Asientos Contables establecidos en el Subsistema de Nómina.

When comparing the configuration of the loadings of the 4-D output from the PCA at 25 m (section 3.2.3) with that obtained at the different grain sizes (50 m, 100 m, 250 m, 500 m and 1000 m) by means of a Procrustes Rotation, the residual sums of squares (RSS) between the pair of configurations increases as the difference in grain sizes being compared increases (Table 3.7). The RSS obtained for each pairwise comparison to 25 m is higher than the RSS obtained from the pairwise comparisons starting from any other scale (Table 3.7). This increase in RSS with scale and high RSS when comparing scales to 25 m is the same pattern to that obtained when comparing the 4D configuration of the NCAs (scores) (see section 3.2.4; Table 3.5). Metric values obtained for grain sizes 250 m and 500 m are most similar, and, relative similarity is also high when comparing 25 m to 50 m, 50 m to 100 m, and 500 m to1000 m (Table 3.7).

When considering the behaviour of individual metrics, as grain size increases from 25 m, a number of metrics including AREA_RA, COHESION, CONTAG, ECON_AM, FRAC_AM, GYRATE_AM, and GYRATE_CV maintain similar values relative to the other metrics with consistently small Procrustes residuals (Figure 3.5a-e – highlighted in green; Table 3.9). Procrustes residuals are obtained when comparing the 4-D configuration of the loadings at two different scales (Appendix A6), and the projected Procrustes residuals (distance) between metric loadings for the first two PCs at each scale comparison are highlighted in Figures 3.5a-e. Several metrics also maintain small Procrustes residuals when comparing 25 m to 50 m and 100 m but not when comparing to larger scales; CWED, LSI, SIMI_AM, SIMI_CV and SIDI (Figure 3.5a-e – highlighted in orange; Table 3.9). In contrast the metrics FRAC_CV, CONTIG_MN, PROX_CV and CONTIG_RA maintain consistently high Procrustes residuals indicating variation in metric value with scale (Figure3.5a-e – highlighted in red; Table 3.9).

Relative Euclidean similarities between the metrics obtained from the 4D output from the PCA at each scale are significantly correlated with each other at each scale combination (Table 3.8). Similarly to the pattern observed from the Procrustes Rotation RSS (Table 3.7) the correlation coefficient decreases as the difference in the scale comparison increases (Table 3.8). Correlation coefficients are lowest when

80 comparing each scale to 25 m in comparison to any other starting scale. For example, the correlation when comparing 100 m to 25 m is lower than that obtained when comparing 100 m to 50 m (r = 0.8418, p<0.001; r = 0.9472, p<0.001 respectively). 25 - 50 0.1130 - 100 0.2596 0.1038 - 250 0.3630 0.3206 0.2177 - 500 0.4408 0.3284 0.2098 0.0915 - 1000 0.5036 0.4182 0.3167 0.1421 0.1186 - 25 50 100 250 500 1000

Table 3.7: Residual Sums of Squares (RSS) from the Procrustes Rotation comparison of the 4-dimension configuration of the loadings from the Principal Component Analysis for each scale. 25 - 50 0.9177 - 100 0.8418 0.9472 - 250 0.7244 0.7911 0.8574 - 500 0.6512 0.7538 0.8228 0.9285 - 1000 0.5904 0.6823 0.7403 0.8543 0.8929 - 25 50 100 250 500 1000

Table 3.8: Pearson product-moment correlations between the Euclidean similarity matrices derived from the 4-dimension configuration of the loadings from the Principal Component Analysis for each scale. Pearson product-moment correlations are obtained by Mantel Tests for each pairwise comparison and all correlations are significant (p<0.001).

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Metric Code Metric (cont.) Code

AREA_MN 1 FRAC_AM 17 AREA_RA 2 FRAC_CV 18 CIRCLE_AM 3 GYRATE_AM 19 CIRCLE_MN 4 GYRATE_CV 20 CIRCLE_RA 5 GYRATE_MN 21 COHESION 6 IJI 22 CONTAG 7 LSI 23 CONTIG_AM 8 MESH 24 CONTIG_MN 9 PRD 25 CONTIG_RA 10 PROX_AM 26 CWED 11 PROX_CV 27 ECON_AM 12 SHAPE_CV 28 ECON_CV 13 SHAPE_MN 29 ENN_AM 14 SIDI 30 ENN_CV 15 SIMI_AM 31 ENN_MN 16 SIMI_CV 32

Table 3.9: Metric codes for Procrustes Residual Plots.

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Figure 3.5 a-e: Comparison of the first two principal components (PC-1 and PC-2) of the 4-D configuration of the loadings from the Principal Component Analysis (PCA) between scales; (a) 25 m to 50 m (b) 25 m to 100 m (c) 25 m to 250 m (d) 25 m to 500 m and (e) 25 m to 1000 m. Distance between the same metric loadings plotted at two different scales represents the metric (loadings) projected residuals from the Procrustes Rotation for the first two PCs. Metrics with high Procrustes residuals across scales are circled in red; small Procrustes residuals across scales in green; and small Procrustes residuals from 25 m to 100 m only in orange. See Appendix A6 for 4D Procrustes Residuals.

84 The hierarchal cluster analysis of NCA landscape structure according to the metric values obtained at a scale of 25 m suggests the partitioning of five clusters of metrics at a similarity level of 80% (Figure 3.6a). Cluster group 1 comprises two metrics associated with PC-4 providing measures of patch shape irregularity (Table 3.3). Cluster group 2 comprises six metrics, four of which are associated with PC-1 with high negative loadings on this component, and collectively provide measures of landscape fragmentation (Table 3.3; Figure 3.5a; Figure 3.6a). Cluster group 3 comprises five metrics which are associated with PC-2, and measure patch shape and extent. Cluster group 4 included 14 metrics, all which have the highest positive loadings on PC-1 and collectively measure landscape fragmentation (Table 3.3; Figure 3.5a; Figure 3.6a). Cluster group 5 comprises five metrics, three of which are associated with PC-3 and provide measures of patch boundary configuration (Table 3.3).

Relationships between metrics are maintained across scales most notably for members of group 4, which remain clustered at 50 m (92% similarity) and 100 m (82% similarity), despite addition of some metrics from the other groups at 100 m (Figure 3.6b,c). From 250 m upwards the metrics ENN_MN and SIMI_AM are no longer clustered with group 4, but the similarity between the remaining members of group 4 increases to 82%, 89% and 90% at the scales 250 m, 500 m and 1000 m respectively (Figure 3.6d-f). Group 3 also maintain similarity across scales, with only the metric CONTIG_RA becoming separated at 50 m (Figure 3.6b). The remaining groups change considerably with increasing grain size, most notably group 1 metrics (CIRCLE_RA and PROX_CV), group 5 metrics (ECON_CV, PRD, FRAC_CV and ENN_CV) and the metric CONTIG_RA from group 3 exhibited greatest variability, joining and re-joining groups at different scales (Figures 3.6a-e).

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Figure 3.6: Clustering of landscape structure metrics using complete link algorithm based on Euclidean similarity matrix. The clusters at 25 m (a) are defined at a similarity level of 80 % and the members of these clusters are then identified at the scales (b) 50 m, (c) 100 m, (d) 250 m, (e) 500 m and (f) 1000 m.

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