Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method
Abstract
:1. Introduction
1.1. Research Background and Related Works
1.2. Motivation and Inspiration
2. Preliminaries
2.1. Fuzzy Sets and Fuzzy Relations
- (1)
- Commutativity:
- (2)
- Associativity:
- (3)
- Monotonicity:
- (4)
- Boundary condition:
- (1)
- Reflexivity:
- (2)
- Symmetry:
- (3)
- T-transitivity:
2.2. Set-Valued Information Systems
3. The Distance between Two Objects in a SIS
4. The Fuzzy -Equivalence Relation Induced by a SIS
Algorithm 1: The fuzzy -equivalence relation. |
5. Information Structures in a SIS
5.1. Some Concepts of Information Structures in a SIS
- (1)
- is said to be dependent of , if , , which is written as .
- (2)
- is said to be strictly dependent of , if and , which is written as .
5.2. Properties of Information Structures in a SIS
- (1)
- If , then , ;
- (2)
- If , then , .
6. Measuring Uncertainty of a SIS
6.1. Granulation Measures for a SIS
- (1)
- Non-negativity: , ;
- (2)
- Invariability: , if , then ;
- (3)
- Monotonicity: , if , then .
- (1)
- If , then ;
- (2)
- If , then .
- (1)
- If , then , .
- (2)
- If , then , .
6.2. Entropy Measures for a SIS
- (1)
- If , then , ;
- (2)
- If , then , .
- (1)
- If , then ;
- (2)⊆
- If , then .
- (1)
- If , then , ;
- (2)
- If , then , .
6.3. Information Amounts in a SIS
- (1)
- If , then ;
- (2)
- If , then .
- (1)
- If , then , ;
- (2)
- If , then , .
- (1)
- If monotonicity is only considered, then δ-information granulation and δ-rough entropy are both monotonically increasing with the δ value growth, that means the uncertainty of four subsystems increase as the δ value increases. Meanwhile, δ-information amount and δ-information entropy are both monotonically decreasing with δ value growth, That means the uncertainty of four subsystem decreases as the δ value increases (see Figure 1, Figure 2, Figure 3 and Figure 4).
- (2)
- If , consider δ-information granulation and δ-rough entropy, is got. That shows the larger the subsystem, the smaller the measured value. Pay attention to δ-information amount and δ-information entropy, we have That displays the measured value of the subsystem is larger than the smaller one (see Figure 5).
6.4. Effectiveness Analysis
6.4.1. Dispersion Analysis
- (1)
- if monotonicity is only needed, then G, , H and E can evaluate uncertainty of a SIS.
- (2)
- if the dispersion degree is only considered, then E has better performance for measuring uncertainty of a SIS.
6.4.2. Association Analysis
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Price () | Mileage () | Size () | Max-Speed () | |
---|---|---|---|---|
{high} | {high} | {full} | {high,mid,low} | |
{mid,low} | {high,mid,low} | {compact} | {high,mid,low} | |
{high,low} | {high} | {full} | {high} | |
{high} | {high,low} | {compact} | {low} | |
{mid} | {high,mid} | {full} | {high,low} | |
{high,mid} | {mid} | {compact} | {high} | |
{high,mid,low} | {high} | {full} | {high,low} | |
{low} | {high,low} | {compact} | {low} | |
{high} | {mid} | {full} | {low} | |
{high} | {high,mid,low} | {compact} | {mid} |
r | ||||
---|---|---|---|---|
1 | ||||
−0.99447 | 1 | |||
0.99444 | −1 | 1 | ||
−1 | 0.99446 | 0.99446 | 1 |
G | E | H | ||
---|---|---|---|---|
G | CPC | |||
E | HNC | CPC | ||
HPC | CNC | CPC | ||
H | CNC | HPC | HPC | CPC |
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He, J.; Wang, P.; Li, Z. Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method. Symmetry 2019, 11, 199. https://doi.org/10.3390/sym11020199
He J, Wang P, Li Z. Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method. Symmetry. 2019; 11(2):199. https://doi.org/10.3390/sym11020199
Chicago/Turabian StyleHe, Jiali, Pei Wang, and Zhaowen Li. 2019. "Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method" Symmetry 11, no. 2: 199. https://doi.org/10.3390/sym11020199
APA StyleHe, J., Wang, P., & Li, Z. (2019). Uncertainty Measurement for a Set-Valued Information System: Gaussian Kernel Method. Symmetry, 11(2), 199. https://doi.org/10.3390/sym11020199