# Incorporating Empirical Orthogonal Function Analysis into Machine Learning Models for Streamflow Prediction

^{*}

## Abstract

**:**

^{2}, RMSE, MAE, NSE and PBIAS were equal to 0.68, 9.40 m

^{3}/s, 5.18 m

^{3}/s, 0.68 and −0.03 for the daily streamflow in the Taolai River Watershed of the UHRB, respectively. Additionally, the LSTM method could provide physically based hydrological explanations of climate predicators in streamflow generation. Therefore, this study demonstrated the unique capability and functionality of incorporating EOF analysis into ML models for streamflow prediction, which could make better use of the readily available gridded climate data in hydrological simulations.

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Study Area

^{2}. The main stream of the Heihe River is approximately 821 km long [28]. The upstream HRB consists of four neighboring watersheds, including the Fengle River Watershed (FRW), Hongshuiba River Watershed (HRW), Taolai River Watershed (TRW) and Yingluoxia Watershed (YW). The elevations of the four watersheds range from 1681 m to 5541 m. The areas of the FRW, HRW, TRW and YW are 574 km

^{2}, 1580 km

^{2}, 6924 km

^{2}and 10,003 km

^{2}, respectively. The four watersheds lie on the northern margin of the Qilian Mountains, where the ecological system is characterized by snow cover, alpine meadows, evergreen needle-leaf forests and streamflow networks [29]. The rivers in the four watersheds are supplied by precipitation, glacier melting and groundwater discharge.

#### 2.2. Data Description

## 3. Methodology

#### 3.1. Empirical Orthogonal Function

#### 3.2. Machine Learning Models

#### 3.2.1. Support Vector Regression

#### 3.2.2. Multilayer Perceptron

#### 3.2.3. Long Short-Term Memory Network

#### 3.2.4. Gradient Boosting Regression Tree

#### 3.3. Integration of the EOF and ML Models

#### 3.4. Variable Importance Analysis in ML Models

#### 3.5. Performance Measurements

## 4. Results and Discussion

#### 4.1. Selection of Reliable Predictors

#### 4.2. Comparison of ML Model Performance

^{2}values.

#### 4.3. Role of EOF Analysis for Improving Streamflow Prediction

#### 4.4. Variable Importance in the Four ML Models

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The study area. The two top maps show the location of the study area in China. Four selected watersheds in the upper HRB are marked with different colors. The bottom map is a DEM map with hydrological stations and stream networks.

**Figure 2.**The spatial heterogeneity of the four hydrometeorological data in the study area: (

**a**) mean annual precipitation, (

**b**) mean daily net solar radiation, (

**c**) mean daily temperature and (

**d**) mean daily wind speed.

**Figure 3.**Schematic diagrams of the four selected ML models: (

**a**) the importance of the kernel function in the SVR model, (

**b**) the structure of the MLP and LSTM models used in this study, (

**c**) the difference between neurons and LSTM cells, and (

**d**) the principle of the GBRT.

**Figure 5.**Cumulative explained variance ratio (cumulative EVR) of the PCs for the four climatic datasets in (

**a**) YW, (

**b**) FRW, (

**c**) HRW and (

**d**) TRW. The grey dotted line refers to the threshold of 85%. PC_P, PC_R, PC_T and PC_W represent the PCs of the precipitation, net solar radiation, temperature and wind speed, respectively.

**Figure 6.**A comparison of the observations and streamflows simulated using (

**a**) GBRT, (

**b**) SVR, (

**c**) MLP and (

**d**) LSTM in the YW for the testing period.

**Figure 7.**Scatter diagrams of the observed and predicted daily streamflows used by four types of ML models in the YW for the testing period.

**Figure 8.**Comparison of model performance between the arithmetic method and EOF analysis for building streamflow prediction models in the (

**a**) YW, (

**b**) FRW, (

**c**) HRW and (

**d**) TRW.

**Figure 9.**Contribution of climatic elements to the daily streamflow in the LSTM model in 2012 for the YW: (

**a**) contribution of P, (

**b**) contribution of R, (

**c**) contribution of T and (

**d**) contribution of W.

**Table 1.**Multisource variables for the data-driven models: P (precipitation), R (net solar radiation), T (surface temperature), W (wind speed) and Q (streamflow).

Category | Variable (Unit) | Spatial Resolution | Timeframe | File Format |
---|---|---|---|---|

Predictors | P (mm) | 3 km × 3 km | 1990–2012 (daily) | NetCDF file |

R (W/m^{2}) | 3 km × 3 km | 1990–2012 (daily) | NetCDF file | |

T (°C) | 3 km × 3 km | 1990–2012 (daily) | NetCDF file | |

W (m/s) | 3 km × 3 km | 1990–2012 (daily) | NetCDF file | |

Responses | Q (m^{3}/s) | 4 stations | 1990–2012 (daily) | Excel file |

Watershed | Data | Grid Points | Number of PCs | Cumulative EVR |
---|---|---|---|---|

YW | P | 1103 | 4 | 87.5% |

R | 1103 | 1 | 93.7% | |

T | 1103 | 1 | 96.0% | |

W | 1103 | 5 | 85.2% | |

FRW | P | 66 | 1 | 86.6% |

R | 66 | 1 | 96.5% | |

T | 66 | 1 | 96.3% | |

W | 66 | 2 | 86.7% | |

HRW | P | 177 | 2 | 86.8% |

R | 177 | 1 | 94.5% | |

T | 177 | 1 | 96.4% | |

W | 177 | 3 | 87.3% | |

TRW | P | 767 | 3 | 85.5% |

R | 767 | 1 | 94.8% | |

T | 767 | 1 | 97.4% | |

W | 767 | 3 | 85.8% |

Watersheds | Metrics | Models | |||
---|---|---|---|---|---|

SVR | GBRT | MLP | LSTM | ||

YW | RMSE | 34.91 | 21.81 | 29.94 | 30.69 |

MAE | 24.47 | 8.35 | 17.64 | 19.72 | |

NSE | 0.57 | 0.83 | 0.68 | 0.66 | |

Pbias | 0.27 | −0.02 | 0.05 | 0.18 | |

FRW | RMSE | 3.43 | 1.64 | 2.73 | 2.45 |

MAE | 2.45 | 0.57 | 1.36 | 1.13 | |

NSE | 0.52 | 0.89 | 0.70 | 0.76 | |

Pbias | 0.45 | −0.02 | −0.16 | 0.03 | |

HRW | RMSE | 8.59 | 5.37 | 7.61 | 6.86 |

MAE | 6.22 | 2.00 | 3.13 | 2.88 | |

NSE | 0.59 | 0.84 | 0.68 | 0.74 | |

Pbias | 0.44 | −0.04 | −0.14 | −0.10 | |

TRW | RMSE | 10.45 | 4.64 | 8.54 | 8.28 |

MAE | 6.51 | 1.93 | 4.10 | 3.97 | |

NSE | 0.50 | 0.90 | 0.67 | 0.69 | |

Pbias | 0.17 | −0.01 | −0.10 | −0.07 |

**Table 4.**Daily performances of the four ML models in the selected watersheds for the testing period.

Watersheds | Metric | Models | |||
---|---|---|---|---|---|

SVR | GBRT | MLP | LSTM | ||

YW | RMSE | 31.57 | 29.72 | 29.87 | 28.92 |

MAE | 22.97 | 18.75 | 20.04 | 19.82 | |

NSE | 0.69 | 0.73 | 0.73 | 0.74 | |

Pbias | 0.09 | −0.13 | −0.07 | 0.06 | |

FRW | RMSE | 4.48 | 3.85 | 4.11 | 3.82 |

MAE | 2.73 | 1.25 | 1.56 | 1.33 | |

NSE | 0.47 | 0.60 | 0.55 | 0.62 | |

Pbias | 0.42 | −0.10 | −0.22 | −0.05 | |

HRW | RMSE | 9.52 | 7.48 | 9.51 | 8.59 |

MAE | 6.63 | 3.41 | 4.24 | 3.90 | |

NSE | 0.63 | 0.78 | 0.64 | 0.70 | |

Pbias | 0.28 | −0.13 | −0.24 | −0.23 | |

TRW | RMSE | 10.30 | 9.40 | 10.30 | 9.78 |

MAE | 6.54 | 5.18 | 5.52 | 5.59 | |

NSE | 0.62 | 0.68 | 0.62 | 0.66 | |

Pbias | 0.09 | −0.03 | −0.12 | −0.08 |

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## Share and Cite

**MDPI and ACS Style**

Wu, Y.; Chen, Y.; Tian, Y.
Incorporating Empirical Orthogonal Function Analysis into Machine Learning Models for Streamflow Prediction. *Sustainability* **2022**, *14*, 6612.
https://doi.org/10.3390/su14116612

**AMA Style**

Wu Y, Chen Y, Tian Y.
Incorporating Empirical Orthogonal Function Analysis into Machine Learning Models for Streamflow Prediction. *Sustainability*. 2022; 14(11):6612.
https://doi.org/10.3390/su14116612

**Chicago/Turabian Style**

Wu, Yajie, Yuan Chen, and Yong Tian.
2022. "Incorporating Empirical Orthogonal Function Analysis into Machine Learning Models for Streamflow Prediction" *Sustainability* 14, no. 11: 6612.
https://doi.org/10.3390/su14116612