## 3 Results  
### Study Area Predictor Variable Distributions  
Area solar radiation derived from the DSM and the point-cloud and imagery metrics gridded to raster yielded 144 data points for analysis as predictors. The histograms in Figure 7 depict distributions of the predictor variable data points used in the regression equation. A collinearity analysis of selected predictor variables is shown in Figure 8. It displays coefficient values in the upper half of the matrix and non-parametric trend lines (loess smoothed fits) and correlation ellipses below the diagonal. Modeled irradiation (_ASR_) within the study area dataset exhibited a correlation with canopy height. A significant positive correlation coefficient (r) of 0.61 provided a credibility test of the irradiation data since canopy height should reasonably have some association to WH/m\({}^{2}\) due to the direct relationship between photosynthetic processes to the quantity of radiant energy in one area of space relative to another area of space. Effective leaf area and vegetation density relationships were not significant. It is worth noting that _LAI_ and _ASR_ were negatively correlated, and although TVD and _ASR_ exhibited minimal correlation, _LAI_ and TVD interestingly did were negatively correlated as well. The acquisition date for the multi-spectral data was early in the growth season, and although the forest canopy is dominated by evergreen species, the full lower strata canopy was not at a seasonal peak. Much of the analysis extent exhibited significant canopy height and structural variation with dead biomass in all strata. A seasonal analysis of _LAI_ would be useful to validate data to use for further canopy studies. However, other controlling factors such as soil and water limitations would also require review. Collinearity analysis was subsequently performed on all predictors and the four dependent variables as well. Abundance of individuals (A) was correlated with the texture of heights (TH, r = 0.59) and the intensity of the returns (IR, r = 0.70). Taxonomic diversity (_MDI_) was significantly correlated with leaf area (_LAI_, r = 0.69). Taxonomic diversity adjusted for evenness (_SDI_) was modestly correlated with canopy height (CH, r = 0.42) and the intensity of the returns (IR, r = 0.24). Figure 6: Distribution of plot values for species richness (**a**), abundance (**b**), the MDI (**c**) and SDI (**d**). Figure 7: Study area predictor variable distributions for effective leaf area index (**a**), canopy height (**b**), area solar radiation (**c**), total vegetation density (**d**), canopy relief ratio (**e**), texture of heights (**f**), foliage height diversity (**g**) and average intensity of return (**h**). The regression model was created by first identifying the best variable subset using cross-validation statistics (the \"regsubsets\" function in the R \"leaps\" library), and then selecting predictors with high r\({}^{2}\). To summarize, the cross-validation statistical notation is:  
\[\left( \begin{array}{c}p\\ k\end{array} \right)=\frac{p(p-1)}{k} \tag{18}\]  
where:  
* \(p\) is the specific predictor variable;
* \(k\) is the total number of predictor variables. If \(M_{0}\) is the null model containing 0 predictor values, the best model (\(M_{k}\)) from a series of models, denoted as \(M_{0}\dots M_{p}\). is derived by fitting \(\left( \begin{array}{c}p\\ k\end{array} \right)\) models containing \(k\) predictors. Biodiversity is analyzed as a function of all the variables in the canopy metrics data frame. The function evaluates the predictor variables using an exhaustive selection algorithm, selecting a maximum of four predictors for each variable subset. Using R linear modeling functions, several variable subsets with significant r\({}^{2}\) values were evaluated to estimate the parameters of the linear model for all measures of diversity.