# Machine Learning Framework with Feature Importance Interpretation for Discharge Estimation: A Case Study in Huitanggou Sluice Hydrological Station, China

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Study Area and Data Sources

_{u}, Z

_{d}, e, n, B

_{q}, and A

_{q}are the upstream stage, downstream stage, sluice gate opening height, sluice gate opening number, discharge measurement cross-section width, and discharge measurement cross-section area, respectively. Among them, the measurement position of Z

_{u}was 120 m away from the upstream of the sluice, and the measurement position of Z

_{d}was 380 m away from the downstream of the sluice. In total, 1281 sets of Z

_{u}, Z

_{d}, e, n, B

_{q}, A

_{q}, and measured discharge (Q

_{m}) data covered the period from 2008 to 2022. The dataset was divided into the training set and the testing set at a ratio of 7:3. The measured data were derived from the Suzhou Hydrology and Water Resources Bureau of Anhui Province. The descriptive statistics of these variables are summarized in Table 1. The variation amplitude of Q

_{m}and A

_{q}was higher than that of Z

_{u}and Z

_{d}. The minimum and maximum values of Z

_{u}, Z

_{d}, A

_{q}, and Q

_{m}in the testing set fell within the range in the training set. This shows that the training set with a wider data spectrum can guarantee a robust model for estimating discharge in a wider range, and can overcome the problem of estimating extreme discharge values.

#### 2.2. Methods

#### 2.2.1. Exploratory Data Analysis (EDA)

#### 2.2.2. Conventional Method: Stage–Discharge Rating Curve (SDRC)

#### 2.2.3. Ensemble Learning Random Forest (ELRF) Algorithm

#### 2.2.4. Ensemble Learning Gradient Boosting Decision Tree (ELGBDT) Algorithm

_{1}) was trained based on W

_{1}, and the weight of BL

_{1}was updated based on its learning performance. The poorly performing data from BL

_{1}were recorded and their weights were set higher in W

_{2}to make them more important to BL

_{2}. (2) The BL

_{2}was trained based on the knowledge from the previous step, and this iteration continued until the number of learning machines reached the set number. In this study, the number of learning machines was set to 100. (3) Based on ensemble learning, by integrating multiple learning machines into a machine learning algorithm, we obtained a machine learning algorithm with greater generalization ability.

#### 2.2.5. SHAP Algorithm

#### 2.2.6. Bayesian Optimization Algorithm

#### 2.3. Performance Evaluation Methods

^{2}), root mean squared error (RMSE), coefficient of correlation (CC), and relative error (RE) (Equations (11)–(14)) were used as the evaluation standards. They, respectively, represent the fitting degree, relative deviation degree, correlation degree between estimated values and measured values, and the stability of models.

## 3. Results

#### 3.1. Data Exploration and Analysis

_{u}and Q

_{m}is the weakest, and the correlation degree of n and Q

_{m}is the strongest. Except for Z

_{u}, the absolute value of SRCC between the variables and Q

_{m}is greater than 0.5, which indicates that the selected variables have a high correlation with Q

_{m}. Therefore, the input variables for the ELRF and ELGBDT models include Z

_{u}, Z

_{d}, e, n, B

_{q}, and A

_{q}.

#### 3.2. Conventional SDRC Fitting

#### 3.3. Model Estimation

#### 3.3.1. Bayesian Hyperparameter Optimization

#### 3.3.2. Model Evaluation

^{2}, RMSE, and CC values of the ELRF are similar to those of the ELGBDT model. This indicates that the ELRF and ELGBDT models estimate discharge with almost identical accuracy. The R

^{2}of the ensemble ML model increased by 13.86% compared to that of the SDRC model. This shows that the fitting degree of the ensemble ML model is high. The RMSE of the ensemble ML model is 41.30% lower than that of the SDRC model. This reveals that the stability of the ensemble ML model has strong. The CC of the ensemble ML model increased by 2.53% compared with the SDRC model. It can be seen that the correlation degree of the ensemble ML model is high. Therefore, the ensemble ML model has a stronger learning ability.

#### 3.4. Model Feature Importance Interpretation

_{q}, from greater to lesser significance, respectively (Figure 12a). In the ELGBDT model, the influential variables from strong to weak are n, e, Z

_{u}, A

_{q}, Z

_{d}, and B

_{q}, respectively (Figure 12b).

_{u}variables, the values appear more heterogeneous due to their continuous nature (more purple than red and blue), unlike n, which is a dichotomous variable with its color polarization (only red and blue).

^{3}/s. For the y axis, the variables are ranked from highest to lowest according to their impact on the estimation of the model. If the discharge increases the average value of the final estimation of the two models, the line is red. Conversely, if the discharge decreases the average value of the final estimation, the line is blue.

_{d}and the relationship with n of the two models are similar. The blue points are those with lower n values, while the red points are those with higher n values. In these plots, it can be explained that the existence of Z

_{d}has a smaller impact on the model when the n is low. Otherwise, at high values of n, the effect of the absence of Z

_{d}in the model is estimated to increase.

## 4. Discussion

^{2}, RMSE, and CC values of the ELRF are similar to those of the ELGBDT model in the submerged orifice flow state. This demonstrates that bagging and boosting estimate discharge with almost the same accuracy. Therefore, these two algorithms can be selected during modeling.

^{2}, RMSE, CC, and mean RE values of the ELRF are similar to those of the ELGBDT model. This demonstrates that the ELRF model performs equally well as the ELGBDT model. Compared with Table 2, the R

^{2}and CC values of the two models are superior to those of the submerged orifice flow state. The R

^{2}of the two models is 5.27% and 5.59% higher than that of the submerged orifice flow state, respectively. The CC of the two models is 1.13% and 1.76% higher than that of the submerged orifice flow state, respectively. This is because the submerged weir flow data are added to the sample set so that the dataset is increased and distributed evenly. It can also be seen that the ensemble ML model can more effectively decrease the error and average deviation degree of the model, and enhance the generalization ability of the model.

^{2}, RMSE, CC, and mean RE of the ELGBDT model are superior to those of the SVM and KNN models. This shows that the estimation accuracy of the ELGBDT model is better than that of the SVM and KNN models. Compared with Table 4, the R

^{2}, RMSE, CC, and mean RE of the ensemble ML model are superior to those of the SVM and KNN models. This reveals that the stability of the ensemble ML model is superior to that of the SVM and KNN models.

^{2}values of the SDRC, ELRF, and ELGBDT without data cleaning are 0.657, 0.701, and 0.703, respectively. The CC values of the SDRC, ELRF, and ELGBDT without data cleaning are 0.852, 0.865, and 0.877, respectively. The R

^{2}and CC values of the three models with data cleaning are higher than those without data cleaning. Meanwhile, the results for the RMSE values are similar. Therefore, data cleaning can remove outliers and provide a reference for modeling.

## 5. Conclusions

- (1)
- The performance of the model is improved by Bayesian optimization. ELRF and ELGBDT models estimate discharge with almost identical accuracy. The accuracy of the ensemble ML model is superior to that of the SDRC method in the submerged orifice flow state. The R
^{2}, RMSE, and CC values of the ensemble ML model are 0.912, 19.578, and 0.971, respectively. The RE distribution parameter and violin plot of the ensemble ML model are the best, and this model has the strongest generalization ability. - (2)
- The SHAP method reveals the interactions between all variables and how this relationship is reflected in the model. In the ensemble ML model, the sluice gate opening number (n) is the strongest influential variable, and the discharge measurement cross-section width (B
_{q}) is the weakest influential variable. The estimated average discharge of the ensemble ML model is less than 100 m^{3}/s. The variables can be appropriately analyzed, resulting in a better model with higher performance indicators. - (3)
- Compared with the SDRC method and single ML model, the ensemble ML model has higher accuracy and better stability, which indicates that the ensemble ML model can express more complex nonlinear transformations accurately and effectively.
- (4)
- The accuracy of the ensemble ML model is the highest without considering the flow state. The R
^{2}, RMSE, and CC values of the ensemble ML model are 0.963, 31.268, and 0.984, which indicates that the ensemble ML model has a strong adaptive ability.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**The boosting algorithm process. ${f}_{i}\left(x\right)$ is the estimated result in the $i$ base learner, and ${W}_{i}$ is the weight in the $i$ base learner.

**Figure 5.**Diagrammatic representation of SHAP. The variables that increase the estimation of the model the most are shown in red boxes and those that reduce the estimation of the model the most are shown in blue boxes.

**Figure 6.**SRCC heatmap of the six variables and measured discharge. These numbers are SRCC values of the six variables and measured discharge.

**Figure 10.**Distribution plot of estimated discharge of three models in the testing set. The blue dots are the estimated discharge, and the dots on the red line indicate that the estimated discharge is equal to the measured discharge.

**Figure 11.**The violin plots of three models. The red dots are the relative error (RE), the outer curve is the probability density curve (PDC), and the white dot is the median.

**Figure 12.**Absolute summary plot of two models in the testing set. The average absolute value of the SHAP values for each variable is used to obtain a bar chart as a function of the contribution of each variable to the estimation of the model. The y axis is the variable used in the study.

**Figure 13.**Summary plot of two models in the testing set. The horizontal axis indicates the SHAP values of each variable used by the model to estimate the discharge. The left vertical axis represents the influence of variables in the model.

**Figure 14.**Decision plot of two models in the testing set. The y axis is the variable used in the study. The straight vertical line represents the base value of the ELRF and ELGBDT models, and the colored lines are the estimated values. Starting at the bottom of the plot, the estimated line indicates how the SHAP values accumulate from the base value to the final model score at the top of the plot.

**Figure 15.**Dependence plot for SHAP values of Z

_{d}and the relationship with n of two models in the testing set. The vertical axis shows the SHAP value, while the horizontal axis represents the actual value of the variable. In addition, each point in the plot is indicated by a color palette on the right-hand side of the plot, which indicates the scale of the value of the second variable at each point.

Dataset | Number of Cases | Variable | Minimum | Maximum | Mean | Median | Standard Deviation |
---|---|---|---|---|---|---|---|

Training | 897 | Z_{u} (m) | 17.07 | 23.06 | 21.13 | 21.31 | 0.81 |

Z_{d} (m) | 16.77 | 22.97 | 18.26 | 18.16 | 0.87 | ||

e (m) | 0.10 | 6.86 | 0.52 | 0.30 | 0.89 | ||

n | 1 | 7 | 3 | 2 | 2 | ||

B_{q} (m) | 97.5 | 158 | 118.1 | 119 | 7.1 | ||

A_{q} (m^{2}) | 28.1 | 856 | 200.7 | 188 | 109.1 | ||

Q_{m} (m^{3}/s) | 2.49 | 875 | 66.9 | 36.6 | 107.4 | ||

Testing | 384 | Z_{u} (m) | 18.43 | 22.96 | 21.15 | 21.27 | 0.69 |

Z_{d} (m) | 17.08 | 22.92 | 18.52 | 18.18 | 1.20 | ||

e (m) | 0.10 | 6.76 | 0.91 | 0.30 | 1.55 | ||

n | 1 | 7 | 3 | 2 | 2 | ||

B_{q} (m) | 89.1 | 156 | 120.8 | 121 | 8.9 | ||

A_{q} (m^{2}) | 63.5 | 818 | 241.9 | 195 | 153.3 | ||

Q_{m} (m^{3}/s) | 7.75 | 767 | 116.9 | 45.6 | 163.3 |

Model | R^{2} | RMSE | CC |
---|---|---|---|

SDRC | 0.801 | 33.354 | 0.947 |

ELRF | 0.911 | 19.578 | 0.971 |

ELGBDT | 0.912 | 19.955 | 0.967 |

Model | CIUL (%) | UQ (%) | Median (%) | LQ (%) | CILL (%) |
---|---|---|---|---|---|

SDRC | 52.90 | 25.80 | 14.02 | 7.74 | 0 |

ELRF | 50.07 | 24.54 | 13.50 | 7.53 | 0 |

ELGBDT | 49.38 | 24.25 | 14.81 | 7.49 | 0 |

Model | R^{2} | RMSE | CC | Mean RE |
---|---|---|---|---|

ELRF | 0.959 | 31.451 | 0.982 | 0.174 |

ELGBDT | 0.963 | 31.268 | 0.984 | 0.173 |

Model | R^{2} | RMSE | CC | Mean RE |
---|---|---|---|---|

SVM | 0.928 | 42.409 | 0.966 | 0.217 |

KNN | 0.943 | 38.284 | 0.973 | 0.195 |

ELGBDT | 0.963 | 31.268 | 0.984 | 0.173 |

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## Share and Cite

**MDPI and ACS Style**

He, S.; Niu, G.; Sang, X.; Sun, X.; Yin, J.; Chen, H.
Machine Learning Framework with Feature Importance Interpretation for Discharge Estimation: A Case Study in Huitanggou Sluice Hydrological Station, China. *Water* **2023**, *15*, 1923.
https://doi.org/10.3390/w15101923

**AMA Style**

He S, Niu G, Sang X, Sun X, Yin J, Chen H.
Machine Learning Framework with Feature Importance Interpretation for Discharge Estimation: A Case Study in Huitanggou Sluice Hydrological Station, China. *Water*. 2023; 15(10):1923.
https://doi.org/10.3390/w15101923

**Chicago/Turabian Style**

He, Sheng, Geng Niu, Xuefeng Sang, Xiaozhong Sun, Junxian Yin, and Heting Chen.
2023. "Machine Learning Framework with Feature Importance Interpretation for Discharge Estimation: A Case Study in Huitanggou Sluice Hydrological Station, China" *Water* 15, no. 10: 1923.
https://doi.org/10.3390/w15101923