# Multivariate Regression Analysis for Identifying Key Drivers of Harmful Algal Bloom in Lake Erie
## 3 Results and Discussions
### Correlation and Relative Importance  
Table 3 presents the relative importance (RI) values for each predictor variable in relation to Chl-a and TSS. For Chl-a, PON, Turbidity, and POC emerge as the most influential variables, contributing 24.26%, 22.77%, and 18.39% to the model, respectively. These factors play a critical role in explaining the variability of Chl-a concentrations in the water. Other significant contributors include TSS and TP, accounting for 13.00% and 10.01%, respectively. However, the remaining variables, such as SD, DO, and T, provide minimal contributions, and variables like Cond and NOx contribute even less. Table 4 presents the Pearson correlation coefficients between all target variables and regressors. Pearson correlation coefficients indicate the strength and direction of the linear relationship between two variables, ranging from \(-1\) (perfect negative correlation) to 1 (perfect positive correlation). A value close to 0 suggests no correlation. This correlation matrix provides important insights into the relationships between environmental variables and the two target variables, Chl-a and TSS. The strongest drivers for Chl-a are PON, Turbidity, and POC, while TSS is primarily influenced by Turbidity, PON, and POC. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Variables** & **Min** & **Max** & **Mean** & **Std. Dev.** \\ \hline SD & 0 & 5.3 & 0.796 & 0.694 \\ T & 10.1 & 29.7 & 22.417 & 3.651 \\ Cond & 19.9 & 583.3 & 337.586 & 67.828 \\ DO & 4.2 & 13.0 & 7.478 & 1.217 \\ Turb & 0.95 & 1148.0 & 29.599 & 78.295 \\ TP & 14.87 & 2482.2 & 119.144 & 181.173 \\ TDP & 2.67 & 273.6 & 30.909 & 34.865 \\ A & 0.04 & 561.6 & 39.822 & 56.930 \\ NOx & 0 & 9.5 & 1.308 & 1.676 \\ POC & 0.15 & 219.3 & 3.946 & 15.381 \\ PON & 0.03 & 40.9 & 0.677 & 2.759 \\ TSS & 1.25 & 540.8 & 25.489 & 44.275 \\ Chl-a & 0.71 & 6784.0 & 61.232 & 347.307 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical description of the dataset used in this study. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **RI for Chl-a (100\%)** & **RI for TSS (100\%)** \\ \hline PON & 24.26 & 28.06 \\ Turb & 22.77 & 35.57 \\ POC & 18.39 & 24.12 \\ TSS & 13.00 & — \\ TP & 10.01 & 6.20 \\ SD & 3.14 & 0.34 \\ DO & 2.44 & 0.00 \\ T & 2.19 & 0.15 \\ TDP & 2.01 & 0.12 \\ A & 1.25 & 0.11 \\ Cond & 0.50 & 0.00 \\ NOx & 0.05 & 0.01 \\ Chl-a & — & 5.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relative importance (RI) value for two considered targets. Figure 3 also shows a bivariate scatter plot indicating pairwise relationships between seven selected regressors, SD, Turb, TP, POC, PON, TSS, and Chl-a, for this analysis. Clear positive linear relationships are observed between Turbidity and TSS, as well as between Turbidity, POC, PON, and Chl-a, suggesting that these variables are important drivers of both suspended solids and algal bloom concentration. TP also shows moderate positive associations with both turbidity and Chl-a, reinforcing the role of phosphorus in contributing to water turbidity and algal growth. To assess multicollinearity among predictor variables, we calculated the variance inflation factor (VIF) values for all predictors related to the Chl-a and TSS models. While most variables had VIFs below 5, a few (e.g., PON, POC, Turbidity) exhibited VIF values above the commonly accepted thresholds (VIF \(>\) 10). Nevertheless, they were retained in the final models due to their strong predictive performance and ecological importance in driving HABs. Although multicollinearity can affect the precision of individual coefficient estimates, it does not necessarily impair model validity for predictive purposes, especially in environmental applications where predictor variables are often interrelated [62; 63; 64]. We therefore prioritized model robustness and ecological relevance over coefficient orthogonality, consistent with best practices in applied regression modeling.