# Estimation of Time-Series Forest Leaf Area Index (LAI) Based on Sentinel-2 and MODIS
## 3 Results
### Remote Sensing Model of LAI in Sailanba Area
#### 3.1.2 Multiple Linear Regression Analysis  
Four vegetation indices were selected to study the multivariate linear statistical model with different combinations, and the \(R^{2}\) and \(RMSE\) were calculated to find the best combination of vegetation indices; the results are shown in Table 5.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**VI** & **Equation Type** & **Regression Equation** & \(R^{2}\) & **RMSE** \\ \hline \multirow{6}{*}{_NDVI_} & Linear & \(LAI=12.786 NDVI-7.48\) & 0.392 & 0.457 \\  & Quadratic & \(LAI=-24.742 NDVI^{2}+5.5133 NDVI-23.933\) & 0.393 & 0.454 \\  & Logarithm & \(LAI=10.4221 NDVI+4.093\) & 0.393 & 0.454 \\  & Power & \(LAI=6.2433 NDVI^{3.804}\) & 0.423 & 0.443 \\  & Exponential & \(LAI=0.071\)exp(\(4.662 NDVI\)) & 0.42 & 0.444 \\ \hline \multirow{6}{*}{_SR_} & Linear & \(LAI=0.1985R+0.964\) & 0.374 & 0.463 \\  & Quadratic & \(LAI=-0.012\)\(SR^{2}+0.4485R-0.356\) & 0.381 & 0.459 \\  & Logarithm & \(LAI=2.068\)In \(SR+0.865\) & 0.385 & 0.457 \\  & Power & \(LAI=0.493 SR^{0.749}\) & 0.401 & 0.451 \\  & Exponential & \(LAI=1.438\)exp(\(0.066 SR\)) & 0.375 & 0.461 \\ \hline \multirow{6}{*}{_DVI_} & Linear & \(LAI=0.001 DVI+0.825\) & 0.198 & 0.525 \\  & Quadratic & \(LAI=-2.384\)\(\times\)\(10^{-7}DVI^{2}+0.002 DVI+0.151\) & 0.208 & 0.519 \\  & Logarithm & \(LAI=1.375\)In \(DVI-1.48\) & 0.207 & 0.519 \\  & Power & \(LAI=0.088 DVI^{0.459}\) & 0.186 & 0.526 \\  & Exponential & \(LAI=1.953\)exp(\(0.002 DVI\)) & 0.178 & 0.529 \\ \hline \multirow{6}{*}{_ARVI_} & Linear & \(LAI=10.729 ARVI-5.992\) & 0.429 & 0.443 \\  & Quadratic & \(LAI=-2.229 ARVI^{2}+14.381 ARVI-7.479\) & 0.43 & 0.44 \\ \cline{1-1}  & Logarithm & \(LAI=8.777\)In \(ARVI+4.509\) & 0.429 & 0.441 \\ \cline{1-1}  & Power & \(LAI=5.258 ARVI^{3.317}\) & 0.499 & 0.413 \\ \cline{1-1}  & Exponential & \(LAI=0.102\)exp(\(4.043 ARVI\)) & 0.493 & 0.415 \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of Unitary Regression Models.  
\begin{table}
\begin{tabular}{c c c c} \hline
**VIs Combinations** & **Regression Equation** & \(R^{2}\) & **RMSE** \\ \hline \multirow{3}{*}{_NDVI_, \(SR\)} & \(LAI=13.716 NDVI-0.014 SR-8.075\) & 0.519 & 0.403 \\  & \(LAI=13.688 NDVI-0.000074 DVI-8.056\) & 0.521 & 0.402 \\ \cline{1-1}  & \(LAI=11.228 ARVI-0.664 NDVI-5.859\) & 0.528 & 0.399 \\ \cline{1-1}  & \(LAI=0.221 SR+0.00001 DVI+0.544\) & 0.512 & 0.406 \\ \cline{1-1}  & \(ARVI\) & \(LAI=0.026 SR+9.435 ARVI-5.188\) & 0.531 & 0.398 \\ \cline{1-1}  & \(LAI=10.175 ARVI+0.000069 DVI-5.672\) & 0.532 & 0.397 \\ \cline{1-1}  & \(LAI=12.058 NDVI+0.035R-0.000097 DVI-6.981\) & 0.533 & 0.395 \\ \cline{1-1}  & \(ARVI\) & \(LAI=15.109 ARVI+0.205 SR-18.42 NDVI+3.268\) & 0.545 & 0.39 \\ \cline{1-1}  & \(ARVI\) & \(LAI=9.832 ARVI-0.000052 DVI+0.011 SR-5.461\) & 0.533 & 0.395 \\ \cline{1-1}  & \(ARVI\) & \(LAI=-4.713 NDVI+13.201 ARVI+0.00019 DVI-4.496\) & 0.539 & 0.392 \\ \cline{1-1}  & \(ARVI\) & \(LAI=-18.322 NDVI+0.192 SR+0.000043 DVI+15.445 ARVI+2.95\) & 0.561 & 0.381 \\ \hline \end{tabular}
\end{table}
Table 5: Comparison of Multiple Regression Models.  
As shown in Table 5, among the equations of the two vegetation indices, the best effect is the DVI and ARVI (\(R^{2}=0.532\), \(RMSE=0.397\)). Among the equations of the three vegetation indices, the best effect is the NDVI, SR, ARVI (\(R^{2}=0.545\), \(RMSE=0.39\)). The regression equation fitting effect of the four VIs is better than the two VIs and the three VIs (\(R^{2}=0.561\), \(RMSE=0.381\)).