# A Novel Daily Runoff Probability Density Prediction Model Based on Simplified Minimal Gated Memory–Non-Crossing Quantile Regression and Kernel Density Estimation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}< τ

_{2}< 1, the τ

_{1}th quantile is greater than the τ

_{2}th quantile. Quantile crossing seriously affects the reliability and prediction accuracy of the predicted quantiles. Therefore, in order to improve the reliability of QR, this paper proposes a new non-crossing quantile regression (NCQR) approach that can avoid the occurrence of quantile crossing.

- (1)
- This article proposes NCQR, which can avoid quantile crossing in predicted quantiles.
- (2)
- In order to reduce training time and improve model accuracy, this article simplifies the model structure of MGM and proposes SMGM.
- (3)
- The combination model based on SMGM and NCQR can efficiently generate reliable quantiles, which KDE converts into continuous PDFs.
- (4)
- All models in this article are compared from multiple perspectives using multiple evaluation metrics in three daily runoff datasets. The experimental results show that the model can efficiently obtain reliable and accurate probability density prediction results.

## 2. Methods

#### 2.1. Simplified Minimal Gated Memory Network

#### 2.2. Novel Non-Crossing Quantile Regression

#### 2.3. Kernel Density Estimation

#### 2.4. Maximal Information Coefficient

#### 2.5. Framework of the Proposed Combined Model

## 3. Model Evaluation Metrics

#### 3.1. Evaluation Metric of Point Prediction

#### 3.2. Evaluation Metric of Interval Prediction

#### 3.3. Quantifying Indicators of Quantile Crossing Degree

#### 3.4. Evaluation Metric of Probability Density Prediction

## 4. Case Study

#### 4.1. Study Area and Data

^{2}. The upper reaches of the Yangtze River have considerable falls, deep gorges, numerous tributaries and abundant precipitation. The average annual precipitation is about 1100 mm. The average annual water volume at the outlet of Yichang station is 451 billion m

^{3}, accounting for about 50% of the water volume of the Yangtze River flowing into the sea.

#### 4.2. Experimental Design and Parameter Settings

#### 4.3. Experimental Results and Comparative Analysis

#### 4.3.1. Task I: Evaluation of Probability Models in the Combined Model

^{3}/s and 7.29%, respectively. The RMSE and MAPE of GRU-GPR are 1023.14 m

^{3}/s and 6.35%, which are 9.67% and 12.9% higher than those of GRU-DeepAR, respectively. The RMSE and MAPE of GRU-QR are 955.81 m

^{3}/s and 6.06%, respectively, and the RMSE and MAPE of GRU-NCQR are 949.98 m

^{3}/s and 5.86%, respectively. The experimental results indicate that GRU-NCQR has the smallest RMSR and MAPE, which means that GRU-NCQR has the best point prediction accuracy. In dataset 2 and dataset 3, the same conclusion can be obtained.

_{90%}values of GRU-DeepAR, GRU-GPR, GRU-QR and GRU-NCQR are 0.9406, 0.9461, 0.9309 and 0.9448, respectively. The PICP

_{80%}values of GRU-DeepAR, GRU-GPR, GRU-QR and GRU-NCQR are 0.9199, 0.8894, 0.8287 and 0.8240, respectively. The PICP

_{70%}values of GRU-DeepAR, GRU-GPR, GRU-QR and GRU-NCQR are 0.8978, 0.7776, 0.7693 and 0.7127, respectively.

_{90%}, PINAW

_{80%}and PINAW

_{70%}of GRU-GPR are 0.1387, 0.1081 and 0.0874, respectively, which are the maximum values among all compared models, indicating that GRU-GPR has the largest prediction interval. The PINAW

_{90%}, PINAW

_{80%}and PINAW

_{70%}of GRU-GPR are 0.1000, 0.0670 and 0.0631, respectively. Compared with GRU-DeepAR and GRU-GPR, GRU-QR reduces PINAW

_{90%}by 39.64% and 13.27%, PINAW

_{80%}by 53.6% and 25.1% and PINAW

_{70%}by 56.67% and 39.94%, respectively. The results indicate that when predicting an unknown probability distribution, GPR and DeepAR as parametric methods provide a more conservative prediction interval compared to QR as a non-parametric method. That is, at the same confidence level, the prediction intervals of GRU-DeepAR and GRU-GPR are wider than that of GRU-QR. The PINAW

_{90%}, PINAW8

_{0%}and PINAW

_{70%}values of GRU-NCQR are 0.0801, 0.0379 and 0.0369, respectively, and the width of the interval prediction is smaller than those of the other comparison models in this task. In dataset 2 and dataset 3, GRU-NCQR also achieved the narrowest predicted interval and was compared with other models of Task I.

#### 4.3.2. Task II: Evaluation of RNN Models with NCQR-Based Models

^{3}/s, representing reductions of 1.73% and 3.50%, respectively, compared with MGM-NCQR and GRU-NCQR. The MAPE of SMGM-NCQR is 4.87%, representing reductions of 8.11% and 16.89%, respectively, compared with MGM-NCQR and GRU-NCQR. This shows that the point prediction accuracy of SMGM-NCQR is higher than those of MGM-NCQR and GRU-NCQR because SMGM-NCQR achieves the smallest RMSE and smallest MAPE in Task II. The same conclusion can be obtained in dataset 2 and dataset 3. This indicates that compared to MGM and GRU, SMGM not only has a simpler model structure but can also better extract effective information from model inputs. Figure 6 shows the point prediction results of all models in Task II for the test sets of three datasets. In order to compare the point prediction results of different models more clearly, Figure 7 shows the point prediction results of all models in Task II for the first flood season of the three test sets, which is from June 10th to September 10th in the first year of each test set. From Figure 6 and Figure 7, it can be observed that the severity of the lag phenomenon in SMGM-NCQR is lower than those of other comparative models, and the predicted runoff of SMGM-NCQR is closer to the actual observed runoff. These indicate that SMGM-NCQR has better performance in runoff point prediction.

_{90%}of SMGM-NCQR is 0.0660, which is a reduction of 11.76% compared with MGM-NCQR and a reduction of 17.60% compared with GRU-NCQR. This shows that the predicted interval of SMGM-NCQR is narrower than those of MGM-NCQR and GRU-NCQR. In dataset 2, the PINAW

_{90%}, PINAW

_{80%}and PINAW

_{70%}of SMGM-NCQR are 0.0601, 0.0386 and 0.0601, respectively; the PINAW

_{90%}, PINAW

_{80%}and PINAW

_{70%}of MGM-NCQR are 0.0638, 0.0400 and 0.0298, respectively; and the PINAW

_{90%}, PINAW

_{80%}and PINAW

_{70%}of GRU-NCQR are 0.0677, 0.0417 and 0.0322, respectively, which shows that SMGM-NCQR has achieved the smallest PINAW in interval prediction at the three confidence levels. In dataset 3, similar conclusions can be obtained. The SMGM-NCQR achieves the narrowest interval width in the interval prediction in all three datasets while meeting the corresponding confidence level. This indicates that, compared with MGM-NCQR and GRU-NCQR, the prediction interval of SMGM-NCQR can provide more information to decision-makers at the same confidence level.

#### 4.3.3. Task III: Displaying the Probability Density Curve

## 5. Conclusions and Discussion

- (1)
- The NCQR proposed in this article avoids the common quantile crossing observed in QR-based models. At the same time, the prediction performance of the model based on NCQR is superior to that of models based on other probabilistic models.
- (2)
- Among RNN models combined with NCQR, SMGM achieves the best predictive performance with the least training time compared to MGM and GRU. This indicates that SMGM not only has a simpler model structure but also has a better ability to extract effective information from model inputs compared with MGM and GRU.
- (3)
- The new model based on SMGM-NCQR and KDE proposed in this article can efficiently obtain reliable and accurate probability density predictions of future daily runoff. While providing high-precision point predictions, it comprehensively quantifies the uncertainty of predictions, which can provide rich information for decision-makers in water conservancy systems.

- (1)
- The input of the proposed SMGM-NCQR only includes historical runoff and does not consider other factors such as precipitation, daily maximum and minimum temperature. The probability density prediction model for daily runoff considering meteorological data is one of our future research directions.
- (2)
- The parameters of SMGM-NCQR are calibrated based on the relationship between historical runoff and target runoff, ignoring the formation process of runoff. Although SMGM-NCQR has good probability density prediction performance, it is physically unknown and lacks interpretability. Improving the interpretability of SMGM-NCQR while maintaining high prediction accuracy is also one of our future research directions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Point prediction results of four models in Task II during the flood season of three test sets.

**Figure 8.**Predicted intervals of SMGM-NCQR with different confidence levels during the flood season of the three datasets.

Dataset | Station | Time | Mean | Maximum | Minimum | Standard Deviation |
---|---|---|---|---|---|---|

Dataset 1 | Zhutuo | 1 January 2000–31 December 2007 | 8283.83 | 37,400 | 2180 | 6325.06 |

Dataset 2 | Yichang | 1 January 1996–31 December 2003 | 13,781.37 | 61,700 | 2950 | 11,377.50 |

Dataset 3 | Pingshan | 1 January 2003–31 December 2010 | 4516.427 | 20,800 | 1180 | 3612.90 |

Dataset | y_{t−1} | y_{t−2} | y_{t−3} | y_{t−4} | y_{t−5} | y_{t−6} | y_{t−7} | y_{t−8} | y_{t−9} | y_{t−10} |
---|---|---|---|---|---|---|---|---|---|---|

Dataset 1 | 0.969 | 0.940 | 0.909 | 0.887 | 0.874 | 0.861 | 0.852 | 0.841 | 0.827 | 0.815 |

Dataset 2 | 0.977 | 0.937 | 0.909 | 0.881 | 0.866 | 0.842 | 0.818 | 0.799 | 0.777 | 0.752 |

Dataset 3 | 0.960 | 0.931 | 0.905 | 0.884 | 0.870 | 0.855 | 0.841 | 0.829 | 0.822 | 0.819 |

Model | Parameter | Value |
---|---|---|

GRU-GPR | number of GRU layer nodes | 32 |

number of GRU layers | 4 | |

GPR kernel function | Rational quadratic kernel | |

number of output layer nodes | 2 | |

GRU-DeepAR | number of GRU layer nodes | 32 |

number of GRU layers | 4 | |

number of output layer nodes | 2 | |

GRU-QR | number of GRU layer nodes | 32 |

number of GRU layers | 4 | |

number of output layer nodes | 19 | |

GRU-NCQR | number of GRU layer nodes | 32 |

number of GRU layers | 4 | |

number of output layer nodes | 20 | |

MGM-NCQR | number of MGM layer nodes | 32 |

number of MGM layers | 4 | |

number of output layer nodes | 20 | |

SMGM-NCQR | number of SMGM layer nodes | 32 |

number of SMGM layers | 4 | |

number of output layer nodes | 20 | |

KDE | K-fold cross-validation in grid search for KDE bandwidth | 5 |

bandwidth range for KDE in grid search | (400, 450, 1) |

Models | Dataset 1 | Dataset 2 | Dataset 3 | |||
---|---|---|---|---|---|---|

RMSE (m ^{3}/s) | MAPE (%) | RMSE (m ^{3}/s) | MAPE (%) | RMSE (m ^{3}/s) | MAPE (%) | |

GRU-GPR | 1023.14 | 6.35 | 1460.83 | 5.68 | 381.39 | 5.41 |

GRU-DeepAR | 1132.67 | 7.29 | 1536.66 | 6.88 | 412.01 | 5.93 |

GRU-QR | 955.81 | 6.06 | 1411.95 | 5.20 | 357.34 | 5.24 |

GRU-NCQR | 949.98 | 5.86 | 1363.40 | 4.96 | 352.01 | 5.07 |

Models | Metrics | Dataset 1 | Dataset 2 | Dataset 3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

90% | 80% | 70% | 90% | 80% | 70% | 90% | 80% | 70% | ||

GRU-GPR | PICP | 0.9406 | 0.9199 | 0.8978 | 0.9504 | 0.0872 | 0.8829 | 0.9655 | 0.9241 | 0.8966 |

PINAW | 0.1387 | 0.1081 | 0.0874 | 0.1119 | 0.0872 | 0.0705 | 0.1074 | 0.0780 | 0.0631 | |

CWC | 0.1387 | 0.1081 | 0.0874 | 0.1119 | 0.0872 | 0.0705 | 0.1074 | 0.0780 | 0.0631 | |

GRU-DeepAR | PICP | 0.9461 | 0.8894 | 0.7776 | 0.9380 | 0.8815 | 0.8182 | 0.9448 | 0.8993 | 0.7917 |

PINAW | 0.1000 | 0.0670 | 0.0631 | 0.0842 | 0.0656 | 0.0530 | 0.0749 | 0.0583 | 0.0472 | |

CWC | 0.1000 | 0.0670 | 0.0631 | 0.0842 | 0.0656 | 0.0530 | 0.0749 | 0.0583 | 0.0472 | |

GRU-QR | PICP | 0.9409 | 0.8523 | 0.7693 | 0.9159 | 0.8333 | 0.7479 | 0.9434 | 0.8428 | 0.7697 |

PINAW | 0.0826 | 0.0497 | 0.0375 | 0.0658 | 0.0432 | 0.0350 | 0.0723 | 0.0480 | 0.0374 | |

CWC | 0.0826 | 0.0497 | 0.0375 | 0.0658 | 0.0432 | 0.0350 | 0.0723 | 0.0480 | 0.0374 | |

GRU-NCQR | PICP | 0.9448 | 0.8204 | 0.7127 | 0.9187 | 0.8292 | 0.7507 | 0.9641 | 0.8510 | 0.7379 |

PINAW | 0.0801 | 0.0379 | 0.0369 | 0.0677 | 0.0417 | 0.0322 | 0.0714 | 0.0457 | 0.0355 | |

CWC | 0.0801 | 0.0379 | 0.0369 | 0.0677 | 0.0417 | 0.0322 | 0.0714 | 0.0457 | 0.0355 |

Model | CS | ||
---|---|---|---|

Dataset 1 | Dataset 2 | Dataset 3 | |

GRU-GPR | 0 | 0 | 0 |

GRU-DeepAR | 0 | 0 | 0 |

GRU-QR | 267.71 | 425.36 | 124.15 |

GRU-NCQR | 0 | 0 | 0 |

Models | CRPS | ||
---|---|---|---|

Dataset 1 | Dataset 2 | Dataset 3 | |

GRU-GPR | 382.62 | 539.07 | 177.90 |

GRU-DeepAR | 307.30 | 440.29 | 152.99 |

GRU-QR | 275.79 | 401.60 | 141.28 |

GRU-NCQR | 270.49 | 389.40 | 138.42 |

Models | Dataset 1 | Dataset 2 | Dataset 3 | |||
---|---|---|---|---|---|---|

RMSE (m ^{3}/s) | MAPE (%) | RMSE (m ^{3}/s) | MAPE (%) | RMSE (m ^{3}/s) | MAPE (%) | |

GRU-NCQR | 949.98 | 5.86 | 1363.40 | 4.96 | 352.01 | 5.07 |

MGM-NCQR | 933.01 | 5.30 | 1347.63 | 4.77 | 346.78 | 4.66 |

SMGM-NCQR | 916.78 | 4.87 | 1324.00 | 4.54 | 338.20 | 4.54 |

Models | Metrics | Dataset 1 | Dataset 2 | Dataset 3 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

90% | 80% | 70% | 90% | 80% | 70% | 90% | 80% | 70% | ||

GRU-NCQR | PICP | 0.9448 | 0.8204 | 0.7127 | 0.9187 | 0.8292 | 0.7507 | 0.9641 | 0.8510 | 0.7379 |

PINAW | 0.0801 | 0.0379 | 0.0369 | 0.0677 | 0.0417 | 0.0322 | 0.0714 | 0.0457 | 0.0355 | |

CWC | 0.0801 | 0.0379 | 0.0369 | 0.0677 | 0.0417 | 0.0322 | 0.0714 | 0.0457 | 0.0355 | |

MGM-NCQR | PICP | 0.9351 | 0.8412 | 0.7279 | 0.9215 | 0.8168 | 0.7410 | 0.9600 | 0.8524 | 0.7545 |

PINAW | 0.0748 | 0.0478 | 0.0359 | 0.0638 | 0.0400 | 0.0310 | 0.0671 | 0.0429 | 0.0326 | |

CWC | 0.0748 | 0.0478 | 0.0359 | 0.0638 | 0.0400 | 0.0310 | 0.0671 | 0.0429 | 0.0326 | |

SMGM-NCQR | PICP | 0.9006 | 0.8204 | 0.7569 | 0.9229 | 0.8003 | 0.7066 | 0.9462 | 0.8234 | 0.7052 |

PINAW | 0.0660 | 0.0435 | 0.0327 | 0.0601 | 0.0386 | 0.0298 | 0.0654 | 0.0407 | 0.0312 | |

CWC | 0.0660 | 0.0435 | 0.0327 | 0.0601 | 0.0386 | 0.0298 | 0.0654 | 0.0407 | 0.0312 |

Models | CRPS | ||
---|---|---|---|

Dataset 1 | Dataset 2 | Dataset 3 | |

GRU-NCQR | 270.49 | 389.40 | 138.42 |

MGM-NCQR | 260.79 | 372.60 | 132.10 |

SMGM-NCQR | 252.69 | 358.29 | 127.85 |

Models | Training Time (s) | ||
---|---|---|---|

Dataset 1 | Dataset 2 | Dataset 3 | |

GRU-NCQR | 255 | 198 | 225 |

MGM-NCQR | 198 | 155 | 174 |

SMGM-NCQR | 176 | 139 | 159 |

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**MDPI and ACS Style**

Liu, H.; Zhu, S.; Mo, L.
A Novel Daily Runoff Probability Density Prediction Model Based on Simplified Minimal Gated Memory–Non-Crossing Quantile Regression and Kernel Density Estimation. *Water* **2023**, *15*, 3947.
https://doi.org/10.3390/w15223947

**AMA Style**

Liu H, Zhu S, Mo L.
A Novel Daily Runoff Probability Density Prediction Model Based on Simplified Minimal Gated Memory–Non-Crossing Quantile Regression and Kernel Density Estimation. *Water*. 2023; 15(22):3947.
https://doi.org/10.3390/w15223947

**Chicago/Turabian Style**

Liu, Huaiyuan, Sipeng Zhu, and Li Mo.
2023. "A Novel Daily Runoff Probability Density Prediction Model Based on Simplified Minimal Gated Memory–Non-Crossing Quantile Regression and Kernel Density Estimation" *Water* 15, no. 22: 3947.
https://doi.org/10.3390/w15223947