Statistical detection of multiscale landscape patterns - Springer Link

Environmental and Ecological Statistics 8, 253±267, 2001

Statistical detection of multiscale landscape patterns L . G R O S S I , 1 G . Z U R L I N I 2 and O . R O S S I 3 1

Statistical Institute, Faculty of Economics and Department of Environmental Science, Viale delle Scienze, 43100 Parma, Italy E-mail: [email protected] 2 National Council of Research and Department of Environmental Science, Viale delle Scienze, 43100 Parma, Italy E-mail: [email protected] 3 Department of Environmental Science, Viale delle Scienze, 43100 Parma, Italy Received June 1999; Revised May 2000 Detection of discontinuities in landscape patterns is a crucial problem both in ecology and in environmental sciences since they may indicate substantial scale changes in generating and maintaining processes of landscape patches. This paper presents a statistical procedure for detecting distinct scales of pattern for irregular patch mosaics using fractal analysis. The method suggested is based on a piecewise regression model given by ®tting different regression lines to different ranges of patches ordered according to patch size (area). Proper shift-points, where discontinuities occur, are then identi®ed by means of an iterative procedure. Further statistical tests are applied in order to verify the statistical signi®cance of the best models selected. Compared to the method proposed by Krummel et al. (1987), the procedure described here is not in¯uenced by subjective choices of initial parameters. The procedure was applied to landscape pattern analysis of irregular patch mosaics (CORINE biotopes) of a watershed within the Map of the Italian Nature Project. Results for three different CORINE patch types are herein presented revealing different scaling properties with special pattern organizations linked to ecological traits of vegetation communities and human disturbance. Keywords: boundary fractal dimension, CORINE biotopes, landscape pattern, piecewise regression 1352-8505 # 2001

Kluwer Academic Publishers

1. Introduction Hierarchy in ecological theory (Allen and Starr, 1982; O'Neill et al., 1986) implies that distinct levels in the ecological system should be re¯ected in corresponding distinct scales of patterns in space. Thus, lower levels should be clustered in space in order to interact frequently and this should be mirrored in ®ne-scale patterns. Higher levels interact as well to generate coarse-scale patterns. According to theory, such patterns would not change continuously as scale increases. Distinct levels should correspond to distinct scales of pattern. 1352-8505 # 2001

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Efforts have been largely devoted to test this for transect data. In this context, the effect of scaling on spatial patterns of diversity has been shown using fractal dimension combined with variogram analysis (He et al., 1994) and by looking at vegetation variance as a function of transect length (Levin and Buttel, 1986; O'Neill et al., 1991). Multivariate methods for spatial autocorrelation have also been applied to locate multiple scales of pattern (Burrough, 1983; Ver Hoef and Glen-Lewis, 1989), and several statistical tests for detecting multiple scales with transect data have been reviewed by Turner et al. (1991). Starting from the discontinuous frequency distribution of mammal and bird biomass, Holling (1992) demonstrated that landscapes are generally structured according to scaling regions with distinct dimensions connected by transition zones. Krummel et al. (1987) were the ®rst to develop a method (herein referred to as Krummel's method from the ®rst author's name) for detecting distinct scales of pattern for mosaics of irregular patches using fractal analysis. Fractal geometry provides a more realistic picture of the geometry of naturally occurring objects than classical Euclidean geometry. When natural ``objects'' like vegetation are not constrained by human activities and land manipulation, they result in highly irregular shapes, determined by iterative and diffusive growth, which can reproduce at different scales independent of size. A shift in the perimeter fractal dimension, or in related fractal parameters, may indicate a substantial change in processes generating and maintaining landscape patches at different scales (Krummel et al., 1987; Palmer, 1988; Sugihara and May, 1990; Milne, 1991). Understanding this effect can be of paramount importance to provide focus and direction for proactive land management activities. To our knowledge, since Krummel's method, only Loehle and Li (1996) has been made an attempt to develop statistical methods for detecting distinct scales of pattern for mosaics of irregular patches by means of fractal analysis. This paper presents a statistical method to detect scaling discontinuities following the need to improve Krummel's method, in particular as regards the dependence of results on window size adopted in rolling regressions. For illustrative purpose, we describe the procedure for perimeter-area relationships, but it is straightforwardly applicable to all power relations with fractal exponents (cf. Nikora et al., 1999). The procedure suggested is iterative and based on piecewise regressions. In order to detect discontinuities in landscape pattern we propose to ®t the piecewise regression model looking at every possible break of the different lines and to select the models which give statistical signi®cance. The method has been applied to landscape pattern analysis of real patch mosaics of a watershed included in the Map of the Italian Nature Project (Zurlini et al., 1999).

2. Detecting multiscale landscape patterns Assuming a landscape constituted by sharply de®ned polygons, at a certain resolution (see Loehle and Wein, 1994), the complexity degree of a polygon is characterized by the fractal dimension …D† of its perimeter, that for individual patches is given by Nikora et al. (1999): D ˆ l…1 ‡ H†; where l is the exponent in the perimeter-area relation P ˆ cAl (Mandelbrot, 1983) of polygonal patches, and H is the Hurst shape exponent in the area-length relation

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A ˆ c1 L…1 ‡ H† (Mandelbrot, 1983), where c and c1 are constants. H is equal to 1 when patches of different scales are geometrically self-similar. Under self-similarity (Milne, 1991; Hastings and Sugihara, 1993), we have that D ˆ 2l and thus l can provide a direct estimate of D from perimeter-area relationships. A fractal dimension indicates the ability of a set of structures to ®ll the Euclidean space where it is embedded (Mandelbrot, 1983; Milne, 1991). For patch boundaries, D is bound to a plane, thus 1  D  2; D assumes the lower bound for simple Euclidean shapes (circles and rectangles) and increases as the polygon becomes more complex leading to more complex plane-®lling fractal edges. When area and perimeter are measured on a collection of n self-similar patches, given that n is not too small, perimeter fractal dimension …D† can be directly estimated from the slope of a simple regression model (Lovejoy, 1982): log…Pi † ˆ log…c† ‡

D log…Ai † ‡ ei ; 2

i ˆ 1; 2; . . . ; n;

so that D ˆ 2l. Let us suppose that the scale spectrum of patches is not continuous and there are two domains of scale with D roughly constant in each domain. When those natural invariant domains of patches are known a priori, we can estimate the corresponding perimeter fractal dimensions by using the following piecewise regression model (cf. Draper and Smith, 1998, p. 307): Yi ˆ b0 ‡ b1 Xi ‡ a0 Zi ‡ a1 Xi Zi ‡ ei ;

i ˆ 1; 2; . . . ; n;

…1†

where Yi ˆ log…Pi †, Xi ˆ log…Ai †, Zi ˆ 0 when the ith patch belongs to the ®rst domain and Zi ˆ 1 otherwise. Model (1) can be written as two distinct lines, one for each scaling domain, Y i ˆ b 0 ‡ b1 X i ‡ e i

for Zi ˆ 0

Yi ˆ …b0 ‡ a0 † ‡ …b1 ‡ a1 †Xi ‡ ei

…2a† for Zi ˆ 1:

…2b†

Parameters a0 and a1 represent changes in the intercept and the slope in going from the ®rst (2a) to the second model (2b). Perimeter fractal dimensions for the two domains are given by D1 ˆ 2b1 and D2 ˆ 2…b1 ‡ a1 †: To be sure that two fractal domains actually exist, it is necessary to test whether two distinct lines are plausible or not. Suitable statistical tests concern single and joint signi®cance of the regression parameters estimated by model (1). Single parameter signi®cance can be tested by usual t-tests for H0 : a1 ˆ 0 versus H1 : a1 6ˆ 0. The signi®cance of a0 is not relevant since there are two fractal domains also when model (2a) and model (2b) have the same intercept. As far as joint signi®cance of parameters is concerned, we have to test the hypothesis that a single line ®ts all the data. In this case the null hypothesis would be H0 : a0 ˆ a1 ˆ 0 which can be assessed by an F-test based on the so-called extra-sum of squares. The appropriate F-statistic is:

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^ ;b ^ SS…^ a0 ; ^ a1 jb 0 1 †=2 …3† s2 ^ ;b ^ where SS…^ a0 ; ^ a1 jb 0 1 † is the extra-sum of squares obtained by subtracting the regression sum of squares of model (2a) from the regression sum of squares of model (1), and then divided by the corresponding degrees of freedom. In (3), s2 is the residual variance of model (1). For signi®cance tests, the F-ratio (3) must be compared with percentiles of the F distribution with (2, n 4) degrees of freedom. Once the statistical signi®cance of two distinct lines has been established, we need to test the appropriateness of the hypotheses H0 : 1  D1  2 and H0 : 1  D2  2. In fact, when the sample size is too small, point estimates of D might be outside the interval [1, 2]. The above hypotheses may be tested by constructing con®dence intervals (at a-level) for the fractal dimensions D1 and D2 and comparing each interval to the range of the null hypothesis. Distributions of D^1 and D^2 are approximately normal with means and variances given by Draper and Smith (1998): Fˆ

^ †; E…D^1 † ˆ 2E…b 1

^ †; var…D^1 † ˆ 4 var…b 1

^ † ‡ E…^a †Š E…D^2 † ˆ 2‰E…b 1 1

and ^ † ‡ var…^a † ‡ 2cov…b ^ ; ^a †Š. var…D^2 † ˆ 4‰var…b 1 1 1 1 So far, we have supposed that shift-points are known. In practice, they are not known and some of the greatest problems in discontinuity analysis are estimating the change points, determining their signi®cance, and assessing the effect of change-point estimation on subsequent hypothesis tests (i.e., degrees of freedom and null distributions). Krummel et al. (1987) suggested a graphical procedure that starts by splitting the initial sample of patches into two groups, i.e., smaller and larger patches, and regressing log…P† on log…A† for the group of small patches. D is estimated as twice the slope of the resulting regression line. Their method to detect discontinuity can be summarized as follows: *

*

compute successive regressions by removing the smallest and adding the next largest patch (rolling regressions). plot the successive D-values against the average log…A† of the corresponding group of patches.

The ®nal plot should show a distinct shift in fractal dimension between different fractal domains. This method was the ®rst to be proposed to detect discontinuity in irregular patch mosaics, although it presents some dif®culties in interpretation: (1) The size of the smallest patch group (initial window size) is not objectively de®ned and results can be very much affected by this initial choice. (2) The method is graphical and does not employ objective statistical tests in assessing signi®cance of perceived differences in D values. To overcome these dif®culties, we propose to ®t the model (1) for every possible shiftpoint, estimating regression parameters by least squares and restricting attention to those ®ts that are statistically signi®cant in accordance with single and joint signi®cance of

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parameters as well as to plausibility of fractal dimensions. The change-point is then estimated objectively by searching for the ®t that yields the smallest residual sum of squares. When residual sum of squares of the acceptable models are very close to each other, much caution is needed. In that case by carefully examining all the signi®cant models, some patterns may be detected which are in accordance with the data. The choice of the best pattern must then be made on the basis of the environmental knowledge acquired about the type of patches examined. The method suggested is iterative. At each step, model (1) is ®tted with Zi ˆ 0, for i ˆ 1; 2; . . . ; j and Zi ˆ 1 for i ˆ j ‡ 1; j ‡ 2; . . . ; n, so that line (2a) is ®tted to j patches and line (2b) to n-j patches, with j ˆ 3; 4; . . . ; n 3 (we set 3 as the minimum sample size for ®tting a line). When there are more than two fractal domains, the procedure can be applied by modifying model (1) to accomodate the required number of lines. The iterative procedure becomes substantially more complex because more dummy variables must be inserted and the minimization of the residual sum of squares must then be jointly performed on all the dummies. When n is not small and the number of dummies is much greater than two, the computational effort of the iterative procedure might become prohibitive even for the most recent and powerful computers. However, in our experience with the Map of the Italian Nature CORINE data, one shift-point (two perimeter fractal domains) was the most frequent situation encountered. Hence, we decided to limit our analyses to a maximum of two dummies (three fractal domains). For three fractal domains, the model takes the form: Yi ˆ b0 ‡ b1 Xi ‡ g0 Zi1 ‡ g1 Xi Zi1 ‡ d0 Zi2 ‡ d1 Xi Zi2 ‡ ei

i ˆ 1; 2; . . . ; n

…4†

where Yi and Xi have the same meaning as in (1), Zi1 ˆ Zi2 ˆ 0 when the ith patch belongs to the ®rst domain, Zi1 ˆ 1; Zi2 ˆ 0 when the ith patch belongs to the second domain, Zi1 ˆ 1; Zi2 ˆ 1 when the ith patch belongs to the third domain. As in the case of model (1), model (4) can be decomposed into distinct lines: Y i ˆ b 0 ‡ b1 X i ‡ e i

for Zi1 ˆ Zi2 ˆ 0:

Yi ˆ …b0 ‡ g0 † ‡ …b1 ‡ g1 †Xi ‡ ei

…5a†

for Zi1 ˆ 1; Zi2 ˆ 0:

Yi ˆ …b0 ‡ g0 ‡ d0 † ‡ …b1 ‡ g1 ‡ d1 †Xi ‡ ei

for Zi1 ˆ 1; Zi2 ˆ 1:

…5b† …5c†

Here, perimeter fractal dimensions for the different domains are given by: D1 ˆ 2b1 ; D2 ˆ 2…b1 ‡ g1 † and D3 ˆ 2…b1 ‡ g1 ‡ d1 †: Single parameter signi®cance as well as plausibility of perimeter fractal dimensions can then be tested as in the case of two fractal domains. But the hypotheses that must be considered in order to test the joint signi®cance of parameters are different. Indeed, the three ®tted lines would be identical if H0 : g0 ˆ g1 ˆ d0 ˆ d1 ˆ 0 were true. To test this ^ ;b ^ hypothesis versus H1 : H0 not true, the extra-sum of squares SS…^g0 ; ^g1 ; ^d0 ; ^d1 jb 0 1 † with 4 degrees of freedom is needed. Another hypothesis which might be tested is the presence of three parallel lines, that is, H0 : g1 ˆ d1 ˆ 0 versus H1 : H0 not true. This hypothesis can ^ ;b ^ ^ g †. be tested through the following extra-sum of squares: SS…^g1 ; ^d1 jb 0 1 ; d0 ; ^ 0 The iterative procedure is as follows: at each step, model (4) is ®tted with Zi1 ˆ 0, for i ˆ 1; 2; . . . ; j, Zi1 ˆ 1, for i ˆ j ‡ 1; j ‡ 2; . . . ; n and Zi2 ˆ 0 i ˆ 1; 2; . . . ; j ‡ k, Zi2 ˆ 1,

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for i ˆ j ‡ k ‡ 1; j ‡ k ‡ 2; . . . ; n, with j ˆ 3; 4; . . . ; n 6 and k ˆ 3; 4; . . . ; n … j ‡ 3†, so that line (5a) is ®tted to j patches, while line (5b) to k patches and line (5c) to n-j-k patches. Taking one shift point ®xed and varying the other shift point, only the ®t with the highest R-squared is selected among all the signi®cant ones. This is only a method to make graphical visualization of the output clearer. Then, the ®rst shift point is varied and the ®nal ®t is the best of best (in terms of R-squared) among those selected at the previous step. To detect the presence of different fractal domains with respect to a new set of patches we suggest to start by checking whether three domains are acceptable, and then to try with two domains. When there is no statistical evidence of even two domains we could conclude that the scale pattern of the patches considered has no discontinuities. During the iterative procedure, statistical tests are used for thousands of combinatorial possibilities. One of the referees has questioned whether this adversely affects averall signi®cance levels as in multiple comparisons. Here, however, the statistical tests are employed as a screening index to aid in estimating the shift-points. In this paper, ®nal hypothesis tests were conducted as thought the change points were known instead of estimated. The effect of change point estimation on null distributions will be reported in a forthcoming publication.

3. Application to corine biotopes The procedure has been applied to landscape pattern data produced within the Map of the Italian Nature Project (Zurlini et al., 1999). They concern mosaics of contiguous irregular patches corresponding to different habitat types in the CORINE biotopes classi®cation (C.E.C., 1991). In Map of Nature the CORINE biotopes classi®cation is derived from satellite, airborne ( photographs and hyperspectral images) and ®eld data. The CORINE classi®cation permits the identi®cation of habitats from a physical point of view (ecosystems sensu Tansley, 1935), and the integration of abiotic and biotic components that are extremely closely related, such as habitats and syntaxa (Usher, 1991), i.e., plant communities in a phytosociological context. Such ecological land classi®cation developed by the European Community, although still under revision, has been of®cially adopted by all the European state members to provide a standard evaluation basis for conservation management issues. Among the 2327 CORINE patches identi®ed in the Val Baganza watershed, the three most frequent types of patches are considered here for analysis: low land hay meadows (CORINE code 38.2; 378 patches); brachypodium grassland (CORINE code 36.334; 131 patches); northern apennine mesobromion (CORINE code 34.3266; 77 patches) (Fig. 1). Other types of biotopes were not analyzed because the corresponding sample sizes are too small to perform the iterative procedure. The original perimeter-area data are available from the authors on request. Fig. 2 reports the ®nal model selected for low land hay meadows (CORINE code 38.2). As can be noted, variance seems to be constant as area increases, but this stabilization be a result of the logarithmic transform. First we explored the potential presence of three fractal domains. Among all the signi®cant ones, 189 models were found to have the highest Rsquared. Among these 189 models, ®ve stood out from the rest in terms of their R-squared values (Fig. 3). The index reported on the x-axes indicates the order in which the models

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Figure 1. Val Baganza watershed (Italy). Spatial distribution of three most frequent CORINE biotopes: low land hay meadows (code 38.2), brachypodium grassland (code 36.334), northern apennine mesobromion (code 34.3266).

were processed and selected by the iterative procedure. The scatterplot is nearly constant and then suddenly explodes; however, this is an incidental feature of the particular data set and the way the loops were arranged in the computer program. In fact, all points in the initial chain have the same second shift point and a very similar R-squared value. It should Table 1. Boundary fractal dimensions and weighted average human disturbance for val baganza watershed (low land hay meadows biotope, CORINE code 38.2). The average human disturbance is largest in the second domain where boundary fractal dimension is lowest.

D1 D2 D3

Value

Standard Error

Domain …m2 †

Average Disturbance

1.163126 0.778567 1.428583

0.029517 0.133872 0.052241

0 13998 13998 32333 4 32333

0.173 0.202 0.183

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Grossi, Zurlini, Rossi

Figure 2. Val Baganza watershed (Italy): Model selected for the low land hay meadows biotope (CORINE code 38.2). A piecewise regression model with three fractal domains has been ®tted. The slope of each segment gives boundary fractal dimension of the corresponding domain.

Figure 3. Val Baganza watershed (Italy). Plot of R-squared for 189 statistically signi®cant models (low land hay meadows, CORINE code 38.2). The shape of points depend on the way the iterative procedure works: all models belonging to the ®rst part of the plot have the same second shift point.

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be noted that we selected the model with a maximum R-squared even when differences among R-squared values were negligible. Table 1 shows the results for the ®nal model. Perimeter fractal dimension is smallest in the second domain. Here the estimated value of D2 is less than 1, but not statistically different from 1. These results are independently corroborated by a human disturbance index provided by the Map of Italian Nature Project (Zurlini et al., 1999). To compute the index, we relied on GIS capability to read and quantify from digital maps the length of networks given by state and regional roads in vector format. Disturbance indicators are computed as kilometers of road length crossing and/or partially or completely delimiting each CORINE patch. The overall exposure length for each patch in kilometers per unit area was considered as a human disturbance index. The weighted mean of such index (Table 1), obtained using patch area as a weight variable, is largest in the second domain. This tends to con®rm the hypothesis that human activity is associated with reduced fractal dimension. Thus low land hay meadows CORINE biotopes are apparently ``affected'' by higher road access and road proximity at intermediate patch dimensions, making for smoother geometry of patches and lower perimeter fractal dimension. For this higher road accessibility, cattle breeding activities in the watershed may also have contributed to the reduction in patch perimeter fractal dimension. Interestingly, the intermediate fractal domain in Fig. 2 seems to correspond to patch size interval suitable for present cattle breeding activites at the family level in the Val Baganza watershed. When Krummel's method is applied, ®ndings are not so clear and their possible

Figure 4. Val Baganza watershed (Italy). Graphical output of Krummel's method applied to the low land hay meadows biotope (CORINE code 38.2). Window size is 20 patches. Boundary fractal dimensions are compressed toward their maximum value of 2 and there is no evidence of discontinuities.

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interpretation is strongly affected by window size. Fig. 4 reports the graphical output of Krummel's procedure when sample size is 20 patches. In this case perimeter fractal dimensions are compressed towards their maximum values and there is no evidence of discontinuities. There is only a slight decrease where log(area) is approximately within the 10±10.5 log(area) range. When sample size is increased to 100, all the information about the tail of patch area distribution is lost and the interpretation is even more uncertain (Fig. 5). In this case the decrease of fractal dimension is more evident, but the information on D-behavior is completely lost for log(area)-values greater than 10.5. For the CORINE biotopes with code 36.334 there was no statistical evidence of three fractal domains so that the presence of two fractal domains was tested. In Fig. 6 the R-squared index plot for the signi®cant models is reported. Here, only ®ve models are signi®cant and their shift-points are very close to each other (across the ®ve models, the range in the shift point is 19,638±21,425 m2). Fig. 7 superimposes the model with the highest R-squared on the scatterplot of the data. The parameters of this model are given in Table 2. For this type of CORINE patch, the fractal dimension is smaller for smaller spatial scales (lower altitudes). In this case, no human disturbance index was available; however, the association between human disturbance and smaller fractal dimension is con®rmed because of noticeable agricultural activities in the Padana ¯at, making for smoother geometry of patches and lower fractal dimension. On the other hand, patch dynamics on larger scales and at higher altitudes seem more driven by natural processes, leading to more complex plane-®lling fractal edges.

Figure 5. Val Baganza watershed (Italy). Graphical output of Krummel's method applied to the low land hay meadows biotope (CORINE code 38.2). Window size is 100 patches. Information about the tails of patch surface distribution is lost and the interpretation is uncertain.

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Figure 6. Val Baganza watershed (Italy). R-squared index plot for the statistically signi®cant models for the brachypodium grassland biotope (CORINE code 36.334). Only 5 models are statistically signi®cant.

Figure 7. Val Baganza watershed (Italy). Model selected for the brachypodium grassland biotope (CORINE code 36.334). A piecewise regression model with two fractal domains has been ®tted. The slope of each segment gives boundary fractal dimension of the corresponding domain.

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Grossi, Zurlini, Rossi Table 2. Boundary fractal dimensions for Val Baganza watershed (brachypodium grassland biotope, CORINE code 36.334).

D1 D2

Value

Standard Error

1.223 1.594

0.04675 0.05784

Domain …m2 † 0

j 20085 > 20085

Finally a patch type (CORINE code 34.3266) was analyzed leading to detection of no shift-point. In this case there was no plausible model either with three or with two fractal domains. Fig. 8 reports the corresponding scatter plot where only one domain is in accordance to data for a D-value of 1.243 (standard error 0.038). This kind of CORINE biotope (northern apennine mesobromion, CORINE code 34.3266; C.E.C., 1991) does not apparently present any statistically signi®cant variation in perimeter fractal dimension across spatial scales. This may indicate no substantial change in scale as regards generating and maintaining processes of landscape patches (Sugihara and May, 1990). This kind of biotope is indeed present in naturally stressed environments due to drought, and thus already adaptive to relatively extreme environmental conditions and rather insensitive to human disturbances. The number of detected change points is related to the sample size of the analyzed biotope. This is reasonable because statistical signi®cance is the criterion for detection. Furthermore, it is reasonable that more complex models should be considered when there are more patches.

Figure 8. Val Baganza watershed (Italy). Model selected for the northern apennine mesobromion biotope (CORINE code 34.3266). A straight line ®ts all the data, so that only one fractal domain is detected.

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As previously reported, we have limited the analyses to a maximum of three fractal domains because of the increasing complexity of the algorithm in terms of computer time. The computational burden grows with the number of scaling domains and with the number of patches. When the presence of two domains was analyzed using a 266 Mhz pentium II, the procedure took less than 1 second for 131 patches and for 378 patches, but the entire 1616 patches of Val Baganza required about 23 seconds. The time employed by the threedomain procedure was as follows: 6 seconds for 131 patches, 6 minutes for 378 patches and approximately 15 hours and 30 minutes for 1616 patches.

4. Concluding remarks Discontinuities in landscape patterns can be studied with respect to different combinations of patch geometry parameters such as area-perimeter, perimeter-length, area-length (Nikora et al., 1999). This paper presents a statistical method for detecting discontinuities in irregular patch mosaic patterns using the perimeter-area relationship, but it can be applied to other combinations of patch geometry as well. The procedure is iterative and based on the use of dummy variables in linear regression models. In comparison with methods proposed and applied up to now, this procedure is not affected by subjective choices of initial parameters of analysis such as sample size in Krummel's method. Analysis of real data revealed that Krummel's method with small window sizes can produce very smooth plots with fractal dimension near the upper bound of 2. When larger window sizes are used, some shifts are detected, but estimates of shift points depend on the window sizes. In this case, moreover, information about fractal dimension for the extreme patches is lost because D-values are centerd on the average area of patches. Our method is based on statistical tests that can be used to discriminate among different plausible models. This procedure has proven to be analytically powerful and effective in detecting landscape patterns when applied to patch mosaics of the Map of Italian Nature Project. In this context, the procedure was helpful in discriminating among CORINE biotopes with respect to their landscape pattern. CORINE biotope types seem to behave differently according to their scaling properties resulting from the interaction between their ecological properties and human disturbance. For illustrative purposes we applied this method to the detection of at most two discontinuities, but it can be easily extended, with suf®cient computing power, to the identi®cation of more fractal domains. Furthermore, we have not considered in this context the possibility to have connected segments of different slope. The ecological implications of this kind of model are different with respect to the disjunct piecewise function. The comparative analyses for these alternative models will be developed in forthcoming papers. When the procedure was applied to all the CORINE biotopes of the Val Baganza watershed, it was possible to detect sectors where the same ecological processes are presumably predominating. Such ®ndings could be important for a landscape planning approach that takes into account spatial and temporal effects of ecological processes.

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Grossi, Zurlini, Rossi

Acknowledgments This research has been supported by the Map of the Italian Nature project. The authors wish to thank G. P. Patil and C. Taillie for their encouragement during a visit of the ®rst author to the Center for Statistical Ecology and Environmental Statistics of the Penn State University. Furthermore, they are grateful to two anonymous referees for their helpful suggestions which have improved the ®rst version of the paper.

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Biographical sketches Luigi Grossi is a statistician. After the degree in Economics at the Faculty of Economics, University of Parma (Italy), he obtained his Ph.D. in Statistics at the University of Bologna where he worked on the statistical analysis of air pollution data. He is currently a postdoctoral research fellow at the Department of Environmental Science of the University of Parma involved in the statistical aspects implied by the Map of the Italian Nature project. Giovanni Zurlini is an ecologist of the National Council of Research (NCR) and professor of Environmental Information Systems at the Department of Environmental Sciences of the University of Parma (Italy). He obtained his Ph.D. in Mathematical Ecology in the Netherlands on insect pest population modeling. He is responsible of a NCR project on application of air-borne Multispectral Infrared and Visible Imaging Spectrometer (MIVIS) to ecosystem health and integrity assessment and cohordinates the Environmental Information System of the Map of the Italian Nature project. He is author of more than 40 papers published in international journals. Orazio Rossi is professor of Quantitative Ecology at the Department of Environmental Sciences of the University of Parma (Italy) and member of the High Consultant Committee of the Minister of the Environment. He has always been involved with nature conservation problems with particular concern to the Small Islands project of UNESCO. He got the Statistical Ecology Award at the 6th International Congress of Ecology held in Manchester (UK) in 1994. He is the general coordinator of the Map of the Italian Nature project and author of more than 70 papers in international journals.