# A Developed Multiple Linear Regression (MLR) Model for Monthly Groundwater Level Prediction

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## Abstract

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## 1. Introduction

- An advanced machine learning model is developed to predict the monthly GWL. We combine three models to predict monthly groundwater levels. As our proposed model consists of two robust deep learning models, it can be superior to linear models such as MLR models. Since our proposed model uses the capabilities of CNN and LSTM models simultaneously, it outperforms standalone CNN and LSTM models. The CLM model is presented as a reliable tool for predicting GWLs in different basins. Since it can be adapted to different watersheds, it can be a reliable tool for water resource managers around the world.
- A new optimizer is introduced to identify the best input combination. An optimization algorithm determines the best input combination, while the correlation method only determines important inputs. To identify the most optimal set of features, the new method performs an extensive search and explores various combinations. The new optimization method handles high-dimensional feature spaces and can be fine-tuned to maximize model accuracy, minimize error, or enhance specific evaluation metrics. As a result, our study contributes to feature selection.
- An effective method is suggested to quantify uncertainty values. This study contributes to the quantification of output uncertainty values. The study illustrates how model parameters and inputs contribute to model uncertainties. Through the analysis of variance (ANOVA) method, the study provides insight into the sources and values of output uncertainty.
- We use an advanced method to decompose the output uncertainty into different sources of uncertainty. Our study proposes a new method to calculate the percentage contribution of inputs and model parameters to overall uncertainty.

## 2. Materials and Methods

#### 2.1. MLR Model

#### 2.2. Mathematical Model of the LSTM

#### 2.3. Mathematical Model of the CNN Model

#### 2.4. Input Selection

#### 2.4.1. Structure of GOA

_{5}: random numbers; m

_{1}: a random number between −a and a; n

_{1}: a random number between −b and b; $G{H}_{i}\left(t\right)$: the location of the ith gannet; $G{H}_{m}\left(t\right)$: the average location; and $G{H}_{r}\left(t\right)$: the random location of gannet. Based on two scenarios, gannets pursue fish. The first scenario assumes that the gannet has enough energy to chase fish. However, the second scenario assumes that the gannet lacks the energy to chase fish [28]. In order to determine the energy level of gannets, we define a parameter called capture ability. Equation (17) is used to define the capture ability parameter. This equation consists of two controller parameters. Equations (19) and (20) are used to define values of controller parameters.

#### 2.4.2. Bat Algorithm (BA)

#### 2.4.3. Particle Swarm Optimization (PSO)

#### 2.4.4. Genetic Algorithm (GA)

#### 2.4.5. Binary Version of Optimization Algorithms

#### 2.5. Hybrid CLM Model for Predicting GWL

- The parameters of the CNN model and the names of the inputs are initialized as the initial population of the binary GOA. In the modeling process, it is important to determine the training and testing data. K-fold cross-validation (KCV) is one of the suitable methods for partitioning input data into training and testing datasets [37]. The KCV method evaluates model accuracy by dividing data into K folds. Training data are divided into K-1 folds. Finally, the model is run K times to calculate the average error function. The average error function value is used to evaluate model performance and determine the optimal number of folds [38].
- Each possible input scenario is encoded as a binary string. Each bit represents the presence (1) or absence (0) of a specific input variable. An initial population of potential solutions (binary strings) is generated. These strings represent various combinations of input variables. The binary string 11,111 indicates that all inputs are considered during modeling. The algorithm can identify the most influential input combinations based on the defined fitness function by exploring various combinations and evaluating their performance.
- The CNN model is executed by using training data.
- CNN is run in the testing phase if termination criteria are met; otherwise, CNN is connected to GOA. CNN parameters and input names are updated using optimization algorithms.
- LSTM models use CNN outputs as inputs. The LSTM model is run using training data. The LSTM parameters are defined as the initial population of optimizers. The LSTM model is run at the testing level if the termination criteria are met; otherwise, the GOA is connected to the LSTM mode.
- MLR uses the outputs of the LSTM model as inputs. The LSTM model extracts temporal features from time series data. Temporal features can capture relationships between different data points. Then, the extracted features are sent to the MLR model to predict GWLs. The MLR model produces final outputs.

#### 2.6. Quantitation of Uncertainty Values

#### 2.7. Determination of Contribution of Input Parameters and Models to Output Uncertainty

#### 2.8. Trend Analysis

## 3. Case Study

#### 3.1. Details of Case Study

- Root mean square error (RMSE)

- Mean absolute error (MAE)

- Nash Sutcliffe efficiency (NSE)

- Percent bias (PBIAS)

- Reliability index (REI)

- Resiliency index (RES)$$\mathrm{R}ES=\left[\begin{array}{l}100\%\leftarrow if\left(\mathrm{Re}liability=100\%\right)\\ \frac{{\displaystyle \sum _{i=1}^{N-1}{R}_{i}}}{N-{\displaystyle \sum _{i=1}^{N}{J}_{i}}}\times 100\end{array}\right]$$$${R}_{i}=\left[\begin{array}{l}1,if\left(RA{E}_{i}>\delta \left(and\right)RA{E}_{i+1}\le \delta \right)\\ 0\leftarrow else\end{array}\right]$$

- Width interval (WI)

- Prediction interval coverage probability (PICP)$$PICP=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\kappa}_{i}}$$$${\kappa}_{i}=\left[\begin{array}{l}1\leftarrow {y}_{i}\in \left[{L}_{i},{U}_{i}\right]\\ 0,\leftarrow {y}_{i}\notin \left[{L}_{i},{U}_{i}\right]\end{array}\right]$$

#### 3.2. Error Function for Evaluating the Performance of Optimization Algorithms in Choosing Inputs

## 4. Results

#### 4.1. Determination of Algorithm Parameters

#### 4.2. Determination of Number of Folds (K)

#### 4.3. Selection of the Best Optimization Algorithm for Choosing Inputs

^{72}− 1 input combinations. The GOA chose the input combination of GWL (t − 1), RELH (t − 1), WIS (t − 1), GWL (t − 2), RA (t − 1), and TEM (t − 1) as the optimal input scenario for predicting GWLs. Previous studies also reported that lagged climate data and GWLs had high correlation values with GWLs [45]. Seasonal and monthly patterns of climate data can affect GWLs [46].

#### 4.4. Evaluation of the Performance of Predictive Models in Predicting GWL

_{0}was rejected as the p-values were lower than 0.05.

## 5. Discussion

#### 5.1. Evaluation of the Accuracy of Models for Different Horizon Predictions

#### 5.2. The Contribution of Input and Parameter Uncertainty to the Output Uncertainty

#### 5.3. Memory Usage Percentage of Models

#### 5.4. Trend Defection

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**(

**a**): Assessment of the efficiency of models based on NSE and PBIAS values, (

**b**): scatterplots of models, and (

**c**): Taylor diagram for assessment of the precision of models.

**Figure 8.**(

**a**): The evaluation of the accuracy of models for predicting different horizons, and (

**b**): the percentage contribution of uncertainty resources to output uncertainty.

References | Results | Discussion |
---|---|---|

Sahoo and Jha [10] | Developed the MLR model to predict Groundwater Level (GWL). Rainfall, river stage, and temperature were used to predict GWL. They tested different input combinations for predicting GWL. They reported that the MLR model was a reliable tool for groundwater modeling | For predicting GWL, the paper used the original SVM and MLR models. However, the original versions of these models may not fully extract important features and accurately predict GWL. Furthermore, random input selection may not result in high accuracy. |

Ebrahimi and Rajaee [11] | Ebrahimi and Rajaee [11] coupled the wavelet method with the MLR, artificial neural network models (ANNs), and support vector machine models (SVMs) to predict GWL. They used the wavelet method to decompose time series into multiple subtime series. Compared to other wavelets, the Db5 wavelets performed better. The wavelet-MLR model improved the precision of the MLR model for predicting GW. | Wavelet preprocessed data points. Wavelet transforms can be used to extract valuable features from time series data. These features allow predictive models to capture both high-frequency and low-frequency components. |

Bahmani and Ouarda [12] | Bahmani and Ouarda [12] used the decision-tree model, the genetic programming (GEP) model, and the MLR model for predicting Groundwater Level (GWL). The Ensemble Empirical Mode Decomposition (EEMD) was used to preprocess the data. Wavelet-GEP performed better than other models. | Since the MLR model lacked advanced operators for analyzing nonlinear data, its accuracy was lower than that of the GEP and EEMD-GEP models. EEMD decomposes the original signal into IMFs that capture more detailed and local features. |

Poursaeid et al. [13] | Poursaeid et al. [13] used the MLR, the extreme learning machine model (ELM), and the SVM model to predict GWL. The study concluded that the ELM model performed better than the MLR model. | Due to the lack of robust optimizers, the model did not achieve high accuracy. To improve the performance of the ELM, SVM, and MLR models, robust optimizers were needed. |

Parameter | Maximum | Average | Minimum |
---|---|---|---|

(EVA) (mm) | 51 | 31 | 14 |

(RELH)% | 75 | 56 | 44 |

(RA) (mm) | 20 | 8 | 0 |

WIS (m/s) | 14.01 | 11.59 | 2.65 |

(TEM) °C | 41 | 25 | 7.4 |

GWL (m) | 1035.52 | 1033.281 | 1031.23 |

**Table 3.**(

**a**): Determination of optimal values of BA parameters, (

**b**): determination of optimal values of PSO parameters (

**c**): U95 and ASS values of different algorithms, and (

**d**): optimal values of model parameters.

(a) | |||||||

Maximum frequency | MAE value | Minimum frequency | MAE value | Maximum Loudness | MAE value | ||

3 | 0.14 | 0 | 0.15 | 0.5 | 0.15 | ||

5 | 0.12 | 1 | 0.12 | 0.6 | 0.12 | ||

7 | 0.09 | 2 | 0.10 | 0.7 | 0.09 | ||

9 | 0.15 | 3 | 0.14 | 0.8 | 0.12 | ||

(b) | |||||||

${\alpha}_{1}$ | MAE value | ${\alpha}_{\begin{array}{l}2\\ \end{array}}$ | MAE value | ||||

1.6 | 0.17 | 1.6 | 0.18 | ||||

1.8 | 0.15 | 1.8 | 0.16 | ||||

2.00 | 0.12 | 2.00 | 0.12 | ||||

2.2 | 0.14 | 2.2 | 0.15 | ||||

(c) | |||||||

Algorithm | U95 | ASS | Standard deviation | ||||

GOA | 3.45 | 0.04 | 0.32 | ||||

BA | 5.75 | 0.07 | 0.67 | ||||

PSO | 8.25 | 0.08 | 0.78 | ||||

GA | 9.12 | 0.12 | 0.82 | ||||

(d) | |||||||

Model | Optimal values of model parameters | ||||||

CNN | The number of convolution layers: 3, Kernel number of layer 1: 20; Kernel number of layer 2: 20, and Kernel number of layer 3: 15, batch size:30 | ||||||

LSTM | Number of hidden layers:200, The maximum epoch:50, Number of hidden neurons: 100 |

Models | CLM vs. LSM | CLM vs. CNL | CLM vs. LSM | CLM vs. CNM | CLM vs. LSTM | CLM vs. CNN | CLM vs. MLR |
---|---|---|---|---|---|---|---|

Assumptions | H_{0}CLM = LSM | H_{0}CLM = CNL | H_{0},CLM = LSM | H_{0},CLM = CNM | H_{0},CLM = LSTM | H_{0},CLM = CNN | H_{0},CLM = MLR |

p-value | <0.002 | <0.002 | <0.002 | <0.002 | <0.002 | <0.002 | <0.002 |

Winner: CLM | Winner: CLM | Winner: CLM | Winner: CLM | Winner: CLM | Winner: CLM | Winner: CLM |

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## Share and Cite

**MDPI and ACS Style**

Ehteram, M.; Banadkooki, F.B.
A Developed Multiple Linear Regression (MLR) Model for Monthly Groundwater Level Prediction. *Water* **2023**, *15*, 3940.
https://doi.org/10.3390/w15223940

**AMA Style**

Ehteram M, Banadkooki FB.
A Developed Multiple Linear Regression (MLR) Model for Monthly Groundwater Level Prediction. *Water*. 2023; 15(22):3940.
https://doi.org/10.3390/w15223940

**Chicago/Turabian Style**

Ehteram, Mohammad, and Fatemeh Barzegari Banadkooki.
2023. "A Developed Multiple Linear Regression (MLR) Model for Monthly Groundwater Level Prediction" *Water* 15, no. 22: 3940.
https://doi.org/10.3390/w15223940