# Multivariate Regression Analysis for Identifying Key Drivers of Harmful Algal Bloom in Lake Erie
## 3 Results and Discussions
### Multivariate Regression Model Performance  
In this study, the objective is to deal with multivariate regression showing the association between dependent and independent variables. With our data, after evaluating the relative importance and correlation analysis, two regression models were determined as below for the prediction Chl-a and TSS, including the following:  
\[\textbf{Model 1:}\ \text{Chl-a}=0.113\ \text{Tur}\ -\ 0.035\ \text{TP}\ -\ 3.453\ \text{POC}+4.115\ \text{PON}\ -\ 0.03\ \text{ TSS}+0.004 \tag{4}\]  
\[\textbf{Model 2:}\ \text{TSS}=1.477\ \text{Tur}+0.196\ \text{TP}+2.125\ \text{POC}\ -\ 2.705\ \text{PON}\ -\ 0.126\ \text{Chl-a}+0.007 \tag{5}\]  
In Table 5, the multivariate correlation coefficient (r) provides insight into the strength and direction of the relationship between independent and dependent variables in our regression models. For Model 1, the r-value is 0.986, indicating a very strong positive correlation, which suggests a high-quality prediction for the dependent variable, Chl-a. Similarly, in Model 2, the r-value is 0.979, also signifying a strong predictive relationship,  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline  & **SD** & **T** & **Cond** & **DO** & **Turb** & **TP** & **TDP** & **A** & **NOx** & **POC** & **PON** & **TSS** & **Chl-a** \\ \hline
**SD** & 1.00 & 0.12 & \(-\)0.13 & \(-\)0.05 & \(-\)0.23 & \(-\)0.27 & \(-\)0.15 & \(-\)0.06 & \(-\)0.01 & \(-\)0.12 & \(-\)0.11 & \(-\)0.35 & \(-\)0.06 \\
**T** & 0.12 & 1.00 & \(-\)0.01 & \(-\)0.32 & \(-\)0.07 & \(-\)0.06 & \(-\)0.06 & \(-\)0.15 & 0.06 & 0.06 & 0.06 & \(-\)0.16 & 0.07 \\
**Cond** & \(-\)0.13 & \(-\)0.01 & 1.00 & \(-\)0.15 & \(-\)0.04 & 0.09 & 0.40 & 0.39 & 0.50 & \(-\)0.06 & \(-\)0.03 & \(-\)0.05 \\
**DO** & \(-\)0.05 & \(-\)0.32 & \(-\)0.15 & 1.00 & 0.10 & 0.03 & \(-\)0.31 & \(-\)0.32 & \(-\)0.23 & 0.16 & 0.16 & 0.06 & 0.19 \\
**Turb** & \(-\)0.23 & \(-\)0.07 & \(-\)0.04 & 0.10 & 1.00 & 0.88 & 0.11 & 0.06 & 0.05 & 0.89 & 0.91 & 0.93 & 0.75 \\
**TP** & \(-\)0.27 & \(-\)0.06 & 0.09 & 0.03 & 0.88 & 1.00 & 0.28 & 0.14 & 0.19 & 0.76 & 0.78 & 0.83 & 0.69 \\
**TDP** & \(-\)0.15 & \(-\)0.06 & 0.40 & \(-\)0.31 & 0.11 & 0.28 & 1.00 & 0.48 & 0.61 & \(-\)0.08 & \(-\)0.08 & 0.17 & \(-\)0.06 \\
**A** & \(-\)0.06 & \(-\)0.15 & 0.39 & \(-\)0.32 & 0.06 & 0.14 & 0.48 & 1.00 & 0.46 & \(-\)0.09 & \(-\)0.09 & 0.12 & \(-\)0.08 \\
**NOx** & \(-\)0.01 & 0.06 & 0.50 & \(-\)0.23 & 0.05 & 0.19 & 0.61 & 0.46 & 1.00 & \(-\)0.09 & \(-\)0.08 & 0.09 & \(-\)0.07 \\
**POC** & \(-\)0.12 & 0.06 & \(-\)0.06 & 0.16 & 0.89 & 0.76 & \(-\)0.08 & \(-\)0.09 & \(-\)0.09 & 1.00 & 0.99 & 0.78 & 0.71 \\
**PON** & \(-\)0.11 & 0.06 & \(-\)0.06 & 0.16 & 0.91 & 0.78 & \(-\)0.08 & \(-\)0.09 & \(-\)0.08 & 0.99 & 1.00 & 0.76 & 0.79 \\
**TSS** & \(-\)0.35 & \(-\)0.16 & \(-\)0.03 & 0.06 & 0.93 & 0.83 & 0.17 & 0.12 & 0.09 & 0.78 & 0.76 & 1.00 & 0.4 this time for the dependent variable TSS. The coefficients of determination, represented by R\({}^{2}\) and adjusted R\({}^{2}\), further emphasize the model's effectiveness in explaining the variability in the outcome variable. In Model 1, R\({}^{2}\) is 0.973, meaning 97.3% of the variance in Chl-a can be attributed to the independent variables in the regression model. The adjusted R\({}^{2}\), which accounts for the number of predictors in the model, remains at 0.973, confirming that the predictors explain the vast majority of variability without overfitting. Model 2 follows a similar trend, with R\({}^{2}\) at 0.958, indicating that 95.8% of the variability in TSS is explained by the independent variables. The adjusted R\({}^{2}\) for Model 2 also mirrors this value, reflecting the robustness of the model. The standard error (SE) of the estimate provides a measure of the average distance that the observed values fall from the regression line. For Model 1, the SE is 0.008, indicating a very small error in the predictions of Chl-a. In Model 2, the SE is slightly higher at 0.016, which still reflects a reasonably accurate prediction for TSS.