A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production
Abstract
1. Introduction
2. Literature Review
2.1. Introduction to Three Types of Natural Gas Production Prediction Methods
2.2. Research on Grey Prediction Model and Its Application in Natural Gas Production Prediction
2.3. Research Limitations, Contributions, and Structure of This Paper
- Given the spatiotemporal and nonlinear characteristics of natural gas production data, this paper utilizes the advantages of partial differentiation to effectively capture details and features in the data. Fractional order damping accumulation can improve model accuracy and effectively compensate for the phenomenon of inaccurate results caused by data fluctuations. A new fractional order multivariate biased grey prediction model is established.
- In terms of model structure, the classic grey prediction model is extended from ordinary differential form to partial differential form, and the fractional order accumulation principle is integrated into the partial grey prediction model to expand the structure of the classic grey prediction model. This improvement enables the model to more effectively capture various complex features such as time and space of data, improve model accuracy, and greatly broaden the structural framework and scope of application of the grey prediction model.
- In terms of application practice, this study applies the newly constructed partial grey prediction to the field of natural gas production forecasting. Based on the selection of oil, raw coal, and electricity production as the relevant series, the validity of the model is analysed in-depth through seven specific cases in three categories. The results showed that the average relative error of the new model was around 1% in all seven cases, and its prediction performance is significantly better than the other five grey prediction models. In addition, the model has successfully achieved an accurate forecast of natural gas production for the next nine months.
3. Modelling Partial Grey Differential Equations in First-Order Individual Variables
3.1. Grey Differential Equation Model with First-Order Individual Variables
3.2. Modelling Partial Grey Differential Equations in First-Order Individual Variables
- (1)
- (2)
3.3. Modelling Steps and Modelling Process for the DPGMC(1,N,ζ)
4. Effectiveness Analysis of DPGMC(1,N,ζ)
4.1. Analysis of the Effectiveness of the First Category DPGMC(1,N,ζ)
4.2. Effectiveness Analysis of the Second Category DPGMC(1,N,ζ)
4.3. Effectiveness Analysis of the Third Category DPGMC(1,N,ζ)
4.4. Summary of Effectiveness Analysis of Three Categories of DPGMC(1,N,ζ)
5. Application of DPGMC(1,N,ζ)
- (1)
- Dynamically adjust production plans and conduct equipment maintenance in advance from January to February before peak periods occur in these months to ensure that production equipment is in optimal condition during peak months. At the same time, utilise low season months to conduct in-depth maintenance and technological upgrades on equipment. Improve the efficiency of natural gas exploration, control open costs, and optimise development plans.
- (2)
- Data-driven decision-making, dynamically guiding and adjusting production strategies based on forecast results, establishing emergency data-driven mechanisms with surrounding provinces, coordinating exports in months of excess production, and complementing supply in months of shortage. At the same time, responding to seasonal fluctuations in natural gas demand based on forecast results, enhancing natural gas peak shaving capabilities, and ensuring stable supply.
- (3)
- Promote the complementary and coordinated development of natural gas and other energy sources. With the development of clean energy in Qinghai Province, the proportion of natural gas in the energy structure may change. Therefore, the energy structure should be adjusted promptly according to market demand, and the production forecast results provide an important basis for the development planning and investment decisions of the shale gas industry.
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Number | Abbreviation | Definition |
---|---|---|
1 | GM(1,1) | Grey model with one variable and one first order equation |
2 | GMC(1,N) | A first-order n-variable grey differential equation model [34] |
3 | GM(1,N) | Grey model of the first order with n variables [35] |
4 | GMVM(1,N) | The grey multivariable Verhulst model [36] |
5 | NSGM(1,N) | The new structured grey model [37] |
6 | NMGM(1,N) | Novel multi-variable grey model [38] |
7 | DPGMC(1,N,ζ) | The damping fractional order multivariate partial grey prediction model |
Month | Natural Gas Production | Crude Oil Production | Raw Coal Production | Electric Power Generation |
---|---|---|---|---|
Mar. 2021 | 5.6 | 19.9 | 76.4 | 65.6 |
Apr. 2021 | 5.4 | 19.2 | 83.8 | 84.4 |
May. 2021 | 5.5 | 19.9 | 88.3 | 70.3 |
Jun. 2021 | 5.3 | 19.2 | 102.9 | 72.9 |
Jul. 2021 | 5.4 | 19.9 | 86.4 | 89.6 |
Aug. 2021 | 4.7 | 19.9 | 84.0 | 84.5 |
Sep. 2021 | 4.4 | 19.2 | 98.9 | 67.1 |
Oct. 2021 | 4.7 | 19.9 | 116.7 | 72.3 |
Nov. 2021 | 5.1 | 19.2 | 111.5 | 77.3 |
Mar. 2022 | 5.2 | 20.0 | 67.5 | 73.4 |
Apr. 2022 | 5.0 | 19.4 | 53.5 | 78.8 |
May. 2022 | 5.2 | 20.0 | 42.1 | 70.2 |
Jun. 2022 | 5.0 | 19.4 | 73.3 | 77.7 |
Jul. 2022 | 4.8 | 20.0 | 75.1 | 82.8 |
Aug. 2022 | 5.1 | 20.0 | 75.2 | 71.2 |
Sep. 2022 | 4.9 | 19.3 | 74.9 | 55.5 |
Oct. 2022 | 5.1 | 19.9 | 76.5 | 64.0 |
Nov. 2022 | 4.8 | 19.3 | 97.1 | 70.9 |
Mar. 2023 | 5.1 | 19.5 | 67.0 | 76.7 |
Apr. 2023 | 4.9 | 19.2 | 70.7 | 76.7 |
May. 2023 | 5.1 | 20.0 | 63.4 | 73.1 |
Jun. 2023 | 4.9 | 19.5 | 60.9 | 75.9 |
Jul. 2023 | 4.8 | 20.0 | 63.7 | 82.0 |
Aug. 2023 | 5.0 | 20.3 | 67.3 | 82.0 |
Sep. 2023 | 4.9 | 19.4 | 70.5 | 70.2 |
Oct. 2023 | 5.1 | 19.9 | 69.3 | 68.7 |
Nov. 2023 | 4.9 | 19.3 | 73.2 | 72.6 |
Month | ||||||||
---|---|---|---|---|---|---|---|---|
DPGMC(1,1,ζ) ζ = 0.0178 | GMC(1,1) | GM(1,1) | GMVM(1,1) | NSGM(1,1) | NMGM(1,1) | LSTM | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 4.5860 |
Apr. 2021 | 5.4 | 5.4000 | 5.3221 | 1.9171 | 5.3003 | 5.4420 | 5.6043 | 5.9785 |
May. 2021 | 5.5 | 5.5000 | 5.3027 | 4.7737 | 5.2531 | 5.3151 | 5.3132 | 4.5090 |
Jun. 2021 | 5.3 | 5.3000 | 5.2666 | 6.8709 | 5.2309 | 5.1924 | 5.1285 | 6.2710 |
Jul. 2021 | 5.4 | 5.4000 | 5.2309 | 6.6953 | 5.2087 | 5.1218 | 5.0441 | 4.5351 |
Aug. 2021 | 4.7 | 4.7000 | 5.2106 | 6.6617 | 5.2019 | 5.0701 | 4.9989 | 6.5843 |
Sep. 2021 | 4.4 | 4.4000 | 5.1794 | 7.5430 | 5.2437 | 5.0061 | 4.9543 | 4.9369 |
Oct. 2021 | 4.7 | 4.7000 | 5.1194 | 8.6381 | 5.3428 | 4.9279 | 4.9063 | 6.3443 |
Nov. 2021 | 5.1 | 5.1000 | 5.0469 | 8.2215 | 5.4148 | 4.8746 | 4.8854 | 4.8361 |
Mar. 2022 | 5.2 | 5.2211 | 5.0144 | 5.1461 | 5.2924 | 4.9015 | 4.9293 | 6.2642 |
Apr. 2022 | 5.0 | 4.9813 | 5.0312 | 4.0544 | 5.2393 | 4.9446 | 4.9720 | 5.0140 |
May. 2022 | 5.2 | 5.2277 | 5.0703 | 3.1641 | 5.1728 | 4.9963 | 5.0109 | 6.1109 |
Jun. 2022 | 5.0 | 5.0008 | 5.0937 | 5.2229 | 5.4071 | 4.9881 | 4.9935 | 5.1525 |
Jul. 2022 | 4.8 | 4.8233 | 5.0891 | 5.2986 | 5.4849 | 4.9789 | 4.9813 | 6.0543 |
Aug. 2022 | 5.1 | 5.0711 | 5.0827 | 5.2712 | 5.5599 | 4.9715 | 4.9743 | 5.0913 |
Sep. 2022 | 4.9 | 4.9099 | 5.0763 | 5.2260 | 5.6388 | 4.9661 | 4.9706 | 5.9866 |
Oct. 2022 | 5.1 | 5.1209 | 5.0687 | 5.3188 | 5.7489 | 4.9594 | 4.9665 | 5.1058 |
Nov. 2022 | 4.8 | 4.8377 | 5.0421 | 6.7273 | 6.1502 | 4.9219 | 4.9381 | 5.9004 |
Mar. 2023 | 5.1 | 5.2085 | 5.0224 | 4.6431 | 5.8386 | 4.9395 | 4.9599 | 5.1154 |
Apr. 2023 | 4.9 | 5.0095 | 5.0253 | 4.8921 | 5.9923 | 4.9475 | 4.9678 | 5.7983 |
May. 2023 | 5.1 | 5.2083 | 5.0312 | 4.3842 | 5.9578 | 4.9653 | 4.9815 | 5.1734 |
Jun. 2023 | 4.9 | 5.0088 | 5.0457 | 4.2087 | 5.9939 | 4.9831 | 4.9924 | 5.7112 |
Jul. 2023 | 4.8 | 4.8096 | 5.0604 | 4.3999 | 6.1411 | 4.9928 | 4.9952 | 5.2856 |
Aug. 2023 | 5.0 | 5.1101 | 5.0699 | 4.6470 | 6.3259 | 4.9948 | 4.9922 | 5.5445 |
Sep. 2023 | 4.9 | 4.9103 | 5.0738 | 4.8668 | 6.5241 | 4.9913 | 4.9864 | 5.3726 |
Oct. 2023 | 5.1 | 5.1103 | 5.0760 | 4.7834 | 6.6227 | 4.9904 | 4.9846 | 5.4490 |
Nov. 2023 | 4.9 | 4.8110 | 5.0761 | 5.0521 | 6.8713 | 4.9837 | 4.9787 | 5.4437 |
MAPE (%) | 0.9507 | 2.6117 | 10.6134 | 18.2251 | 2.3439 | 2.2724 | 11.0028 | |
MAE | 0.0474 | 0.1288 | 0.5293 | 0.9009 | 0.1174 | 0.1137 | 0.5439 | |
MSE | 0.0040 | 0.0232 | 0.6023 | 1.0985 | 0.0188 | 0.0174 | 0.4775 | |
0.7563 | −0.4029 | −35.4246 | −65.4012 | −0.1382 | −0.0490 | −27.8616 | ||
RMSE | 0.0635 | 0.1523 | 0.7763 | 1.0481 | 0.1372 | 0.1317 | 0.6910 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,1,ζ) ζ = 0.9000 | GMC(1,1) | GM(1,1) | GMVM(1,1) | NSGM(1,1) | NMGM(1,1) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 4.8564 | 4.2758 | 5.2770 | 5.3342 | 5.5273 |
May. 2021 | 5.5 | 5.5000 | 4.8806 | 5.1209 | 5.2388 | 5.2693 | 5.2927 |
Jun. 2021 | 5.3 | 5.3000 | 4.8934 | 4.8830 | 5.2080 | 5.1501 | 5.1076 |
Jul. 2021 | 5.4 | 5.4000 | 4.9088 | 5.0590 | 5.1871 | 5.1268 | 5.0621 |
Aug. 2021 | 4.7 | 4.7000 | 4.9387 | 5.0589 | 5.1739 | 5.1087 | 5.0372 |
Sep. 2021 | 4.4 | 4.4000 | 4.9378 | 4.8810 | 5.1647 | 5.0257 | 4.9672 |
Oct. 2021 | 4.7 | 4.7000 | 4.9427 | 5.0589 | 5.1702 | 5.0304 | 4.9850 |
Nov. 2021 | 5.1 | 5.1000 | 4.9408 | 4.8810 | 5.1738 | 4.9651 | 4.9386 |
Mar. 2022 | 5.2 | 5.1762 | 4.9484 | 5.0843 | 5.1980 | 4.9933 | 4.9773 |
Apr. 2022 | 5.0 | 4.9555 | 4.9545 | 4.9318 | 5.2151 | 4.9560 | 4.9504 |
May. 2022 | 5.2 | 5.1586 | 4.9641 | 5.0843 | 5.2552 | 4.9863 | 4.9838 |
Jun. 2022 | 5.0 | 4.9320 | 4.9665 | 4.9318 | 5.2833 | 4.9506 | 4.9540 |
Jul. 2022 | 4.8 | 4.7799 | 4.9732 | 5.0843 | 5.3409 | 4.9821 | 4.9858 |
Aug. 2022 | 5.1 | 5.0143 | 4.9938 | 5.0843 | 5.3951 | 5.0065 | 5.0032 |
Sep. 2022 | 4.9 | 4.8891 | 4.9859 | 4.9064 | 5.4345 | 4.9564 | 4.9566 |
Oct. 2022 | 5.1 | 5.0602 | 4.9820 | 5.0589 | 5.5176 | 4.9767 | 4.9792 |
Nov. 2022 | 4.8 | 4.8149 | 4.9742 | 4.9064 | 5.5675 | 4.9334 | 4.9434 |
Mar. 2023 | 5.1 | 5.0503 | 4.9595 | 4.9572 | 5.6506 | 4.9195 | 4.9398 |
Apr. 2023 | 4.9 | 4.8916 | 4.9433 | 4.8810 | 5.7183 | 4.8792 | 4.9138 |
May. 2023 | 5.1 | 5.0821 | 4.9503 | 5.0843 | 5.8455 | 4.9267 | 4.9637 |
Jun. 2023 | 4.9 | 4.9338 | 4.9594 | 4.9572 | 5.9153 | 4.9143 | 4.9510 |
Jul. 2023 | 4.8 | 4.7403 | 4.9704 | 5.0843 | 6.0462 | 4.9540 | 4.9841 |
Aug. 2023 | 5.0 | 5.0638 | 5.0019 | 5.1606 | 6.1770 | 5.0143 | 5.0264 |
Sep. 2023 | 4.9 | 4.8218 | 5.0031 | 4.9318 | 6.2298 | 4.9723 | 4.9774 |
Oct. 2023 | 5.1 | 5.0698 | 4.9978 | 5.0589 | 6.3844 | 4.9890 | 4.9906 |
Nov. 2023 | 4.9 | 4.7660 | 4.9863 | 4.9064 | 6.4587 | 4.9429 | 4.9497 |
MAPE (%) | 0.9192 | 2.3101 | 1.7746 | 14.4265 | 2.0889 | 2.1594 | |
MAE | 0.0458 | 0.1156 | 0.0878 | 0.7132 | 0.1048 | 0.1081 | |
MSE | 0.0031 | 0.0178 | 0.0147 | 0.7169 | 0.0154 | 0.0158 | |
0.8147 | −0.0752 | 0.1140 | −42.3333 | 0.0684 | 0.0422 | ||
RMSE | 0.0554 | 0.1334 | 0.1211 | 0.8467 | 0.1241 | 0.1259 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,1,ζ) ζ = 0.7151 | GMC(1,1) | GM(1,1) | GMVM(1,1) | NSGM(1,1) | NMGM(1,1) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 5.3818 | 3.7821 | 5.2706 | 5.3673 | 5.5417 |
May. 2021 | 5.5 | 5.5000 | 5.3835 | 5.3309 | 5.2321 | 5.3095 | 5.3183 |
Jun. 2021 | 5.3 | 5.3000 | 5.4212 | 5.2045 | 5.2021 | 5.2453 | 5.1753 |
Jul. 2021 | 5.4 | 5.4000 | 5.4070 | 6.1805 | 5.1916 | 5.0820 | 5.0129 |
Aug. 2021 | 4.7 | 4.7000 | 5.3547 | 5.7842 | 5.1833 | 4.9836 | 4.9419 |
Sep. 2021 | 4.4 | 4.4000 | 5.3628 | 4.5831 | 5.1605 | 5.0195 | 4.9826 |
Oct. 2021 | 4.7 | 4.7000 | 5.4103 | 4.9346 | 5.1671 | 5.0141 | 4.9818 |
Nov. 2021 | 5.1 | 5.1000 | 5.4318 | 5.2750 | 5.1877 | 4.9764 | 4.9578 |
Mar. 2022 | 5.2 | 5.1762 | 5.4507 | 5.0087 | 5.1964 | 4.9718 | 4.9621 |
Apr. 2022 | 5.0 | 4.9577 | 5.4680 | 5.3772 | 5.2405 | 4.9321 | 4.9392 |
May. 2022 | 5.2 | 5.1451 | 5.4945 | 4.7903 | 5.2394 | 4.9571 | 4.9664 |
Jun. 2022 | 5.0 | 4.9336 | 5.5285 | 5.3021 | 5.3119 | 4.9275 | 4.9468 |
Jul. 2022 | 4.8 | 4.7812 | 5.5279 | 5.6501 | 5.3952 | 4.8696 | 4.9113 |
Aug. 2022 | 5.1 | 5.0219 | 5.5444 | 4.8586 | 5.3794 | 4.8998 | 4.9454 |
Sep. 2022 | 4.9 | 4.8754 | 5.6423 | 3.7872 | 5.3050 | 5.0288 | 5.0394 |
Oct. 2022 | 5.1 | 5.0449 | 5.7727 | 4.3672 | 5.4103 | 5.0769 | 5.0542 |
Nov. 2022 | 4.8 | 4.8085 | 5.8687 | 4.8381 | 5.5285 | 5.0699 | 5.0303 |
Mar. 2023 | 5.1 | 5.0635 | 5.9352 | 5.2339 | 5.6617 | 5.0256 | 4.9890 |
Apr. 2023 | 4.9 | 4.8939 | 5.9897 | 5.2339 | 5.7493 | 4.9898 | 4.9648 |
May. 2023 | 5.1 | 5.1080 | 6.0590 | 4.9882 | 5.7938 | 4.9846 | 4.9676 |
Jun. 2023 | 4.9 | 4.9283 | 6.1373 | 5.1793 | 5.9263 | 4.9618 | 4.9561 |
Jul. 2023 | 4.8 | 4.7343 | 6.1965 | 5.5955 | 6.1299 | 4.9027 | 4.9205 |
Aug. 2023 | 5.0 | 5.0432 | 6.2420 | 5.5955 | 6.2570 | 4.8548 | 4.8997 |
Sep. 2023 | 4.9 | 4.8406 | 6.3247 | 4.7903 | 6.1621 | 4.8944 | 4.9433 |
Oct. 2023 | 5.1 | 5.0804 | 6.4550 | 4.6880 | 6.2352 | 4.9366 | 4.9758 |
Nov. 2023 | 4.9 | 4.7653 | 6.5900 | 4.9541 | 6.4264 | 4.9448 | 4.9765 |
MAPE (%) | 0.8624 | 18.4241 | 7.9274 | 14.1216 | 2.3354 | 2.3263 | |
MAE | 0.0430 | 0.9126 | 0.3934 | 0.6975 | 0.1170 | 0.1164 | |
MSE | 0.0028 | 1.0040 | 0.2438 | 0.6851 | 0.0193 | 0.0174 | |
0.8307 | −59.6868 | −13.7351 | −41.0168 | −0.1651 | −0.0527 | ||
RMSE | 0.0529 | 1.0020 | 0.4937 | 0.8337 | 0.1388 | 0.1320 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,2,ζ) ζ = 0.8734 | GMC(1,2) | GM(1,2) | GMVM(1,2) | NSGM(1,2) | NMGM(1,2) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 5.0003 | 4.2089 | 5.3210 | 5.4388 | 5.6012 |
May. 2021 | 5.5 | 5.5000 | 4.9732 | 5.2639 | 5.2600 | 5.3175 | 5.3160 |
Jun. 2021 | 5.3 | 5.3000 | 4.9329 | 5.0249 | 5.2614 | 5.1875 | 5.1240 |
Jul. 2021 | 5.4 | 5.4000 | 4.9091 | 5.1014 | 5.2382 | 5.1229 | 5.0459 |
Aug. 2021 | 4.7 | 4.7000 | 4.9198 | 5.0895 | 5.2459 | 5.0756 | 5.0043 |
Sep. 2021 | 4.4 | 4.4000 | 4.8962 | 4.9931 | 5.3993 | 5.0034 | 4.9510 |
Oct. 2021 | 4.7 | 4.7000 | 4.8589 | 5.2409 | 5.6992 | 4.9324 | 4.9104 |
Nov. 2021 | 5.1 | 5.1000 | 4.8177 | 5.0514 | 5.9142 | 4.8718 | 4.8820 |
Mar. 2022 | 5.2 | 5.2356 | 4.8299 | 5.0367 | 5.4608 | 4.9050 | 4.9324 |
Apr. 2022 | 5.0 | 4.9979 | 4.8788 | 4.8302 | 5.2478 | 4.9413 | 4.9684 |
May. 2022 | 5.2 | 5.2430 | 4.9390 | 4.9192 | 4.9400 | 4.9979 | 5.0126 |
Jun. 2022 | 5.0 | 5.0176 | 4.9660 | 4.9219 | 5.6115 | 4.9845 | 4.9901 |
Jul. 2022 | 4.8 | 4.8393 | 4.9707 | 5.0719 | 5.7166 | 4.9821 | 4.9849 |
Aug. 2022 | 5.1 | 5.0903 | 4.9883 | 5.0724 | 5.8207 | 4.9800 | 4.9818 |
Sep. 2022 | 4.9 | 4.9238 | 4.9804 | 4.9056 | 5.9710 | 4.9662 | 4.9689 |
Oct. 2022 | 5.1 | 5.1334 | 4.9744 | 5.0548 | 6.1166 | 4.9638 | 4.9694 |
Nov. 2022 | 4.8 | 4.8470 | 4.9489 | 5.0084 | 7.2429 | 4.9200 | 4.9349 |
Mar. 2023 | 5.1 | 5.1215 | 4.9303 | 4.9163 | 6.0758 | 4.9346 | 4.9550 |
Apr. 2023 | 4.9 | 4.9847 | 4.9268 | 4.8626 | 6.3920 | 4.9350 | 4.9571 |
May. 2023 | 5.1 | 5.1087 | 4.9451 | 5.0178 | 5.9908 | 4.9608 | 4.9803 |
Jun. 2023 | 4.9 | 4.9503 | 4.9705 | 4.8881 | 5.9415 | 4.9758 | 4.9883 |
Jul. 2023 | 4.8 | 4.7979 | 4.9933 | 5.0192 | 6.0961 | 4.9925 | 4.9977 |
Aug. 2023 | 5.0 | 5.0994 | 5.0289 | 5.1067 | 6.3659 | 5.0055 | 5.0036 |
Sep. 2023 | 4.9 | 4.9263 | 5.0318 | 4.9089 | 6.8499 | 4.9947 | 4.9884 |
Oct. 2023 | 5.1 | 5.1124 | 5.0261 | 5.0215 | 6.8014 | 4.9970 | 4.9893 |
Nov. 2023 | 4.9 | 4.8727 | 5.0130 | 4.8978 | 7.3875 | 4.9822 | 4.9752 |
MAPE (%) | 0.6530 | 2.6438 | 2.2076 | 23.3142 | 2.3165 | 2.2437 | |
MAE | 0.0325 | 0.1326 | 0.1101 | 1.1527 | 0.1161 | 0.1122 | |
MSE | 0.0017 | 0.0244 | 0.0205 | 1.7536 | 0.0185 | 0.0171 | |
0.8970 | −0.4751 | −0.2371 | −105.0031 | −0.1202 | −0.0317 | ||
RMSE | 0.0413 | 0.1562 | 0.1431 | 1.3242 | 0.1361 | 0.1306 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,2,ζ) ζ = 0.6213 | GMC(1,2) | GM(1,2) | GMVM(1,2) | NSGM(1,2) | NMGM(1,2) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 5.4719 | 3.3740 | 5.3189 | 5.4456 | 5.6010 |
May. 2021 | 5.5 | 5.5000 | 5.4715 | 5.5741 | 5.2632 | 5.3542 | 5.3438 |
Jun. 2021 | 5.3 | 5.3000 | 5.4834 | 5.8037 | 5.2615 | 5.2487 | 5.1658 |
Jul. 2021 | 5.4 | 5.4000 | 5.4570 | 6.2538 | 5.2171 | 5.0943 | 5.0156 |
Aug. 2021 | 4.7 | 4.7000 | 5.4158 | 5.8223 | 5.2126 | 4.9984 | 4.9470 |
Sep. 2021 | 4.4 | 4.4000 | 5.4096 | 5.0243 | 5.3805 | 4.9945 | 4.9539 |
Oct. 2021 | 4.7 | 4.7000 | 5.4072 | 5.5266 | 5.6485 | 4.9426 | 4.9209 |
Nov. 2021 | 5.1 | 5.1000 | 5.3732 | 5.7196 | 5.8065 | 4.8801 | 4.8884 |
Mar. 2022 | 5.2 | 5.2598 | 5.3676 | 4.9060 | 5.4310 | 4.9025 | 4.9281 |
Apr. 2022 | 5.0 | 5.0225 | 5.4018 | 5.0076 | 5.1809 | 4.9089 | 4.9466 |
May. 2022 | 5.2 | 5.2562 | 5.4628 | 4.3851 | 4.9910 | 4.9732 | 5.0006 |
Jun. 2022 | 5.0 | 5.0385 | 5.5195 | 5.2172 | 5.5049 | 4.9486 | 4.9736 |
Jul. 2022 | 4.8 | 4.8683 | 5.5293 | 5.5186 | 5.5480 | 4.8995 | 4.9371 |
Aug. 2022 | 5.1 | 5.1159 | 5.5509 | 4.8901 | 5.7919 | 4.9207 | 4.9574 |
Sep. 2022 | 4.9 | 4.9391 | 5.6322 | 4.0335 | 6.1672 | 5.0212 | 5.0268 |
Oct. 2022 | 5.1 | 5.1402 | 5.7367 | 4.5169 | 6.2535 | 5.0562 | 5.0358 |
Nov. 2022 | 4.8 | 4.8658 | 5.8016 | 5.1721 | 7.1353 | 5.0236 | 4.9950 |
Mar. 2023 | 5.1 | 5.1670 | 5.8501 | 5.0770 | 5.9739 | 5.0025 | 4.9803 |
Apr. 2023 | 4.9 | 5.0092 | 5.9097 | 5.1274 | 6.2233 | 4.9808 | 4.9678 |
May. 2023 | 5.1 | 5.1714 | 5.9840 | 4.8325 | 6.0322 | 4.9909 | 4.9809 |
Jun. 2023 | 4.9 | 4.9893 | 6.0738 | 4.9505 | 5.8608 | 4.9874 | 4.9811 |
Jul. 2023 | 4.8 | 4.8118 | 6.1524 | 5.3199 | 5.8574 | 4.9489 | 4.9562 |
Aug. 2023 | 5.0 | 5.1194 | 6.2194 | 5.3689 | 6.1158 | 4.9133 | 4.9376 |
Sep. 2023 | 4.9 | 4.9502 | 6.3112 | 4.7718 | 6.8594 | 4.9428 | 4.9661 |
Oct. 2023 | 5.1 | 5.1529 | 6.4382 | 4.6740 | 6.9831 | 4.9762 | 4.9898 |
Nov. 2023 | 4.9 | 4.8786 | 6.5692 | 4.9389 | 7.2853 | 4.9781 | 4.9858 |
MAPE (%) | 1.1135 | 17.6279 | 6.8397 | 22.2456 | 2.4293 | 2.3169 | |
MAE | 0.0555 | 0.8728 | 0.3408 | 1.1007 | 0.1216 | 0.1158 | |
MSE | 0.0039 | 0.9352 | 0.1844 | 1.6421 | 0.0193 | 0.0170 | |
0.7623 | −55.5327 | −10.1480 | −98.2617 | −0.1689 | −0.0291 | ||
RMSE | 0.0627 | 0.9671 | 0.4294 | 1.2814 | 0.1391 | 0.1305 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,2,ζ) ζ = 0.4363 | GMC(1,2) | GM(1,2) | GMVM(1,2) | NSGM(1,2) | NMGM(1,2) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 5.0016 | 4.2726 | 5.2852 | 5.3840 | 5.5468 |
May. 2021 | 5.5 | 5.5000 | 4.9679 | 5.0992 | 5.2560 | 5.3742 | 5.3696 |
Jun. 2021 | 5.3 | 5.3000 | 4.9701 | 4.8934 | 5.2252 | 5.2243 | 5.1514 |
Jul. 2021 | 5.4 | 5.4000 | 4.9009 | 5.2378 | 5.1370 | 5.0864 | 5.0153 |
Aug. 2021 | 4.7 | 4.7000 | 4.8372 | 5.1756 | 5.0911 | 5.0140 | 4.9633 |
Sep. 2021 | 4.4 | 4.4000 | 4.8439 | 4.8180 | 5.1755 | 4.9744 | 4.9398 |
Oct. 2021 | 4.7 | 4.7000 | 4.8912 | 5.0271 | 5.1695 | 5.0204 | 4.9900 |
Nov. 2021 | 5.1 | 5.1000 | 4.8846 | 4.9421 | 5.0545 | 4.8971 | 4.8953 |
Mar. 2022 | 5.2 | 5.2054 | 4.8899 | 5.0613 | 5.1767 | 4.9652 | 4.9702 |
Apr. 2022 | 5.0 | 4.9861 | 4.8863 | 5.0020 | 5.0031 | 4.8739 | 4.9017 |
May. 2022 | 5.2 | 5.1713 | 4.9067 | 5.0224 | 5.3136 | 4.9719 | 4.9931 |
Jun. 2022 | 5.0 | 4.9622 | 4.9145 | 4.9886 | 5.0297 | 4.8884 | 4.9221 |
Jul. 2022 | 4.8 | 4.8054 | 4.8807 | 5.1757 | 4.8970 | 4.8819 | 4.9305 |
Aug. 2022 | 5.1 | 5.0522 | 4.9063 | 5.0346 | 5.4149 | 4.9704 | 5.0046 |
Sep. 2022 | 4.9 | 4.8996 | 4.9980 | 4.6977 | 6.2207 | 5.0496 | 5.0474 |
Oct. 2022 | 5.1 | 5.0727 | 5.0891 | 4.9261 | 6.1322 | 5.1496 | 5.1046 |
Nov. 2022 | 4.8 | 4.8351 | 5.0995 | 4.8850 | 5.7034 | 5.0726 | 5.0164 |
Mar. 2023 | 5.1 | 5.1278 | 5.0490 | 4.9973 | 5.4233 | 4.9968 | 4.9558 |
Apr. 2023 | 4.9 | 4.9455 | 4.9837 | 4.9348 | 5.3394 | 4.8825 | 4.8781 |
May. 2023 | 5.1 | 5.1576 | 4.9718 | 5.0577 | 6.0024 | 4.9555 | 4.9616 |
Jun. 2023 | 4.9 | 4.9664 | 4.9686 | 4.9876 | 5.6272 | 4.9067 | 4.9273 |
Jul. 2023 | 4.8 | 4.7729 | 4.9363 | 5.1660 | 5.2298 | 4.9035 | 4.9384 |
Aug. 2023 | 5.0 | 5.0747 | 4.9219 | 5.2285 | 5.3558 | 4.9525 | 4.9861 |
Sep. 2023 | 4.9 | 4.8775 | 4.9295 | 4.8974 | 6.4082 | 4.9332 | 4.9624 |
Oct. 2023 | 5.1 | 5.1000 | 4.9708 | 4.9833 | 7.0872 | 5.0156 | 5.0252 |
Nov. 2023 | 4.9 | 4.7903 | 4.9833 | 4.9057 | 6.3451 | 4.9484 | 4.9582 |
MAPE (%) | 0.7072 | 2.5213 | 2.4869 | 13.3693 | 2.1880 | 2.0976 | |
MAE | 0.0352 | 0.1263 | 0.1233 | 0.6642 | 0.1096 | 0.1048 | |
MSE | 0.0020 | 0.0237 | 0.0275 | 0.7785 | 0.0174 | 0.0155 | |
0.8785 | −0.4299 | −0.6649 | −46.0577 | −0.0531 | 0.0615 | ||
RMSE | 0.0448 | 0.1538 | 0.1660 | 0.8823 | 0.1320 | 0.1246 |
Month | |||||||
---|---|---|---|---|---|---|---|
DPGMC(1,3,ζ) ζ = 0.9900 | GMC(1,3) | GM(1,3) | GMVM(1,3) | NSGM(1,3) | NMGM(1,3) | ||
Mar. 2021 | 5.6 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 | 5.6000 |
Apr. 2021 | 5.4 | 5.4000 | 5.0734 | 4.2107 | 5.3169 | 5.4243 | 5.5829 |
May. 2021 | 5.5 | 5.5000 | 5.0184 | 5.2433 | 5.2642 | 5.3824 | 5.3694 |
Jun. 2021 | 5.3 | 5.3000 | 4.9901 | 5.0157 | 5.2599 | 5.2324 | 5.1535 |
Jul. 2021 | 5.4 | 5.4000 | 4.9095 | 5.1678 | 5.2066 | 5.0926 | 5.0165 |
Aug. 2021 | 4.7 | 4.7000 | 4.8462 | 5.1325 | 5.1962 | 5.0145 | 4.9596 |
Sep. 2021 | 4.4 | 4.4000 | 4.8345 | 4.9569 | 5.3640 | 4.9728 | 4.9356 |
Oct. 2021 | 4.7 | 4.7000 | 4.8469 | 5.2098 | 5.6078 | 4.9779 | 4.9483 |
Nov. 2021 | 5.1 | 5.1000 | 4.8149 | 5.0582 | 5.7339 | 4.8635 | 4.8710 |
Mar. 2022 | 5.2 | 5.2238 | 4.8217 | 5.0325 | 5.4092 | 4.9267 | 4.9451 |
Apr. 2022 | 5.0 | 4.9915 | 4.8455 | 4.8682 | 5.1519 | 4.8759 | 4.9181 |
May. 2022 | 5.2 | 5.2089 | 4.8953 | 4.9115 | 5.0251 | 4.9765 | 5.0058 |
Jun. 2022 | 5.0 | 5.0082 | 4.9183 | 4.9453 | 5.4484 | 4.9108 | 4.9473 |
Jul. 2022 | 4.8 | 4.8182 | 4.8896 | 5.1092 | 5.4635 | 4.8955 | 4.9408 |
Aug. 2022 | 5.1 | 5.0906 | 4.9118 | 5.0540 | 5.7654 | 4.9629 | 4.9931 |
Sep. 2022 | 4.9 | 4.8925 | 4.9881 | 4.8234 | 6.2331 | 5.0395 | 5.0370 |
Oct. 2022 | 5.1 | 5.1072 | 5.0653 | 5.0028 | 6.2924 | 5.1176 | 5.0764 |
Nov. 2022 | 4.8 | 4.8299 | 5.0644 | 4.9896 | 7.0310 | 5.0450 | 4.9984 |
Mar. 2023 | 5.1 | 5.0779 | 5.0188 | 4.9363 | 5.9176 | 4.9909 | 4.9609 |
Apr. 2023 | 4.9 | 4.9408 | 4.9698 | 4.8857 | 6.1296 | 4.9064 | 4.9083 |
May. 2023 | 5.1 | 5.1072 | 4.9679 | 5.0140 | 6.0482 | 4.9669 | 4.9718 |
Jun. 2023 | 4.9 | 4.9078 | 4.9770 | 4.9071 | 5.8288 | 4.9364 | 4.9527 |
Jul. 2023 | 4.8 | 4.7862 | 4.9588 | 5.0580 | 5.7566 | 4.9298 | 4.9554 |
Aug. 2023 | 5.0 | 5.0431 | 4.9532 | 5.1390 | 6.0054 | 4.9595 | 4.9823 |
Sep. 2023 | 4.9 | 4.9376 | 4.9596 | 4.8976 | 6.8459 | 4.9506 | 4.9712 |
Oct. 2023 | 5.1 | 5.1095 | 4.9930 | 4.9954 | 7.0521 | 5.0171 | 5.0183 |
Nov. 2023 | 4.9 | 4.8625 | 5.0012 | 4.8984 | 7.2136 | 4.9670 | 4.9705 |
MAPE (%) | 0.3821 | 2.6781 | 2.3823 | 21.5136 | 2.2198 | 2.1290 | |
MAE | 0.0189 | 0.1343 | 0.1188 | 1.0649 | 0.1111 | 0.1064 | |
MSE | 0.0005 | 0.0265 | 0.0229 | 1.5653 | 0.0177 | 0.0157 | |
0.9683 | −0.5996 | −0.3883 | −93.6203 | −0.0716 | 0.0533 | ||
RMSE | 0.0229 | 0.1627 | 0.1513 | 1.2511 | 0.1331 | 0.1251 |
Month | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|
Nature Gas Production | 7.6703 | 7.0827 | 7.3308 | 7.3726 | 7.2158 | 7.1693 | 7.2791 | 7.5310 | 7.3757 |
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Share and Cite
Li, H.; Duan, H.; Chen, H. A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production. Fractal Fract. 2025, 9, 422. https://doi.org/10.3390/fractalfract9070422
Li H, Duan H, Chen H. A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production. Fractal and Fractional. 2025; 9(7):422. https://doi.org/10.3390/fractalfract9070422
Chicago/Turabian StyleLi, Hui, Huiming Duan, and Hongli Chen. 2025. "A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production" Fractal and Fractional 9, no. 7: 422. https://doi.org/10.3390/fractalfract9070422
APA StyleLi, H., Duan, H., & Chen, H. (2025). A Novel Fractional Order Multivariate Partial Grey Model and Its Application in Natural Gas Production. Fractal and Fractional, 9(7), 422. https://doi.org/10.3390/fractalfract9070422