Visibility Graph Power Geometric Aggregation Operator and Its Application in Water, Energy and Food Efficiency Evaluation
Abstract
:1. Introduction
- (1)
- How to determine the weights of the time series values and that the weights are obtained in a more objective way?
- (2)
- How to consider the relationships of in time series values?
- (1)
- A new aggregation operator, namely the visibility graph power geometric (VGPG) aggregation operator, is proposed to aggregate the time series values.
- (2)
- A new support function is proposed to denote the support of the two values when they have visibility.
- (3)
- Based on the support function and visibility matrix, a new objective weight determination method for the VGPG operator is developed.
2. Preliminaries
2.1. The OWA, PA, and PG Operators
- (1)
- Sup(a,b) ∈ [0,1];
- (2)
- Sup(a,b) = Sup(b,a);
- (3)
- Sup(a,b) ≥ Sup(x,y), if |a − b|<|x − y|.
2.2. The Visibility Graph
- (1)
- Connected: each node connects with its nearest left and right neighbor nodes.
- (2)
- Undirected: there is no direction defined in the links.
- (3)
- Invariant under affine transformations of the series data: the visibility criteria is invariant under the rescaling of both the horizontal and vertical axes.
2.3. Visibility Graph Power Averaging Operator
3. Visibility Graph Power Geometric Operator
4. Application in Water, Energy, and Food Efficiency Evaluation
4.1. The Case Study
4.2. Comparative Analysis
5. Conclusions
- (1)
- The time series data are transformed into a VG, and a visibility matrix is developed to denote the links of different data, while the other methods do not consider the relationships of different data.
- (2)
- The support function is developed to measure the similarity of two linked values, while the power aggregation measures the similarity between any two values. It does not consider whether the two values are linked.
- (3)
- The weights determined by the VG and support function are more objective and reasonable, while the weights of OWA obtained by various methods are stationary when the parameters are specified. These methods do not consider the relationship of the input arguments.
Author Contributions
Funding
Conflicts of Interest
References
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Province | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 |
---|---|---|---|---|---|---|---|---|---|---|
Beijing | 0.239 | 0.336 | 0.335 | 0.412 | 0.36 | 0.16 | 0.184 | 0.087 | 0.132 | 0.122 |
Tianjin | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.277 | 0.308 | 0.348 |
Hebei | 0.726 | 0.672 | 0.987 | 0.761 | 0.917 | 0.501 | 0.517 | 0.174 | 0.17 | 0.159 |
Shanxi | 1 | 1 | 1 | 0.817 | 0.941 | 0.896 | 0.942 | 1 | 1 | 1 |
InnerMongolia | 0.546 | 0.584 | 0.554 | 0.462 | 0.593 | 0.642 | 0.604 | 0.946 | 0.991 | 0.961 |
Liaoning | 0.517 | 0.527 | 0.48 | 0.359 | 0.497 | 0.298 | 0.32 | 0.155 | 0.182 | 0.194 |
Jilin | 0.39 | 0.358 | 0.36 | 0.245 | 0.383 | 0.282 | 0.18 | 0.105 | 0.102 | 0.094 |
HeiLongJiang | 0.728 | 0.663 | 0.655 | 0.493 | 0.571 | 0.373 | 0.31 | 0.206 | 0.206 | 0.189 |
Shanghai | 0.607 | 0.693 | 0.551 | 0.674 | 1 | 0.371 | 0.42 | 0.176 | 0.175 | 0.157 |
Jiangsu | 1 | 1 | 1 | 1 | 1 | 0.713 | 0.524 | 0.209 | 0.204 | 0.242 |
Zhejiang | 0.597 | 0.79 | 0.907 | 0.966 | 0.939 | 0.427 | 0.423 | 0.162 | 0.2 | 0.197 |
Anhui | 0.324 | 0.438 | 0.582 | 0.487 | 0.615 | 0.403 | 0.353 | 0.348 | 0.365 | 0.346 |
Fujian | 0.399 | 0.379 | 0.399 | 0.348 | 0.438 | 0.346 | 0.387 | 0.164 | 0.174 | 0.175 |
Jiangxi | 0.511 | 0.558 | 0.395 | 0.472 | 0.48 | 0.183 | 0.161 | 0.126 | 0.099 | 0.115 |
Shandong | 1 | 1 | 1 | 0.961 | 0.963 | 0.541 | 0.549 | 0.267 | 0.264 | 0.262 |
Henan | 0.583 | 0.544 | 0.577 | 0.695 | 0.788 | 0.314 | 0.293 | 0.267 | 0.193 | 0.161 |
Hebei | 0.633 | 0.758 | 0.69 | 0.665 | 0.706 | 0.279 | 0.268 | 0.174 | 0.14 | 0.132 |
Hunan | 0.39 | 0.374 | 0.476 | 0.424 | 0.426 | 0.2 | 0.176 | 0.132 | 0.105 | 0.099 |
Guangdong | 0.763 | 0.757 | 0.697 | 0.669 | 0.86 | 0.453 | 0.281 | 0.144 | 0.144 | 0.138 |
Guangxi | 0.345 | 0.505 | 0.501 | 0.484 | 0.467 | 0.261 | 0.181 | 0.101 | 0.085 | 0.084 |
Hainan | 0.618 | 0.561 | 0.449 | 0.467 | 0.36 | 0.197 | 0.215 | 0.084 | 0.096 | 0.106 |
Chongqing | 0.289 | 0.372 | 0.341 | 0.302 | 0.343 | 0.182 | 0.198 | 0.15 | 0.166 | 0.169 |
Sichuan | 0.523 | 0.592 | 0.606 | 0.49 | 0.484 | 0.382 | 0.274 | 0.164 | 0.167 | 0.188 |
Guizhou | 0.677 | 0.668 | 0.788 | 0.573 | 0.426 | 0.433 | 0.271 | 0.392 | 0.362 | 0.368 |
Yunnan | 0.388 | 0.481 | 0.46 | 0.389 | 0.445 | 0.448 | 0.48 | 0.228 | 0.219 | 0.248 |
Xizang | 0.118 | 0.097 | 0.088 | 0.07 | 0.083 | 0.067 | 0.052 | 0.019 | 0.025 | 0.044 |
Shaanxi | 0.77 | 0.781 | 0.897 | 0.829 | 0.857 | 1 | 1 | 1 | 1 | 1 |
Gansu | 0.438 | 0.436 | 0.323 | 0.251 | 0.351 | 0.334 | 0.273 | 0.124 | 0.126 | 0.124 |
Qinghai | 0.897 | 1 | 1 | 1 | 1 | 1 | 1 | 0.363 | 0.269 | 0.292 |
Ningxia | 0.524 | 0.566 | 0.492 | 0.471 | 0.595 | 0.603 | 0.704 | 0.426 | 0.407 | 0.468 |
Xinjiang | 0.929 | 1 | 1 | 0.763 | 0.819 | 1 | 1 | 1 | 1 | 1 |
Province | w1 | w2 | w3 | w4 | w5 | w6 | w7 | w8 | w9 | w10 |
---|---|---|---|---|---|---|---|---|---|---|
Beijing | 0.0494 | 0.0992 | 0.0758 | 0.1169 | 0.1325 | 0.072 | 0.145 | 0.0741 | 0.121 | 0.1141 |
Tianjin | 0.0676 | 0.1014 | 0.1014 | 0.1014 | 0.1014 | 0.1014 | 0.0991 | 0.1073 | 0.1094 | 0.1094 |
Hebei | 0.0813 | 0.0796 | 0.125 | 0.0792 | 0.1348 | 0.0777 | 0.1373 | 0.0803 | 0.1178 | 0.087 |
Shanxi | 0.0519 | 0.0779 | 0.148 | 0.0699 | 0.1483 | 0.0988 | 0.1256 | 0.15 | 0.0779 | 0.0519 |
InnerMongolia | 0.0634 | 0.1362 | 0.1326 | 0.0687 | 0.1341 | 0.1352 | 0.0649 | 0.1436 | 0.0724 | 0.0488 |
Liaoning | 0.052 | 0.1022 | 0.0996 | 0.0716 | 0.1781 | 0.0725 | 0.1402 | 0.0733 | 0.1177 | 0.0929 |
Jilin | 0.0826 | 0.0829 | 0.1015 | 0.0578 | 0.1652 | 0.1104 | 0.098 | 0.0939 | 0.1146 | 0.0931 |
HeiLongJiang | 0.0675 | 0.069 | 0.1102 | 0.0651 | 0.1478 | 0.1035 | 0.1261 | 0.0833 | 0.1261 | 0.1013 |
Shanghai | 0.0715 | 0.1261 | 0.0932 | 0.1001 | 0.1368 | 0.0658 | 0.131 | 0.0831 | 0.1109 | 0.0815 |
Jiangsu | 0.0571 | 0.0856 | 0.0856 | 0.0856 | 0.1051 | 0.1011 | 0.1261 | 0.1041 | 0.1236 | 0.1261 |
Zhejiang | 0.0589 | 0.0877 | 0.092 | 0.095 | 0.1129 | 0.081 | 0.1555 | 0.088 | 0.1303 | 0.0987 |
Anhui | 0.0633 | 0.066 | 0.1076 | 0.0668 | 0.1584 | 0.1118 | 0.1124 | 0.1121 | 0.1363 | 0.0653 |
Fujian | 0.1043 | 0.0783 | 0.1294 | 0.0756 | 0.1505 | 0.0759 | 0.1392 | 0.0732 | 0.0999 | 0.0737 |
Jiangxi | 0.0453 | 0.1074 | 0.0641 | 0.0889 | 0.1439 | 0.0839 | 0.1281 | 0.1062 | 0.1048 | 0.1275 |
Shandong | 0.0561 | 0.0842 | 0.1101 | 0.0831 | 0.1326 | 0.0721 | 0.1326 | 0.1041 | 0.1126 | 0.1124 |
Henan | 0.1158 | 0.1132 | 0.0959 | 0.113 | 0.1434 | 0.0625 | 0.0863 | 0.1067 | 0.0824 | 0.0808 |
Hebei | 0.0455 | 0.1132 | 0.0945 | 0.0933 | 0.1543 | 0.0622 | 0.1261 | 0.1044 | 0.1035 | 0.103 |
Hunan | 0.069 | 0.0686 | 0.1122 | 0.0702 | 0.1555 | 0.0655 | 0.1318 | 0.1096 | 0.109 | 0.1085 |
Guangdong | 0.0671 | 0.1098 | 0.0868 | 0.0855 | 0.146 | 0.1039 | 0.1119 | 0.0889 | 0.1119 | 0.0883 |
Guangxi | 0.045 | 0.0694 | 0.0729 | 0.0726 | 0.1311 | 0.1067 | 0.1311 | 0.1106 | 0.1304 | 0.1303 |
Hainan | 0.0657 | 0.0879 | 0.0675 | 0.1564 | 0.1192 | 0.0663 | 0.1465 | 0.0672 | 0.1229 | 0.1004 |
Chongqing | 0.0461 | 0.0928 | 0.0945 | 0.0703 | 0.1944 | 0.0679 | 0.1379 | 0.0901 | 0.1148 | 0.0913 |
Sichuan | 0.0467 | 0.0706 | 0.1047 | 0.0696 | 0.1247 | 0.106 | 0.111 | 0.0934 | 0.1291 | 0.1442 |
Guizhou | 0.0619 | 0.0618 | 0.1419 | 0.1269 | 0.0749 | 0.134 | 0.0584 | 0.1335 | 0.0931 | 0.1136 |
Yunnan | 0.0419 | 0.1277 | 0.1287 | 0.0631 | 0.1286 | 0.108 | 0.1574 | 0.0817 | 0.0813 | 0.0817 |
Xizang | 0.0915 | 0.0748 | 0.0934 | 0.0561 | 0.1662 | 0.0927 | 0.1113 | 0.0744 | 0.1106 | 0.1288 |
Shaanxi | 0.0832 | 0.0836 | 0.1613 | 0.1086 | 0.1102 | 0.1333 | 0.0872 | 0.0872 | 0.0872 | 0.0582 |
Gansu | 0.051 | 0.1408 | 0.0967 | 0.093 | 0.1218 | 0.1177 | 0.1148 | 0.0728 | 0.0983 | 0.0929 |
Qinghai | 0.0648 | 0.099 | 0.1025 | 0.1025 | 0.1025 | 0.1025 | 0.0999 | 0.1093 | 0.1077 | 0.1093 |
Ningxia | 0.0466 | 0.1338 | 0.1092 | 0.0895 | 0.1339 | 0.0688 | 0.1578 | 0.0871 | 0.0862 | 0.0871 |
Xinjiang | 0.0601 | 0.0913 | 0.1429 | 0.1082 | 0.1117 | 0.1429 | 0.0935 | 0.0935 | 0.0935 | 0.0624 |
w1 | w2 | w3 | w4 | w5 | w6 | w7 | w8 | w9 | w10 | |
---|---|---|---|---|---|---|---|---|---|---|
α = 0.1 | 0.0007 | 0.0014 | 0.0029 | 0.0061 | 0.0127 | 0.0268 | 0.0563 | 0.1186 | 0.2495 | 0.5250 |
α = 0.4 | 0.0576 | 0.0644 | 0.0720 | 0.0804 | 0.0899 | 0.1005 | 0.1123 | 0.1256 | 0.1403 | 0.1569 |
α = 0.5 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
α = 0.9 | 0.5250 | 0.2495 | 0.1186 | 0.0563 | 0.0267 | 0.0127 | 0.0061 | 0.0029 | 0.0014 | 0.0007 |
Province | VGPG | Rank | VGPA | Rank | α = 0.1 | Rank | α = 0.4 | Rank | α = 0.5 | Rank | α = 0.9 | Rank |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Beijing | 0.2135 | 29 | 0.2409 | 30 | 0.1126 | 27 | 0.2033 | 30 | 0.2367 | 30 | 0.3751 | 29 |
Tianjin | 0.6824 | 4 | 0.7753 | 4 | 0.3704 | 6 | 0.7075 | 4 | 0.7933 | 4 | 0.9965 | 4 |
Hebei | 0.4743 | 11 | 0.5775 | 9 | 0.2057 | 15 | 0.4684 | 10 | 0.5584 | 10 | 0.9065 | 10 |
Shanxi | 0.9594 | 1 | 0.9609 | 1 | 0.8677 | 1 | 0.9427 | 1 | 0.9596 | 1 | 0.9991 | 2 |
InnerMongolia | 0.6595 | 6 | 0.6794 | 6 | 0.5096 | 4 | 0.6369 | 6 | 0.6883 | 7 | 0.9383 | 8 |
Liaoning | 0.3303 | 20 | 0.3600 | 20 | 0.1852 | 20 | 0.3101 | 21 | 0.3529 | 21 | 0.5080 | 22 |
Jilin | 0.2214 | 28 | 0.2549 | 29 | 0.1116 | 28 | 0.2143 | 29 | 0.2499 | 29 | 0.3754 | 28 |
HeiLongJiang | 0.3773 | 18 | 0.4236 | 18 | 0.2156 | 13 | 0.3787 | 16 | 0.4394 | 17 | 0.6783 | 16 |
Shanghai | 0.4298 | 13 | 0.5126 | 12 | 0.1934 | 17 | 0.4043 | 14 | 0.4824 | 14 | 0.8352 | 11 |
Jiangsu | 0.52 | 9 | 0.6347 | 8 | 0.2604 | 11 | 0.5865 | 7 | 0.6892 | 6 | 0.9895 | 6 |
Zhejiang | 0.4478 | 12 | 0.5458 | 10 | 0.2102 | 14 | 0.4672 | 11 | 0.5608 | 9 | 0.9184 | 9 |
Anhui | 0.4262 | 14 | 0.4381 | 17 | 0.3378 | 8 | 0.3988 | 15 | 0.4261 | 18 | 0.5698 | 19 |
Fujian | 0.3189 | 21 | 0.3376 | 22 | 0.1883 | 18 | 0.2908 | 22 | 0.3209 | 22 | 0.4161 | 26 |
Jiangxi | 0.2371 | 26 | 0.2938 | 24 | 0.1200 | 25 | 0.2570 | 24 | 0.3100 | 24 | 0.5184 | 21 |
Shandong | 0.5654 | 7 | 0.6568 | 7 | 0.3033 | 10 | 0.5854 | 8 | 0.6807 | 8 | 0.9846 | 7 |
Henan | 0.4224 | 15 | 0.4783 | 15 | 0.2029 | 16 | 0.3778 | 17 | 0.4415 | 16 | 0.7102 | 15 |
Hebei | 0.3609 | 19 | 0.4482 | 16 | 0.1630 | 22 | 0.3695 | 18 | 0.4445 | 15 | 0.7162 | 13 |
Hunan | 0.2318 | 27 | 0.275 | 26 | 0.1184 | 26 | 0.2371 | 27 | 0.2802 | 26 | 0.4426 | 25 |
Guangdong | 0.3967 | 16 | 0.4972 | 14 | 0.1700 | 21 | 0.4064 | 13 | 0.4906 | 13 | 0.7969 | 12 |
Guangxi | 0.2095 | 30 | 0.2682 | 27 | 0.1042 | 30 | 0.2491 | 25 | 0.3014 | 25 | 0.4879 | 23 |
Hainan | 0.2491 | 24 | 0.3088 | 23 | 0.1074 | 29 | 0.2575 | 23 | 0.3153 | 23 | 0.5591 | 20 |
Chongqing | 0.2414 | 25 | 0.2556 | 28 | 0.1630 | 23 | 0.2267 | 28 | 0.2512 | 28 | 0.3505 | 30 |
Sichuan | 0.3177 | 22 | 0.359 | 21 | 0.1876 | 19 | 0.3355 | 20 | 0.3870 | 19 | 0.5758 | 18 |
Guizhou | 0.4775 | 10 | 0.5013 | 13 | 0.3222 | 9 | 0.4484 | 12 | 0.4958 | 12 | 0.7152 | 14 |
Yunnan | 0.384 | 17 | 0.3993 | 19 | 0.2443 | 12 | 0.3479 | 19 | 0.3786 | 20 | 0.4724 | 24 |
Xizang | 0.0581 | 31 | 0.0659 | 31 | 0.0280 | 31 | 0.0569 | 31 | 0.0663 | 31 | 0.1045 | 31 |
Shaanxi | 0.9073 | 3 | 0.9116 | 3 | 0.7934 | 3 | 0.8852 | 3 | 0.9134 | 3 | 0.9968 | 3 |
Gansu | 0.2587 | 23 | 0.5847 | 25 | 0.1405 | 24 | 0.2428 | 26 | 0.2780 | 27 | 0.4134 | 27 |
Qinghai | 0.6745 | 5 | 0.7676 | 5 | 0.3582 | 7 | 0.6943 | 5 | 0.7821 | 5 | 0.9959 | 5 |
Ningxia | 0.5332 | 8 | 0.5412 | 11 | 0.4283 | 5 | 0.4996 | 9 | 0.5256 | 11 | 0.6478 | 17 |
Xinjiang | 0.9455 | 2 | 0.9499 | 2 | 0.8220 | 2 | 0.9284 | 2 | 0.9511 | 2 | 0.9993 | 1 |
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Liu, L.; Huang, J.; Wang, H. Visibility Graph Power Geometric Aggregation Operator and Its Application in Water, Energy and Food Efficiency Evaluation. Int. J. Environ. Res. Public Health 2020, 17, 3891. https://doi.org/10.3390/ijerph17113891
Liu L, Huang J, Wang H. Visibility Graph Power Geometric Aggregation Operator and Its Application in Water, Energy and Food Efficiency Evaluation. International Journal of Environmental Research and Public Health. 2020; 17(11):3891. https://doi.org/10.3390/ijerph17113891
Chicago/Turabian StyleLiu, Lihua, Jing Huang, and Huimin Wang. 2020. "Visibility Graph Power Geometric Aggregation Operator and Its Application in Water, Energy and Food Efficiency Evaluation" International Journal of Environmental Research and Public Health 17, no. 11: 3891. https://doi.org/10.3390/ijerph17113891