# Fractional-Order Grey Prediction Method for Non-Equidistant Sequences

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theories

#### 2.1. r-AGO

#### 2.2. The Traditional NGM(1,1)

## 3. The Proposed r-NGM(1,1)

#### 3.1. r-NAGO

#### 3.2. Negative Fractional Order

#### 3.3. r-NGM(1,1)

#### 3.4. Optimization of the Accumulated Order

#### 3.5. Statistical Indicators

## 4. Computational Cases

^{−6}, and the factor of is 0.1 in the optimization of the fractional order.

#### 4.1. Case 1

_{−1}, MPa) of titanium alloy with different temperatures (T

_{e}, °C) under the action of a symmetrical cyclic load, shown in Table 1. It is the monotonic decreasing non-equidistant sequence. All data are used to construct different grey models. The sampling interval is set as T = 10 in r-NGM(1,1). Actual and estimated values of several grey models are presented in Table 1 and Table 2.

#### 4.2. Case 2

#### 4.3. Case 3

## 5. Discussion

#### 5.1. Applicability of the Proposed Method

#### 5.2. The Sampled Interval

#### 5.3. Comparison with Other Method

## 6. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AGO | Accumulated generating operation |

1-AGO | First-order accumulated generating operation |

r-AGO | Fractional-order accumulated generating operation |

1-NAGO | First-order non-equidistant accumulated generating operation |

r-NAGO | Fractional-order non-equidistant accumulated generating operation |

GM(1,1) | Grey model with first-order differential equation and a single variable |

NGM(1,1) | Non-equidistant grey model with first-order differential equation and single variable |

r-NGM(1,1) | Fractional-order non-equidistant grey model with 1-order differential equation and single variable |

APD | Absolute percent deviation |

RMSE | Root mean square error |

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**Figure 4.**Errors of the estimated sequence with two methods: (

**a**) Case 1; (

**b**) Case 2; and (

**c**) Case 3.

k | t_{k} | n_{k} | X^{(0)} | ${\widehat{\mathit{X}}}^{\mathbf{(}\mathbf{0}\mathbf{)}}$ | |||||
---|---|---|---|---|---|---|---|---|---|

NGM(1,1) | −0.017-NGM(1,1) | 0.995-NGM(1,1) | Ref. [8] | Ref. [9] | Ref. [29] | ||||

1 | 100 | 1 | 560.00 | 560.00 | 560.00 | 560.00 | 560.00 | 560.00 | 562.87 |

2 | 130 | 4 | 557.54 | 557.21 | 556.79 | 555.37 | 554.82 | 557.73 | 551.39 |

3 | 170 | 8 | 536.10 | 538.35 | 537.87 | 538.57 | 536.49 | 538.73 | 535.23 |

4 | 210 | 12 | 516.10 | 517.55 | 517.25 | 518.14 | 516.28 | 517.82 | 518.06 |

5 | 240 | 15 | 505.60 | 500.01 | 501.95 | 500.52 | 499.20 | 500.21 | 504.47 |

6 | 270 | 18 | 486.10 | 485.45 | 486.99 | 485.74 | 485.02 | 485.58 | 490.25 |

7 | 310 | 22 | 467.40 | 469.02 | 467.74 | 468.97 | 469.01 | 469.04 | 470.24 |

8 | 340 | 25 | 453.80 | 453.11 | 454.01 | 452.70 | 453.49 | 453.08 | 454.41 |

9 | 380 | 29 | 436.40 | 437.78 | 436.47 | 436.95 | 438.52 | 437.65 | 432.14 |

Grey Model | Modeling Data | |
---|---|---|

RMSE | APD (%) | |

NGM(1,1) | 2.34 | 0.35 |

−0.017-NGM(1,1) | 1.55 | 0.22 |

0.995-NGM(1,1) | 2.37 | 0.38 |

Ref. [8] | 2.67 | 0.37 |

Ref. [9] | 2.34 | 0.35 |

Ref. [29] | 3.31 | 0.55 |

k | t_{k} | n_{k} | X^{(0)} | ${\widehat{\mathit{X}}}^{\mathbf{(}\mathbf{0}\mathbf{)}}$ | |||
---|---|---|---|---|---|---|---|

−0.13-NGM(1,1) | 0.93-NGM(1,1) | NGM(1,1) | Ref. [30] | ||||

1 | 50 | 1 | 0.7660 | 0.7660 | 0.7660 | 0.7660 | 0.7660 |

2 | 55 | 6 | 0.8192 | 0.8192 | 0.8198 | 0.8311 | 0.8361 |

3 | 65 | 16 | 0.9063 | 0.9062 | 0.9055 | 0.8871 | 0.8949 |

4 | 80 | 31 | 0.9848 | 0.9848 | 0.9852 | 0.9890 | 0.9908 |

5 | 86 | 37 | 0.9976 | 1.0050 | 1.0360 | 1.0826 | 1.0320 |

Grey Model | Modeling Data | Predicting Data | ||
---|---|---|---|---|

RMSE | APD (%) | RMSE | APD (%) | |

−0.13-NGM(1,1) | 0.0001 | 0.002 | 0.007 | 0.75 |

0.93-NGM(1,1) | 0.001 | 0.068 | 0.068 | 3.85 |

NGM(1,1) | 0.013 | 1.33 | 0.085 | 8.52 |

Ref. [30] | 0.011 | 1.31 | 0.034 | 3.45 |

k | t_{k} | n_{k} | X^{(0)} | ${\widehat{\mathit{X}}}^{\mathbf{(}\mathbf{0}\mathbf{)}}$ | ||
---|---|---|---|---|---|---|

NGM(1,1) | 1.01-NGM(1,1) | −0.01-NGM(1,1) | ||||

1 | February 2007 | 1 | 2.214517 | 2.214517 | 2.214517 | 2.214517 |

2 | May 2007 | 4 | 2.209514 | 2.191818 | 2.206821 | 2.205258 |

3 | August 2007 | 7 | 2.180164 | 2.190058 | 2.188831 | 2.190021 |

4 | February 2008 | 13 | 2.180396 | 2.187422 | 2.180613 | 2.176111 |

5 | August 2008 | 19 | 2.180480 | 2.183911 | 2.176442 | 2.174253 |

6 | November 2008 | 22 | 2.180469 | 2.181282 | 2.175648 | 2.175123 |

7 | February 2009 | 25 | 2.180391 | 2.179531 | 2.175813 | 2.176498 |

8 | October 2009 | 33 | 2.170843 | 2.176325 | 2.177042 | 2.180976 |

9 | April 2010 | 39 | 2.180387 | 2.172250 | 2.179622 | 2.184262 |

10 | July 2010 | 42 | 2.190126 | 2.169635 | 2.181756 | 2.185791 |

Model | Modeling Data | Predicting Data | ||
---|---|---|---|---|

RMSE | APD (%) | RMSE | APD (%) | |

NGM(1,1) | 0.008 | 0.30 | 0.020 | 0.94 |

1.01-NGM(1,1) | 0.005 | 0.18 | 0.008 | 0.38 |

−0.01-NGM(1,1) | 0.006 | 0.27 | 0.004 | 0.20 |

- | T | n_{m} | Running Time (s) | r | Modeling Data | Predicting Data | ||
---|---|---|---|---|---|---|---|---|

RMSE | APD (%) | RMSE | APD (%) | |||||

Case 1 | 10 | 29 | 1.3 | −0.017 | 1.55 | 0.22 | - | - |

1 | 281 | 5.9 | −0.007 | 1.52 | 0.20 | - | - | |

0.1 | 2801 | 275.6 | −0.004 | 1.48 | 0.21 | - | - | |

Case 2 | 1 | 37 | 2.9 | −0.13 | 0.0001 | 0.01 | 0.008 | 0.75 |

0.1 | 361 | 5.4 | −0.13 | 0.0005 | 0.05 | 0.008 | 0.84 | |

0.01 | 3601 | 148.7 | −0.13 | 0.0006 | 0.05 | 0.008 | 0.85 | |

Case 3 | 1 | 42 | 1.5 | −0.01 | 0.007 | 0.27 | 0.004 | 0.20 |

0.1 | 411 | 4.6 | −0.01 | 0.007 | 0.28 | 0.001 | 0.04 | |

0.01 | 4101 | 338.8 | −0.01 | 0.008 | 0.28 | 0.001 | 0.03 |

- | Case 1 | Case 2 | Case 3 | ||
---|---|---|---|---|---|

r-NGM(1,1) | r | −0.017 | −0.13 | −0.01 | |

Modeling data | RMSE | 1.5531 | 0.0001 | 0.0064 | |

APD (%) | 0.2159 | 0.0037 | 0.2744 | ||

Predicting data | RMSE | - | 0.0074 | 0.0043 | |

APD (%) | - | 0.7418 | 0.1979 | ||

ANN | Number of hidden nodes | 2 | 2 | 4 | |

Modeling data | RMSE | 1.3495 | 0 | 0.0006 | |

APD (%) | 0.1868 | 0 | 0.0252 | ||

Predicting data | RMSE | - | 0.0042 | 0.0025 | |

APD (%) | - | 0.4210 | 0.1146 |

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**MDPI and ACS Style**

Shen, Y.; He, B.; Qin, P.
Fractional-Order Grey Prediction Method for Non-Equidistant Sequences. *Entropy* **2016**, *18*, 227.
https://doi.org/10.3390/e18060227

**AMA Style**

Shen Y, He B, Qin P.
Fractional-Order Grey Prediction Method for Non-Equidistant Sequences. *Entropy*. 2016; 18(6):227.
https://doi.org/10.3390/e18060227

**Chicago/Turabian Style**

Shen, Yue, Bo He, and Ping Qin.
2016. "Fractional-Order Grey Prediction Method for Non-Equidistant Sequences" *Entropy* 18, no. 6: 227.
https://doi.org/10.3390/e18060227