Quantifying Coral Reef Stress Exposure

Environmental variables were grouped into three categories based on the role that they play as coral stressors: (1) radiation variables, consisting of variables derived from temperature (mean SST, TSA and WSSTA magnitude and duration), UV-erythermal and wind speed data (doldrums index); (2) stress reinforcing variable (TSM and chlorophyll-a), representing sedimentation and eutrophication; and (3) stress reducing variables, consisting of SST variability and tidal range. Values of each variable that correspond with the approximately 4000 reef locations were extracted, and examined for normality and log10-transformations applied where necessary (Appendix S3). For each variable, a membership function with similar behavior pattern to a normal cumulative distribution function was used to characterize the relationship between coral exposure and a stress variable. Membership functions capture the degree to which the variable x is a member of a fuzzy set A using a suitably chosen function μ(x) [48] . Here we used spline-based logistic functions:
where x_a and x_b are control values and correspond to the lower and upper bound of a stressor values, respectively (Table 1). These were calculated for each variable as the mean value of minus or plus two standard deviations, respectively. Radiation and reinforcing variables were normalized using an increasing curve (Eq. 1) and stress reducing variables were normalized using a decreasing curve (Eq. 2) (Fig. 1).
Spatial Principal Component Analyses (SPCA) was used to combine the standardized variables within each category. Principal Component Analysis transforms each variable into a linear combination of orthogonal common components (output layers), or latent variables with decreasing variation. The linear transformation assumes the components will explain all of the variance in each variable. Hence, for each output the latent component layer carries different information, which is uncorrelated with other components. This enables a reduction of output maps because the last transformed map(s) may be discarded as they have little or no variation left and may be virtually constant. The component weightings were calculated using coefficients of linear correlation to weigh the contribution of factors in spatial principal component analysis [67] . SPCA was performed to synthesize the standardized variables within radiation, stress reducing, and stress reinforcing categories. A final composite map from each of these three groups was computed by summing PC's with contribution ratio >1, weighted by their respective contribution ratio (Equation 3; [68] , [16] ). where Y_i is the i^th principal component, while α_i is its corresponding contribution ratio.
The output maps were standardized between zero and one, representing low and high exposure respectively. To combine the stress reducing and radiation variables, SPCA procedure described above was repeated with standardized radiation and reducing variables as the input variables. The output PC's were synthesized using a weighted sum equation (Eq. 3) to yield a layer with estimates of exposure to radiation taking into account the contribution from reducing variables. Fuzzy-integration-based approach was used to integrate the output from this procedure with the reinforcing variables into a single composite layer. [69] lists five fuzzy operators that are most useful for combining fuzzy data (AND, OR, sum, product and gamma). Given two fuzzy sets (standardized layers) A and B, the fuzzy sum operator produces a layer whose values are equal to or greater than each of the input layers A and B and results in an increased effect [69] . We therefore used fuzzy sum operator to reflect the reinforcing behaviour of sediment and eutrophication to radiation stress: where .is the membership value for i-th map, and i = A, B, n maps.
Coral reef location data was obtained from the Reef Base website (http://reefgis.reefbase.org/) and the Wildlife Conservation Society monitoring sites in the western Indian Ocean [70] (link). The location data were grouped into eleven oceanic provinces [9] (link) (Fig. 2). For the respective locations, exposure metrics as described above were extracted for the corresponding locations. Box plots of exposure metrics by stressors against the coral reef provinces were plotted.