# Assessing the Forecasting Accuracy of a Modified Grey Self-Memory Precipitation Model Considering Scale Effects

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Study Area

^{4}km

^{2}(Figure 1) [22] and is part of the Songhua River system, with large air temperature differences throughout the year. The lowest air temperature can reach below −30 °C in winter, the highest air temperature can reach above 35 °C in summer, the annual average temperature is 2–6 °C, and the frost-free period is between 100 and 160 days. Water surface evaporation and precipitation average 600–950 mm and 380–520 mm each year, respectively. Affected by the semiarid monsoon climate and the temperate continental monsoon climate, the overall precipitation across the region is relatively limited, especially in the western semiarid region, where annual precipitation has totaled less than 300 mm in recent years. To cope with drought, groundwater in parts of the Songnen Plain has been severely overexploited, and groundwater funnels have formed, affecting regional ecological and water security to a certain extent.

#### 2.2. Data Sources

#### 2.3. The Modified Grey Self-Memory Precipitation Forecasting Theory

#### 2.3.1. The Modified Grey Self-Memory Model

- (1)
- Assuming the rainfall time series is ${x}^{\left(0\right)}=\left[{x}^{\left(0\right)}\left(1\right),{x}^{\left(0\right)}\left(2\right),\dots ,{x}^{\left(0\right)}\left(n\right)\right]$, the first-order accumulation series is ${x}^{\left(1\right)}=\left[{x}^{\left(1\right)}\left(1\right),{x}^{\left(1\right)}\left(2\right),\text{}\dots ,{x}^{\left(1\right)}\left(n\right)\right]$. According to grey forecasting theory, the whitening equation in GM(1,1) is constructed as follows:$$\frac{\mathrm{d}{x}^{\left(1\right)}}{\mathrm{d}t}+a{x}^{\left(1\right)}=b$$
- (2)
- The dynamic core of the self-memory model is constructed as follows:$$F\left(x,t\right)=-a{x}^{\left(1\right)}+b$$
- (3)
- Taking $T=\left[{t}_{-p},{t}_{-p+1},\dots ,{t}_{-1},{t}_{0},t\right]$ as a time series, ${t}_{-p},{t}_{-p+1},\dots ,{t}_{-1}$ represent historical observation moments, ${t}_{0}$ represents the initial forecasting moment, t represents a future forecasting moment, and p is the backtracking order. According to self-memory theory, Equation (3) can be transformed into:$${{\displaystyle \int}}_{t-p}^{t-p+1}\beta \left(\tau \right)\frac{\partial x}{\partial \tau}d\tau +{{\displaystyle \int}}_{t-p+1}^{t-p+2}\beta \left(\tau \right)\frac{\partial x}{\partial \tau}d\tau +\dots +{{\displaystyle \int}}_{{t}_{0}}^{t}\beta \left(\tau \right)\frac{\partial x}{\partial \tau}d\tau ={{\displaystyle \int}}_{t-p}^{t}\beta \left(\tau \right)F\left(x,\tau \right)d\tau $$Using the mean value theorem, inner product, and integration by parts, Equation (4) can be transformed into:$${\beta}_{t}{x}_{t}-{\beta}_{-p}{x}_{-p}-{\displaystyle \sum}_{i=-p}^{0}{x}_{i}^{m}\left({\beta}_{i+1}-{\beta}_{i}\right)-\underset{t-p}{\overset{t}{{\displaystyle \int}}}\beta \left(\tau \right)F\left(x,\tau \right)d\tau =0$$
- (4)
- Let ${x}_{-p}\equiv {x}_{-p-1}^{m}$ and ${\beta}_{-p-1}\equiv 0$; then, the p-order self-memory equation can be transformed into:$${x}_{t}=\frac{1}{{\beta}_{t}}{\displaystyle \sum}_{i=-p-1}^{0}{x}_{i}^{m}\left({\beta}_{i+1}-{\beta}_{i}\right)+\frac{1}{{\beta}_{t}}{{\displaystyle \int}}_{t-p}^{t}\beta \left(\tau \right)F\left(x,t\right)d\tau $$$${x}_{t}={\displaystyle \sum}_{i=-p-1}^{-1}{\alpha}_{i}{y}_{i}+{\displaystyle \sum}_{i=-p}^{0}{\theta}_{i}F\left(x,i\right)$$$$F={\displaystyle \sum}_{t=1}^{n}(\left(\left|\widehat{{x}_{t}}-{x}_{t}\right|/{x}_{t}\right)\times 100\%)$$
- (5)
- According to grey forecasting theory, ${x}_{t}$ was reduced, and the reduced value of precipitation was obtained as follows:$${\hat{x}}^{\left(0\right)}\left(t+1\right)={\hat{x}}^{\left(1\right)}\left(t+1\right)-{x}^{\left(1\right)}\left(t\right)$$

#### 2.3.2. Evaluation of Model Accuracy

_{p}(t), SYQ

_{p}(t), and ${M}_{p}^{i}\left(t\right)$, respectively, where t = 1, 2..., and 58 and i = 5, 6, 7, 8, and 9. Thus, the following relationship was obtained:

## 3. Results and Discussion

#### 3.1. Construction of the Precipitation Forecasting Model

#### 3.2. Evaluation of the MGSM and Comparison with Other Grey System Models

#### 3.3. Precipitation Forecasts

## 4. Conclusions

- (1)
- The MGSM model constructed in this paper yields higher fitting accuracy at different scales than both the GM(1,1) model and the GSM. The NSE of the precipitation forecasting results at various scales was greater than 0.69, the MARE was between 0.28% and 9.36%, and the RMSE was between 8.5 and 31.81 mm.
- (2)
- Based on the time scale effects of precipitation, the accuracy of the precipitation forecasting results from 2019 to 2023 was tested. The growth period and annual NSE values both exceeded 0.5, and the average relative error was within 5%. The RMSE was also within 30 mm, and the accuracy of estimates in the forecasting stage met the relevant requirements. The proposed method can overcome the shortcomings of traditional methods in which the forecasting accuracy cannot be assessed because of the lack of available measurements.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The simulated and observed precipitation values at different scales in the study area. (Note: Because the backtracking order used in the precipitation forecasting model varied at different scales, the starting year of the simulations also varied.) Figures show the results at different scales. (

**a**) Year. (

**b**) Crop growth period. (

**c**) May. (

**d**) June. (

**e**) July. (

**f**) August. (

**g**) September.

**Figure 3.**Comparative analysis of the precision of different precipitation forecasting models at different time scales. Figures show the results at different scales. (

**a**) Year. (

**b**) Crop growth period. (

**c**) May. (

**d**) June. (e) July. (

**f**) August. (

**g**) September. A, B and C represent the methods of MGSM, GSM, and GM(1,1) respectively. REF is the reference point.

**Figure 4.**Forecasting results at different scales from 2019 to 2023. (

**a**) Annual and crop growth period scales; (

**b**) Monthly scale during the crop growth period.

Serial Number | Station Number | Station Name | Longitude | Latitude | Starting Date (year month) | Ending Date (year month) | Date(s) of the Missing |
---|---|---|---|---|---|---|---|

1 | 50557 | Nenjiang | 125.23 | 49.17 | 1951 01 | 2018 12 | |

2 | 50646 | Nehe | 124.85 | 48.48 | 1961 01 | 2018 12 | |

3 | 50655 | Dedu | 126.18 | 48.5 | 1966 08 | 2018 12 | 1961 01–1966 07 |

4 | 50656 | Beian | 126.51 | 48.28 | 1958 09 | 2018 12 | |

5 | 50658 | Keshan | 125.88 | 48.05 | 1951 01 | 2018 12 | |

6 | 50659 | Kedong | 126.25 | 48.03 | 1959 01 | 2018 12 | 1995 04–1998 12 |

7 | 50739 | Longjiang | 123.18 | 47.33 | 1958 01 | 2018 12 | |

8 | 50741 | Gannan | 123.5 | 47.93 | 1954 11 | 2018 12 | |

9 | 50742 | Fuyu | 124.48 | 47.8 | 1956 10 | 2018 12 | |

10 | 50745 | Qiqihaer | 123.92 | 47.38 | 1951 01 | 2018 12 | |

11 | 50749 | Lindian | 124.83 | 47.18 | 1956 12 | 2018 12 | |

12 | 50750 | Yian | 125.3 | 47.9 | 1956 12 | 2018 12 | |

13 | 50755 | Baiquan | 126.1 | 47.6 | 1956 12 | 2018 12 | |

14 | 50756 | Hailun | 126.97 | 47.43 | 1952 07 | 2018 12 | |

15 | 50758 | Mingshui | 125.9 | 47.16 | 1953 01 | 2018 12 | |

16 | 50767 | Suiling | 127.1 | 47.23 | 1961 01 | 2018 12 | 1995 04–1998 12 |

17 | 50842 | Dumeng | 124.43 | 46.87 | 1959 01 | 2018 12 | |

18 | 50844 | Tailai | 123.42 | 46.4 | 1958 01 | 2018 12 | |

19 | 50851 | Qingang | 126.1 | 46.68 | 1956 12 | 2018 12 | |

20 | 50852 | Wangkui | 126.48 | 46.87 | 1956 12 | 2018 12 | |

21 | 50853 | Suihua | 126.96 | 46.61 | 1952 07 | 2018 12 | |

22 | 50854 | Anda | 125.32 | 46.38 | 1952 07 | 2018 12 | |

23 | 50858 | Zhaodong | 125.97 | 46.07 | 1959 01 | 2018 12 | |

24 | 50859 | Lanxi | 126.27 | 46.25 | 1956 11 | 2018 12 | 1995 04–1998 12 |

25 | 50861 | Qingan | 127.48 | 46.88 | 1956 12 | 2018 12 | |

26 | 50867 | Bayan | 127.35 | 46.08 | 1960 01 | 2018 12 | |

27 | 50950 | Zhaozhou | 125.25 | 45.7 | 1961 01 | 2018 12 | |

28 | 50953 | Haerbin | 126.77 | 45.75 | 1951 01 | 2018 12 | |

29 | 50954 | Zhaoyuan | 125.08 | 45.5 | 1959 01 | 2018 12 | |

30 | 50955 | Shuangcheng | 126.3 | 45.38 | 1956 12 | 2018 12 | |

31 | 50956 | Hulan | 126.6 | 46 | 1955 01 | 2018 12 | 1961 01–2004 12 |

32 | 50958 | Acheng | 126.95 | 45.52 | 1959 05 | 2018 12 | |

33 | 50960 | Bixnian | 127.45 | 45.78 | 1958 01 | 2018 12 | |

34 | 50962 | Mulan | 128.03 | 45.95 | 1956 12 | 2018 12 | |

35 | 54080 | Wuchang | 127.15 | 44.9 | 1957 12 | 2018 12 |

Parameters | Annual | Crop Growth Period | Monthly | |||||
---|---|---|---|---|---|---|---|---|

May | June | July | August | September | ||||

Dynamic core parameters | a | 0.00 | 0.00 | −0.01 | −0.01 | 0.00 | 0.00 | 0.01 |

b | 471.38 | 418.65 | 29.77 | 65.53 | 148.47 | 113.51 | 59.43 | |

Backtracking order | p | 6 | 6 | 4 | 6 | 5 | 5 | 4 |

Self-memory model parameters | ${\alpha}_{-6}$ | −0.50 | −0.50 | - | −0.50 | - | - | - |

${\alpha}_{-5}$ | 1.00 | 1.00 | - | 1.00 | 0.50 | 0.50 | - | |

${\alpha}_{-4}$ | −1.50 | −1.50 | −0.67 | −1.50 | −1.17 | −1.17 | −0.67 | |

${\alpha}_{-3}$ | 2.00 | 2.00 | 1.33 | 2.00 | 1.67 | 1.67 | 1.33 | |

${\alpha}_{-2}$ | −2.50 | −2.50 | −2.00 | −2.50 | −2.33 | −2.33 | −2.00 | |

${\alpha}_{-1}$ | 3.00 | 3.00 | 2.66 | 3.00 | 2.83 | 2.83 | 2.67 | |

${\alpha}_{0}$ | −3.50 | −3.50 | −3.33 | −3.50 | −3.50 | −3.50 | −3.33 | |

${\alpha}_{1}$ | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | 2.00 | |

${\theta}_{-6}$ | 202.59 | 700.25 | - | 36.25 | - | - | - | |

${\theta}_{-5}$ | −202.59 | −700.26 | - | −36.23 | 579.56 | 97.18 | - | |

${\theta}_{-4}$ | 202.59 | 700.26 | 47.16 | 36.27 | −772.75 | −129.59 | −57.46 | |

${\theta}_{-3}$ | −202.59 | −700.26 | −47.14 | −36.26 | 579.56 | 97.19 | 57.45 | |

${\theta}_{-2}$ | 202.60 | 700.26 | 47.19 | 36.32 | −772.75 | −129.60 | −57.49 | |

${\theta}_{-1}$ | −202.60 | −700.26 | −47.19 | −36.32 | 579.56 | 97.21 | 57.49 | |

${\theta}_{0}$ | 202.62 | 700.26 | 47.25 | 36.39 | −772.76 | −129.63 | −57.54 | |

${\theta}_{1}$ | −202.62 | −700.27 | −47.27 | −36.41 | 579.57 | 97.24 | 57.55 |

Parameter | Annual | Crop Growth Period | May | June | July | August | September |
---|---|---|---|---|---|---|---|

NSE | 0.80 | 0.82 | 0.76 | 0.79 | 0.72 | 0.80 | 0.69 |

MARE | 0.28 | 0.33 | 4.88 | 3.13 | 3.48 | 3.26 | 9.36 |

RMSE | 31.81 | 28.20 | 8.50 | 13.07 | 21.37 | 18.27 | 11.46 |

**Table 4.**Forecasting results and precision analysis of precipitation at different time scales from 2019 to 2023.

Years/Parameters | Year (mm) | Crop Growth Period (mm) | May (mm) | June (mm) | July (mm) | August (mm) | September (mm) | $\hat{\mathit{S}\mathit{Y}{\mathit{Q}}_{\mathit{p}}^{\prime}}$ | $\hat{{\mathit{Y}}_{\mathit{p}}^{\prime}}$ |
---|---|---|---|---|---|---|---|---|---|

2019 | 578 | 465 | 38 | 73 | 180 | 119 | 67 | 477 | 538 |

2020 | 480 | 410 | 36 | 90 | 150 | 84 | 41 | 402 | 474 |

2021 | 495 | 415 | 43 | 115 | 121 | 102 | 47 | 427 | 480 |

2022 | 578 | 515 | 32 | 128 | 103 | 140 | 61 | 464 | 596 |

2023 | 580 | 484 | 30 | 102 | 147 | 161 | 73 | 514 | 560 |

2019–2023 Mean value | 542 | 461 | 36 | 102 | 142 | 121 | 58 | / | / |

1961–2018 Mean value | 492 | 427 | 39 | 83 | 145 | 107 | 52 | / | / |

NSE | / | / | / | / | / | / | / | 0.520 | 0.745 |

MARE | / | / | / | / | / | / | / | 4.758 | 3.527 |

RMSE | / | / | / | / | / | / | / | 28.01 | 22.68 |

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**MDPI and ACS Style**

Meng, F.; Sun, Z.; Yang, L.; Yu, K.; Wang, Z.
Assessing the Forecasting Accuracy of a Modified Grey Self-Memory Precipitation Model Considering Scale Effects. *Water* **2022**, *14*, 1647.
https://doi.org/10.3390/w14101647

**AMA Style**

Meng F, Sun Z, Yang L, Yu K, Wang Z.
Assessing the Forecasting Accuracy of a Modified Grey Self-Memory Precipitation Model Considering Scale Effects. *Water*. 2022; 14(10):1647.
https://doi.org/10.3390/w14101647

**Chicago/Turabian Style**

Meng, Fanxiang, Zhimin Sun, Long Yang, Kui Yu, and Zongliang Wang.
2022. "Assessing the Forecasting Accuracy of a Modified Grey Self-Memory Precipitation Model Considering Scale Effects" *Water* 14, no. 10: 1647.
https://doi.org/10.3390/w14101647