Algebraic approaches to granular computing
Tóm tắt
We study the granular structures in granular computing from algebraic views. We model a granular structure based on an algebra that consists of a universe and a closure operator. Based on this formulation, we define the basic granules in a granular structure through the notion of subalgebras. Using formal concept analysis and rough set analysis as two examples, we demonstrate that the proposed formulation can unify a number of current studies on the related areas of granular computing. Moreover, by modelling the granular structures in rough sets with algebraic approaches, we generalize the notion of approximations in rough sets into a general granular structure. The algebraic approaches provide a mathematical formulation of granular structures, which may assist us in the construction and interpretation of granules and granular structures.
Tài liệu tham khảo
Birkhoff G (1979) Lattice theory, 3rd edn. American Mathematical Society, Providence
Chen ZH, Lin TY, Xie G (2013) Knowledge approximations in binary relation: granular computing approach. Int J Intell Syst 28:843–864. https://doi.org/10.1002/int.21607
Chen LS, Wang JY, Li L (2016) The models of granular system and algebraic quotient space in granular computing. Chin J Electron 25:1109–1113. https://doi.org/10.1049/cje.2016.08.001
Chibbaro S, Rondoni L, Vulpiani A (2014) Reductionism, emergence and levels of reality: the importance of being borderline. Springer, Switzerland
Fan J, Xie W, Pei J (1999) Subsethood measure: new definitions. Fuzzy Sets Syst 106:201–209. https://doi.org/10.1016/S0165-0114(97)00275-3
Ganter B, Wille R (1999) Formal concept analysis: mathematical foundations. Springer, Berlin
Grätzer G (1979) Universal algebra, 2nd edn. Springer, New York
Hońko P (2017) Properties of a granular computing framework for mining relational data. Int J Intell Syst 32:227–248. https://doi.org/10.1002/int.21839
Hu MJ, Yao YY (2019) Structured approximations as a basis for three-way decisions in rough set theory. Knowl Based Syst 165:92–109. https://doi.org/10.1016/j.knosys.2018.11.022
Hu MJ, Deng XF, Yao YY (2019) On the properties of subsethood measures. Inf Sci 494:208–232. https://doi.org/10.1016/j.ins.2019.04.038
Kang XP, Miao DQ (2016) A study on information granularity in formal concept analysis based on concept-bases. Knowl Based Syst 105:147–159. https://doi.org/10.1016/j.knosys.2016.05.005
Kok VJ, Chan CS (2017) GrCS: Granular computing-based crowd segmentation. IEEE Trans Cybern 47:1157–1168. https://doi.org/10.1109/TCYB.2016.2538765
Li XP, Fang LX, Lu ZG, Zhang JF, Zhao H (2017) A line flow granular computing approach for economic dispatch with line constraints. IEEE Trans Power Syst 32:4832–4842. https://doi.org/10.1109/TPWRS.2017.2665583
Liang JY, Wang F, Dang CY, Qian YH (2012) An efficient rough feature selection algorithm with a multi-granulation view. Int J Approx Reason 53:912–926. https://doi.org/10.1016/j.ijar.2012.02.004
Lin TY (1998) Granular computing on binary relations 1: data mining and neighborhood systems. In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery 1. Physica, Heidelberg, pp 107–121
Mckenzie RN, McNulty GF, Taylor WF (1987) Algebras, lattices, varieties, vol I. Wadsworth Inc., Bemont
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:341–356. https://doi.org/10.1007/BF01001956
Pawlak Z (1984) Rough classification. Int J Man Mach Stud 20:469–483. https://doi.org/10.1016/S0020-7373(84)80022-X
Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic, Boston
Qian YH, Zhang H, Li FJ, Hu QH, Liang JY (2014) Set-based granular computing: a lattice model. Int J Approx Reason 55:834–852. https://doi.org/10.1016/j.ijar.2013.11.001
Skowron A, Stepaniuk J (2007) Modeling of high quality granules. In: Kryszkiewicz M, Peters JF, Rybinski H, Skowron A (eds) Rough sets and intelligent systems paradigms (RSEISP 2007). Springer, Heidelberg, pp 300–309. https://doi.org/10.1007/978-3-540-73451-2_32
Wang Y, Miao DQ (2012) Comparison of granular computing models in a set-theoretic framework. In: Zanzotto FM, Tsumoto S, Taatgen N, Yao YY (eds) Brain Informatics. Springer, Heidelberg, pp 332–337. https://doi.org/10.1007/978-3-642-35139-6_31
Wang GY, Xu J (2014) Granular computing with multiple granular layers for brain big data processing. Brain Inform 1:1–10. https://doi.org/10.1007/s40708-014-0001-z
Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival I (ed) Ordered sets. Springer, Dordrecht, pp 445–470. https://doi.org/10.1007/978-94-009-7798-3_15
Xu WH, Li WT (2016) Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Trans Cybern 46:366–379. https://doi.org/10.1109/TCYB.2014.2361772
Yao YY (2000a) Information granulation and rough set approximation. Int J Intell Syst 16:87–104. https://doi.org/10.1002/1098-111X(200101)16:1<87::AID-INT7>3.0.CO;2-S
Yao YY (2000b) Granular computing: Basic issues and possible solutions. In: Proceedings of the 5th joint conference on information sciences, pp 186–189
Yao YY (2004) A partition model of granular computing. In: Peters JF, Skowron A, Grzymała-Busse JW, Kostek B, Świniarski RW, Szczuka MS (eds) Transactions on rough sets I. Springer, Heidelberg, pp 232–253. https://doi.org/10.1007/978-3-540-27794-1_11
Yao YY (2016) A triarchic theory of granular computing. Granul Comput 1:145–157. https://doi.org/10.1007/s41066-015-0011-0
Yao YY, Zhang N, Miao DQ, Xu FF (2012) Set-theoretic approaches to granular computing. Fund Inform 115:247–264. https://doi.org/10.3233/FI-2012-653
Young VR (1996) Fuzzy subsethood. Fuzzy Sets Syst 77:371–384. https://doi.org/10.1016/0165-0114(95)00045-3
Zadeh LA (1979) Fuzzy sets and information granularity. In: Gupta N, Ragade R, Yager R (eds) Advances in fuzzy set theory and applications. North-Holland, Amsterdam, pp 3–18. https://doi.org/10.1142/9789814261302_0022
Zadeh LA (1997) Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 19:111–127. https://doi.org/10.1016/S0165-0114(97)00077-8
Zeng Y, Zhong N (2008) On granular knowledge structures. In: Proceedings of 2008 international conference on advanced intelligence, Posts and Telecommunications Press, Beijing, pp 28–33
Zhi HL, Li JH (2016) Granule description based on formal concept analysis. Knowl Based Syst 104:62–73. https://doi.org/10.1016/j.knosys.2016.04.011
Zhong N, Huang JJ (2015) Granular structures induced by interval sets and rough sets. In: Yao YY, Hu QH, Yu H, Grzymala-Busse JW (eds) Rough sets, fuzzy sets, data mining, and granular computing (RSFDGrC 2015). Springer, Cham, pp 49–60. https://doi.org/10.1007/978-3-319-25783-9_5
Zhu W (2009) Relationships among basic concepts in covering-based rough sets. Inf Sci 179:2478–2486. https://doi.org/10.1016/j.ins.2009.02.013
Zhu XB, Pedrycz W, Li ZW (2009) Granular data description: designing ellipsoidal information granules. IEEE Trans Cybern 47:1157–1168. https://doi.org/10.1109/TCYB.2016.2612226