Determine the most influencing stressors and the most susceptible resources for environmental integrated assessment

Ecological Modelling 220 (2009) 2335–2340

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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Determine the most influencing stressors and the most susceptible resources for environmental integrated assessment Liem T. Tran a,∗ , Robert V. O’neill b , Elizabeth R. Smith c a b c

Department of Geography, University of Tennessee at Knoxville, 1000 Phillip Fulmer Way, Knoxville, TN 37996, United States T N & Associates, Oak Ridge, TN, United States U.S. Environmental Protection Agency, Office of Research and Development, National Exposure Research Laboratory, Research Triangle Park, NC, United States

a r t i c l e

i n f o

Article history: Received 30 September 2008 Received in revised form 19 May 2009 Accepted 1 June 2009 Available online 1 July 2009 Keywords: Stressors Resources Limiting priorities Integrated environmental assessment

a b s t r a c t The paper presents a method to determine the most influencing stressors and the most susceptible resources for complex assessment problems involving multiple stressors impacting multiple resources over a region. The method is based on the concept of limiting priorities in a square matrix which capture the transmission of influence along all paths between stressors and resources in the matrix. The proposed method allows the relationship between stressors and resources to be looked at in both univariate and multivariate fashion, taking into account the interactions among the variables. Hypothetical and case study examples are given for illustration purpose. It shows that the proposed method is suitable for the determination of the most important stressors and the most susceptible resources, a common (but often uneasy) task in integrated environmental assessment. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In the context of cumulative effects of multiple stressors on multiple natural resources in a region, what are the most influencing stressors and what are the most susceptible resources? The paper is to introduce a quantitative method to address these two questions. The most influencing stressor as defined in this study is the stressor which shows the strongest connection/association with the multiple resources over the region given the interactions within and between stressors and resources. Note that the most influencing stressor might or might not be the stressor which causes the most damage to a specific resource at a particular location in the study area. However, a change to the most influencing stressor as defined in this case is more likely to have an evident impact on multiple resources across the region. In a similar fashion, the most susceptible resource is defined as the most sensible resource with respect to the synergistic effects of multiple stressors and resources. Getting the answers for the two questions listed above is valuable and crucial to environmental managers who must focus limited resources and/or prioritize actions (e.g., mitigation, adaptation, or conservation) on the most influencing stressors and the most susceptible resources while taking into account the cumulative effects of the stressors and resources across the whole the region.

∗ Corresponding author. Tel.: +1 865 974 6034. E-mail address: [email protected] (L.T. Tran). 0304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2009.06.013

Complex assessment problems often involve multiple stressors impacting multiple resources. To deal with this problem, the assessment community has developed a matrix procedure (Bain et al., 1986; Ferenc and Foran, 2000). The matrix represents stressors as the rows and the resources as the columns. The approach was originally proposed by Leopold et al. (1971) and a number of variations are reviewed by Canter (1977). This matrix approach has been used in a number of applications. Bain et al. (1986) analyzed the impact of multiple human developments on multiple resources. All possible combinations of stressors are considered with the impact of each combination computed as the sum of all project-specific impacts, adjusted for the effect of interactions among projects. This results in a matrix representing the relative impact of every possible combination of stressors on each resource. The matrix is then searched for combinations that minimize the impact summed across all resources. Harris et al. (1994) developed an impact matrix of stressors and resources in the form of impaired use criteria. Experts then filled in the matrix with values from 0 (no impact) to 3 (major impact). The row sums point out the stressor with the greatest impacts summed across the suite of resources. In the typical application, quantitative information is not available for the individual cells of the matrix. A panel of experts is asked to assess the individual impacts and supply a qualitative value. In the EPA Regional Vulnerability Assessment Program (ReVA), data are available for stressors and resources and a unique opportunity exists to apply the matrix approach quantitatively. This paper develops a methodology for determining the most important stressors

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Fig. 1. Conceptual graphs to derive (a) the most important stressors and (b) the most susceptible resources.

and most susceptible resources when data are available for multiple watersheds across a region. 2. Methodology Note that the units used in this study are watersheds with n attributes: l resources and m stressors (l + m = n). 2.1. Determine the most influencing stressors To determine the most influencing stressors, the system is conceptualized as a graph with nodes organized in three blocks (Fig. 1a). Nodes in the first block represent the resources in terms of their importance with respect to the whole system. Nodes in the second block also stand for the resources but in terms of their interdependence (i.e., there is a cycle within the resources). Nodes in the third block symbolize the stressors. Such graph corresponds to supermatrix V (Diestel, 1997; Saaty, 2001):

⎛

0

0

0

⎞

⎜ ⎟ V = ⎝ V1 V2 0 ⎠ 0

V3

I

where V1 is the column vector (size l) representing the connections from nodes in the first block to nodes in the second block, V2 is the (square) matrix (l × l) standing for the connections among nodes in the second block (i.e., the interdependence among the resources), V3 is the matrix (l × m) denoting the connections from nodes in the second block to those in the third block (i.e., the dependence of the resources on the stressors), and I is the identity matrix (m × m) characterizing the stressors. Entries of vector V1 are squares of the coefficient of multiple determination of each resource by all other stressors and resources. V2 is comprised of squares of the correlation coefficient between the resources. Entries of V3 are squares of the coefficient of single determination of each resource by each stressor. Note that there is no path from the third block back to the first block of the graph. This feature corresponds to the irreducibility of supermatrix V (Meyer, 2000).

Our intention is to capture the transmission of influence along all paths of the supermatrix (i.e., the graph). It can be done by raising the supermatrix to powers. To be able to do that, first the supermatrix needs to be transformed to a column stochastic matrix (each of its columns sums to unity). Then we need to find lim V k . Note that k→∞

the existence of lim V k depends upon the concepts of reducibility, k→∞

primitivity, and cyclicity of the supermatrix V. Appendix A shows basic explanation on matrix setting used in the paper. However, readers are advised to explore more details of the issue in textbooks on matrix analysis, such as those of Minc (1988) and Meyer (2000)). In our case, lim V k exists (Meyer, 2000) and is given by: k→∞

 k

lim V =

k→∞

0 0 V3 (I − V2 )−1 V1

0 0 V3 (I − V2 )−1

0 0 I



where the limiting priorities of the stressors coincide with the last m elements of the first column (i.e., those in V3 (I − V2 )−1 V1 ) (Diestel, 1997; Saaty, 2001). Practically lim V k is calculated by sufficiently k→∞

raising matrix V to large powers with a simple matrix algorithm on a computer (see Appendix A). For practical purpose, a Microsoft Excel file which facilitates those matrix calculations will be provided upon request (via email to the first author). 2.2. Determine the most susceptible resources Similar to the setup in previous section, the system is conceptualized as a three-block graph (Fig. 1b). Nodes in the first block stand for the stressors in terms of their importance with respect to the whole system. Nodes in the second block denote the stressors with respect to their interdependence (i.e., there is a cycle within the stressors). Nodes in the third block symbolize the resources. Let:

⎛ ⎜

0

W = ⎝ W1 0

0

0

⎞ ⎟

W2

0⎠

W3

I

L.T. Tran et al. / Ecological Modelling 220 (2009) 2335–2340 Table 1 Hypothetical example: (a) raw data matrix for the six attributes of 15 hypothetical watersheds (S: stressor, R: resource); (b) the correlation coefficient matrix; (c) matrix V; and (d) matrix W.

S1

S2

S3

R1

R2

R3

0.847 0.334 0.102 0.934 0.420 0.563 0.953 0.631 0.459 0.002 0.710 0.474 0.554 0.192 0.958

0.138 0.404 0.647 0.307 0.067 0.657 0.833 0.821 0.260 0.415 0.465 0.468 0.128 0.878 0.962

0.739 0.892 0.298 0.430 0.965 0.278 0.334 0.270 0.799 0.568 0.616 0.901 0.189 0.322 0.209

0.254 0.313 0.007 0.910 0.333 0.197 0.710 0.349 0.041 0.002 0.648 0.010 0.023 0.030 0.765

0.004 0.009 0.002 0.020 0.001 0.018 0.033 0.054 0.004 0.030 0.033 0.006 0.006 0.052 0.018

0.017 0.413 0.101 0.043 0.297 0.050 0.034 0.077 0.258 0.220 0.122 0.310 0.076 0.093 0.013

(b)

S1 S2 S3 R1 R2 R3

S2

S3

1.000

0.087 1.000

−0.194 −0.606 1.000

R1

R2

R1 R2 R3 S1 S2 S3

0.796 0.217 −0.171 1.000

0.031 0.652 −0.457 0.210 1.000

−0.541 −0.388 0.798 −0.381 −0.340 1.000

R1

R2

R3

S1

S2

S3

0.000 1.000 0.044 0.145 0.634 0.047 0.029

0.000 0.044 1.000 0.116 0.001 0.425 0.209

0.000 0.145 0.116 1.000 0.293 0.150 0.637

0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 1.000

S1

S2

S3

R1

R2

R3

0.000 1.000 0.008 0.037 0.634 0.001 0.293

0.000 0.008 1.000 0.368 0.047 0.425 0.150

0.000 0.037 0.368 1.000 0.029 0.209 0.637

0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 1.000

(d)

S1 S2 S3 R1 R2 R3

0.000 0.768 0.526 0.903 0.000 0.000 0.000

k→∞

k→∞

2.3. A hypothetical example The proposed method was applied to a hypothetical data set of 15 watersheds with six attributes, three stressors (S1 , S2 , and S3 ) and three resources (E1 , E2 , and E3 ) (Table 1, part (a)). Data were arranged to make the six attributes interdependent to different extents. Among the three stressors S1 was the most independent attribute while S2 and S3 are interdependent to each other to some extent (Table 1, part (b)). On the other hand, while the three resources are quite independent to each other, their dependence upon the three stressors is different from one to another (Table 1, part (b)). Table 1, parts (c) and (d), represent the matrix V and W, respectively, of the hypothetical example.

R3

(c)

0.000 0.694 0.480 0.816 0.000 0.000 0.000

where the limiting priorities of the resources coincide with the last l elements of the first column (i.e., those in W3 (I − W2 )−1 W1 ). Similar to lim V k , lim W k is calculated by raising matrix W to large powers with a matrix algorithm on a computer.

(a)

S1

2337

2.4. Regional vulnerability assessment (ReVA) case study The proposed method was applied to 134 watersheds (8-digit hydrologic unit: USGS 1982) in the Mid-Atlantic region (Fig. 2) whose conditions were described via a set of 10 EQ-indicators (Table 2, part (a)). The EQ-indicators were collected on a watershedby-watershed basis by the U.S. Environmental Protection Agency’s ReVA program (Jones et al., 1997; Smith et al., 2003). Data were normalized and inverted if necessary to make all indicators range from 0 to 1, where 0 and 1 represent environmentally desirable (good) and undesirable (poor) conditions, respectively. Details on the data set (e.g., metadata, source) and other data-related issues (e.g., normalization, arrangement, skewness, etc.) are discussed in Smith et al. (2003) and on the ReVA website (www.epa.gov/epa/). Table 2, parts (b) and (c), represent the matrices V and W, respectively, of the hypothetical example.

3. Results 3.1. Hypothetical example Results of the lim V k and lim W k of the hypothetical example k→∞

k→∞

are shown in Table 3, parts (a) and (b), respectively. Elements in the first column of lim V k shows that S1 is the most influencing stressor k→∞

be the supermatrix representing such graph where W1 is the column vector (size l) of the stressors with respect to the overall system, W2 is the (square) matrix (m × m) representing the interdependence among the stressors, W3 is the matrix (m × l) of the influence of the stressors on resources, and I is the identity matrix (l × l) denoting the resources. Entries of vector W1 are squares of the canonical correlation between each stressor and all of other stressors and all of the endpoints. W2 is comprised of squares of the correlation coefficient between the stressors. Entries of W3 are square of the coefficient of single determination between a stressor and a resource. Similar to the analysis of the supermatrix V, lim W k exists k→∞

(Meyer, 2000) and is given by:

 lim W k =

k→∞

0 0 W3 (I − W2 )−1 W1

0 0 W3 (I − W2 )−1

0 0 I



followed by S3 and then S2 . Results from lim W k indicates that R3 is k→∞

the most susceptible resource while R1 and R2 are at a similar level of being influenced.

3.2. ReVA case study Results of the lim V k and lim W k of the ReVA case study are k→∞

k→∞

shown in Table 4, parts (a) and (b), respectively. Elements in the first column of lim V k shows that INSECTICIDE and RDDENS are k→∞

the most influencing stressors followed by POPDENS and AGSL. DAMS is the least influencing stressor with respect to the five resources included in the case study. lim W k indicates that NATk→∞

COVERPCT is the most susceptible resource followed by INT2 and MIGSCENARIO. Relatively, TERRNATIVE and AQUANATIVE are the least influenced resources by the five stressors included in the study.

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Fig. 2. The Mid-Atlantic region and its 8-digit HUC watersheds (source: Smith et al., 2003).

4. Discussion

In general, the limiting priorities of the three resources adequately reflect their impact by the three stressors.

4.1. Hypothetical example 4.2. ReVA case study Results shown in Tables 1(c) and 3(a), demonstrates that the limiting priorities of the three stressors adequately reflect their influence on the three resources: as S1 and S3 exert strong influence on R1 and R3 , they receive very high limiting priorities (0.412 and 0.343, respectively). On the other hand, S2 mainly influence R2 but at a weaker magnitude compared with those of S1 and S3 on R1 and R3 . As a consequence, S2 has the lowest limiting priority among the three stressors. In general, the more influence on resources a stressor has, the higher limiting priority the stressor obtains. Among the three resources, R3 is strongly influenced by S3 and S1 and by S2 to some extent (Tables 1(d) and 3(b)). As a result, R3 receives the highest limiting priority (0.459) (i.e., R3 is the most influenced resource among the three resources). Compared with R3 , R1 is influenced at a lesser magnitude and mainly by S1 . Accordingly, R1 has the second highest limiting priority (0.273). As the least influenced resource, R2 receives the lowest limiting priority (0.268).

The pattern of limiting priorities derived for the five stressors in the ReVA case study is similar to those in the hypothetical example. That is higher limiting priority for more influencing stressor. On the same token, resources which are influenced more stressors obtain higher limiting priorities. Note that the results should be viewed in the context of cumulative effects and multiple interactions between and within the stressors and resources under study. Furthermore, the proposed method does not imply any direct causal relationship between the most influencing stressors and the most susceptible resources. For example, among the five resources being explored in the ReVA case study, NATCOVERPCT and INT2 are the most susceptible ones with respect to not only the two most influencing stressors – INSECTICIDE and RDDENS – but the cumulative effects of the five stressors under study as well as the interactions among the resources. On the other hand, among the five stressors being ana-

L.T. Tran et al. / Ecological Modelling 220 (2009) 2335–2340

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Table 2 ReVA case study: (a) environmental indicators of the Mid-Atlantic region used in the case studya ; (b) matrix V, and (c) matrix W. (a) No. Stressors 1 2 3 4 5 Resources 6 7 8

9 10

Indicators

Abbreviations

Proportion of watershed with agriculture land cover on slopes that are greater than three percent Impoundment density (number of dams per 1000 kilometers of stream length) Annual O-P insecticides loadings 1990–1993 Population density The density numbers are meters of road per hectare of area

AGSL DAMS INSECTICIDE POPDENS RDDENS

Count of native aquatic – fish and mussels – species Percentage of forest habitat called interior (2 ha scale) Number of migratory scenarios for long-distance forest migrants that use a particular HUC or hexagon. Scenarios are defined by a combination of compass heading, landfall location along the Gulf Coast and southern Atlantic Coast, and nightly flight distance Percent coverage with FOREST that matches potential vegetation designated by Kuchler (1964) Count of native birds, mammals, butterflies, amphibians, and reptiles

AQUANATIVE INT2 MIGSCENARIO

NATCOVERPCT TERRNATIVE

(b)

0.000 0.133 0.589 0.548 0.423 0.255 0.000 0.000 0.000 0.000 0.000

AQUANATIVE (1) INT2 (2) MIGSCENARIO (3) NATCOVERPCT (4) TERRNATIVE (5) AGSL (6) DAMS (7) INSECTICIDE (8) POPDENS (9) RDDENS (10)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0.000 1.000 0.002 0.072 0.002 0.023 0.000 0.002 0.007 0.001 0.010

0.000 0.002 1.000 0.375 0.050 0.008 0.000 0.001 0.080 0.091 0.336

0.000 0.072 0.375 1.000 0.087 0.002 0.008 0.002 0.151 0.076 0.143

0.000 0.002 0.050 0.087 1.000 0.056 0.216 0.020 0.264 0.017 0.000

0.000 0.023 0.008 0.002 0.056 1.000 0.010 0.004 0.074 0.055 0.002

0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0.000 1.000 0.055 0.151 0.008 0.002 0.000 0.000 0.008 0.216 0.010

0.000 0.055 1.000 0.044 0.012 0.059 0.002 0.001 0.002 0.020 0.004

0.000 0.151 0.044 1.000 0.002 0.004 0.007 0.080 0.151 0.264 0.074

0.000 0.008 0.012 0.002 1.000 0.226 0.001 0.091 0.076 0.017 0.055

0.000 0.002 0.059 0.004 0.226 1.000 0.010 0.336 0.143 0.000 0.002

0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

(c)

0.000 0.381 0.138 0.507 0.325 0.536 0.000 0.000 0.000 0.000 0.000

AGSL (1) DAMS (2) INSECTICIDE (3) POPDENS (4) RDDENS (5) AQUANATIVE (6) INT2 (7) MIGSCENARIO (8) NATCOVERPCT (9) TERRNATIVE (10) a

Detailed information of the indicators can be found in Jones et al. (1997) and Smith et al. (2003).

Table 3 Hypothetical example: (a) lim V k and (b) lim W k . k→∞

k→∞

(a)

R1 R2 R3 S1 S2 S3

0.000 0.000 0.000 0.000 0.412 0.245 0.343

R1

R2

R3

S1

S2

S3

0.000 0.000 0.000 0.000 0.759 0.108 0.133

0.000 0.000 0.000 0.000 0.088 0.566 0.346

0.000 0.000 0.000 0.000 0.308 0.173 0.519

0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 1.000

(b)

S1 S2 S3 R1 R2 R3

0.000 0.000 0.000 0.000 0.273 0.268 0.459

S1

S2

S3

R1

R2

R3

0.000 0.000 0.000 0.000 0.655 0.018 0.328

0.000 0.000 0.000 0.000 0.076 0.544 0.380

0.000 0.000 0.000 0.000 0.064 0.320 0.616

0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 1.000

lyzed, INSECTICIDE and RDDENS have a more evident influence not just only on NATCOVERPCT and INT2 but on all of the five resources under study. The use of the coefficients derived from linear relationships among variables (e.g., coefficients of single/multiple determination) is to include all of the direct and indirect interactions among the variables. Note that the coefficients used in this study are derived from a spatial dataset but not a time series one. Hence the limiting priorities of stressors and resources in this case denote the role of each variable in spatial equilibrium condition (i.e., they are not overall priorities over time). In comparison with some common stressor-resource matrix methods as those mentioned in the introduction section, the proposed method has a solid and sound mathematical foundation (e.g., graph theory and matrix analysis). Furthermore, the interpretation of the contributing factors in other matrix methods is not always straightforward, often equipped with subjective judgment, and consequently subject to disagreement among different experts. The proposed method, in contrast, does not suffer from this problem. In addition, by including the interdependence and capturing the transmission of influence between and/or within stressors and resources along all paths of the matrix, the proposed method suc-

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Table 4 ReVA case study: (a) lim V k and (b) lim W k . k→∞

k→∞

(a)

AQUANATIVE (1) INT2 (2) MIGSCENARIO (3) NATCOVERPCT (4) TERRNATIVE (5) AGSL (6) DAMS (7) INSECTICIDE (8) POPDENS (9) RDDENS (10)

0.000 0.000 0.000 0.000 0.000 0.000 0.123 0.019 0.361 0.167 0.330

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0.000 0.000 0.000 0.000 0.000 0.000 0.070 0.031 0.368 0.170 0.360

0.000 0.000 0.000 0.000 0.000 0.000 0.044 0.008 0.251 0.173 0.525

0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.012 0.341 0.175 0.408

0.000 0.000 0.000 0.000 0.000 0.000 0.324 0.033 0.468 0.080 0.095

0.000 0.000 0.000 0.000 0.000 0.000 0.129 0.030 0.474 0.278 0.089

0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

(b)

AGSL (1) DAMS (2) INSECTICIDE (3) POPDENS (4) RDDENS (5) AQUANATIVE (6) INT2 (7) MIGSCENARIO (8) NATCOVERPCT (9) TERRNATIVE (10)

0.000 0.000 0.000 0.000 0.000 0.000 0.013 0.321 0.235 0.340 0.091

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.091 0.127 0.701 0.075

0.000 0.000 0.000 0.000 0.000 0.000 0.021 0.262 0.200 0.427 0.091

0.000 0.000 0.000 0.000 0.000 0.000 0.011 0.140 0.233 0.500 0.115

0.000 0.000 0.000 0.000 0.000 0.000 0.010 0.464 0.297 0.088 0.140

0.000 0.000 0.000 0.000 0.000 0.000 0.018 0.583 0.285 0.062 0.051

0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000

cessfully takes into account the synergistic effect among stressors and resources. Although the proposed method has several advantages over other existing matrix methods, it has some weaknesses. One of them is its inability to accommodate nonlinearity (i.e., it is based on the assumption of linear relationships among stressors and resources reflecting via entries for the supermatrix). Consequently results from the proposed method might be affected by problems common to other multivariate statistical analyses, such as data skewness, outliers, and data discontinuity. The proposed method requires numerical data at interval or ratio scale. This can be a limitation as data for various stressors and resources are only available at nominal or ordinal scale. Spatial autocorrelation is another potential problem to the proposed method which is based on a non-spatial statistics. The implications of those potential problems on the proposed method are beyond the scope of this paper and warrant a comprehensive analysis in future study. For conclusion, the proposed method allows the determination of the most influencing stressors and the most susceptible resources in a complex data set involving multiple stressors impacting multiple resources. Such feature is fundamental for various environmental studies, such as integrated environmental assessment and ecological vulnerability assessment. Appendix A. Matrix V can be irreducible or reducible (see textbooks on matrix analysis and linear algebra, such as Schott (1997) or Meyer (2000), for detailed definition and explanation of reducible and irreducible matrices). V is irreducible if there is a group of attributes that are correlated within group, but independent of other attributes in the data set. If V is reducible, it can be reduced by permutation of indices to a block-diagonal matrix of irreducible matrices (Meyer, 2000). In that context, without sacrificing generality, V can be assumed as irreducible (if V is reducible, the calculation presented below can be applied for each irreducible matrix on the diagonal). Furthermore, V is primitive because its diagonal elements are positive

(Meyer, 2000). To allow the application of the Perron–Frobenius theorem, V is converted to a (row or column) stochastic matrix whose individual row (or column) sum is equal to 1. According to the Perron–Frobenius theorem applied to an irreducible, primitive, stochastic matrix (Meyer, 2000): • V has a real positive simple eigenvalue (max of 1; • lim V k = ev where v is a positive row vector, v = (v1 , v2 , . . . , vn ), k→∞ n



vi = 1, and e = (1, 1,. . ., 1)T .

i=1

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