## 3 Results
### Study Area Predictor Variable Distributions
A cross-validation analysis is shown in Figure 9 with the coefficient of determination (r\({}^{2}\)) values on the y-axis, canopy parameters on the x-axis and values indicating the variables included in the validation. Results indicate that the greater portion of the variance (0.9) is associated with two of the variables for richness (R). The results indicate a similar portion of the variance (0.91) is associated with two of the variables for abundance (A), but the variables differ. The MDI diversity cross-validation indicates that much of the variance (0.99) is associated with three of the variables, while the SDI diversity analysis indicates that much of the variance (0.99) is attributed to three of the predictors and a similar proportion (0.94) is associated with two of the variables. Figure 8: Density matrix of area solar radiation (\(ASR\)), leaf area index (\(LAI\)), canopy height (CH) and total vegetation density (TVD). Development of the prediction model focused the two diversity indexes, _MDI_ and _SDI_. Table 3 compares the results of three _SDI_ models (Lm0,1,2) to see if the available data would yield a good predictive model. The Lm0 F statistic and \(p\)-value do not indicate the model itself is significant, while Lm1 was created with two variables having the best \(p\)-values from the first model, and a third model (Lm2) contains an interaction term that increases the total variance associated with the predictors. Linear model 3 (Lm3) for _MDI_ summary statistics present the best scenario for predictive modeling given the available data. The model \(p\)-value is below 0.05 and the adjusted \(r^{2}\) value is greater than 0.96. In addition, each of the predictor variables strongly contributes to the significance of the model without adjustment. The validity of linear regression modelling for prediction rests on four assumptions:  
* A linear relationship between the dependent and predictor variables;
* The model errors are independent;
* The model errors are normally distributed;
* The model errors have a constant variance with respect to the predictor variables [53]. The plot in Figure 9(a) graphs the model errors (residuals) vs. the predicted values (fitted) and tests for non-linearity as well as heteroscedacity (non-constant variance) in errors. The plotted points should be symmetrically distributed around zero (a horizontal line) \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Model**}} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}{c} **LAI + CH + IR** \\ **(Lm0)** \\ \end{tabular} }} & \multicolumn{1}{c}{\multirow{2}{*}{\begin{tabular}{c} **LAI + CH** \\ **(Lm1)** \\ \end{tabular} }} & \multicolumn{1}{c}{\multirow{2}{*}{\
\begin{tabular}{c} **CH + IR + CH:IR** \\ **(Lm2)** \\ \end{tabular} }} & \multicolumn{1}{c}{\multirow{2}{*}{
\begin{tabular}{c} **CRR + TH + TVD** \\ **(Lm3)** \\ \end{tabular} }} \\ \hline Adjusted r\({}^{2}\) & 0.94 & \(-\)0.3252 & 0.693 & 0.9687 \\ F statistic & 22.24 & 0.38 & 4.763 & 52.53 \\ \(p\)-value & 0.1544 & 0.709 & 0.1784 & 0.01874 \\ Coefficient significance & CH/0.05 & N/ A & CH, IR, CH:IR/0.05 & CRR, TVD/0.001; TH/0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model Result Summary. Figure 9: Cross-validation graphic displaying variance relationships for predictor variables as applied to richness (**a**), abundance (**b**), the _MDI_ (**c**) and _SDI_ (**d**). indicating that the model doesn't make systematic errors [53]. The QQ plot (Figure 10b) displays a satisfactory positive diagonal trendline for the available data. Coefficients from the model were used to display the prediction spatially over the study area extent (Figure 11):  
Multicollinearity, such as that exhibited between CRR and TVD (r\({}^{2}\) = 0.80), makes it difficult to assess the relative importance of independent variables if they are both used in the model, but it does not impact the usefulness of the regression equation for prediction. Even when multicollinearity is great, the least-squares regression equation can be highly predictive.