Aggregation metrics

Background.--Aggregation refers to the tendency of patch types to be spatially aggregated; that is, to occur in large, aggregated or "contagious" distributions. This property is also often referred to as landscape texture. We use the term "aggregation" as an umbrella term to describe several closely related concepts: 1) dispersion, 2) interspersion, 3) subdivision, and 4) isolation. Each of these concepts relates to the broader concept of aggregation, but is distinct from the others in subtle but important ways, as described below. The concept of aggregation and its subsumed component concepts (listed above) is also indirectly related to the concept of connectivity, since the aggregation of the patch mosaic can affect the connectivity of the mosaic depending on how connectivity is measured. However, here we treat connectivity separately from aggregation and the other groups of metrics that can directly or indirectly affect connectivity but which may also be used to describe pattern and process that is unrelated to connectivity.

• Dispersion and Interspersion -- Many of the aggregation metrics deal explicitly with the spatial properties of dispersion and interspersion, and thus it is important to distinguish these two distinct components. Dispersion refers to the spatial distribution of a patch type (class) without explicit reference to any other patch types. Dispersion deals with how spread out or dispersed a patch type is, whereby the greater the dispersion, the greater the disaggregation of the class or landscape. Interspersion, on the other hand, refers to the spatial intermixing of different patch types (classes) without explicit reference to the dispersion of any patch type. Interspersion deals solely with how often each patch type is adjacent to each other patch type and not by the size, contiguity or dispersion of patches. Dispersion and interspersion are both aspects of landscape texture; they both deal with the adjacency of patch types, but do so in a different manner. Dispersion reflects the spatial distribution of a particular patch type and is based on how often cells of a patch type are adjacent to cells of the same patch type, whereas interspersion reflects the intermixing of patch types and is based on how often cells along the perimeter of patches are adjacent to other patch types. These two spatial properties are often highly confounded in real landscapes; as patch types become more dispersed they also tend to be more well interspersed among other patch types. Thus, an aggregated landscape tends to exhibit low dispersion and interspersion, whereas a disaggregated landscape tends to exhibit high dispersion and interspersion. Nevertheless, these two components can be measured independently or jointly, as described below.
  
  
• Subdivision -- Subdivision is closely related to the concept of dispersion; 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). Whereas dispersion deals with the aggregation or disaggregation of cells of the same patch type and is based on cell adjacencies independent of patch membership, subdivision deals explicitly with the subdivision of patch types into disjunct patches. Thus, two distributions can have identical levels of dispersion (e.g., if there are no like cell adjacencies, as in the case of a checkboard-like distribution), but they can have very different levels of subdivision. Of course, these two components of aggregation are often highly confounded in real landscapes; as patch types become more dispersed they also tend be more subdivided.
  
  The subdivision of a particular habitat type may affect a variety of ecological processes, depending on the landscape context. For example, the number or density of patches may determine the number of subpopulations in a spatially-dispersed population, or metapopulation, for species exclusively associated with that habitat type. The number of subpopulations could influence the dynamics and persistence of the metapopulation (Gilpin and Hanski 1991). The number or density of patches also can alter the stability of species interactions and opportunities for coexistence in both predator-prey and competitive systems (Kareiva 1990). The number or density of patches in a landscape mosaic (pooled across patch types) can have the same ecological applicability, but more often serves as a general index of spatial heterogeneity of the entire landscape mosaic. A landscape with a greater number or density of patches has a finer grain; that is, the spatial heterogeneity occurs at a finer resolution. Although the number or density of patches in a class or in the landscape may be fundamentally important to a number of ecological processes, often it does not have any interpretive value by itself because it conveys no information about the area or distribution of patches. Number or density of patches is probably most valuable, however, as the basis for computing other, more interpretable, metrics, but is often use in combination with other metrics to characterize subdivision.
  
  
• Isolation -- Isolation is closely related to the concept of subdivision; both refer to the subdivision per se of patch types, but isolation deals explicitly with the degree to which patches are spatially isolated from each other, whereas subdivision doesn't address the distance between patches, only that they are disjunct. Thus, two distributions can have identical levels of subdivision (e.g., identical patch size distributions), but they can have very different levels of isolation, for example if the patches are farther apart in one landscape compared to the other. Of course, these two components of aggregation are often highly confounded in real landscapes; as patch types become more subdivided they also tend be more isolated, but this isn't always the case. Consider the case when large contiguous patches get subdivided by roads; the level of patch subdivision goes up but the patches may or may not be more isolated from each other as a result.

The texture of a landscape is a fundamental aspect of landscape pattern and is important in many ecological processes. Interspersion is presumed to affect the quality of habitat for many species that require different patch types to meet different life history requisites, as in the process of landscape complementation (Dunning et al. 1992). Indeed, the notion of habitat interspersion has had a preeminent role in wildlife management during the past century. Wildlife management efforts are often focused on maximizing habitat interspersion because it is believed that the juxtaposition of different habitats will increase species diversity (Leopold 1933).

The disaggregation of a patch type of course plays a crucial role in the process of habitat loss and fragmentation. Specifically, habitat loss and fragmentation generally involves the disaggregation of contiguous habitat into more dispersed habitat and/or disjunct (i.e., subdivided) and more isolated patches. As habitat loss and fragmentation proceeds, habitat becomes disaggregated and eventually ecological function is impaired (Saunders et al.1991). Specifically, the subdivision and isolation of populations caused by this habitat loss and fragmentation can lead to reduced dispersal success and patch colonization rates which may result in a decline in the persistence of individual populations and an enhanced probability of regional extinction for entire populations across the landscape (e.g., Lande 1987; With and King 1999a,b; With 1999). In addition, the disaggregation of patch types may affect the propagation of disturbances across a landscape (Franklin and Forman 1987). Specifically, a patch type that is highly disaggregated and/or subdivided may be more resistant to the propagation of some disturbances (e.g., disease, fire, etc.), and thus more likely to persist in a landscape than a patch type that is highly aggregated and/or contiguous. Conversely, highly disaggregated and/or subdivided patch types may suffer higher rates of disturbance for some disturbance types (e.g. windthrow) than more aggregated and/or contiguous distributions.

Isolation of habitat patches is a critical factor in the dynamics of spatially structured populations. For example, there has been a proliferation of mathematical models on population dynamics and species interactions in spatially subdivided populations (Kareiva 1990), and results suggest that the dynamics of local plant and animal populations in a patch are influenced by their proximity to other subpopulations of the same or competing species. Patch isolation plays a critical role in island biogeographic theory (MacArthur and Wilson 1967) and metapopulation theory (Levins 1970, Gilpin and Hanski 1991). The role of patch isolation (e.g., as measured by interpatch distance) in metapopulations has had a preeminent role in conservation efforts for endangered species (e.g., Lamberson et al. 1992, McKelvey et al. 1992).

Isolation is particularly important in the context of habitat loss and fragmentation. Several authors have claimed, for example, that patch isolation explains why fragmented habitats often contain fewer bird species than contiguous habitats (Moore and Hooper 1975, Forman et al. 1976, Helliwell 1976, Whitcomb et al. 1981, Hayden et al. 1985, Dickman 1987). Specifically, as habitat is lost and fragmented, residual habitat patches become more isolated from each other in space and time. One of the more immediate consequence of this is the disruption of movement patterns and the resulting isolation of individuals and local populations. This has important metapopulation consequences. As habitat is fragmented, it is broken up into remnants that are isolated to varying degrees. Because remnant habitat patches are relatively small and therefore support fewer individuals, there will be fewer local (within patch) opportunities for intra-specific interactions. This may not present a problem for individuals (and the persistence of the population) if movement among patches is largely unimpeded by intervening habitats in the matrix and connectivity across the landscape can be maintained. However, if movement among habitat patches is significantly impeded or prevented, then individuals (and local populations) in remnant habitat patches may become functionally isolated. The degree of isolation for any fragmented habitat distribution will vary among species depending on how they perceive and interact with landscape patterns (Dale et al. 1994, With and Crist 1995, Pearson et al. 1996, With et al. 1997, With 1999); less vagile species with very restrictive habitat requirements and limited gap-crossing ability will likely be most sensitive to isolation effects.

Habitat patches can become functionally isolated in several ways. First, the patch edge may act as a filter or barrier that impedes or prevents movement, thereby disrupting emigration and dispersal from the patch (Wiens et al. 1985). Some evidence for this exists for small mammals (e.g., Wegner and Merriam 1979, Chasko and Gates 1982, Bendell and Gates 1987, Yahner 1986), but the data are scarce for other vertebrates. Thus, subdivision per se can lead to increased isolation. Whether edges themselves can limit movement presumably depends on what species are trying to cross the edge and on the structure of the edge habitat (Kremsater and Bunnell 1999). Second, the distance from remnant habitat patches to other neighboring habitat patches may influence the likelihood of successful movement of individuals among habitat patches. Again, the distance at which movement rates significantly decline will vary among species depending on how they scale the environment. In general, larger organisms can travel longer distances. Therefore, a 100 m-wide agricultural field may be a complete barrier to dispersal for small organisms such as invertebrates (e.g., Mader 1984), yet be quite permeable for larger and more vagile organisms such as birds. Lastly, the composition and structure of the intervening landscape mosaic may determine the permeability of the landscape to movements. Note that under an island biogeographic perspective, habitat patches exist in a uniform sea that is hostile to both survival and dispersal. In this case, the matrix is presumed to contain no meaningful structure and isolation is influenced largely by the distance among favorable habitat patches. However, under a landscape mosaic perspective, habitat patches are bounded by other patches that may be more or less similar (as opposed to highly contrasting and hostile) and connectivity is assessed by the extent to which movement is facilitated or impeded through different habitat types across the landscape. Each habitat may differ in its "viscosity" or resistance to movement, facilitating movement through certain elements of the landscape and impeding it in others. Again, the degree to which a given landscape structure facilitates or impedes movement will vary among organisms. Regardless of how habitat patches become isolated, whether it be due to properties of the edges themselves, the distance between patches, or properties of the intervening matrix, the end result is the same - fewer individual movements among habitat patches.

FRAGSTATS Metrics.--There are several different approaches for measuring aggregation. 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 the Interspersion and juxtaposition index (IJI) that isolates the interspersion aspect of aggregation; it increases in value as patches tend to be more evenly interspersed in a "salt and pepper" mixture. Unlike the previous contagion index that is 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 and juxtaposition 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 and juxtaposition 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.

FRAGSTATS computes a suite of metrics from the cell adjacency matrix that isolate the dispersion aspect of aggregation. 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 aggregation that isolates the dispersion (as opposed to interspersion) component of aggregation. However, this index requires careful interpretation because it varies in relation to the proportion of the landscape comprised of the focal class (Pi). It has been shown that PLADJ for class i will equal Pi 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 < Pi. If the focal class is more contagiously distributed, then PLADJ > Pi. 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 Pi.

FRAGSTATS computes two indices based on PLADJ that adjust for Pi in different ways. The Aggregation index (AI) is computed as a percentage based on the ratio of the observed number of like adjacencies (eii), based on the single-count method, to the maximum possible number of like adjacencies (max_ei,i) given Pi (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 ei,i for any Pi,. He et al. (2000) provide the formula for computing max_ei,i. The index ranges from 0 when there is no like adjacencies (i.e., when the class is maximally dissagregated) to 1 when ei,i reaches the maximum (i.e., when the class is maximally aggregated). However, AI is partially confounded with Pi because the minimum value of the index varies with Pi when Pi > 0.5; specifically, the minimum value > 0 when Pi > 0.5 and asymptotically approaches 1 as Pi ?1. Thus, AI does not account for the expected value under a spatially random distribution when Pi > 0.5; e.g., AI could equal 0.8 and yet the distribution could be more disaggregated than expected under a random distribution if Pi > 0.8. Thus, caution must be exercised in interpreting this metric. The Clumpiness index (CLUMPY) is a class-level only metric 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 Pi. 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 aggregation 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.

FRAGSTATS computes a few metrics based on the number of unlike cell adjacencies (i.e., edges or patch perimeters). As the proportion of like cell adjacencies increases, the number of unlike cell adjacencies decreases. Unlike cell adjacencies represent the edges between patch types. Thus, there is an inverse relationship between the proportion of like cell adjacencies (the basis for PLADJ, AI and CLUMPY) and the length of edge. The Landscape shape index (LSI) index measures the perimeter-to-area ratio for the landscape as a whole. This index is identical to the habitat diversity index proposed by Patton (1975), except that we apply the index at the class level as well. LSI is identical to the shape index at the patch level (SHAPE), except that it treats the entire landscape as if it were one patch and any patch edges (or class edges) as though they belong to the perimeter. Like the shape index, it can be interpreted as a measure of the overall geometric complexity of the landscape or of a focal class; however, it can also be interpreted as a measure of landscape disaggregation - the greater the value of LSI, the more dispersed are the patch types. The landscape boundary must be included as edge in the calculation in order to use a square standard for comparison. Unfortunately, this may not be meaningful in cases where the landscape boundary does not represent true edge and/or the actual shape of the landscape is of no particular interest. In this case, the total amount of true edge, or some other index based on edge, would probably be more meaningful. If the landscape boundary represents true edge or the shape of the landscape is particularly important, then LSI can be a useful index, especially when comparing among landscapes of varying sizes. At the class level, the landscape shape index suffers from confounding with the extent of the class, similar to PLADJ and AI, but the confounding is nonlinear making interpretation even more difficult. Part of the difficulty lies in the fact that the minimum and maximum length of edge varies with the proportion of the landscape comprised of the focal patch, Pi. The Normalized Landscape shape index (nLSI) isolates the aggregation effect from the landscape composition effect by attempting to scale the index between the theoretical minimum and maximum values for any given level of Pi, but it can be biased when Pi is quite large (e.g., Pi >> .5) and when the landscape shape is not rectangular. Nevertheless, nLSI, like CLUMPY, provides a more useful index of dispersion that isolates the configuration component from the composition component. FRAGSTATS also computes the Patch cohesion index (COHESION) proposed by Schumaker (1996) to quantify the connectivity of habitat as perceived by organisms dispersing in binary landscapes. COHESION is computed from the information contained in patch area and perimeter; briefly, it is proportional to the area-weighted mean perimeter-area ratio divided by the area-weighted mean patch shape index (i.e., standardized perimeter-area ratio). COHESION is similar to the Perimeter-to-area ratio (PARA, see Shape metrics) and thus is also confounded with Pi, like PLADJ, AI, and LSI, but it is invariant to changes in the cell size and is bounded 0-1, which makes it easier to interpret and robust to changes in the grain. It is well known that, on random binary maps, patches gradually coalesce as the proportion of habitat cells increases, forming a large, highly connected patch (termed a percolating cluster) that spans that lattice at a critical proportion (pc) that varies with the neighbor rule used to delineate patches (Staufer 1985, Gardner et al. 1987). Patch cohesion has the interesting property of increasing monotonically until an asymptote is reached near the critical proportion.

FRAGSTATS computes a suite of metrics that focus on the subdivision aspect of aggregation. The simplest measure of subdivision is the Number of patches (NP) or Patch density (PD). However, these simple measures of subdivision and other measures of aggregation 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. 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, 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.

FRAGSTATS computes several metrics that focus on the isolation aspect of aggregation. Unfortunately, because of the many factors that influence the functional isolation of a patch, it is a difficult thing to capture in a single measure. In the context of habitat fragmentation, for example, isolation can be measured as the time since the habitat was physically subdivided, but this is fraught with practical difficulties, because rarely do we have accurate historical data from which to determine when each patch was isolated. Moreover, given that fragmentation is an ongoing process, it can be difficult to objectively determine at what point the habitat becomes subdivided, since this is largely a function of scale. Isolation can be measured in the spatial dimension in several ways, depending on how one views the concept of isolation. The simplest measures discussed below are based on Euclidean distance between nearest neighbors (McGarigal and Marks 1995) or the cumulative area of neighboring habitat patches weighted by nearest neighbor distance within some ecological neighborhood (Gustafson and Parker 1992). These measures adopt an island biogeographic perspective, as they treat the landscape as a binary mosaic consisting of habitat patches and uniform matrix. Thus, the context of a patch is defined by the proximity and area of neighboring habitat patches; the role of the matrix is ignored. However, these measures can be modified to take into account other habitat types in the so-called matrix and their affects on the insularity of the focal habitat. For example, simple Euclidean distance can be modified to account for functional differences among organisms. The functional distance between patches clearly depends on how each organism scales and interacts with landscape patterns (With 1999); in other words, the same gap between patches may not be perceived as a relevant disconnection for some organisms, but may be an impassable barrier for others.

FRAGSTATS computes three isolation metrics that adopt an island biogeographic perspective on patch isolation. Euclidean nearest-neighbor distance (ENN) is perhaps the simplest measure of patch isolation. Here, nearest neighbor distance is defined using simple Euclidean geometry as the shortest straight-line distance between the focal patch and its nearest neighbor of the same class, based on the distance between the cell centers of the two closest cells from the respective patches. At the class and landscape levels, FRAGSTATS computes the Mean Euclidean nearest-neighbor distance (ENN_MN). At the class level, ENN_MN can only be computed if there are at least two patches of the corresponding type. At the landscape level, ENN_MN considers only patches that have neighbors. Thus, there could be 10 patches in the landscape, but eight of them might belong to separate patch types and therefore have no neighbor within the landscape. In this case, ENN_MN would be based on the distance between the two patches of the same type. These two patches could be close together or far apart. In either case, the mean nearest-neighbor distance for this landscape may not characterize the entire landscape very well. For this reason, these metrics should be interpreted carefully when landscapes contain rare patch types. In addition to these first-order statistics, the variability in ENN provides a measure patch dispersion. Specifically, a small standard deviation (SD) in ENN (ENN_SD) relative to the mean implies a fairly uniform or regular distribution of patches across landscapes, whereas a large SD relative to the mean implies a more irregular or uneven distribution of patches. The distribution of patches may reflect underlying natural processes or human-caused disturbance patterns. In absolute terms, the magnitude of ENN_SD is a function of the mean nearest-neighbor distance and variation in nearest-neighbor distance among patches. Thus, while SD does convey information about nearest neighbor variability, it is a difficult parameter to interpret without doing so in conjunction with the mean nearest-neighbor distance. For example, two landscapes may have the same ENN_SD, e.g., 100 m; yet one landscape may have a mean nearest-neighbor distance of 100 m, while the other may have a mean nearest-neighbor distance of 1,000 m. In this case, the interpretations of landscape pattern would be very different, even though the absolute variation is the same. Specifically, the former landscape has a more irregular but concentrated pattern of patches, while the latter has a more regular but dispersed pattern of patches. For these reasons, coefficient of variation (CV) often is preferable to SD for comparing variability among landscapes. Coefficient of variation measures relative variability about the mean (i.e., variability as a percentage of the mean), not absolute variability, and is akin to the familiar indices of dispersion in point patterns based on the variance to mean ratio in nearest neighbor distance (e.g., Clark and Evans 1954). Thus, it is not necessary to know the mean nearest-neighbor distance to interpret this metric. Even so, ENN_CV can be misleading with regards to landscape structure without also knowing the number of patches or patch density and other structural characteristics. For example, two landscapes may have the same ENN_CV, e.g., 100%; yet one landscape may have 100 patches with a mean nearest-neighbor distance of 100 m, while the other may have 10 patches with a mean nearest-neighbor distance of 1,000 m. In this case, the interpretations of overall landscape pattern could be very different, even though ENN_CV is the same; although the identical CV's indicate that both landscapes have the same regularity or uniformity in patch distribution. Finally, both SD and CV assume a normal distribution about the mean. In a real landscape, nearest-neighbor distribution may be highly irregular. In this case, it may be more informative to inspect the actual distribution itself (e.g., plot a histogram of the nearest neighbor distances for the corresponding patches), rather than relying on summary statistics such as SD and CV that make assumptions about the distribution and therefore can be misleading.

FRAGSTATS also computes the Connectance index (CONNECT) as the proportion of functional joinings among all patches, where each pair of patches is either connected or not based on some criterion. FRAGSTATS computes connectance using a threshold distance specified by the user and reports it as a percentage of the maximum possible connectance given the number of patches. The threshold distance in FRAGSTATS is based on Euclidean distance, but it could be based on some other measure of functional distance, such as the least cost path distance.

Even though nearest-neighbor distance is often used to evaluate patch isolation, it is important to recognize that the single nearest patch may not fully represent the ecological neighborhood of the focal patch. For example, a neighboring patch 100 m away that is 1 ha is size may not be as important to the effective isolation of the focal patch as a neighboring patch 200 m away, but 1000 ha in size. To overcome this limitation, the Proximity index (PROX) was developed by Gustafson and Parker (1992)[see also Gustafson and Parker 1994, Gustafson et al. 1994, Whitcomb et al. 1981]. This index considers the size and proximity of all patches whose edges are within a specified search radius of the focal patch. The index is computed as the sum, over all patches of the corresponding patch type whose edges are within the search radius of the focal patch, of each patch size divided by the square of its distance from the focal patch. Note that FRAGSTATS uses the distance between the focal patch and each of the other patches within the search radius, similar to the isolation index of Whitcomb et al. (1981), rather than the nearest-neighbor distance of each patch within the search radius (which could be to a patch other than the focal patch), as in Gustafson and Parker (1992). The proximity index quantifies the spatial context of a (habitat) patch in relation to its neighbors of the same class; specifically, the index distinguishes sparse distributions of small habitat patches from configurations where the habitat forms a complex cluster of larger patches. All other things being equal, a patch located in a neighborhood (defined by the search radius) containing more of the corresponding patch type than another patch will have a larger index value. Similarly, all other things being equal, a patch located in a neighborhood in which the corresponding patch type is distributed in larger, more contiguous, and/or closer patches than another patch will have a larger index value. Thus, the proximity index measures both the degree of patch isolation and the degree of fragmentation of the corresponding patch type within the specified neighborhood of the focal patch.

FRAGSTATS computes a single isolation metric that adopts a landscape mosaic perspective on patch isolation. The Similarity index (SIMI) is a modification of the proximity index, the difference being that similarity considers the size and proximity of all patches, regardless of class, whose edges are within a specified search radius of the focal patch. SIMI quantifies the spatial context of a (habitat) patch in relation to its neighbors of the same or similar class; specifically, the index distinguishes sparse distributions of small and insular habitat patches from configurations where the habitat forms a complex cluster of larger, hospitable (i.e., similar) patches. All other things being equal, a patch located in a neighborhood (defined by the search radius) deemed more similar (i.e., containing greater area in patches with high similarity) than another patch will have a larger index value. Similarly, all other things being equal, a patch located in a neighborhood in which the similar patches are distributed in larger, more contiguous, and/or closer patches than another patch will have a larger index value. Essentially, the similarity index performs much the same way as the proximity index, but instead of focusing on only a single patch type (i.e., island biogeographic perspective), it considers all patch types in the mosaic (i.e., landscape mosaic perspective). Thus, the similarity index is a more comprehensive measure of patch isolation than the proximity index for organisms and processes that perceive and respond to patch types differentially.

Limitations.--All measures based on the adjacency matrix (i.e., the number of adjacencies between each pair of patch types) that include like-adjacencies (i.e., PLADJ, AI, CLUMPY, and CONTAG) are strongly affected by the grain size or resolution of the image. Given a particular patch mosaic, a smaller grain size will result in a proportional increase in like adjacencies. Given this scale dependency, these metrics are best used if the scale is held constant. Note, IJI, LSI, nLSI, and COHESION are not affected by resolution directly because only patch edges are considered. In addition, there are alternative ways to consider cell adjacencies. Adjacencies may include only the 4 cells sharing a side with the focal cell, or they may include the diagonal neighbors as well. FRAGSTATS uses the 4-neighbor approach for the purpose of calculating these metrics. Further, there are at least two basic approaches for counting cell adjacencies, referred to as the single count and double count methods. As noted above, FRAGSTATS adopts the double count method in which pixel order is preserved. In this method, all non-background cells inside the landscape (i.e., positively-valued cells) are visited and the four sides of each cell are tallied in the adjacency matrix. As a result, all cell sides involving non-background classes inside the landscape are tallied twice (hence the term double count), but all cell sides involving background or landscape border (i.e., negatively-valued cells) are only counted once, as those cells are themselves not visited.

There are significant limitations associated with the use of isolation metrics that must be understood before they are used. The most important limitation of these particular metrics is that nearest-neighbor distances are computed solely from patches contained within the landscape boundary. If the landscape extent is small relative to the scale of the organism or ecological processes under consideration and the landscape is an "open" system relative to that organism or process, then nearest-neighbor results can be misleading. For example, consider a small subpopulation of a bird species occupying a patch near the boundary of a somewhat arbitrarily defined (from a bird's perspective) landscape. The nearest neighbor within the landscape boundary might be quite far away, yet in reality the closest patch might be very close, but just outside the designated landscape boundary. The magnitude of this problem is a function of scale. Increasing the size of the landscape relative to the scale at which the organism under investigation perceives and responds to the environment will decrease the severity of this problem.

Similarly, the proximity and similarity indices involve a search window around the focal patch. Thus, these metrics may be biased low for patches located within the search radius distance from the landscape boundary because a portion of the search area will be outside the area under consideration. The magnitude is of this problem is also a function of scale. Increasing the size of the landscape relative to the average patch size and/or decreasing the search radius will decrease the severity of this problem at the class and landscape levels. However, at the patch level, regardless of scale, individual patches located within the search radius of the boundary will have biased indices. In addition, these indices evaluate the landscape context of patches at a specific scale of analysis defined by the size of the search radius. Therefore, these indices are only meaningful if the specified search radius has some ecological relevance to the phenomenon under consideration. Otherwise, the results will be arbitrary and therefore meaningless.

Lastly, the similarity index is a functional metric in that it requires additional parameterization, in this case, similarity coefficients that are unique to the ecological phenomenon under consideration. Consequently, as with any functional metric, its meaning depends entirely on the meaningfulness of the similarity coefficients applied. If these are arbitrary assignments or based on weak observational data, results will be arbitrary and therefore meaningless.