Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares
Abstract
:1. Introduction
- This paper constructed a statistical invariant with symmetric uncertainty distribution based on the observation data and disturbance term, and applied the least squares principle to estimate unknown parameters and the uncertain disturbance term in the uncertain regression model.
- A numerical algorithm was designed to solve the specific estimator.
- Two numerical examples and an empirical case study of forecasting of electrical power output were provided to illustrate the method proposed in this paper.
2. Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Model
2.1. Least Squares Estimator
2.2. Least Squares Estimation of Uncertain Regression Model
Algorithm 1: Numerical solutions of least squares estimators. |
Step 0: Input observational data
|
Step 2: For each and , compute by
|
Step 4: Set
|
Step 5: If , then go to Step 4. |
Step 6: Find and such that reaches its minimum value. |
Step 7: Output and . |
3. Hypothesis Test and Forecast
3.1. Uncertain Hypothesis Test
3.2. Point Forecast and Interval Forecast
4. Numerical Examples
5. Case Study: Forecast of Electrical Power Output
5.1. Uncertain Electrical Power Output Model
5.2. Uncertain Hypothesis Test and Electrical Power Output Forecast
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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x | y | x | y | x | y | x | y |
---|---|---|---|---|---|---|---|
0.0350 | 1.5892 | 1.1663 | 3.9409 | 2.1875 | 8.6692 | 3.0388 | 16.5826 |
0.1450 | 3.3379 | 1.1963 | 3.3124 | 2.2113 | 7.8528 | 3.0463 | 14.4552 |
0.2550 | 4.8082 | 1.2450 | 3.3465 | 2.2775 | 8.8732 | 3.0688 | 14.2875 |
0.3163 | 1.7829 | 1.2488 | 4.1678 | 2.2950 | 9.0323 | 3.1650 | 14.0064 |
0.3925 | 3.0347 | 1.2925 | 5.3190 | 2.3038 | 8.5864 | 3.2775 | 13.9808 |
0.4825 | 3.0839 | 1.3613 | 6.5189 | 2.3738 | 7.7935 | 3.3425 | 16.4002 |
0.5000 | 3.6518 | 1.4313 | 5.0478 | 2.4975 | 9.9483 | 3.4488 | 17.9623 |
0.5725 | 4.1642 | 1.4388 | 6.0847 | 2.5175 | 9.0674 | 3.4775 | 18.2491 |
0.5963 | 3.3266 | 1.5113 | 4.1250 | 2.5363 | 8.1565 | 3.5313 | 20.5180 |
0.6038 | 4.0595 | 1.5713 | 6.8535 | 2.5563 | 10.7031 | 3.5775 | 19.6079 |
0.6113 | 5.8057 | 1.6538 | 5.8316 | 2.5650 | 12.2296 | 3.6650 | 22.3669 |
0.7288 | 5.3459 | 1.7525 | 7.3063 | 2.6550 | 9.5657 | 3.7838 | 23.4053 |
0.7938 | 5.6176 | 1.7763 | 6.7783 | 2.7113 | 10.7657 | 3.8113 | 23.9019 |
0.8125 | 4.4627 | 1.8975 | 5.9172 | 2.7750 | 12.4397 | 3.9063 | 23.3548 |
0.8938 | 3.3543 | 1.9650 | 5.8426 | 2.8513 | 9.6549 | 3.9713 | 26.7689 |
0.9663 | 4.5373 | 2.0800 | 4.9724 | 2.9625 | 10.7734 | 4.0663 | 28.5248 |
1.0050 | 4.9135 | 2.1613 | 7.5202 | 2.9775 | 12.5799 | 4.1763 | 29.5484 |
1.0888 | 5.4231 | 2.1800 | 7.5710 | 3.0138 | 14.1928 | 4.2050 | 31.2873 |
1.1388 | 5.2447 | 2.1863 | 7.3537 | 3.0213 | 16.5998 | 4.3225 | 34.2073 |
x | y | x | y | x | y | x | y |
---|---|---|---|---|---|---|---|
0.0030 | 29.5059 | 0.6885 | 54.1126 | 1.1620 | 58.4385 | 1.6830 | 58.0711 |
0.0395 | 32.0362 | 0.7110 | 54.4404 | 1.1915 | 53.4652 | 1.6955 | 54.6119 |
0.0780 | 36.4589 | 0.7530 | 54.4186 | 1.2355 | 52.9707 | 1.7000 | 58.9852 |
0.1230 | 38.9807 | 0.8025 | 52.3946 | 1.2630 | 55.1536 | 1.7355 | 57.7782 |
0.1615 | 40.0327 | 0.8210 | 55.5801 | 1.2780 | 55.4998 | 1.7785 | 56.8902 |
0.2090 | 40.0993 | 0.8295 | 55.7937 | 1.3230 | 55.9606 | 1.8180 | 52.0411 |
0.2360 | 43.6076 | 0.8710 | 54.8597 | 1.3630 | 54.2075 | 1.8555 | 54.1480 |
0.2840 | 46.4421 | 0.9005 | 54.0925 | 1.4020 | 54.6567 | 1.8605 | 53.0196 |
0.3330 | 46.7559 | 0.9490 | 53.7172 | 1.4320 | 54.4987 | 1.8865 | 55.2992 |
0.3590 | 45.9164 | 0.9735 | 52.2714 | 1.4600 | 52.0097 | 1.9290 | 53.4092 |
0.4015 | 47.9307 | 0.9860 | 54.6858 | 1.4750 | 53.7355 | 1.9715 | 54.3374 |
0.4465 | 51.3303 | 1.0200 | 55.6078 | 1.5025 | 55.0737 | 1.9905 | 57.2144 |
0.4935 | 50.9886 | 1.0660 | 56.1994 | 1.5525 | 54.3553 | 2.0040 | 53.1554 |
0.5165 | 52.0612 | 1.0970 | 54.6051 | 1.5765 | 56.2438 | 2.0100 | 52.2278 |
0.5545 | 52.5257 | 1.1130 | 54.0854 | 1.6110 | 55.8744 | 2.0445 | 53.8768 |
0.6015 | 53.7737 | 1.1375 | 57.4118 | 1.6350 | 55.4344 | 2.0725 | 54.6140 |
0.6420 | 54.4962 | 1.1580 | 57.4150 | 1.6725 | 55.9188 | 2.0726 | 56.0243 |
y | y | ||||||||
---|---|---|---|---|---|---|---|---|---|
14.96 | 41.76 | 1024.07 | 73.17 | 463.26 | 10.54 | 34.03 | 1018.71 | 74 | 478.77 |
25.18 | 62.96 | 1020.04 | 59.08 | 444.37 | 27.71 | 74.34 | 998.14 | 71.85 | 434.2 |
5.11 | 39.4 | 1012.16 | 92.14 | 488.56 | 23.11 | 68.3 | 1017.83 | 86.62 | 437.91 |
20.86 | 57.32 | 1010.24 | 76.64 | 446.48 | 7.51 | 41.01 | 1024.61 | 97.41 | 477.61 |
10.82 | 37.5 | 1009.23 | 96.62 | 473.9 | 26.46 | 74.67 | 1016.65 | 84.44 | 431.65 |
26.27 | 59.44 | 1012.23 | 58.77 | 443.67 | 29.34 | 74.34 | 998.58 | 81.55 | 430.57 |
15.89 | 43.96 | 1014.02 | 75.24 | 467.35 | 10.32 | 42.28 | 1008.82 | 75.66 | 481.09 |
9.48 | 44.71 | 1019.12 | 66.43 | 478.42 | 22.74 | 61.02 | 1009.56 | 79.41 | 445.56 |
14.64 | 45 | 1021.78 | 41.25 | 475.98 | 13.48 | 39.85 | 1012.71 | 58.91 | 475.74 |
11.74 | 43.56 | 1015.14 | 70.72 | 477.5 | 25.52 | 69.75 | 1010.36 | 90.06 | 435.12 |
17.99 | 43.72 | 1008.64 | 75.04 | 453.02 | 21.58 | 67.25 | 1017.39 | 79 | 446.15 |
20.14 | 46.93 | 1014.66 | 64.22 | 453.99 | 27.66 | 76.86 | 1001.31 | 69.47 | 436.64 |
24.34 | 73.5 | 1011.31 | 84.15 | 440.29 | 26.96 | 69.45 | 1013.89 | 51.47 | 436.69 |
25.71 | 58.59 | 1012.77 | 61.83 | 451.28 | 12.29 | 42.18 | 1016.53 | 83.13 | 468.75 |
26.19 | 69.34 | 1009.48 | 87.59 | 433.99 | 15.86 | 43.02 | 1012.18 | 40.33 | 466.6 |
21.42 | 43.79 | 1015.76 | 43.08 | 462.19 | 13.87 | 45.08 | 1024.42 | 81.69 | 465.48 |
18.21 | 45 | 1022.86 | 48.84 | 467.54 | 24.09 | 73.68 | 1014.93 | 94.55 | 441.34 |
11.04 | 41.74 | 1022.6 | 77.51 | 477.2 | 20.45 | 69.45 | 1012.53 | 91.81 | 441.83 |
14.45 | 52.75 | 1023.97 | 63.59 | 459.85 | 15.07 | 39.3 | 1019 | 63.62 | 464.7 |
13.97 | 38.47 | 1015.15 | 55.28 | 464.3 | 32.72 | 69.75 | 1009.6 | 49.35 | 437.99 |
17.76 | 42.42 | 1009.09 | 66.26 | 468.27 | 18.23 | 58.96 | 1015.55 | 69.61 | 459.12 |
5.41 | 40.07 | 1019.16 | 64.77 | 495.24 | 35.56 | 68.94 | 1006.56 | 38.75 | 429.69 |
7.76 | 42.28 | 1008.52 | 83.31 | 483.8 | 18.36 | 51.43 | 1010.57 | 90.17 | 459.8 |
27.23 | 63.9 | 1014.3 | 47.19 | 443.61 | 26.35 | 64.05 | 1009.81 | 81.24 | 433.63 |
27.36 | 48.6 | 1003.18 | 54.93 | 436.06 | 25.92 | 60.95 | 1014.62 | 48.46 | 442.84 |
27.47 | 70.72 | 1009.97 | 74.62 | 443.25 | 8.01 | 41.66 | 1014.49 | 76.72 | 485.13 |
14.6 | 39.31 | 1011.11 | 72.52 | 464.16 | 19.63 | 52.72 | 1025.09 | 51.16 | 459.12 |
7.91 | 39.96 | 1023.57 | 88.44 | 475.52 | 20.02 | 67.32 | 1012.05 | 76.34 | 445.31 |
5.81 | 35.79 | 1012.14 | 92.28 | 484.41 | 10.08 | 40.72 | 1022.7 | 67.3 | 480.8 |
30.53 | 65.18 | 1012.69 | 41.85 | 437.89 | 27.23 | 66.48 | 1005.23 | 52.38 | 432.55 |
23.87 | 63.94 | 1019.02 | 44.28 | 445.11 | 23.37 | 63.77 | 1013.42 | 76.44 | 443.86 |
26.09 | 58.41 | 1013.64 | 64.58 | 438.86 | 18.74 | 59.21 | 1018.3 | 91.55 | 449.77 |
29.27 | 66.85 | 1011.11 | 63.25 | 440.98 | 14.81 | 43.69 | 1017.19 | 71.9 | 470.71 |
27.38 | 74.16 | 1010.08 | 78.61 | 436.65 | 23.1 | 51.3 | 1011.93 | 80.05 | 452.17 |
24.81 | 63.94 | 1018.76 | 44.51 | 444.26 | 10.72 | 41.38 | 1021.6 | 63.77 | 478.29 |
12.75 | 44.03 | 1007.29 | 89.46 | 465.86 | 29.46 | 71.94 | 1006.96 | 62.26 | 428.54 |
24.66 | 63.73 | 1011.4 | 74.52 | 444.37 | 8.1 | 40.64 | 1020.66 | 89.04 | 478.27 |
16.38 | 47.45 | 1010.08 | 88.86 | 450.69 | 27.29 | 62.66 | 1007.63 | 58.02 | 439.58 |
13.91 | 39.35 | 1014.69 | 75.51 | 469.02 | 17.1 | 49.69 | 1005.53 | 81.82 | 457.32 |
23.18 | 51.3 | 1012.04 | 78.64 | 448.86 | 11.49 | 44.2 | 1018.79 | 91.14 | 475.51 |
22.47 | 47.45 | 1007.62 | 76.65 | 447.14 | 23.69 | 65.59 | 1010.85 | 88.92 | 439.66 |
13.39 | 44.85 | 1017.24 | 80.44 | 469.18 | 13.51 | 40.89 | 1011.03 | 84.83 | 471.99 |
9.28 | 41.54 | 1018.33 | 79.89 | 482.8 | 9.64 | 39.35 | 1015.1 | 91.76 | 479.81 |
11.82 | 42.86 | 1014.12 | 88.28 | 476.7 | 25.65 | 78.92 | 1010.83 | 86.56 | 434.78 |
10.27 | 40.64 | 1020.63 | 84.6 | 474.99 | 21.59 | 61.87 | 1011.18 | 57.21 | 446.58 |
22.92 | 63.94 | 1019.28 | 42.69 | 444.22 | 27.98 | 58.33 | 1013.92 | 54.25 | 437.76 |
16 | 37.87 | 1020.24 | 78.41 | 461.33 | 18.8 | 39.72 | 1001.24 | 63.8 | 459.36 |
21.22 | 43.43 | 1010.96 | 61.07 | 448.06 | 18.28 | 44.71 | 1016.99 | 33.71 | 462.28 |
13.46 | 44.71 | 1014.51 | 50 | 474.6 | 13.55 | 43.48 | 1016.08 | 67.25 | 464.33 |
9.39 | 40.11 | 1029.14 | 77.29 | 473.05 | 22.99 | 46.21 | 1010.71 | 60.11 | 444.36 |
31.07 | 73.5 | 1010.58 | 43.66 | 432.06 | 23.94 | 59.39 | 1014.32 | 74.55 | 438.64 |
12.82 | 38.62 | 1018.71 | 83.8 | 467.41 | 13.74 | 34.03 | 1018.69 | 67.34 | 470.49 |
32.57 | 78.92 | 1011.6 | 66.47 | 430.12 | 21.3 | 41.1 | 1001.86 | 42.75 | 455.13 |
8.11 | 42.18 | 1014.82 | 93.09 | 473.62 | 27.54 | 66.93 | 1017.06 | 55.2 | 450.22 |
13.92 | 39.39 | 1012.94 | 80.52 | 471.81 | 24.81 | 63.73 | 1009.34 | 83.61 | 440.43 |
23.04 | 59.43 | 1010.23 | 68.99 | 442.99 | 4.97 | 42.85 | 1014.02 | 88.78 | 482.98 |
27.31 | 64.44 | 1014.65 | 57.27 | 442.77 | 15.22 | 50.88 | 1014.19 | 100.12 | 460.44 |
5.91 | 39.33 | 1010.18 | 95.53 | 491.49 | 23.88 | 54.2 | 1012.81 | 64.52 | 444.97 |
25.26 | 61.08 | 1013.68 | 71.72 | 447.46 | 33.01 | 68.67 | 1005.2 | 51.41 | 433.94 |
27.97 | 58.84 | 1002.25 | 57.88 | 446.11 | 25.98 | 73.18 | 1012.28 | 85.78 | 439.73 |
26.08 | 52.3 | 1007.03 | 63.34 | 442.44 | 28.18 | 73.88 | 1005.89 | 75.41 | 434.48 |
29.01 | 65.71 | 1013.61 | 48.07 | 446.22 | 21.67 | 60.84 | 1017.93 | 81.63 | 442.33 |
12.18 | 40.1 | 1016.67 | 91.87 | 471.49 | 17.67 | 45.09 | 1014.26 | 51.92 | 457.67 |
13.76 | 45.87 | 1008.89 | 87.27 | 463.5 | 21.37 | 57.76 | 1018.8 | 70.12 | 454.66 |
25.5 | 58.79 | 1016.02 | 64.4 | 440.01 | 28.69 | 67.25 | 1017.71 | 53.83 | 432.21 |
28.26 | 65.34 | 1014.56 | 43.4 | 441.03 | 16.61 | 43.77 | 1012.25 | 77.23 | 457.66 |
21.39 | 62.96 | 1019.49 | 72.24 | 452.68 | 27.91 | 63.76 | 1010.27 | 65.67 | 435.21 |
7.26 | 40.69 | 1020.43 | 90.22 | 474.91 |
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Wang, H.; Liu, Y.; Shi, H. Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares. Symmetry 2024, 16, 1182. https://doi.org/10.3390/sym16091182
Wang H, Liu Y, Shi H. Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares. Symmetry. 2024; 16(9):1182. https://doi.org/10.3390/sym16091182
Chicago/Turabian StyleWang, Han, Yang Liu, and Haiyan Shi. 2024. "Estimating Unknown Parameters and Disturbance Term in Uncertain Regression Models by the Principle of Least Squares" Symmetry 16, no. 9: 1182. https://doi.org/10.3390/sym16091182