Material and Methods
The database and the macroecological variables used for the analyses
were the same considered in Vieira et al (Vieira et al. 2018), January
evapotranspiration (ETJan), June evapotranspiration (ETJune); annual
rainfall variation (ARV), primary productivity (PP), annual temperature
variation (TempVar) and annual rainfall variation (ARV) and can be
accessed by https://doi.org/10.1371/journal.pone.0204114.s005. The
stationarity’s quantification of the fish richness-macroecologial
variables relationship was done using a Geographically Weighted
Regression (GWR) following the protocol indicated in Figure 1. This
analysis establishes local estimates of adjustments and regression
coefficients using subsets of the database considered and defined
according to a Spatial Weighting Function. This function attributes a
weight (or importance) to each site, which will be used in the
coefficient estimates of a focal point. Thus, sites close to each other
(given a connectivity criterion) will have greater importance than sites
further away from the focal point, considering the close and far
threshold defined by the chosen bandwidth, which in our case is
represented by the connectivity between sites. This allows the
specification of heterogeneity in relationships and identify regions
where the model is more robust, as well as which variables are more
important to explain the observed pattern. In this study, connectivity
between sampling sites was defined in three ways: i) Euclidian distance
between all sites; ii) Euclidian distance between all sites (W Global)
present in a same hydrographic basin (W Basin), so the sites located in
different basins have a connectivity of zero; and iii) Euclidian
distance between all sites present in the same ecoregion (W FEOW), thus
the sites in different ecoregions have a connectivity of zero. The
number of sites used in local estimates was defined as fixed and the
radius (664.05 km to Global; 403.14 km to Basin and 486.706 km to FEOW)
that minimized spatial self-correlation was chosen (Figure 1).
Ecoregions were considered as defined by Abell et al. (2008).
A way to quantify non-stationarity is using local estimate techniques
such as the Geographically Weighted Regression – GWR. It calculates all
regression coefficients for each point present in a database,
contrasting from the Ordinate Least Square – OLS, which calculates the
average parameters for the entire database considered in the analysis.
The calculation is done by partitioning the data set into subsets, given
a connectivity criterion between points(C. Brunsdon, Fotheringham, and
Charlton 1998; Chris Brunsdon et al. 1998). A way of quantifying
stationarity level is by using local estimate techniques such as
Geographically Weighted Regression – GWR, which calculates and adjusts
the coefficients of determination for each data point in the database.
If the relationship is stationary, the GWR presents the same
coefficients throughout the entire geographical extension. Although GWR
presents advantages over OLS regression models, it should not be used as
an alternative, but as a supplement to OLS(Osborne et al. 2007). While
OLS offers an average global estimate of the relationships, GWR shows
the peculiarities present in the database, therefore improving the power
to predict and explain mechanisms and processes(Osborne et al. 2007).
To quantify spatial autocorrelation, there was considered the W Global
matrix (connectivity criterion) and 17 distance classes (each composed
by an equal number of sites). For the W Basin and W FEOW matrices,
classes that maintained equal distances between the classes’ centroids
were defined. Afterwards, a GWR was generated for each class (using its
respective W matrix as the sites’ connectivity criterion) and the Akaike
Information Criterion (AIC) was calculated for each model. The Moran’s I
and AIC distance class values were plotted on a graph, and the classes
with the lowest value for the AIC and Moran’s I equal or close to zero
were selected. This procedure was performed for each W matrix, allowing
the selection of three GWR models, one W Global model, one W Basin
model, and one W FEOW model (Figure 1). The autocorrelation of each
model was evaluated with a Moran’s scatterplot. For the best GWR, the
global adjustment for the model (r²) was calculated, the spatial
autocorrelation of the residuals was measured, and the variable
determination coefficients were specified. The GWR was run on the
program Spatial Analysis for Macroecology (SAM(Rangel, Diniz-Filho, and
Bini 2010)) using the Gaussian Spatial Weighting Function, all models
present the Moran’s I value and the Akaike Information Criterion.