Filling index
To assess if abundance-environment plots show a polygon-shaped point
cloud characterized by an upper boundary, we developed a procedure namedfilling index . This tool can be applied to any point cloud
representing the relationship between two variables. Polygon-shaped
point clouds are expected to present a continuum of points from
zero-abundance to the upper boundary, i.e., a polygon filled with points
(Fig 1A). Conversely, line-shaped patterns are expected to show empty
space between the point cloud and the bottom of the plot - the X axis
(Fig 1B). Overall, polygon-shaped
point clouds are expected to have larger filling of the space, from
zero-abundance to the upper boundary of points, than line-shaped point
clouds.
To calculate the filling index for each plot, we first rasterize
the observed point cloud, by adding a grid of 50x50 to the
abundance-environment plot and identifying the cells overlapping with at
least one data point (purple cells in Fig 3). Then, for each column of
the raster, we identify the cells with the highest value of abundance
(dark purple cells) and connect them to get the upper boundary of the
data (dark purple line). Finally, we calculate the number of cells that
included at least one observation in relation to the total number of
cells bellow the upper boundary (i.e., sum of colored cells/sum of all
cells bellow the line). The resulting filling index varies from 0
to 1, where a value of 1 indicates that the area below the upper
boundary is fully filled and thus the plot represents a polygon.
To determine the probability of
the observed point cloud being polygon-shaped, the observedfilling index is compared with a null distribution offilling indexes resulting from 100 simulated line-shaped plots
(Fig 3B). Line-shaped plots are generated to have the same sample size
and environmental variable values (X-axis) than the observed data.
Abundance values (Y-axis) are simulated to resemble the general trend of
the observed data, though following a line-shaped pattern, using the
following procedure: i) we fitted a median QR (τ =0.5) to estimate the
general trend of observed data; ii) we predicted the median abundance
based on the observed environmental gradient (Ŷ|X); iii) we
added variation to predicted median abundance (Ŷ) (we simulated
variation under a normal distribution, with mean equals to zero and
standard deviation equals to the square root of the maximum predicted
abundance (σ = \(\sqrt{\max(Y)}\))). Finally, we calculated thefilling index of the simulated abundance-environment plot. We
repeated the procedure 999 times, and then calculated the probability of
occurrence of the observed filling index considering the
distribution of null filling index es. If the observedfilling index is significantly higher than the simulated
distribution, i.e., probability is lower than 0.05, then we concluded
that the shape of the observed pattern is significantly different from a
line.
We tested the performance of the filling index procedure in
differentiating line- and polygon-shaped patterns by applying it to
simulated point clouds (Fig 3D-F and Appendix S6). To do that, we
generated 100 bivariate plots with different number of observations
(n=400 and n=100) and dispersal values (σ = 1.1 to simulate line-shaped
patterns, and σ = 2 to simulate polygon-shaped patterns). Moreover, we
tested the performance of the procedure using two alternative grid
resolutions for rasterization (plots rasterized by 50x50 cells or 25x25
cells). Finally, we tested whether the filling index procedure
described above correctly classified the simulated point clouds as line-
or polygon-shaped under each sample size and grid resolution scenario
(Appendix S6).
We also exemplified the application of the filling index with the
observational data used in this study, thus assessing whether the point
clouds for tree and bird species were polygon-shaped.