# A Machine Learning Method to Estimate Reference Evapotranspiration Using Soil Moisture Sensors
## 3 Results and Discussion
### Accuracy Analysis of the Selected Model  
As a result of the previous experiments, the optimal method selected for the estimation of ETo using the moisture sensors and meteorological data was the combination of RFC and the K* algorithm. Recall that this method consists of two steps: firstly, the input tuples are projected into a random \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **(i) Daily Data** & **(ii) Data at 6-h** & **(iii) Data at 6-h** \\
**Intervals** & & **intervals** & **intervals** \\
**RT** & 0.7000 & 0.6700 & 0.6600 \\
**Best Weka model** & 0.4973 & 0.3567 & 0.1829 \\
**SVR** & 0.5354 & 0.4994 & 0.4043 \\
**ANN** & 0.5481 & 0.3037 & 0.2972 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Root-mean-squared error (RMSE) in mm/day, for the prediction of the daily ETo using the different regression algorithms on the three datasets defined. RT: regression tree; SVR: support vector regression; ANN: artificial neural network. subspace of lower dimensionality; then, the K nearest training samples to the input tuple are used to estimate the value of ETo. In this subsection, the results of this method are analyzed in more detail and compared with those of the ANN. The main accuracy parameters are presented in Table 3. These measures used the dataset at 6-h intervals removing sensor 3. Figure 6 shows a comparison of the ETo values estimated using the two best regression models found, compared to the actual values obtained with the Penman-Monteith method in the agroclimatic station of La Murada, for a random subset of evapotranspiration measurements in dataset (iii). The predictions in Figure 6a are clearly closer to the expected values than those in Figure 6b, in coherence with the error measures. The regression line of the plot in Figure 6a corresponds to the following equation:  
\[y=0.9518x+0.1699. \tag{6}\]  
As indicated in Table 3, the correlation of the regression line is 0.9936. Finally, Figure 7 shows a histogram showing the error distribution of the RFC + K* model for the same dataset. Most of the errors were within \(-1\) and \(+1\), with almost no samples having a greater error. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **RMSE** & **MAE** & **MRE** & **R** \\ \hline
**RFC + K*** & 0.1829 & 0.0899 & 6.52\% & 0.9936 \\
**ANN** & 0.2972 & 0.1521 & 11.03\% & 0.9470 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Root-mean-squared error (RMSE) in mm/day, mean absolute error (MAE) in mm/day, mean relative error (MRE), and correlation coefficient (_R_) for the prediction of the daily ETo using the two best regression algorithms, randomizable filtered classifier with K* algorithm (RFC + K*) and artificial neural network (ANN). Figure 6: Scatter plots of the reference evapotranspiration (ETo) obtained with the Penman–Monteith method, and the generated predictions using the dataset at 6-h intervals removing sensor 3, applying the two best models. (**a**) randomizable filtered classifier and K* algorithm; (**b**) artificial neural network. The accuracy measures prove that the proposed method is able to achieve a very precise estimation of ETo, with an average error below 6.5% in relative terms. Thus, a good estimation of ETo can be obtained with a cost-effective system that would only require moisture sensors and agroclimatic data. Since the selected RFC method relies on subspace projections, we performed a principal component analysis (PCA) of the independent variables. It was observed that, from the 43 input variables, the first principal component (PC) accounted for only a 32% of the variance, and the second PC accounted for 13%. This indicates a high complexity in the distribution of the input data. A plot of the dataset (iii) projection in these two components is depicted in Figure 8. The first 10 PCs were required to capture 89% of the variance. The predicted ETo values are very accurate compared to other estimation methods based on meteorological information of remote imagery.