# A New GM(1,1) Model Based on Cubic Monotonicity-Preserving Interpolation Spline

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. ${\mathit{C}}^{\mathbf{1}}$ Monotonicity-Preserving Piecewise Cubic Interpolation Spline

## 3. Establish New GM(1,1) Model

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | The Model in [20] | |||
---|---|---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

2.9836 | 2.9836 | 0 | 2.9836 | 0 | 2.9836 | 0 |

4.4511 | 4.3804 | 1.5816 | 4.3531 | 2.2021 | 4.4561 | 0.1123 |

6.6402 | 6.5006 | 2.0903 | 6.5222 | 1.7835 | 6.6132 | 0.4066 |

9.9061 | 9.6469 | 2.5994 | 9.7720 | 1.3569 | 9.8146 | 0.9237 |

14.7781 | 14.3162 | 3.1039 | 14.6413 | 0.9344 | 14.5657 | 1.4373 |

22.0464 | 21.2454 | 3.6069 | 21.9368 | 0.5013 | 21.6168 | 1.9486 |

32.8893 | 31.5285 | 4.1065 | 32.8675 | 0.0793 | 32.0812 | 2.4570 |

$\overline{\epsilon}$ (%) | 2.8481 | 1.1429 | 1.2143 |

${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | The Model in [21] | |||
---|---|---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

21.1 | 21.1 | 0 | 21.1 | 0 | 21.1000 | 0 |

26.6 | 24.4166 | 8.2083 | 24.0779 | 9.4816 | 23.3606 | 12.1782 |

36.1 | 35.7198 | 1.0531 | 32.5648 | 9.7928 | 35.7858 | 0.8704 |

52.3 | 52.2557 | 0.0847 | 50.3116 | 3.8018 | 54.8198 | 4.8180 |

80.1 | 76.4466 | 4.5611 | 77.7300 | 2.9588 | 83.9777 | 4.8411 |

126.8 | 111.8361 | 11.8012 | 120.0906 | 5.2913 | 128.6443 | 1.4545 |

196.3 | 163.6087 | 16.6537 | 185.5365 | 5.4832 | 197.0684 | 0.3914 |

$\overline{\epsilon}$ (%) | 7.0604 | 6.1349 | 4.0923 |

${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | The Model in [7] | |||
---|---|---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

110,852 | 110,852 | 0 | 110,852 | 0 | 110,852 | 0 |

135,175 | 117,980 | 12.72 | 130,078 | 3.77 | 127,821 | 5.41 |

153,647 | 119,117 | 22.47 | 128,574 | 16.32 | 126,664 | 17.66 |

120,296 | 128,264 | 6.62 | 127,087 | 5.64 | 125,830 | 4.68 |

96,362 | 121,422 | 26.27 | 125,617 | 30.36 | 124,380 | 29.23 |

90,798 | 122,592 | 35.01 | 124,164 | 36.75 | 123,253 | 35.70 |

102,591 | 123,773 | 20.65 | 122,728 | 19.63 | 122,137 | 19.11 |

150,534 | 124,965 | 16.99 | 121,308 | 19.41 | 121,031 | 19.63 |

175,123 | 126,168 | 27.95 | 119,905 | 31.53 | 119,934 | 31.52 |

127,148 | 113,383 | 10.83 | 118,518 | 6.79 | 114,848 | 9.76 |

102,085 | 128,610 | 25.98 | 117,147 | 14.75 | 117,772 | 15.47 |

97,103 | 116,705 | 20.19 | 115,792 | 19.25 | 116,705 | 20.21 |

$\overline{\epsilon}$ (%) | 21.73 | 18.56 | 18.91 |

${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | The Model in [9] | |||
---|---|---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

5.08 | 5.08 | 0 | 5.08 | 0 | 5.08 | 0 |

4.80 | 3.55 | 26.04 | 4.24 | 11.51 | 3.65 | 23.9 |

4.67 | 4.52 | 3.21 | 5.25 | 12.57 | 4.61 | 1.28 |

4.50 | 5.76 | 28.00 | 5.51 | 22.44 | 5.84 | 29.78 |

7.12 | 7.34 | 3.09 | 8.05 | 13.09 | 7.42 | 4.21 |

9.67 | 9.34 | 3.41 | 9.97 | 3.05 | 9.45 | 2.28 |

12.80 | 11.90 | 7.03 | 12.33 | 3.64 | 12.04 | 5.94 |

15.88 | 15.15 | 4.59 | 15.26 | 3.88 | 15.37 | 3.21 |

19.49 | 19.30 | 0.97 | 18.89 | 3.07 | 19.65 | 0.82 |

23.07 | 24.57 | 6.50 | 23.38 | 1.34 | 24.57 | 8.92 |

26.86 | 31.29 | 16.49 | 28.93 | 7.72 | 31.29 | 19.73 |

$\overline{\epsilon}$ (%) | 9.03 | 7.48 | 8.29 |

${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | The Model in [11] | |||
---|---|---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

1610.71 | 1610.71 | 0 | 1610.71 | 0 | 1610.71 | 0 |

1245.28 | 1363.91 | 9.524 | 1365.22 | 9.632 | 1342.76 | 7.833 |

1347.71 | 1274.95 | 5.402 | 1275.75 | 5.340 | 1280.32 | 5.004 |

1382.45 | 1191.79 | 13.795 | 1192.14 | 13.766 | 1204.58 | 12.872 |

1018.45 | 1114.06 | 9.381 | 1114.01 | 9.383 | 1122.64 | 10.203 |

1014.96 | 1041.39 | 2.597 | 1041.00 | 2.566 | 1040.18 | 2.476 |

949.46 | 973.47 | 2.519 | 972.78 | 2.456 | 960.34 | 1.147 |

$\overline{\epsilon}$ (%) | 6.174 | 6.163 | 5.648 |

${\mathit{x}}^{(0)}$ | Classical GM(1,1) | New GM(1,1) | ||
---|---|---|---|---|

Prediction Data | $\mathit{\epsilon}$ (%) | Prediction Data | $\mathit{\epsilon}$ (%) | |

64,832.05 | 64,832.05 | 0 | 64,832.05 | 0 |

71,847.09 | 57,476.77 | 20.001 | 57,308.50 | 20.235 |

78,646.30 | 67,165.21 | 14.598 | 67,022.83 | 14.779 |

86,293.10 | 78,486.76 | 9.046 | 78,383.83 | 9.166 |

93,887.95 | 91,716.70 | 2.312 | 91,670.62 | 2.362 |

105,557.09 | 107,176.71 | 1.534 | 107,209.65 | 1.566 |

125,761.85 | 125,242.71 | 0.413 | 125,382.70 | 0.301 |

143,143.63 | 146,353.96 | 2.243 | 146,636.24 | 2.440 |

168,850.20 | 171,023.78 | 1.287 | 171,492.46 | 1.565 |

198,739.27 | 199,852.01 | 0.560 | 200,562.04 | 0.917 |

245,352.80 | 233,539.60 | 4.815 | 234,559.18 | 4.400 |

278,541.09 | 272,905.57 | 2.023 | 274,319.16 | 1.516 |

334,839.41 | 318,907.39 | 4.758 | 320,818.83 | 4.187 |

386,086.72 | 372,663.28 | 3.477 | 375,200.63 | 2.820 |

$\overline{\epsilon}$ (%) | 4.791 | 4.732 |

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## Share and Cite

**MDPI and ACS Style**

Zhu, Y.; Jian, Z.; Du, Y.; Chen, W.; Fang, J.
A New GM(1,1) Model Based on Cubic Monotonicity-Preserving Interpolation Spline. *Symmetry* **2019**, *11*, 420.
https://doi.org/10.3390/sym11030420

**AMA Style**

Zhu Y, Jian Z, Du Y, Chen W, Fang J.
A New GM(1,1) Model Based on Cubic Monotonicity-Preserving Interpolation Spline. *Symmetry*. 2019; 11(3):420.
https://doi.org/10.3390/sym11030420

**Chicago/Turabian Style**

Zhu, Yuanpeng, Zehua Jian, Yurui Du, Wenqing Chen, and Jiwei Fang.
2019. "A New GM(1,1) Model Based on Cubic Monotonicity-Preserving Interpolation Spline" *Symmetry* 11, no. 3: 420.
https://doi.org/10.3390/sym11030420