# Study on Forecasting Break-Up Date of River Ice in Heilongjiang Province Based on LSTM and CEEMDAN

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}. There are three mountain ranges (Greater Khingan Mountains, Lesser Khingan Mountains, and the northern remnants of Changbai Mountain named Wanda Mountains) and two plains (Songnen plain and Sanjiang plain). The geographical location and topography of Heilongjiang Province are shown in Figure 1. Heilongjiang Province is located in cold and middle temperate zones with an average annual temperature of 3–5 °C. Heilongjiang province has a long and cold winter with an average temperature of −17.1 °C, and a windy and dry spring with an average temperature of 4.8 °C. The rivers in Heilongjiang province freeze up at the end of October and break up in April of the next year.

#### 2.2. CEEMDAN

_{j}(t) to the break-up date series x(t), and obtain the signal x

_{j}(t) used to decompose:

_{j}(t) = x(t) + p

_{i}n

_{j}(t)(i = 1, 2, ⋯, M; j = 1, 2, ⋯, N)

_{k}(t), the value of j in the hundreds will lead to a very good result [36], is set to 500 in this paper; p

_{i}is the standard deviation in the noise, which controls the signal-to-noise ratio of n

_{j}(t) to x(t), the value of p

_{i}is set to 0.2 times the standard deviation of the break-up date series, which is recommended by Colominas, Schlotthauer, and Torres [37].

_{j}(t) to obtain the IMF component $IM{F}_{i}^{j}\left(t\right)$. $IM{F}_{i}^{j}\left(t\right)$ is the ith IMF component of x

_{j}(t) after EMD decomposition.

_{j}(t) each time. The maximum iteration, which is not limited [39], is set to 5000 by experience to ensure that all extracted IMFs are valid before the EMD stops.

_{1}(t) is defined as a process of the first IMF component $\overline{IM{F}_{1}\left(t\right)}$ by EMD and the sequence r

_{1}(t) + p

_{2}e

_{1}(n

_{j}(t)) to obtain the second IMF component as follows:

#### 2.3. LSTM Network

_{t}represents the hidden state; ${W}_{f}$, ${U}_{f}$ and ${b}_{f}$ represent the learnable parameters for the forget gate, ${W}_{f}$ and ${U}_{f}$ represent two adjustable weight matrices, and ${b}_{f}$ represents a bias vector.

#### 2.4. Fundamental Frameworks of CEEMDAN-LSTM

- (1)
- Obtain the break-up date series and decompose it into subsequences (multiple IMFs and a residual series) by CEEMDAN.
- (2)
- Divide the IMFs and residual series into training data and forecast data, and normalize them.
- (3)
- Calculate the maximum autocorrelation order of each IMF and residual series (shown in Table 3), and build the LSTM to forecast the value of IMFs and residual series.
- (4)
- Denormalize and evaluate the forecast result.

#### 2.5. Performance Evaluation

_{t}) [46]. The Taylor diagram is constructed by the relationship between the four statistical quantities above shown in Formula (16). The SD, RMSD, and R can be calculated by Formula (17)–(20).

^{2}= SD

_{o}

^{2}+ SD

_{f}

^{2}− 2SD

_{o}SD

_{f}R

_{max}is the maximum value of R between the forecast break-up date and the observed break-up date.

## 3. Result

#### 3.1. Results of Decomposing the Observed Break-Up Date Series Using CEEMDAN

#### 3.2. Result of LSTM Applied to Forecast the Break-Up Date

_{f}or SD

_{o}), and the distance from the reference point (azimuth angle = 0 and radius = SD

_{o}), being in full agreement with observed break-up date series, represents the RMSD. In a Taylor diagram, the smaller the azimuth angle, the closer the radius to SD

_{o}, and the closer the distance to the reference point, the better the LSTM performance.

_{o}, and the points representing the forecast break-up date series of FJ, MDJ, and YC are closer to the corresponding reference points. The Taylor skill score (S) values of FJ, MDJ, and YC are all 0.99 (Table 5). The azimuth angle of the points representing the forecast break-up date series of QQHR is the largest, the radius of points representing the forecast break-up date series of QQHR is the farthest from the corresponding SD

_{o}, and the points representing the forecast break-up date series of QQHR are the farthest from the corresponding reference points. The S value of QQHR is 0.87 (Table 5).

_{f}values of FJ, YC, and MDJ are 5.23, 5.27, and 5.65, respectively, which are close to the SD

_{o}values of FJ, YC, and MDJ with 5.21, 5.20, and 5.44, respectively. The difference between the SD

_{f}and the SD

_{o}of FJ, YC, and MDJ are 0.02, 0.07, and 0.21, respectively. The SD

_{f}of QQRH is 4.64, which is quite different from the SD

_{o}of QQHR with 6.82. The difference between the SD

_{f}and the SD

_{o}of QQHR is 2.18.

#### 3.3. Result of CEEMDAN-LSTM Applied to Forecast the Break-Up Date

_{o}values, and the points representing the forecast break-up date series are closer to the corresponding reference points. As shown in Table 7, the Taylor skill score of each station obtained by CEEMDAN-LSTM was improved to 0.99. The Taylor skill score of QQHR increased the most, by 0.12. The RMSD values obtained by CEEMDAN-LSTM range from 0.95 to 1.69. The RMSD value of QQHR is the largest with 1.69, followed by BQ with the RMSD value of 1.55. The R values obtained by CEEMAN-LSTM range from 0.97 to 0.98. The R values of QQHR and BQ are smaller with 0.97. Compared with Table 5, the RMSD values obtained by CEEMDAN-LSTM decrease by 0.27–4.28. The RMSD value of QQHR decreases the most with a decrease of 4.28. The R values obtained by CEEMDAN-LSTM improved by 0.01–0.46. The R value of QQHR improved the most with an increase of 0.46. Based on the analysis for Figure 6 and Table 7 above, the LSTM-CEEMDAN performs better than LSTM.

## 4. Discussion

#### 4.1. Reasons for the Improvement of the Forecasting Accuracy of Break-Up Date by CEEMDAN-LSTM

^{−1}. The IMF5, IMF6, and residual series have a low fluctuation degree and weak randomness. The MAE values of IMF5, IMF6, and the residual series are small with the order of magnitude about 10

^{−2}to 10

^{−3}. The MAE value with an order of magnitude about 10

^{−2}to 10

^{−3}has negligible influence on the forecasting accuracy for break-up date. The Taylor diagram, which comprehensively shows the forecast accuracy of each subsequence of each station, is shown in Figure A1 in the Appendix A.

#### 4.2. Reasons for the Lower Forecasting Accuracy of BQ and QQHR by LSTM

#### 4.3. The Performance of LSTM in Forecasting the Black Swan Events

## 5. Conclusions

- (1)
- CEEMDAN decomposed the observed break-up date series into subsequences according to different fluctuations and frequencies. The observed break-up date series with a larger standard deviation, compared with a similar break-up date series in length, had relatively more decomposed subsequences. With the decomposition processing, the frequency and fluctuation degree of subsequence decreased, and the sample values of subsequence increased. The residual series had the lowest fluctuation degree and frequency, which was close to linear and varied slightly around the long-term average.
- (2)
- The subsequence decomposed by CEEMDAN with a lower fluctuation degree or smaller sample values compared with the observed series for LSTM obtained a higher forecasting accuracy. The IMF1 and IMF2 had smaller values, the MAE values for forecasting results of IMF1 and IMF2 were small with the order of magnitude of 10
^{−1}. The IMF5, IMF6, and residual series had lower fluctuation degrees, and the MAE values of forecasting results of IMF5, IMF6, and residual series were small with the order of magnitude of 10^{−2}and 10^{−3}. - (3)
- Among the performance evaluation of the LSTM for all seven stations, the absolute error ranged from −13 to 12, the MAE values ranged from 0.80 to 6.40, the QR values were above 60%, the RMSD values ranged from 1.37 to 5.97, the R values ranged from 0.51 to 0.97, and the S values ranged from 0.87 to 0.99.
- (4)
- The forecasting accuracy was obviously improved by LSTM coupled with CEEMDAN. CEEMDAN-LSTM performed better than LSTM. In the performance evaluation of the CEEMDAN-LSTM for all seven stations, the absolute error ranged from −6 to 4, the MAE values ranged from 0.75 to 3.40, the QR values improved to 100%, the RMSD values ranged from 0.95 to 1.69, the R values ranged from 0.97 to 0.98, and the S values improved to 0.99.
- (5)
- CEEMDAN can reduce the influence of the few samples on the forecasting accuracy of LSTM. The forecasting accuracy by LSTM was obviously improved after decomposing the observed break-up date series of QQHR with a short length by CEEMDAN. The MAE value of forecasting results for QQHR decreased from 6.33 to 1.83, the QR value was improved from 80% to 100%, and the S value was improved from 0.87 to 0.99.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 2.**Observed break-up date series and decomposition results of CEEMDAN at each station in Heilongjiang Province (the unit of the break-up date is the “day”).

**Figure 3.**The absolute error (AE) of the LSTM at each station in Heilongjiang Province (the black dashed line represents the maximum allowable error stipulated by the standard for hydrological information and hydrological forecasting, and the yellow dashed line divides the training period and the forecast period).

**Figure 5.**The absolute error (AE) of the simulation results of CEEMDAN-LSTM at each station in Heilongjiang Province (the black dashed line represents the maximum allowable error stipulated by the standard for hydrological information and hydrological forecasting, and the yellow dashed line divides the training period and the forecast period).

**Figure 7.**The forecast results and the absolute error of IMFs and residual series of each station by CEEMDAN-LSTM (the blue line represents the absolute error, and the red line represents the forecasting value).

Station | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | (6) | (7) | |

Time span | 1951–2019 | 1961–2019 | 1960–2019 | 1958–2019 | 1960–2019 | 1984–2019 | 1973–2019 |

Series length (year) | 69 | 59 | 69 | 62 | 60 | 36 | 47 |

Mean | Day 98 | Day 97 | Day 107 | Day 119 | Day 101 | Day 98 | Day 107 |

Standard deviation (day) | 5.14 | 6.84 | 5.21 | 5.07 | 5.44 | 6.82 | 5.20 |

Range (day) | 25 | 45 | 23 | 25 | 25 | 34 | 19 |

**Table 2.**The maximum autocorrelation order of the break-up date series at each hydrological station.

Model | Station | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|---|

LSTM | Maximum autocorrelation order | 14 | 3 | 1 | 16 | 3 | 6 | 2 |

**Table 3.**The maximum autocorrelation order of each IMF and residual series of the break-up date series of each hydrological station.

Model | Station | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|---|

CEEMDAN-LSTM | IMF1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 |

IMF2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |

IMF3 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | |

IMF4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

IMF5 | 1 | 1 | 1 | 1 | 1 | - | 1 | |

IMF6 | - | 1 | - | - | - | - | - | |

Residual | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Period | Performance Evaluation Index | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|---|

Training | Range of AE | [−8, 9] | [−9, 9] | [−6, 5] | [−13, 9] | [−5, 4] | [−11, 12] | [−8, 9] |

MAE | 2.53 | 2.61 | 1.69 | 2.48 | 0.80 | 3.97 | 2.71 | |

QR(%) | 95.31 | 96.30 | 100.00 | 91.23 | 100.00 | 75.19 | 95.24 | |

Forecast | Range of AE | [−6, 0] | [−10, 8] | [0, 4] | [−2, 4] | [1, 2] | [−13, −1] | [0, 3] |

MAE | 1.80 | 6.40 | 2.00 | 2.67 | 2.17 | 6.33 | 1.67 | |

QR(%) | 100.00 | 60.00 | 100.00 | 100.00 | 100.00 | 80.00 | 100.00 |

Station | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|

S | 0.93 | 0.95 | 0.99 | 0.95 | 0.99 | 0.87 | 0.99 |

RMSD | 3.58 | 3.96 | 2.27 | 3.87 | 1.37 | 5.97 | 3.10 |

R | 0.72 | 0.82 | 0.91 | 0.66 | 0.97 | 0.51 | 0.83 |

SD_{o} | 5.14 | 6.84 | 5.21 | 5.07 | 5.44 | 6.82 | 5.20 |

SD_{f} | 3.95 | 5.50 | 5.23 | 4.05 | 5.65 | 4.64 | 5.27 |

Period | Performance Evaluation Index | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|---|

Training | Range of AE | [−3, 3] | [−4, 4] | [−3, 4] | [−2, 3] | [−4, 3] | [−3, 3] | [−3, 4] |

MAE | 0.75 | 1.97 | 0.79 | 0.77 | 0.87 | 1.35 | 1.01 | |

QR(%) | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | |

Forecast | Range of AE | [−1, 1] | [−6, −1] | [−2, 1] | [−2, 2] | [−1, 1] | [−4, 2] | [−2, 2] |

MAE | 0.80 | 3.40 | 0.83 | 1.17 | 0.83 | 1.83 | 1.00 | |

QR(%) | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

Station | HRB | BQ | FJ | MH | MDJ | QQHR | YC |
---|---|---|---|---|---|---|---|

S | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 |

RMSD | 0.95 | 1.55 | 1.05 | 1.02 | 1.10 | 1.69 | 1.27 |

R | 0.98 | 0.97 * | 0.98 | 0.98 | 0.98 | 0.97 ** | 0.97 *** |

SD_{o} | 5.14 | 6.84 | 5.21 | 5.07 | 5.44 | 6.82 | 5.20 |

SD_{f} | 5.23 | 6.45 | 5.31 | 5.15 | 5.42 | 6.73 | 5.51 |

Station | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | Residual |
---|---|---|---|---|---|---|---|

HRB | 0.293 | 0.155 | 0.126 | 0.095 | 0.042 | - | 0.037 |

BQ | 0.741 | 0.511 | 0.288 | 0.265 | 0.073 | 0.035 | 0.053 |

FJ | 0.342 | 0.213 | 0.105 | 0.089 | 0.004 | - | 0.043 |

MH | 0.362 | 0.233 | 0.134 | 0.100 | 0.071 | - | 0.054 |

MDJ | 0.368 | 0.157 | 0.185 | 0.082 | 0.029 | - | 0.043 |

QQHR | 0.569 | 0.298 | 0.158 | 0.170 | - | 0.163 | |

YC | 0.455 | 0.222 | 0.163 | 0.083 | 0.086 | - | 0.005 |

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## Share and Cite

**MDPI and ACS Style**

Liu, M.; Wang, Y.; Xing, Z.; Wang, X.; Fu, Q. Study on Forecasting Break-Up Date of River Ice in Heilongjiang Province Based on LSTM and CEEMDAN. *Water* **2023**, *15*, 496.
https://doi.org/10.3390/w15030496

**AMA Style**

Liu M, Wang Y, Xing Z, Wang X, Fu Q. Study on Forecasting Break-Up Date of River Ice in Heilongjiang Province Based on LSTM and CEEMDAN. *Water*. 2023; 15(3):496.
https://doi.org/10.3390/w15030496

**Chicago/Turabian Style**

Liu, Mingyang, Yinan Wang, Zhenxiang Xing, Xinlei Wang, and Qiang Fu. 2023. "Study on Forecasting Break-Up Date of River Ice in Heilongjiang Province Based on LSTM and CEEMDAN" *Water* 15, no. 3: 496.
https://doi.org/10.3390/w15030496