Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables
Abstract
:1. Motivation
2. Introduction
3. Preliminaries
3.1. Fuzzy Implication
3.2. Fuzzy Negations
3.3. Triangular Norms (Conjunctions)
4. Main Results
4.1. Construction of the Table of the Empiristic Implication
- Low temperature means High humidity.
- Medium temperature means Medium humidity.
- High temperature means Low humidity.
4.2. Generated Parametric Fuzzy Implications
4.3. Construction of the Table of the Parametric Fuzzy Implication
- Low temperature means High humidity.
- Medium temperature means Medium humidity.
- High temperature means Low humidity.
4.4. Find the Most Appropriate Parametric Fuzzy Implications
4.4.1. The Control of the Norm of the Empiristic and Three Well Known Implications (see Example 1).
- The squared error of the two implications (Empiristic and Kleen-Dienes) gives the result is 8.4922.
- The squared error of the two implications (Empiristic and Lukasiewicz) gives the result is 9.1654.
- The squared error of the two implications (Empiristic and Reichenbach) gives the result is 8.6690.
4.4.2. The Control of the Norm of the Empiristic and Parametric Implication
Description the MATLAB code for the Control between Empiristic and Parametric Implication
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Designation | Equation |
---|---|
Minimum | |
Algebraic product | |
Lukasiewicz | |
Active product | |
Nilpotent mimimum |
NaN | 0.5857 | 1 | 1 | 1 | 0.4000 | 0.1564 | 0.3682 | 0.5800 | 0.8382 | 1 | 1 |
1 | 2 | 0 | 1 | 3 | 2 | 4 | 8 | 8 | 13 | 14 | 16 |
0.6020 | 1 | 0 | 1 | 5 | 4 | 6 | 14 | 5 | 9 | 15 | 11 |
0.3967 | 0 | 0 | 0 | 4 | 3 | 8 | 7 | 10 | 8 | 10 | 21 |
1 | 0 | 1 | 3 | 4 | 5 | 10 | 7 | 10 | 13 | 12 | 6 |
0.8567 | 1 | 3 | 6 | 3 | 9 | 7 | 6 | 11 | 9 | 8 | 8 |
0.4033 | 3 | 2 | 10 | 6 | 14 | 4 | 3 | 12 | 4 | 8 | 5 |
0.8220 | 4 | 4 | 9 | 9 | 9 | 12 | 8 | 6 | 5 | 3 | 2 |
1 | 6 | 9 | 13 | 14 | 6 | 7 | 6 | 3 | 5 | 1 | 1 |
1 | 8 | 16 | 10 | 8 | 8 | 9 | 6 | 2 | 4 | 0 | 0 |
1 | 19 | 12 | 11 | 11 | 5 | 3 | 5 | 4 | 1 | 0 | 0 |
1 | 27 | 24 | 7 | 4 | 6 | 1 | 1 | 0 | 0 | 0 | 0 |
6.31 | 1 | 75.87 | 10 |
1.58 | 1 | 76.69 | 10 |
5.05 | 1 | 76.11 | 10 |
5.83 | 1 | 74.92 | 10 |
2.26 | 1 | 75.09 | 10 |
4.55 | 1 | 75.30 | 10 |
6.37 | 1 | 77.58 | 10 |
5.42 | 1 | 77.71 | 10 |
4.87 | 1 | 77.89 | 10 |
5.66 | 1 | 77.25 | 10 |
6.20 | 1 | 75.52 | 10 |
4.40 | 1 | 76.21 | 10 |
6.20 | 1 | 75.93 | 10 |
5.33 | 1 | 76.53 | 10 |
0.0282 | 0 | 0.0141 | 0.0423 | 0.0282 | 0.0563 | 0.1127 | 0.1127 | 0.1831 | 0.1972 | 0.2286 |
0.0141 | 0 | 0.0141 | 0.0704 | 0.0563 | 0.0845 | 0.1972 | 0.0704 | 0.1268 | 0.2113 | 0.1571 |
0 | 0 | 0 | 0.0563 | 0.0423 | 0.1127 | 0.0986 | 0.1408 | 0.1127 | 0.1408 | 0.3000 |
0 | 0.0141 | 0.0423 | 0.0563 | 0.0704 | 0.1408 | 0.0986 | 0.1408 | 0.1831 | 0.1690 | 0.0857 |
0.0141 | 0.0423 | 0.0845 | 0.0423 | 0.1268 | 0.0986 | 0.0845 | 0.1549 | 0.1268 | 0.1127 | 0.1143 |
0.0423 | 0.0282 | 0.1408 | 0.0845 | 0.1972 | 0.0563 | 0.0423 | 0.1690 | 0.0563 | 0.1127 | 0.0714 |
0.0563 | 0.0563 | 0.1268 | 0.1268 | 0.1268 | 0.1690 | 0.1127 | 0.0845 | 0.0704 | 0.0423 | 0.0286 |
0.0845 | 0.1268 | 0.1831 | 0.1972 | 0.0845 | 0.0986 | 0.0845 | 0.0423 | 0.0704 | 0.0141 | 0.0143 |
0.1127 | 0.2254 | 0.1408 | 0.1127 | 0.1127 | 0.1268 | 0.0845 | 0.0282 | 0.0563 | 0 | 0 |
0.2676 | 0.1690 | 0.1549 | 0.1549 | 0.0704 | 0.0423 | 0.0704 | 0.0563 | 0.0141 | 0 | 0 |
0.3803 | 0.3380 | 0.0986 | 0.0563 | 0.0845 | 0.0141 | 0.0141 | 0 | 0 | 0 | 0 |
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Makariadis, S.; Souliotis, G.; Papadopoulos, B. Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables. Symmetry 2021, 13, 509. https://doi.org/10.3390/sym13030509
Makariadis S, Souliotis G, Papadopoulos B. Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables. Symmetry. 2021; 13(3):509. https://doi.org/10.3390/sym13030509
Chicago/Turabian StyleMakariadis, Stefanos, Georgios Souliotis, and Basil Papadopoulos. 2021. "Parametric Fuzzy Implications Produced via Fuzzy Negations with a Case Study in Environmental Variables" Symmetry 13, no. 3: 509. https://doi.org/10.3390/sym13030509