An information system as a database that shows relationships between objects and attributes is a crucial mathematical model in the field of artificial intelligence. A real-valued information system is an information system where information function values of each attribute are real numbers. This paper explores information structures in an incomplete real-valued information system. Distances between two objects in a given subsystem of an incomplete real-valued information system is first constructed. Then, the fuzzy Tcos-equivalence relation, induced by this subsystem by using Gaussian kernel method, is obtained, where Gaussian kernel is based on this distance. Next, information structure of this subsystem is proposed. Moreover, relationships between two information structures are studied from the two aspects of dependence and separation. Finally, the dependence between two information structures is studied by using inclusion degree. These results will be helpful for establishing a framework of granular computing.
BlaszczynskiJ., SlowinskiR., SzelagM., Sequential covering rule induction algorithm for variable consistency rough set approaches, Information Sciences181(5) (2011), 987–1002.
2.
CornelisC., JensenR., MartinG.H., SlezakD., Attribute selection with fuzzy decision reducts, Information Sciences180 (2010), 209–224.
3.
ChenY.M., XueY., MaY., XuF.F., Measures of uncertainty for neighborhood rough sets, Knowledge-Based Systems120 (2017), 226–235.
4.
DuboisD., PradeH., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems17(2-3) (1990), 191–209.
5.
HuQ.H., PedryczW., YuD.R., LangJ., Selecting discrete and continuous features based on neighborhood decision error minimization, IEEE Transactions on Systems, Man and Cybernetics (Part B)40 (2010), 137–150.
6.
HuQ.H., XieZ.X., YuD.R., Hybrid attribute reduction based on a novel fuzzy-rough model and information granulation, Pattern Recognition40 (2007), 3509–3521.
7.
HuQ.H., ZhangL., ChenD.G., PedryczW., YuD.R., Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications, International Journal of Approximate Reasoning51 (2010), 453–471.
8.
JensenR., ShenQ., Semantics-preserving dimensionality reduction: Rough and fuzzy rough based approaches, IEEE Transactions on Snowledge and Data Engineering16 (2004), 1457–1471.
9.
JensenR., ShenQ., New approaches to fuzzy-rough feature selection, IEEE Transactions on Fuzzy Systems17 (2009), 824–838.
10.
KryszkiewiczM., Rules in incomplete information systems, Information Sciences113 (1999), 271–292.
11.
LinT.Y., Granular computing on binary relations I: Data mining and neighborhood systems, In: Rough Sets In Knowledge Discovery, SkowronA. and PolkowskiL. (eds), Physica-Verlag, 1998, pp. 107–121.
12.
LinT.Y., Granular computing on binary relations II: Rough set representations and belief functions, In: Rough Sets In Knowledge DiscoverySkowronA. and PolkowskiL. (eds), Physica-Verlag, 1998, pp. 121–140.
13.
LinT.Y., Granular computing: Fuzzy logic and rough sets, In: Computing with words in information intelligent systemsZadehL.A. and KacprzykJ., Physica-Verlag, 1999, pp. 183–200.
14.
LiZ.W., CuiR.C., Similarity of fuzzy relations based on fuzzy topologies induced by fuzzy rough approximation operators, Information Sciences305 (2015), 219–233.
15.
LiZ.W., CuiR.C., T-similarity of fuzzy relations and related algebraic structures, Fuzzy Sets and Systems275 (2015), 130–143.
16.
LiC.R., DuanG.D., ZhongF.J., Rotation invariant texture retrieval considering the scale dependence of gabor wavelet, IEEE Transactions on Image Processin24(8) (2015), 2344–2354.
17.
LiZ.W., LiuX.F., ZhangG.Q., XieN.X., WangS.C., A multi-granulation decision-theoretic rough set method for distributed fc-decision information systems: An application inmedical diagnosis, Applied Soft Computing56 (2017), 233–244.
18.
LiZ.W., LiuY.Y., LiQ.G., QinB., Relationships between knowledge bases and related results, Knowledge and Information Systems49 (2016), 171–195.
19.
LiZ.W., LiQ.G., ZhangR.R., XieN.X., Knowledge structures in a knowledge base, Expert Systems33 (2016), 581–591.
20.
MoserB., On the T-transitivity of kernels, Fuzzy Sets and Systems157 (2006), 1787–1796.
21.
MoserB., On representing and generating kernels by fuzzy equivalence relations, Journal of Machine Learning Research7 (2006), 2603–2630.
22.
MiJ.S., LeungY., WuW.Z., An uncertainty measure in partition-based fuzzy rough sets, International Journal of General Systems34 (2005), 77–90.
23.
MaJ., ZhangW., LeungY., SongX., Granular computing and dual Galois connection, Information Sciences177 (2007), 5365–5377.
24.
PawlakZ., Rough sets, International Journal of Computer Information Science11 (1982), 341–356.
25.
PawlakZ., Rough Sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Dordrecht, 1991.
26.
PawlakZ., Rough sets, International Journal of Computer and Information Sciences11(5) (1982), 341–356.
27.
PawlakZ., SkowronA., Rough sets and boolean reasoning, Information Sciences177 (2007), 41–73.
28.
QianY.H., LiangJ.Y., PedryczW., DangC.Y., An accelerator for attribute reduction in rough set theory, Artificial Intelligence174 (2010), 597–618.
29.
QianY.H., LiangJ.Y., WuW.Z., DangC.Y., Information granularity in fuzzy binary GrC model, IEEE Transactions on Fuzzy Systems19(2) (2011), 253–264.
30.
Shawe-TayorJ., CristianiniN., Kernel methods for pattern analysis, Cambridge University Press, 2004.
31.
SwiniarskiR.W., SkowronA., Rough set methods in feature selection and recognition, Pattern Recognition Letters24 (2003), 833–849.
32.
SlowinskiR., VanderpootenD., A generalized definition of rough approximations based on setilarity, IEEE Transactions on Snowledge and Data Engineering12 (2000), 331–336.
33.
ThangavelS., PethalakshmiA., Dimensionality reduction based on rough set theory: A review, Applied Soft Computing9 (2009), 1–12.
34.
WiermanM.J., Measuring uncertainty in rough set theory, International Journal of General Systems28 (1999), 283–297.
35.
WuW.Z., LeungY., MiJ., Granular computing and knowledge reduction in formal contexts,(10), IEEE Transactions on Knowledge and Data Engineering21 (2009), 1461–1474.
36.
WangX.Z., TsangE.C.C., ZhaoS.Y., ChenD.G., YeungD.S., Learning fuzzy rules from fuzzy samples based on rough set technique, Information Sciences177 (2007), 4493–4514.
37.
XieN.X., LiuM., LiZ.W., ZhangG.Q., New measures of uncertainty for an interval-valued information system, Information Sciences470 (2019), 156–174.
38.
XiaM.M., XuZ.S., Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52 (2011), 395–407.
39.
YaoY.Y., Relational interpretations of neighborhood operators and rough set approximation operators, Information Sciences111 (1998), 239–259.
40.
YaoY.Y., Information granulation and rough set approximation, International Journal of Intelligent Systems16 (2001), 87–104.
41.
YaoY.Y., Probabilistic approaches to rough sets, Expert Systems20 (2003), 287–297.
42.
YaoY.Y., Perspectives of Granular computing, Proceedings of 2005 IEEE International Conference on Granular Computing1 (2005), 85–90.
43.
YuH., LiuZ.G., WangG.Y., An automatic method to determine the number of clusters using decision-theoretic rough set, International Journal of Approximate Reasoning55 (2014), 101–115.
44.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.
ZadehL.A., Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems90 (1997), 111–127.
47.
ZadehL.A., Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information intelligent systems, Soft Computing2 (1998), 23–25.
48.
ZadehL.A., A new direction in AI-Toward a computational theory of perceptions, Ai Magazine22(1) (2001), 73–84.
49.
ZhangG.Q., LiZ.W., WuW.Z., LiuX.F., XieN.X., Information structures and uncertainty measures in a fully fuzzy information system, International Journal of Approximate Reasoning101 (2018), 119–149.
50.
ZhangW.X., QiuG.F., Uncertain decision making based on rough sets, Tsinghua University Publishers, Beijing, 2005.
51.
ZhangL., ZhangB., Theory and application of problem solving-theory and application of granular computing in quotient spaces, Tsinghua University Publishers, Beijing, 2007.
