## 3 Results
### Study Area Predictor Variable Distributions
Since predicting beyond the ranges of the original data will result in model extrapolation and subsequent prediction errors, areas where the model extrapolated beyond the range of observation were \"masked out\". Average CRR at a 30 m resolution ranged in value from 0.27 to 0.40, the average TH from 0.23 to 0.31 and TVD from 656 to 1704 per meter square. The Con tool in ArcGIS creates a raster where \"1\" is the assigned value if the canopy parameters are met, and if false, \"0\" is the assigned value. Combining three mask extrapolation layers--one for each variable--by deriving the product (using the Times tool) of raster layers with 0 or 1 pixel values creates an output raster consisting of pixels where variables are within the range of observations (a value of 1). Finally, these raster data are used to create a combined raster extrapolation mask applied to the original prediction file with the Set Null tool to derive a new predicted range of values [54] shown in Figure 12.  
Figure 11: Graphic displaying predictive scatterplot for _MDI_ (Lm3). Figure 10: MDI linear model (Lm3) residuals vs. fitted values (**a**). Some row values are flagged for a discrepancy in the fitted vs. residual relationship (1,4–5). QQ plot of linear model (Lm3) errors (**b**). Variables 1, 5–6 are flagged. Table 4 provides a comparative summary for the Moran's I and Getis-Ord General G values. A 100 m Euclidian distance parameter was estimated for spatial statistics, although an estimated value based on nearest neighbor distances is considered as a default practice when not specified. The resulting statistics imply that the data as a whole are non-random with respect to spatial autocorrelation--the Global Moran's I \(p\)-value indicates significance, so the null hypothesis is rejected. However, the Getis-Ord General G \(p\)-value affirms the null hypothesis that there is randomness in the data with respect to the clustering not being attributable to either high or low MDI biodiversity values. The Getis-Ord Gi* statistic is used to determine statistically significant spatial clustering of data with an influence threshold distance. The resulting data can be used to create an MDI biodiversity hotspot analysis map from the predicted raster values (Figure 13). Hotspots derived from this statistic are both positive and negative, like the General G, and both types have significance for biodiversity prediction, based on the study parameters. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Global Moran’s I** & \multicolumn{3}{c}{**Getis-Ord General G**} \\ \hline Moran’s Index & 0.261149 & Observed General G & 0.002234 \\ Expected Index & \(-\)0.016667 & Expected General G & 0.002245 \\ Variance & 0.004745 & Variance & 0.000000 \\ z-score & 4.033045 & z-score & \(-\)0.162268 \\ \(p\)-value & 0.000055 & \(p\)-value & 0.871095 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Global Moran’s I and General G Results for Predicted MDI. Figure 12: Map depicting spatial prediction raster of _MDI_ in 30 m resolution, adjusted for predictor variable ranges. The notation for the local Gi-star (z-score) statistic is:  
\[G_{i}^{*}=\frac{\sum_{j=1}^{n}w_{i,j}x_{j}-\overline{X}\ \sum_{j=1}^{n}w_{i,j}}{S \sqrt{\frac{\left[n\sum_{j=1}^{n}w_{i,j}^{2}-\left(\sum_{j=1}^{n}w_{i,j}\right)^{ 2}\right]}{n-1}}}\]  
where:  
* \(x_{j}\) is the attribute for feature \(j\);
* \(w_{ij}\) is the spatial weight between features \(i\) and \(j\);
* \(n\) is the number of features in the dataset and:  
\[S=\sqrt{\frac{\sum_{j=1}^{n}x_{j}^{2}}{n}}-\left(\overline{X}\right)^{2}\]  
\[\overline{X}=\frac{\sum_{j=1}^{n}x_{j}}{n} \tag{19}\]  
The statistic in this case used a fixed distance band of 100 m to impose a \"sphere of influence\" or \"moving window\" conceptual model of spatial interactions onto the data. Each feature is analyzed within the context of those neighboring features located within the specified distance for the \"distance band\" and neighbors within that distance are weighted equally.