4, when ai > n(1+n). If A is the landscape area, including all internal background (in terms of number of cells), B = number of cells on the boundary (perimeter) of the landscape, Z = total length of landscape boundary (perimeter) given in number of cell surfaces, and Pi = proportion of the landscape comprised of the corresponding class, then the maximum edge or perimeter of class i, max-ei, will take one of the three forms: max-ei = 4ai, when Pi ≦ 0.5, or max-ei = 3A - 2ai, when A is even; 0.5 < Pi ≦ (0.5A + 0.5B)/A, or max-ei = 3A - 2ai + 3, when A is odd; 0.5 < Pi ≦ (0.5A + 0.5B)/A, or max-ei = Z + 4(A - ai), when Pi > (0.5A + 0.5B)/A Note, the formula for max-ei recognizes the fact that as Pi increases beyond 0.5, the maximum total length of edge is achieved when the cells of the focal class fill in first along the boundary of the landscape. Unfortunately, the formulas given above for Pi > 0.5 are only an approximation for this effect. An analytical solution is not possible given the infinite number of landscape shapes possible. In addition, the formula for min-ei assumes that the maximally aggregated class is a single square or almost square patch. However, if the landscape shape is highly irregular, then as the proportional class area Pi approaches 1, the shape of the landscape will constrain the minimum class edge possible (i.e., the actual min-ei << the theoretical min-ei) and nLSI will biased high (i.e., the class will appear to be relatively less aggregated than it actually is). However, for square or rectangular landscapes, or classes with Pi << 1, there is either no bias or it is trivial. Units None Range 0 ≦ nLSI ≦ 1nLSI = 0 when the landscape consists of a single square or maximally compact (i.e., almost square) patch of the corresponding type; LSI increases as the patch type becomes increasingly disaggregated and is 1 when the patch type is maximally disaggregated (i.e., a checkerboard when Pi ≦ 0.5). Note, nLSI is undefined and reported as N/A in the output files whenever max-ei = min-ei, which exists 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 Normalized Landscape shape index is the normalized version of the landscape shape index (LSI) and, as such, provides a simple measure of class aggregation or clumpiness. The normalization essentially rescales LSI to the minimum and maximum values possible for any class area. When the patch type is relatively rare (say Pi < 0.1) or relative dominant (say Pi > 0.5), the range between the minimum and maximum total edge (or perimeter) is relatively small; whereas when the patch type is intermediate in abundance (say Pi = 0.5), the range is quite large. nLSI essentially measures the degree of aggregation given this variable range. Note, just as LSI and the Aggregation Index (AI) are closely related, the normalized versions of these metrics are related, in fact perfectly so. For this reason, the normalized version of AI is not computed since it is completely redundant with nLSI. In addition, given the considerations given above regarding the computational method that assumes a square or almost square shape for a maximally compact class and the bias this creates if the landscape is highly irregular and the percentage of the landscape comprised of the focal class is high, it is advisable to avoid using this metric under these conditions of bias. Also, for these reasons, this metric is not available in the Moving Window analysis mode when a circular window shape is selected.