FRAGSTATS Metrics.–There are several different approaches for measuring contagion and interspersion. One popular index that subsumes both dispersion and interspersion is the contagion index (CONTAG) based on the probability of finding a cell of type i next to a cell of type j. This index was proposed first by O'Neill et al. (1988) and subsequently it has been widely used (Turner and Ruscher 1988, Turner 1989, Turner et al. 1989, Turner 1990a and b, Graham et al. 1991, Gustafson and Parker 1992). Li and Reynolds (1993) showed that the original formula was incorrect; they introduced 2 forms of an alternative contagion index that corrects this error and has improved performance. FRAGSTATS computes one of the contagion indices proposed by Li and Reynolds (1993). This contagion index is based on raster “cell" adjacencies, not "patch" adjacencies, and consists of the sum, over patch types, of the product of 2 probabilities: (1) the probability that a randomly chosen cell belongs to patch type i (estimated by the proportional abundance of patch type i), and (2) the conditional probability that given a cell is of patch type i, one of its neighboring cells belongs to patch type j (estimated by the proportional abundance of patch type i adjacencies involving patch type j). The product of these probabilities equals the probability that 2 randomly chosen adjacent cells belong to patch type i and j. This contagion index is appealing because of the straightforward and intuitive interpretation of this probability.

The contagion index has been widely used in landscape ecology because it seems to be an effective summary of overall clumpiness on categorical maps (Turner 1989). In addition, in many landscapes, it is highly correlated with indices of patch type diversity and dominance (Ritters et al. 1995) and thus may be an effective surrogate for those important components of pattern (O’Neill et al. 1996). Contagion measures both patch type interspersion (i.e., the intermixing of units of different patch types) as well as patch dispersion (i.e., the spatial distribution of a patch type) at the landscape level. All other things being equal, a landscape in which the patch types are well interspersed will have lower contagion than a landscape in which patch types are poorly interspersed. Contagion measures the extent to which patch types are aggregated or clumped (i.e., dispersion); higher values of contagion may result from landscapes with a few large, contiguous patches, whereas lower values generally characterize landscapes with many small and dispersed patches. Thus, holding interspersion constant, a landscape in which the patch types are aggregated into larger, contiguous patches will have greater contagion than a landscape in which the patch types are fragmented into many small patches. Contagion measures dispersion in addition to patch type interspersion because cells, not patches, are evaluated for adjacency. Landscapes consisting of large, contiguous patches have a majority of internal cells with like adjacencies. In this case, contagion is high because the proportion of total cell adjacencies comprised of like adjacencies is very large and the distribution of adjacencies among edge types is very uneven.

Unfortunately, as alluded to above, there are alternative procedures for computing contagion, and this has contributed to some confusion over the interpretation of published contagion values (see Ritters et al. 1996 for a discussion). Briefly, to calculate contagion, the adjacency of patch types is first summarized in an adjacency or co-occurrence 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 (note, FRAGSTATS includes only the 4 orthogonal neighbors, not diagonal neighbors, regardless of the choice of neighbor rules for defining patches). Although this would seem to be a simple task, it is the source of differences among procedures for calculating contagion. The difference arises out of the option to count each immediately-adjacent pixel pair once or twice. In the single-count method, each pixel adjacency is counted once and the order of pixels is not preserved; whereas, in the double-count method, each pixel adjacency is counted twice and the order of pixels is preserved. Ritters et al. (1996) discuss the merits of both approaches. FRAGSTATS adopts the double-count method in which pixel order is preserved, with two exceptions. If a landscape border is present, the adjacencies along the landscape boundary (i.e., those between cells inside the landscape and those in the border) are only counted once, and they are tallied for the cells inside the landscape. For example, an adjacency on the landscape boundary between class 2 (inside the landscape) and class -3 (in the landscape border) is recorded as a 2-3 adjacency in the adjacency matrix, not a 3-2. Thus, if a landscape border is present, the adjacency matrix includes double-counts for all internal cell adjacencies and single-counts for all adjacencies on the landscape boundary not involving background. In effect, this gives double the weight to the internal adjacencies than those on the boundary, although the effect will be trivial in most landscapes because the boundary edges will represent a relative minor proportion of the total adjacencies. Similarly, all adjacencies involving background (both internal, i.e., inside the landscape, and external, i.e., on the landscape boundary) are counted only once, and they are tallied for the non-background cells. Essentially, each non-background cell inside the landscape (i.e., positively valued cell) is visited and the four cell sides are evaluated and tallied in the adjacency matrix. Since background cells and all cells in the landscape border, if present, are not visited per se, the edges involving these cells only get tallied once in association with the non-background cell inside the landscape.

McGarigal and Marks (1995) introduced a complementary interspersion and juxtaposition index (IJI) that increases in value as patches tend to be more evenly interspersed in a "salt and pepper" mixture. Unlike the earlier contagion indices that are based on raster cell adjacencies, this index is based on patch adjacencies; only the patch perimeters are considered in determining the total length of each unique edge type. Each patch is evaluated for adjacency with all other patch types; like adjacencies are not possible because a patch can never be adjacent to a patch of the same type. Because this index is a measure of patch adjacency and not cell adjacency, the interpretation is somewhat different than the contagion index. The interspersion index measures the extent to which patch types are interspersed (not necessarily dispersed); higher values result from landscapes in which the patch types are well interspersed (i.e., equally adjacent to each other), whereas lower values characterize landscapes in which the patch types are poorly interspersed (i.e., disproportionate distribution of patch type adjacencies). The interspersion index is not directly affected by the number, size, contiguity, or dispersion of patches per se, as is the contagion index. Consequently, a landscape containing 4 large patches, each a different patch type, and a landscape of the same extent containing 100 small patches of 4 patch types will have the same index value if the patch types are equally interspersed (or adjacent to each other based on the proportion of total edge length in each edge type); whereas, the value of contagion would be quite different. Like the contagion index, the interspersion index is a relative index that represents the observed level of interspersion as a percentage of the maximum possible given the total number of patch types.

It is important to note the differences between the contagion index and the interspersion and juxtaposition index. Contagion is affected by both interspersion and dispersion. The interspersion and juxtaposition index, in contrast, is affected only by patch type interspersion and not necessarily by the size, contiguity, or dispersion of patches. Thus, although often indirectly affected by dispersion, the interspersion and juxtaposition index directly measures patch type interspersion, whereas contagion measures a combination of both patch type interspersion and dispersion. In addition, contagion and interspersion are typically inversely related to each other. Higher contagion generally corresponds to lower interspersion and vice versa. Finally, in contrast to the interspersion and juxtaposition index, the contagion index is strongly affected by the grain size or resolution of the image. Given a particular patch mosaic, a smaller grain size will result in greater contagion because of the proportional increase in like adjacencies from internal cells. The interspersion and juxtaposition index is not affected in this manner because it considers only patch edges. This scale effect should be carefully considered when attempting to compare results from different studies.

Other contagion-like metrics can be generated from the matrix of pairwise adjacencies between patch types. FRAGSTATS computes the percentage of like adjacencies (PLADJ), which is computed as the sum of the diagonal elements (i.e., like adjacencies) of the adjacency matrix divided by the total number of adjacencies. A landscape containing greater aggregation of patch types (e.g., larger patches with compact shapes) will contain a higher proportion of like adjacencies than a landscape containing disaggregated patch types (e.g., smaller patches and more complex shapes). In contrast to the contagion index, this metric measures only patch type dispersion, not interspersion, and is unaffected by the method used to summarize adjacencies. At the class level, this metric is computed as the percentage of like adjacencies of the focal class. A highly contagious (aggregated) patch type will contain a higher percentage of like adjacencies. Conversely, a highly fragmented (disaggregated) patch type will contain proportionately fewer like adjacencies. As such, this index provides an effective measure of class-specific contagion that isolates the dispersion (as opposed to interspersion) component of configuration. However, this index requires careful interpretation because it varies in relation to the proportion of the landscape comprised of the focal class (P_i). It has been shown that PLADJ for class i will equal P_i for a completely random map (Gardner and O’Neill 1991). If the focal class is more dispersed than is expected of a random distribution (i.e., overdispersed), then PLADJ < P_i. If the focal class is more contagiously distributed, then PLADJ > P_i. Thus, although PLADJ provides an absolute measure of aggregation of the focal class, it is difficult to interpret as a measure of contagion without adjusting for P_i.

FRAGSTATS computes two class-specific indices based on PLADJ that adjust for P_i in different ways. The clumpiness index (CLUMPY) introduced here is computed such that it ranges from -1 when the patch type is maximally disaggregated to 1 when the patch type is maximally clumped. It returns a value of zero for a random distribution, regardless of P_i. Values less than zero indicate greater dispersion (or disaggregation) than expected under a spatially random distribution, and values greater than zero indicate greater contagion. Hence, this index provides a measure of class-specific contagion that effectively isolates the configuration component from the area component and, as such, provides an effective index of fragmentation of the focal class that is not confounded by changes in class area. The aggregation index (AI) is computed as a percentage based on the ratio of the observed number of like adjacencies (e_i,i), based on the single-count method, to the maximum possible number of like adjacencies (max_e_i,i) given P_i (He et al. 2000). Note, the single-count method of tallying adjacencies is employed to be consistent with the published algorithm. The maximum number of like adjacencies is achieved when the class is clumped into a single compact patch, which does not have to be a square. The trick here is in determining the maximum value of e_i,i for any P_i,. He et al. (2000) provide the formula for computing max_e_i,i. The index ranges from 0 when there is no like adjacencies (i.e., when the class is maximally dissagregated) to 1 when e_i,i reaches the maximum (i.e., when the class is maximally aggregated).

There are alternative methods for calculating class-specific contagion using fractal geometry (Gardner and O’Neill 1991). FRAGSTATS computes the mass fractal dimension (MFRAC) for each class, which is based on the scaling relationship between box mass (i.e., the number of pixels of a focal class within a window) and the size of the box defining the window (r). Specifically, a range of box sizes is used to delineate windows, from 3 pixels on a side (r=3) to a maximum of approximately ⅓ of the landscape. For each box size, the mean number of pixels of the focal class is determined by centering the box on every pixel of that class and counting the number of pixels of that class in the box sample. Mass fractal dimension is equal to the slope derived from regressing the log of the mean number of pixels for each box size on the log of the box lengths (Voss 1988, Milne 1991). Mass fractal dimension decreases as the percentage of the landscape comprised of the focal class decreases. After accounting for this relationship, higher values of mass fractal dimension are associated with higher contagion.

Lacunarity is another method borrowed from fractal geometry by which class-specific contagion can be characterized across a range of spatial scales (Plotnick et al. 1993 and 1996, Dale 2000). Consider a binary ("on/off") raster map of dimension m. The technique involves using a moving window and is concerned with the frequency with which one encounters the focal class in a window of different sizes, similar to the mass fractal dimension. To do this, a gliding box is constructed of dimension r, and a frequency distribution of tallies of S=1,2, ..., r² "on" cells is constructed. In this gliding, the box is moved over one cell at a time, so that the boxes overlap during the sliding. Define n(S,r) as the number of boxes that tally S "on" cells for a box of size r. There are N(r)=(m-r+1)² boxes of size r, and the probability associated with n(S,r) is Q(S,r)=n(S,r)/N(r). The first and second moments of Q are used to define lacunarity L (lambda) for this box size as a variance to mean-square ratio. This is repeated for a range of box sizes r. A log-log plot of lacunarity against window size expresses the contagion of the map, or its tendency to aggregate into discrete patches, across a range of spatial scales. Note that as r increases, the tally n(S,r) converges on the mean and its variance converges on 0. Thus, for very large boxes L=1 and ln(L)=0. At the smallest box size (r=1), L(1) = 1/p, where p is the proportion of the map occupied ("on"). Thus, for the finest resolution of a map, L depends solely on p. At intermediate scales, lacunarity expresses the contagion of the map, or its tendency to clump into discrete patches. So the lacunarity plot summarizes the contagion of the map across all scales.

As noted in the background discussion, contagion and subdivision are closely related concepts. Both refer to the aggregation of patch types, but subdivision deals explicitly with the degree to which patch types are broken up (i.e., subdivided) into separate patches (i.e., fragments). Subdivision can be evaluated using a wide variety of metrics already described; for example, the number, density, and average size of patches and the degree of contagion all indirectly relate to subdivision. However, these metrics have been criticized for their insensitivity and inconsistent behavior across a wide range of subdivision patterns. Jaeger (2000) discussed the limitations of these metrics for evaluating habitat fragmentation and concluded that most of these metrics do not behave in a consistent and logical manner across all phases of the fragmentation process. He introduced a suite of metrics derived from the cumulative distribution of patch sizes that provide alternative and more explicit measures of subdivision. When applied at the class level, these metrics can be used to measure the degree of fragmentation of the focal patch type. Applied at the landscape level, these metrics measure the graininess of the landscape; i.e., the tendency of the landscape to exhibit a fine- versus coarse-grain texture. A fine-grain landscape is characterized by many small patches (highly subdivided); whereas, a coarse-grain landscape is characterized by fewer large patches.

FRAGSTATS computes three of the subdivision metrics proposed by Jaeger (2000). All of these metrics are based on the notion that two animals, placed randomly in different areas somewhere in a region, will have a certain likelihood of being in the same undissected area (i.e., the same patch), which is a function of the degree of subdivision of the landscape. The landscape division index (DIVISION) is based on the degree of coherence (C), which is defined as the probability that two animals placed in different areas somewhere in the region of investigation might find each other. Degree of coherence is based on the cumulative patch area distribution and is represented graphically as the area above the cumulative area distribution curve. Degree of coherence represents the probability that two animals, which have been able to move throughout the whole region before the landscape was subdivided, will be found in the same patch after the subdivision is in place. The degree of landscape division is simply the complement of coherence and is defined as the probability that two randomly chosen places in the landscape are not situated in the same undissected patch. Graphically, the degree of landscape division is equal to the area below the cumulative area distribution curve.

The splitting index (SPLIT) is defined as the number of patches one gets when dividing the total landscape into patches of equal size in such a way that this new configuration leads to the same degree of landscape division as obtained for the observed cumulative area distribution. The splitting index can be interpreted to be the “effective mesh number” of a patch mosaic with a constant patch size dividing the landscape into S patches, where S is the splitting index. The effective mesh size (MESH) simply denotes the size of the patches when the landscape is divided into S areas (each of the same size) with the same degree of landscape division as obtained for the observed cumulative area distribution. Thus, all three subdivision metrics are easily computed from the cumulative patch area distribution. These measures have the particular advantage over other conventional measures of subdivision (e.g., mean patch size, patch density) in that they are insensitive to the omission or addition of very small patches. In practice, this makes the results more reproducible as investigators do not always use the same lower limit of patch size. Jaeger (2000) argues that the most important and advantageous feature of these new measures is that effective mesh size is ‘area-proportionately additive’; that is, it characterizes the subdivision of a landscape independently of its size. In fact, these three measures are closely related to the area-weighted mean patch size (AREA_AM) discussed previously, and under certain circumstances are perfectly redundant. The distinctions are discussed below for each metric.