A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models
Abstract
1. Introduction
2. Crisp Linear Regression
3. Fuzzy Linear Regression
4. Proposed Method
5. Case Study
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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29.50 | 31.30 | 37.60 | 39.90 | 39.90 | 40.30 | 41.50 | 43.60 | 45.70 | 47.80 | 49.50 | |
79.99 | 75.63 | 69.25 | 62.75 | 64.66 | 63.09 | 61.51 | 60.07 | 58.22 | 58.43 | 60.57 | |
133.60 | 137.63 | 147.86 | 196.76 | 220.53 | 223.25 | 233.19 | 265.67 | 335.16 | 411.29 | 460.68 |
50.10 | 50.20 | 49.90 | 50.00 | 50.00 | 50.00 | 50.90 | 53.10 | 55.20 | |
58.23 | 58.03 | 57.53 | 55.68 | 55.24 | 54.51 | 50.08 | 50.05 | 49.72 | |
477.96 | 474.02 | 466.80 | 466.16 | 469.80 | 468.95 | 476.24 | 499.39 | 521.20 |
2 | 4 | 6 | 8 | 10 | 12 | 16 | 18 | |
14 | 16 | 14 | 18 | 18 | 22 | 18 | 22 |
Hvalue | A | Total Credibility |
---|---|---|
0.3148 | y ∗ = (−610.8251, 0.0000) + (19.1484, 0.0000)x1 + (1.4018, 1.2301)x2 | 0.1371 |
0 | y = (−610.8233, 0.0000) + (19.1484, 0.0000)x1 + (1.4017, 0.8429)x2 | 0.1082 |
0.5 | y = (−610.8240, 0.0000) + (19.1484, 0.0000)x1 + (1.4017, 1.6857)x2 | 0.1271 |
Confidence Level | Linear Model Based on Fuzzy Prediction Intervals (Data from Liu and Chen, 2013) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.2153 | 0 |
95% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.2056 | 0.0169 |
99% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.1949 | 0.0566 |
Proportion | Linear Model Based on Fuzzy Tolerance Interval (Data from Liu and Chen 2013) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.2029 | 0 |
95% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.1871 | 0.0793 |
99% | y = (−1590.9) + (29.6726)x1 + (9.9906)x2 | 0.1631 | 0.1794 |
Hvalue | Fuzzy Model (Data from Liu and Chen 2013) | Total Credibility |
---|---|---|
0.2666 | y ∗ = (12.0000, 1.3635) + (0.6250,0.1704)x | 1.4960 |
0 | y = (12.0000, 1.000) + (0.6250,0.1300)x | 1.2984 |
0.7 | y = (12.0000, 3.330) + (0.6250,0.4200)x | 0.9736 |
Confidence Level | Linear Model Based on Fuzzy Prediction Interval (Data from Liu and Chen 2013) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (12.93) + (0.54)x | 1.5612 | 0.1371 |
95% | y = (12.93) + (0.54)x | 1.4698 | 0.1825 |
99% | y = (12.93) + (0.54)x | 1.3651 | 0.2155 |
Proportion | Linear Model Based on Fuzzy Tolerance Interval (Data from Liu and Chen 2013) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (12.93) + (0.54)x | 1.3813 | 0.2921 |
95% | y = (12.93) + (0.54)x | 1.1850 | 0.4034 |
99% | y = (12.93) + (0.54)x | 0.8866 | 0.5627 |
Hvalue | Fuzzy Model (Car Data UCI Repository) | Total Credibility |
---|---|---|
0.3 | y = (45.4865, 13.8711) + (−0.0026, 0.0000)x1 + (−1.091, 0.0000)x2 | 4.4629 |
0 | y * = (45.4865, 9.7097) + (−0.0026, 0.0000)x1 + (−1.091, 0.0000)x2 | 5.4725 |
0.5 | y = (45.4865, 19.4195) + (−0.0026, 0.0000)x1 + (−1.091, 0.0000)x2 | 3.7626 |
Confidence Level | Linear Model Based on Fuzzy Prediction Intervals (Car data UCI Repository) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.9395 | 0 |
95% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.6792 | 0 |
99% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.3592 | 0 |
Proportion | Linear Model Based on Fuzzy Tolerance Interval (Car data UCI Repository) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.7065 | 0 |
95% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.4172 | 0 |
99% | y = (47.7694) + (−0.0066)x1 + (−0.0420)x2 | 7.0796 | 0 |
Hvalue | Fuzzy Model (Hald Cement Dataset [17]) | Total Credibility |
---|---|---|
0.2487 | y ∗ = (79.3955, 0.0000) + (1.5212, 0.2474)x1 + (0.3238, 0.0000)x2 + (−0.0839, 0.1453)x3 + (0.3187, 0.0000)x4 | 1.9208 |
0 | y = (79.3955, 0.0000) + (1.5212, 0.1859)x1 + (0.3238, 0.0000)x2 + (−0.0839, 0.1091)x3 + (0.3187, 0.0000)x4 | 1.7102 |
0.5 | y ∗ = (79.3955, 0.0000) + (1.5212, 0.3718)x1 + (0.3238, 0.0000)x2 + (−0.0839, 0.2183)x3 + (0.3187, 0.0000)x4 | 1.7059 |
Confidence Level | Linear Model Based on Fuzzy Prediction Intervals (Hald Cement Dataset [17]) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.8326 | 0.0762 |
95% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.7302 | 0.1248 |
99% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.6160 | 0.1602 |
Proportion | Linear Model Based on Fuzzy Tolerance Interval (Hald Cement Dataset [17]) | Total Credibility | Lowest Membership Value |
---|---|---|---|
90% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.7655 | 0.1870 |
95% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.6683 | 0.2298 |
99% | y = (62.4054) + (1.5511)x1 + (0.5102)x2 + (0.1091)x3 + (−0.1441)x4 | 1.5724 | 0.2609 |
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Al-Kandari, M.; Adjenughwure, K.; Papadopoulos, K. A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models. Mathematics 2020, 8, 1422. https://doi.org/10.3390/math8091422
Al-Kandari M, Adjenughwure K, Papadopoulos K. A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models. Mathematics. 2020; 8(9):1422. https://doi.org/10.3390/math8091422
Chicago/Turabian StyleAl-Kandari, Maryam, Kingsley Adjenughwure, and Kyriakos Papadopoulos. 2020. "A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models" Mathematics 8, no. 9: 1422. https://doi.org/10.3390/math8091422
APA StyleAl-Kandari, M., Adjenughwure, K., & Papadopoulos, K. (2020). A Fuzzy-Statistical Tolerance Interval from Residuals of Crisp Linear Regression Models. Mathematics, 8(9), 1422. https://doi.org/10.3390/math8091422