# Improving Forecasting Accuracy of Multi-Scale Groundwater Level Fluctuations Using a Heterogeneous Ensemble of Machine Learning Algorithms

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## Abstract

**:**

## 1. Introduction

- Performance evaluation of the seven ML-based individual models to forecast multi-step ahead GWL fluctuations.
- Development of a heterogeneous ensemble of the GWL forecast models using the BMA approach and comparison of the performance of the ensemble with that of the standalone forecast models.

## 2. Materials and Methods

#### 2.1. Study Area and the Data

^{2}. The area falls in the extensive Gangetic floodplain, which has a typical climatic pattern with very cold winters (below 6 °C) and very dry and hot summers (up to 45 °C) [34]. It experiences little annual rainfall compared to other parts of the country. Groundwater recharge from rainfall is hindered by a thick clayey layer of around 18 m at the top surface.

#### 2.2. Machine Learning-Based Models

#### 2.2.1. Adaptive Neuro-Fuzzy Inference System (ANFIS)

#### 2.2.2. Bagged and Boosted RF

#### 2.2.3. Gaussian Process Regression (GPR)

#### 2.2.4. Bidirectional Long Short-Term Memory (Bi-LSTM) Network

- A sequence input layer, which matched the number of input variables or features.
- A Bi-LSTM layer, whose units corresponded to the number of hidden units.
- A fully connected layer, tailored to the number of output variables or response variables.
- Finally, a regression layer.

#### 2.2.5. Multivariate Adaptive Regression Spline (MARS)

#### 2.2.6. Support Vector Regression (SVR)

#### 2.3. Modeling Techniques

#### 2.3.1. Data Preprocessing

#### 2.3.2. Selection of Input Variables

#### 2.3.3. Standardization of Data

#### 2.3.4. Development of Individual Models

#### 2.3.5. Development of Ensemble Models

#### 2.4. Model Performance Evaluation

^{20}− index:

#### 2.5. Variable Importance

## 3. Results and Discussion

#### 3.1. Performance of the Individual Forecasting Models during Training and Validation

#### 3.2. Performance of the Standalone and Ensemble Models on the Independent Test Dataset

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Variable importance based on MRMR scores for one-, two-, and three-step ahead predictions at different observation wells.

**Figure 5.**Train and validation RMSE during the training and validation phases of model development: observation wells GT8134017 and GT8134020.

**Figure 6.**Train and validation RMSE during the training and validation phases of model development: observation wells GT8134021 and GT8134022.

**Figure 7.**Error of the models developed to forecast the weekly GWL of observation wells GT8134017, GT8134020, GT8134021, and GT8134022.

**Figure 8.**Performance of the models developed to forecast the weekly GWL of observation wells GT8134017, GT8134020, GT8134021, and GT8134022.

**Table 1.**Values of the statistical parameters computed on the GWL data (m) at the designated observation wells.

Observation Well | Mean | STD | Skewness | Kurtosis |
---|---|---|---|---|

GT8134017 | 6.796 | 2.797 | −0.172 | −0.446 |

GT8134020 | 7.735 | 2.683 | −0.043 | −0.596 |

GT8134021 | 6.612 | 2.555 | −0.457 | −0.535 |

GT8134022 | 6.534 | 2.274 | −0.218 | −0.557 |

Model | Parameters |
---|---|

ANFIS | Number of clusters: GT8134017-GWL (t + 1) = 6, GT3330001-GWL (t + 2) = 3, GT3330001-GWL (t + 3) = 3 GT8134020-GWL (t + 1) = 3, GT3330002-GWL (t + 2) = 2, GT3330002-GWL (t + 3) = 4 GT8134021-GWL (t + 1) = 6, GT3330020-GWL (t + 2) = 3, GT3330020-GWL (t + 3) = 5 GT813402-GWL (t + 1) = 5, GT3330020-GWL (t + 2) = 4, GT3330020-GWL (t + 3) = 3 Initial FIS: Fuzzy partition matrix exponent = 2 Maximum number of iterations = 500 Minimum improvement = 1 × 10 ^{−5}ANFIS: Maximum number of epochs: 500 Error goal = 0 Initial step size = 0.01 Step size decrease rate = 0.9 Step size increase rate = 1.1 |

Bagged RF | Number of variables to sample = all Predictor selection = interaction-curvature Method = bag Number of learning cycles = 200 Learn rate = 1 |

Boosted RF | Method = LSBoost Minimum number of parents = 10 Minimum number of leafs = 5 Maximum splits = 12 Number of learning cycles = 57 Learn rate = 0.1929 |

GPR | Basis function = Linear Kernel function = Rational Quadratic Fit method = Exact, predict method = Exact Beta = 0, Sigma = 0.4081 Optimizer = quasinewton |

MARS | Number of Basis functions at the forward pass = 100 Number of Basis functions at the backward pass = 50 Minimum number of observations between the knots = 3 No penalty is added to the variables to give equal priority to all input variables |

Bi-LSTM | Gradient decay factor = 0.9, Epsilon = 1 × 10^{−8}, Initial learn rate = 0.01Learn rate drop factor = 0.1, Learn rate drop period = 10, Gradient threshold = 1 L2 regularization = 1 × 10 ^{−4}, Gradient threshold method = l2norm,Maximum number of epochs = 1000, Mini batch size = 150 |

SVR | Kernel function = linear, Box constraint = 25.4335, Epsilon = 0.1021 Delta gradient tolerance = 0, Gap tolerance = 1 × 10 ^{−3}, Kernel scale = 7.4663Solver = SMO, Bias = 6.7549, Iteration limit = 1,000,000 |

**Table 3.**Performance of the models in forecasting weekly groundwater levels of GT8134017, GT8134020, GT8134021, and GT8134022.

Model | GWL (t + 1) | GWL (t + 2) | GWL (t + 3) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | MAE | NS | MBE | RMSE | MAE | NS | MBE | RMSE | MAE | NS | MBE | |

GT8134017 | ||||||||||||

ANFIS | 1.85 | 0.93 | 0.40 | −0.25 | 1.57 | 0.95 | 0.57 | −0.29 | 1.82 | 1.16 | 0.43 | −0.44 |

BaggedRF | 1.59 | 1.01 | 0.56 | −0.86 | 1.83 | 1.33 | 0.41 | −1.17 | 2.03 | 1.56 | 0.28 | −1.41 |

BoostedRF | 1.67 | 1.12 | 0.51 | −0.92 | 2.20 | 1.63 | 0.16 | −1.48 | 2.09 | 1.59 | 0.24 | −1.40 |

GPR | 1.29 | 0.67 | 0.71 | −0.16 | 2.16 | 1.48 | 0.18 | −1.14 | 3.43 | 2.60 | −1.04 | −2.42 |

BiLSTM | 1.38 | 0.80 | 0.67 | −0.67 | 1.49 | 0.98 | 0.61 | −0.68 | 1.47 | 0.98 | 0.63 | −0.59 |

MARS | 1.95 | 1.10 | 0.33 | −0.67 | 8.65 | 5.40 | −12.06 | −5.23 | 8.13 | 5.49 | −10.46 | −5.22 |

SVR | 1.42 | 0.70 | 0.64 | −0.17 | 1.46 | 0.88 | 0.63 | −0.33 | 1.85 | 1.31 | 0.40 | −0.80 |

BMA | 0.87 | 0.50 | 0.87 | 0.00 | 1.13 | 0.68 | 0.78 | 0.00 | 1.23 | 0.77 | 0.74 | 0.00 |

GT8134020 | ||||||||||||

ANFIS | 0.76 | 0.45 | 0.52 | −0.10 | 0.80 | 0.51 | 0.47 | −0.15 | 0.98 | 0.67 | 0.20 | −0.17 |

BaggedRF | 1.17 | 0.89 | −0.15 | −0.84 | 1.25 | 1.00 | −0.31 | −0.96 | 1.42 | 1.19 | −0.70 | −1.16 |

BoostedRF | 1.25 | 0.96 | −0.30 | −0.94 | 1.39 | 1.15 | −0.61 | −1.13 | 1.63 | 1.40 | −1.23 | −1.39 |

GPR | 0.67 | 0.41 | 0.63 | −0.21 | 0.77 | 0.54 | 0.50 | −0.36 | 2.08 | 1.86 | −2.62 | −1.86 |

BiLSTM | 1.46 | 1.16 | −0.77 | −0.98 | 1.21 | 0.90 | −0.23 | −0.83 | 1.40 | 1.08 | −0.64 | −1.00 |

MARS | 1.32 | 0.68 | −0.46 | −0.01 | 1.12 | 0.67 | −0.05 | −0.17 | 1.61 | 0.92 | −1.16 | 0.06 |

SVR | 0.71 | 0.41 | 0.58 | −0.18 | 0.84 | 0.58 | 0.41 | −0.39 | 1.07 | 0.81 | 0.04 | −0.63 |

BMA | 0.59 | 0.33 | 0.71 | 0.00 | 0.64 | 0.38 | 0.66 | 0.00 | 0.71 | 0.47 | 0.58 | 0.00 |

GT8134021 | ||||||||||||

ANFIS | 0.85 | 0.60 | 0.79 | −0.35 | 1.15 | 0.88 | 0.62 | −0.65 | 1.30 | 0.97 | 0.52 | −0.59 |

BaggedRF | 1.10 | 0.80 | 0.66 | −0.69 | 1.47 | 1.15 | 0.38 | −1.03 | 1.85 | 1.51 | 0.02 | −1.39 |

BoostedRF | 1.07 | 0.79 | 0.68 | −0.65 | 1.53 | 1.19 | 0.33 | −1.05 | 1.83 | 1.47 | 0.05 | −1.31 |

GPR | 0.78 | 0.54 | 0.83 | −0.29 | 1.11 | 0.85 | 0.65 | −0.66 | 1.42 | 1.13 | 0.43 | −0.96 |

BiLSTM | 0.58 | 0.40 | 0.91 | −0.23 | 1.02 | 0.75 | 0.70 | −0.29 | 1.08 | 0.84 | 0.67 | −0.31 |

MARS | 0.83 | 0.60 | 0.80 | −0.31 | 1.22 | 0.96 | 0.58 | −0.39 | 1.64 | 1.26 | 0.24 | −0.60 |

SVR | 0.78 | 0.51 | 0.82 | −0.12 | 0.98 | 0.73 | 0.73 | −0.46 | 1.28 | 1.02 | 0.54 | −0.82 |

BMA | 0.44 | 0.27 | 0.95 | 0.00 | 0.75 | 0.50 | 0.84 | 0.00 | 0.83 | 0.61 | 0.81 | 0.00 |

GT8134022 | ||||||||||||

ANFIS | 0.63 | 0.45 | 0.92 | −0.23 | 1.02 | 0.73 | 0.81 | −0.42 | 1.35 | 1.07 | 0.66 | −0.61 |

BaggedRF | 0.63 | 0.44 | 0.93 | −0.20 | 1.01 | 0.72 | 0.81 | −0.44 | 1.39 | 1.08 | 0.65 | −0.83 |

BoostedRF | 0.70 | 0.48 | 0.91 | −0.26 | 1.07 | 0.78 | 0.79 | −0.55 | 1.57 | 1.19 | 0.55 | −0.95 |

GPR | 0.82 | 0.63 | 0.87 | −0.49 | 1.41 | 1.15 | 0.63 | −1.00 | 1.70 | 1.37 | 0.47 | −1.20 |

BiLSTM | 0.34 | 0.22 | 0.98 | −0.17 | 0.70 | 0.53 | 0.91 | −0.34 | 0.82 | 0.64 | 0.88 | −0.23 |

MARS | 0.69 | 0.49 | 0.91 | −0.14 | 1.10 | 0.79 | 0.78 | −0.28 | 1.40 | 1.08 | 0.64 | −0.53 |

SVR | 0.62 | 0.45 | 0.93 | −0.23 | 0.99 | 0.80 | 0.82 | −0.52 | 1.26 | 1.02 | 0.71 | −0.66 |

BMA | 0.28 | 0.18 | 0.98 | 0.00 | 0.55 | 0.38 | 0.94 | 0.00 | 0.77 | 0.56 | 0.89 | 0.00 |

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## Share and Cite

**MDPI and ACS Style**

Roy, D.K.; Munmun, T.H.; Paul, C.R.; Haque, M.P.; Al-Ansari, N.; Mattar, M.A.
Improving Forecasting Accuracy of Multi-Scale Groundwater Level Fluctuations Using a Heterogeneous Ensemble of Machine Learning Algorithms. *Water* **2023**, *15*, 3624.
https://doi.org/10.3390/w15203624

**AMA Style**

Roy DK, Munmun TH, Paul CR, Haque MP, Al-Ansari N, Mattar MA.
Improving Forecasting Accuracy of Multi-Scale Groundwater Level Fluctuations Using a Heterogeneous Ensemble of Machine Learning Algorithms. *Water*. 2023; 15(20):3624.
https://doi.org/10.3390/w15203624

**Chicago/Turabian Style**

Roy, Dilip Kumar, Tasnia Hossain Munmun, Chitra Rani Paul, Mohamed Panjarul Haque, Nadhir Al-Ansari, and Mohamed A. Mattar.
2023. "Improving Forecasting Accuracy of Multi-Scale Groundwater Level Fluctuations Using a Heterogeneous Ensemble of Machine Learning Algorithms" *Water* 15, no. 20: 3624.
https://doi.org/10.3390/w15203624