Quantyfiying Landscape

Royal Swedish Academy of Sciences

Quantifying Spatial Patterns of LandscapesAuthor(s): Magnus EkströmSource: Ambio, Vol. 32, No. 8, Remote Sensing for the Environment (Dec., 2003), pp. 573-576Published by: Springer on behalf of Royal Swedish Academy of SciencesStable URL: http://www.jstor.org/stable/4315442Accessed: 09-06-2016 06:26 UTC

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Magnus Ekstrom

Quantifying Spatial Patterns of Landscapes

In this paper we discuss information theoretical landscape indices based on data from digitized maps in grid format: measures based on Shannon's entropy, e.g. the measures of diversity and contagion; and measures based on conditional entropy, e.g. a new index which can be seen as an alternative to the measure of contagion that does not have the disadvantage of being highly correlated to the measure of diversity. We also introduce a measurement on how much information is contained in a coarse-scale map about a fine-scale map.

INTRODUCTION

The spatial pattern of ecosystems influences many ecological phenomena. An underlying assumption of many environmental decisions is that some patterns are more desirable than others (1), and 'before the interaction between landscape structure and ecological processes can be understood, landscape patterns must be identified and quantified in meaningful ways' (2). Therefore, tools are needed to understand and describe the spatial structures of landscapes. One idea is to use a set of indices that captures important aspects of landscape structure in a few numbers.

Since the late 1980s many landscape indices have been proposed; see, e.g. the review by Haines-Young and Chopping (1) where a selection of 56 different indices, introduced since 1988, are listed. The choice of which indices to use seems overwhelming, and it is clear that there is a good deal of redundancy in the indices available.

In this paper the focus will be on indices, based on data from digitized maps in grid format, which are derived from information theory.

Indices Based on Shannon's Entropy

Consider a raster map, where each picture element, or pixel, can take any one of K distinct land-cover types. For example, the map might be a Thematic Mapper satellite image with a 30 m x 30 m pixel resolution, classified into different land- cover categories, e.g. land and water.

The first index is Shannon's entropy:

K

EK kp lok 1g02pk

where Pk iS the probability that a randomly chosen pixel is of land-cover category k (we define xlog2 x = 0 if x = 0, thus extending xlog2 x to the origin by continuity). This index is also known as the measure of diversity; see e.g. Turner et al. (3). A well-known property of the entropy is that

EK < 10g2 K, Eq. 1

with equality if and only if Pk =1/K for all k. Thus, large values of EK indicate that the land-cover categories are represented in approximately equal proportions, and low

values indicate that the landscape is dominated by one or a few cover categories.

In O'Neill et al. (4) the deviation from the maximum possible entropy was introduced as an index, the measure of domination:

K

DK = 10g2 K +k1 Pk 092 Pk-

Moreover, in the same paper another landscape index based on the entropy was introduced, the measure of contagion. Here we consider the (corrected) version discussed by Li and Reynolds (5):

K K

CK= 2 10g2 K + E E Pij0g2 Pij 1

where pj is the probability that a randomly chosen pixel is of category i, and that a pixel, chosen randomly from the pixels adjacent to the first pixel, is of category j. Here, a pixel is considered to be adjacent only to the neighboring pixels on the 4 sides, i.e. upper, lower, left, and right (pixels touching at corners are not considered as adjacent in this paper). CK is a nonnegative number, and is always less or equal to 210g2 K (which is a direct consequence of the inequality in Eq. 1).

According to O'Neill et al. (4) higher values of CK indicate that a few large patches are found on the landscape, whereas lower values indicate that the landscape is disserted into many small patches. Unfortunately, with this in mind, one is easily misled by this index. For example, compare the simulated landscapes (c) and (g) in Figure 1; landscape (c) clearly has fewer and larger patches than landscape (g), but according to O'Neill et al. the values of CK indicate the opposite. From Figure 1 it is clear that the measure of contagion decreases as the landscape becomes more diverse, i.e. less dominated by one or a few land-cover categories, and as the landscape becomes more dissected into many small patches. Note that CK and EK (or DK) do not give independent information, i.e. CK changes when the diversity changes. See also Riitters et al. (6) for more detailed comments on the measure of contagion.

For any K, the function EK = EK (P1,... ,p) is a continuous and symmetric function with respect to all its arguments. Furthermore, for every K, we have

EK+1 (PI' 'PK'O) = EK (PI ...'PK), Eq. 2

i.e., the adding of a nonobserved land-cover category does not change the measure of divergence. These 2 properties together imply that the entropy is nonsensitive to the presence of rare land-cover categories. The measures of dominance and contagion, however, do not satisfy a property like Eq. 2. Thus, when K is defined as the number of observed distinct land-cover categories on the landscape, it is clear that the contagion index, just as the measure of dominance, is sensitive to the presence of rare land-cover categories (3, 5), which may cause problems in the interpretation, e.g. when comparing

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I'Or A~~~~~

(a) E5 = 1.54 C5 = 3.05 (b) E5 = 2.02 C5 = 2.56 (c) E5 = 2.31 C5 = 2.26 Cl= 0.15 Cl = 1.15 Cl = 0.20 Cl - 1.52 CZ = 0.22 Cl 2=1.74 10- 252 ~~~5.10 5,252 I -0.2 2521 7 W,f510%.10 W1 5= 0.76 H = -010 H 2=0.75 H5' -0.10 Ii -076 ,7io 1?39 j25. = 0.38 H1 O182 t 5 0 50 5-1 2 09 0,252 056 5,10 5,252 5,10 -5,252 -5,10 5,252-

"7~~~~7

1

A~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A

(d) Es= 1.52 C5=3.04 (e) E5= 2.05 C = 2.49 (f) E5= 2.30 Cs= 2.22 CI. =0.24C1 = 1.40 CI = 0.32 C 2= 1.84 C/510_0.37 Cl 1ss- 2.08 -,=0.16 H5 = 0.92 W1.=00016 W125= 0.90 H = -016 i- -090 5,0128 j.,252..= 0.12 110:= 1.73 1. -5 0.21 71,10 1 93 P252 -0 -23 15,10 - 5,252 5,10 -5,252 -5,10 -5,252 -

(g) E5 = 1.55 C5 =2.95 (h) E5 = 2.04 C5 = 2.42 (i) E5 = 2.32 C5 =2.11 CI. -0=041 CZ =1.51 CI -052 CI 521.98 CI -060 CI, -220 ,10. 26 _l5 I= 0.98 fl10 026 142- 9 I 11006 f252 =0.95 15'10-1.14 1'252-004 11015 2/52 -007 ,110=172 =52 012 7 11 " I5 52200775,10 - - 5,252 5,10 - 5,52-5,10 -5,252-

Figure 1. The images above are based on 3 different vector images; one image for each row. The coloring of the polygons in each image was done by random; with probabilities 0.6 (white), 0.25 (light grey), 0.1 (grey), 0.045 (dark grey) and 0.005 (black), for the images to the left; with probabilities 0.4, 0.3, 0.15, 0.1 and 0.05, for the images in the middle; and with equal probabilities for the 3 at the right-hand side. Thereafter, each image was transformed to a raster image of 2520 x 2520 pixels, and the different indices presented above were calculated based on the pixel data given in each raster image.

contagion indices from different landscapes and/or spatial resolution. For instance, in Turner et al. (3) the relationship of the measures of dominance and contagion to different spatial resolutions, i.e. to different pixel sizes, was examined by using simple linear regression. When the spatial resolution becomes increasingly coarser, rare land-cover categories disappear and K, defined as the number of observed distinct land-cover categories on the landscape, decreases. Thus,

since neither DK nor CK satisfy a property like Eq. 2, the benefit of using simple linear regression, i.e. of fitting a straight line, is limited, e.g. for extrapolation of the measures DK and CK from one level of resolution to another. However, if K is defined as the number of land-cover categories of interest, rather than the number of observed categories - which can be less than the number of categories of interest - then methods like linear regression could be more useful, since K does not change as the resolution changes in this case.

574 C Royal Swedish Academy of Sciences 2003 Ambio Vol. 32 No. 8, Dec. 2003 http://www.ambio.kva.se


(a) H 20.75 /1 = .50 (b) H =0.88 1 = 0.24 (c) H2= 0.00 I2= 2.00 H 40 75 / = 0.50 H 40 96 / = 0.07 H = 0.00 I = 2.00 H- =0.75 /4, 8= 050 H=0.99 / 8=0.02 H -8 ??? / 8=2.00

(d) =4=0 /20.00 (e) H4=.4= 2.00 (f) H = 0.58 12= 0.83 H'4 1.00 I' = 0.00 H'4 =1.00 /' = 0.00 H' = 0.98 14' = 0.04 H 8=1.0/ ?-? H = 1.00 / = ?-?? H 8 = 0-99 14 = 0.01

()H =0.67 14 = 0.60 (h) H =0.79 14 = 0.39 (i) H4 = 0.60 14 = 0.79 H' =0 67 / = 0.61 H' =0 83 / = 0.32 H = 0.76 / = 0.48 H- =0.66 /4 8= 0.63 H- =0.80 / 8 0.37 H- 8=0.96 / 8= 0.08

4 U... 48=4U 1,

Figure 2. Nine different images represented on a 16 x 16 grid.

INDICES BASED ON EXTENSIONS OF ENTROPY

Studies into the way landscape analysis is affected by scale considerations are numerous in the literature, and it has been recognized that scale has a significant effect on most measurements of different aspects of spatial patterns. Therefore, one should seek for measurements that can incorporate a range of scales. In Johnson and Patil (7) one such measurement is discussed, the conditional entropy, which can be applied over a range of resolutions.

Again, consider a raster map composed of a large number of square pixels of equal size, where each data pixel represents one of K distinct land-cover categories. From this original map of, let us say, M x N pixels, we will, in order to define conditional entropy, derive a map at a coarser resolution by aggregating neighborhoods of m x m pixels into new larger pixels. Thus, if we for simplicity assume that both M and N are divisible by m, we obtain a coarser-scale map of M/m x N/m pixels. The assignment of a land-cover category to an aggregate can be done in different ways; e.g. by choosing a category from the original pixels that formed the aggregate, which can, for instance, be done by choosing a fixed position of the original pixels (like the middle, the upper right, etc.), by choosing one of the original pixels by random, or by choosing the mode of the original pixels categories (in the case when the mode is not unique, one of the modes is chosen by random). Below, we will use the latter technique.

For the original map, let Pk be the probability that a pixel

picked by random is of category k, and let qk be the probability that a pixel picked by random from the coarse-scale map will be assigned land-cover category k. Further, let P]jk be the probability that a randomly chosen pixel from the fine-scale map is of category j, given that this is one of a group of m x m pixels that defines an aggregate of category k. Then the conditional entropy between the 2 maps is

K K

CIK m =- XI qk Pjlk 10_2 Pjlk -

In the definition of CIKmX i.e. of qk and PjlkI' we do not assume that the values of all data pixels on the coarse- scale map are known; that is, it is not assumed that the outcomes of the assignments in the multimodal cases are known. One could, however, define CIKm from one realization of the coarse-scale map, but we do not do that here, since that would make CIKm dependent on realizations of random variables that have nothing to do with the landscape under study. The conditional entropy is a nonnegative quantity

bounded from above by EK = EK (P' ... ' PK)' i.e.

K

0 < CIK m EK= - Pk1g2 Pk1 Eq.3

The lower bound is attained when for all k with qk > 0, there exists j such that plk= and pi = 0 for all i ? k, i.e. when each group ofm x m pixels, that defines an aggregate, consists of one land-cover category only, e.g. as in Figure 2 (c) for m = 2, 4 and 8. The upper bound is obtained when each group of m x m pixels, that defines an aggregate, contains the land-cover categories 1,.. .,K in equal proportions (e.g. as in Fig. 2 (d) for m = 2, 4 and 8, and Fig. 2 (e) for m = 4 and 8). For more details about conditional entroDv and

related quantities in information theory, see e.g. Guiasu (8). Here we propose a new index, a normed variant of the

conditional entropy:

CIKm HKm- EK

K

which gives a measurement with quite different properties than the conditional entropy itself. From the inequalities in Eq. 3 we see that HK,m is bounded from below by zero, and from above by one. When the landscape, represented by the fine-scale map, is grouped into disjoint neighborhoods (of m x m pixels), and if each neighborhood contains land-cover categories in approximately the same proportions as for the whole fine-scale map, then HKm will be large. If most neighborhoods contain only one of the different land-cover categories found on the map, however, then HKm will be small. Therefore, we call HK,m the measure of homogeneity, since the larger HKm, the more equal the neighborhoods are to each other (with respect to proportions of land-cover categories).

Even though we have assumed that each neighborhood should be of the same size and shape, i.e. a square of m x m pixels, we want to stress that this is not a necessary assumption. The neighborhoods can, for instance, be naturally defined regions on the landscape. Observe that we get information about the landscape on a local level by choosing small m (neighborhoods), and on a regional level by choosing large m (neighborhoods). HKm will increase, as the value of m increases, if the proportions of land-cover categories 1,...,K

Ambio Vol. 32 No. 8, Dec. 2003 ? Royal Swedish Academy of Sciences 2003 575 http://www.ambio.kva.se


in the aggregates get closer and closer to the proportions

PI' 'PK (see e.g. Fig. 1 or Fig. 2 (b) and (i)). However, it should be noted that if there is a large regional variety, i.e. if some land-cover categories are rare, sparse or absent in some regions, but dominate in others, HKm might have a more irregular behavior as m increases (Fig. 2 (g) and (h)). Further, it is clear that HKm increases as the landscape becomes more dissected into many small patches, as illustra- ted in Figure 1, and therefore one may consider HKm as a measure of fragmentation as well.

An important criterion for a set of useful indices is that they should be relatively independent of each other (4, 9). Figure 1 indicates that this is the case for the 2 indices EK and HK m Also, it should be noted that the measure of homogeneity (fragmentation), just as conditional entropy, could be applied over a range of resolutions as described in Johnson and Patil (7).

According to Johnson and Patil (7) the conditional entropy (with m = 2) is closely related to the measure of contagion. However, if one considers 210g 2 K- CK, rather than the measure of contagion itself, the relationship is even stronger, e.g. both measurements increase both with diversity and fragmentation, and neither of them is sensitive to the presence of rare land-cover categories. Therefore, it is possible to consider the quantity

21og2K- CK

EK

as an alternative to HKm. However, this alternative measure does not have a natural upper bound, and is therefore more difficult to interpret. Moreover, by using HK m we can study both local (by choosing small m) and regional phenomena (by choosing large m).

A MEASUREMENT OF THE INFORMATION CONTAINED IN A COARSE-SCALE MAP

Studies on how the landscape analysis is affected by scale considerations are numerous in the literature. A well-known problem is that important information is often lost as the resolution of spatial data becomes increasingly coarser (1-3, 9,10). In Turner et al. (3) '... the development of methods that will ... quantify the loss of information with changing scales' was asked for, since 'such methods are necessary before ecological insights can be extrapolated between spatial and temporal scales'. Here we propose a related quantity, i.e.

IK,m =EK CIK mK

which can be seen as a measurement of the amount of information which is contained in the coarse-scale map about the fine-scale map. From the inequalities in Eq. 1 and Eq. 3 it follows that

O < IK m < EK< 10g2 K

If the landscape consists of land-cover categories in approximately equal proportions, and if the patches found on the landscape are "large" (compact and extensive) relative to mn, then the information contained in the coarse-scale map about the fine-scale map is large, i.e. little information is lost when the resolution becomes more coarse. However, the more the landscape is dominated by one or a few land-cover categories, or the more the landscape becomes more dissected

into many small patches (here "small" includes elongated

and thin patches of small area), the smaller value of IKf. In particular, if the pixel size in the coarse-scale map is larger than many of the patches found on the map at the fine scale, then it is clear that substantial information is lost, e.g. see Figure 1 where the values of I5252 are very low for images (d)-(h), reflecting that a considerable amount of information is lost when-transforming the raster images of 2520 x 2520

pixels into images of only 10 x 10 pixels. In most cases, IKm is a nondecreasing function of m, reflecting that more and more information is lost as the resolution becomes increasingly more coarse. When the landscape has large local/regional

differences (as in Fig. 2 (a), (c), (g) and (h)), however, IK,m sometimes decrease with m (as in Fig. 2 (g) and (h)), and should therefore be used with caution.

DISCUSSION

Even though it is impossible to capture the complexity of spatial structures of landscapes in just a few numbers, the use of landscape indices can help us understand and describe important features of it, just like quantities like an average or a standard deviation can describe important aspects of a data set.

We believe that indices like the measures of diversity and contagion do provide important information about a landscape's spatial patterning. A problem however, is that these 2 indices are highly correlated (6). But the main characteristics attached to the measure of contagion in O'Neill et al. (4) are captured by our proposed index HKm as well. Thus, both a large CK and a small HKm may indicate that a few large patches are found on the landscape, while both a small CK and a large HKm may indicate that the landscape is dissected into many small patches (note, what is considered to be a small or a large patch when dealing with HKm depends on the choice of

m). But in addition, HK,m has the advantage of not being highly correlated to the measure of diversity, as indicated in Figure 1.

References and Notes

1. Haines-Young, R. and Chopping, M. 1996. Quantifying landscape structure: a review of landscape indices and their application to forested landscapes. Prog. Phys. Geogr 20, 418-445.

2. Turner, M.G. 1990. Spatial and temporal analysis of landscape patterns. Landscape Ecol. 4, 21-30.

3. Turner, M.G., O'Neill, R.V, Gardner, R.H. and Milne, B.T. 1989. Effects of changing spatial scale on the analysis of landscape pattern. Landscape Ecol. 3, 153-162.

4. O'Neill, R.V, Krummel, J.R., Gardner, R.H., Sugihara, G., Jackson, B., DeAngelis, D.L., Milne, B.T., Turner, M.G., Zygmunt, B., Christensen, SW, Dale, VH. and Graham, R.L. 1988. Indices of landscape pattern. Landscape Ecol. 1, 153-162.

5. Li, H. and Reynolds, J.F. 1993. A new contagion index to quantify spatial patterns of landscapes. Landscape Ecology 8, 155-162.

6. Riitters, K.H., O'Neill, R.V, Wickham, J.D. and Jones, K.B. 1996. A note on contagion indices for landscape analysis. Landscape Ecol. 11, 197-202.

7. Johnson, G.D. and Patil, G.P. 1998. Quantitative multiresolution characterization of landscape patterns for assessing the status of ecosystem health in watershed management areas. Ecosyst. Health 4, 177-187.

8. Guiasu, S. 1977. Information Theory with Applications. New York, McGraw-Hill. 9. O'Neill, R.V, Hunsaker, C.T., Timmins, S.P., Jackson, B.L., Jones, K.B., Riitters, K.H.

and Wickham, J.D. 1996. Scale problems in reporting landscape pattern at the regional scale. Landscape Ecol. 11, 169-180.

10. Benson, B.J. and MacKenzie, M.D. 1995. Effects of sensor spatial resolution on landscape structure parameters. Landscape Ecol. 10, 113-120.

Magnus Ekstrom, PhD, is an associate professor at the Center of Biostochastics, Swedish University of Agricultural Sciences. His research interests include estimation theory, resampling techniques, spatial statistics, and classification theory. His address: Center of Biostochastics, Swedish University of Agricultural Sciences, SE-901 83 Umea, Sweden. [email protected]

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Quantyfiying Landscape

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