## 2 Background Fundamentals
### Randomized Low-Rank Approximations
\begin{table}
\begin{tabular}{c c c c} \hline
**Contingency Factors** & **Attribute Names** & **Notation** & **Possible Range** & **Max Value** \\ \hline \multirow{10}{*}{Geological} & Depth of water & \(a_{2}\) & (0–4.21) & 4.21 \\  & extractable GW & \(a_{5}\) & (0.005–66.88) & 66.88 \\  & Total GW extraction & \(a_{6}\) & (0.005–46.03) & 46.03 \\  & GW availability & \(a_{7}\) & (0.002–21.53) & 21.53 \\  & Aquatic veg land & \(a_{20}\) & (0–228,174) & 228,174 \\  & Cultivable land & \(a_{21}\) & (1.5–25,503.7) & 25,503.7 \\  & Nitrogen & \(a_{22}\) & (1–2.98) & 2.98 \\  & Phosphorus & \(a_{23}\) & (1.02–2.54) & 2.54 \\  & Potassium & \(a_{24}\) & (1.03–2.69) & 2.69 \\  & Organic carbon & \(a_{25}\) & (1.12–2.99) & 2.99 \\  & Boron & \(a_{26}\) & (1–2) & 2 \\  & Copper & \(a_{27}\) & (1.38–2) & 2 \\  & Iron & \(a_{28}\) & (1.34–2) & 2 \\  & Manganese & \(a_{29}\) & (1.28–2) & 2 \\  & Sulpher & \(a_{30}\) & (1–2) & 2 \\  & Zinc & \(a_{31}\) & (1.38–1.99) & 1.99 \\  & GWL decision & \(a_{32}\) & (1–3) & 3 \\ \hline \multirow{10}{*}{Topographic} & Land degradation & \(a_{12}\) & (42–18,034,066) & 18,034,066 \\  & Wasteland & \(a_{13}\) & (81.27–175,697) & 175,697 \\  & Geo area & \(a_{16}\) & (3000–34,223,900) & 34,223,900 \\  & Forest ecosystem & \(a_{17}\) & (2591–84,147) & 84,147 \\  & Wetland & \(a_{18}\) & (350–3,474,950) & 3,474,950 \\ \hline \multirow{10}{*}{Hydrological} & Rainfall & \(a_{1}\) & (351.8–4489.5) & 4489.5 \\  & GW recharge & \(a_{3}\) & (0.01–72.2) & 72.2 \\  & Natural GW discharge & \(a_{4}\) & (0.01–5.32) & 5.32 \\  & GW temperature & \(a_{8}\) & (0–39) & 39 \\  & GW dissolved O2 & \(a_{9}\) & (0–9.1) & 9.1 \\  & GW pH & \(a_{10}\) & (0–9.7) & 9.7 \\  & GW nitrate & \(a_{11}\) & (0–50) & 50 \\  & Soil depth & \(a_{14}\) & (25–50) & 50 \\  & soil pH & \(a_{15}\) & (5–8.4) & 8.4 \\  & Open water & \(a_{19}\) & (242–1,150,755) & 1,150,755 \\ \hline \end{tabular}
\end{table} 
Table 5: Notation table. ### Attribute Approximation Using Randomized Low-Rank Approximation Method

After the attribute selection through the VIF number of the corresponding column, the rank of the reduced dataset was checked using the rank function of MATLAB. The rank function identified that the reduced dataset was of rank 18. The basics of RLRA are explained in Section 2.2 with Algorithm 2. The sample dataset is depicted in Figure 8.

The real rank of the dataset was 18, but it was approximated by the RLRA technique, which obtained 3 as the reduced rank. Figure 9 shows the approximated dataset of the sample dataset. To depict the data authenticity, a heat map and similarity representation were created for both the original sample dataset and approximated data, as shown in Figures 10 and 11. It can be seen that the data distribution on the knowledge base is almost the same for both datasets. Thus, data authenticity is maintained. The approximated and non-approximated dataset is further applied to the deep ensemble learning process. Figure 8: Sample dataset. Figure 7: Attribute reduction. Figure 9: Approximated dataset using RLRA. ### Proposed Deep Ensemble Model

The main idea of the proposed ensemble method is to combine the assorted base classifiers into different layers. The first layer (Level 0) trains the group of deep base learners on a different partition of the training dataset to form a so-called team. In the next layer, Level 1, a group of meta-classifiers is trained on the prediction of the team of Level 0. In the final layer, the predictions of the previous group of classifiers are combined with the meta-classifier to produce the results. Figure 12 shows the stacking mechanism of the proposed model, as discussed in Section 2.3.1. Figure 11: Similarity between sample dataset and approximated dataset. Figure 12: Stacking mechanism of proposed model. #### 3.4.1 Proposed Deep Ensemble Architecture

The proposed deep ensemble model encompasses random forest (RF) as the base learner and double-edged bi-directional LSTM (DEBi-LSTM) as the meta-classifier [33; 34].