New Topological Approaches for Data Granulation


Data granulation is a good tool of decision making in various types of real life applications. The basic ideas of data granulation have appeared in many fields, such as interval analysis, quantization, rough set theory, Dempster-Shafer theory of belief functions, divide and conquer, cluster analysis, machine learning, databases, information retrieval, and many others. In this paper, we initiate some new topological tools for data granulation using rough set approximations. Moreover, we define some topological measures of data granulation in topological I formation systems. Topological generalizations using δβ-open sets and their applications of information granulation are developed.

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A. S. Salama and O. G. Elbarbary, "New Topological Approaches for Data Granulation," Journal of Software Engineering and Applications, Vol. 6 No. 7B, 2013, pp. 1-6. doi: 10.4236/jsea.2013.67B001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] D. Andrijevic, Semi-preopen sets and Mat. Vesnik, Vol. 38, 1986, pp. 24-32.
[2] A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb, “On Pre-Continuous and Week Precontinuous Mappings,” Proc. Math. & phys. Soc. Egypt, Vol. 53, 1982, pp. 47-53.
[3] T. Nishino, M. Nagamachi and H. Tanaka, “Variable Precision Bayesian Rough Set Model and Its Application to Human Evaluation Data,” RSFDGrC 2005, LNAI 3641, Springer Verlag, 2005, pp. 294-303.
[4] T. Nishino, M. Sakawa, K. Kato, M. Nagamachi and H. Tanak, “Probabilistic Rough Set Model and Its Application to Kansei Engineering, Transactions on Rough Sets V (Inter. J. of Rough Set Society)”, LNCS 4100, Springer, 2006, pp. 190-206.
[5] O. Njasted, “On Some Classes of Nearly Open Sets,” Pro. J. Math. Vol. 15, 1965, pp. 961-970.
[6] N. Levine, “Semi Open Sets and Semi Continuity Topological Spaces, “The American Mathematical Monthly,” Vol. 70, 1963, pp. 24-32. doi:10.2307/2312781
[7] J. Y. Liang, J. H. Wang and Y. H. Qian, “A New Measure of Uncertainty Based on Knowledge Granulation for Rough Sets,” Information Sciences, Vol. 179, 2009, pp. 458-470. doi:10.1016/j.ins.2008.10.010
[8] E. Lashein,, A. M. Kozae, A. A. Khadra and T. Medhat, “Rough Set Theory for Topological Spaces,” International Journal of Approximate Reasoning, Vol. 40, 2005, pp. 35-43. doi:10.1016/j.ijar.2004.11.007
[9] T.Y. Lin, Granular Computing on Binary Relations I: data mining and neighborhood systems, II: rough set representations and belief functions, In: Rough Setsin Knowledge Discovery 1, L. Polkowski, A. Skowron (Eds.), Phys.-Verlag, Heidelberg, 1998, pp. 107-14.
[10] T. Y. Lin, Y. Y. Yao and L. A. Zadeh, “Data Mining, Rough Sets and Granular Computing (Studies in Fuzziness and Soft Computing),” Physica-Verlag, Heidelberg ,2002.doi:10.1007/978-3-7908-1791-1
[11] G. L. Liu and Y. Sai, “A Comparison of Two Types of Rough Sets Induced by Coverings,” International Journal of Approximate Reasoning, Vol. 50, 2009, pp. 521-528. doi:10.1016/j.ijar.2008.11.001
[12] Y. Leung, M. M. Fischer, W.-Z. Wu and J.-S. Mi, “A Rough Set Approach for the Discovery of Classification Rules in Interval-Valued Information Systems,” International Journal of Approximate Reasoning, Vol. 47, 2008, pp. 233-246. doi:10.1016/j.ijar.2007.05.001
[13] G. L. Liu, “Axiomatic Systems for Rough Sets and Fuzzy Rough Sets,” International Journal of Approximate Reasoning, Vol. 48, 2008, pp. 857-867. doi:10.1016/j.ijar.2008.02.001
[14] Z. Pei, D. W. Pei and L. Zheng, “Topology vs Generalized Rough Sets,” International Journal of Approximate Reasoning, Vol. 52, No. 2, 2011, pp. 231-239. doi:10.1016/j.ijar.2010.07.010
[15] Z. Pei, D. W. Pei and L. Zheng, “Covering Rough Sets Based on Neighborhoods an Approach without Using Neighborhoods,” International Journal of Approximate Reasoning, Vol. 52 , 2011, pp. 461-472. doi:10.1016/j.ijar.2010.07.010
[16] L. Polkowski and A. Skowron, “Towards Adaptive Calculus of Granules,” Proceedings of 1998 IEEE Inter. Conf. on Fuzzy Sys., 1998, pp. 111-116.
[17] Z. Pawlak and A. Skowron, “Rough Sets and Boolean Reasoning,” Information Sciences, Vol. 177, 2007, pp. 41-73. doi:10.1016/j.ins.2006.06.007
[18] Z. Pawlak and A. Skowron, “Rough Sets: Some Extensions,” Information Sciences, Vol. 177, 2007, pp. 28-40. doi:10.1016/j.ins.2006.06.006
[19] Z. Pawlak and A. Skowron, “Rudiments of Rough Sets,” Information Sciences, Vol. 177, No.1, 2007, pp. 3-27. doi:10.1016/j.ins.2006.06.003
[20] Z. Pawlak, “Rough sets,” International Journal of Computer&Information Sciences, Vol. 11, 1981, pp. 341-356. doi:10.1007/BF01001956
[21] Y. H. Qian, J. Y. Liang and C. Y. Dang, “Knowledge Structure, Knowledge Granulation and Knowledge Distance in a knowledge base,” International Journal of Approximate Reasoning, Vol. 50 , 2009, pp. 174-188. doi:10.1016/j.ijar.2008.08.004
[22] Y. H. Qian, J. Y. Liang, Y. Y. Yao and C. Y. Dang, “MGRS: A Multi-Granulation Rough Set,” Information Sciences, Vol. 180, 2010, pp. 949-970. doi:10.1016/j.ins.2009.11.023
[23] Q. H. Hu, J. F. Liu and D. R. Yu, “Mixed Feature Selection Based on Granulation and Approximation,” Knowledge-based system, Vol. 21, 2008, pp. 294-304.
[24] A. S. Salama, “Topologies Induced by Relations with Applications,” Journal of Computer Science, Vol. 4, 2008, pp. 879-889. doi:10.3844/jcssp.2008.877.887
[25] A. S. Salama, “Two New Topological Rough Operators,” Journal of Interdisciplinary Math,” Vol. 11, No. 1, New Delhi Taru Publications, INDIA, 2008, pp. 1-10.
[26] A. S. Salama, “Topological Solution for Missing Attribute Values in Incomplete Information Tables,” Information Sciences, Vol. 180, 2010, pp. 631-639. doi:10.1016/j.ins.2009.11.010
[27] D. J. Spiegelhalter, K. R. Abrams and J. P. Myles, “Bayesian Approaches to Clinical Trials and Health-Care Evaluation,” John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, England, 2004.
[28] D. Slezak and W. Ziarko, “Attribute Reduction in the Bayesian Version of Variable Precision Rough Set Model,” In: Proc. of RSKD, ENTCS, Vol. 82, 2003, pp. 4-14.
[29] D. Slezak, W. Ziarko, “The Investigation of the Bayesian Rough Set Model,” International Journal of Approximate Reasoning, Vol. 40, 2005, pp. 81-91. doi:10.1016/j.ijar.2004.11.004
[30] D. Slezak, “The Rough Bayesian Model for Distributed Decision Systems,” RSCT 2004, LNAI 3066, Springer Verlag, 2004, pp. 384-393.
[31] D. Slezak, “Rough Sets and Bayes factors,” Transactions on Rough Set III, Lecture Notes Computer Science, Vol. 3400, 2005, pp. 202-229. doi:10.1007/11427834_10
[32] D. Slezak and W. Ziarko, Bayesian Rough Set Model, In: Proc. of the Int. Workshop on Foundation of Data Mining (FDM 2002), December 9, Maebashi, Japan, 2002, pp. 131-135.
[33] D. Slezak and W. Ziarko, “Variable Precision Bayesian Rough Set Model,” RSFDGrC 2003, LNAI 2639, Springer Verlag, 2003, pp. 312-315.
[34] R. R. Yager, “Comparing Approximate Reasoning and Probabilistic Reasoning Using the Dempster-Shafer Framework,” International Journal of Approximate Reasoning, Vol. 50, 2009, pp. 812-821. doi:10.1016/j.ijar.2009.03.003
[35] E. A. Rady, A. M. Kozae and M. M. E. Abd El-Monsef, “Generalized Rough Sets,” Chaos, Solitons, & Fractals, Vol. 21, 2004, pp. 49-53.doi:10.1016/j.chaos.2003.09.044
[36] Y. Y. Yao, “Constructive and Algebraic Methods of Theory of Rough Sets,” Information Sciences, Vol. 109, 1998, pp. 21-47. doi:10.1016/S0020-0255(98)00012-7
[37] Y. Y. Yao, “Relational Interpretations of Neighborhood Operators and Rough Set Approximation Operators,” Information Sciences, Vol. 111, 1998, pp. 239-259. doi:10.1016/S0020-0255(98)10006-3
[38] Y. Yang and R. I. John, “Generalizations of Roughness Bounds in Rough Set Operations,” International Journal of Approximate Reasoning, Vol. 48, 2008, pp. 868-878. doi:10.1016/j.ijar.2008.02.002
[39] Y. Yao and Y. Zhao, “Attribute Reduction in Decision-Theoretic Rough Set Models,” Information Sciences, Vol. 178, 2008, pp. 3356-3373. doi:10.1016/j.ins.2008.05.010
[40] Y. Y. Yao, “Granular Computing using Neighborhood Systems,” in: Advances in Soft Computing: Engineering Design and Manufacturing, R. Roy, T. Furuhashi, and P. K. Chawdhry (Eds.), Springer-Verlag, London, 1999, pp. 539-553.
[41] A. M. Zahran, “Regularly Open Sets and a Good Extension on Fuzzy Topological Spaces,” Fuzzy Sets and Systems, Vol. 116, 2000, pp. 353-359. doi:10.1016/S0165-0114(98)00139-0
[42] L. A. Zadeh, “Fuzzy Sets and Information Granularity,” In: Advances in Fuzzy Set Theory and Applications, Gupta, N., Ragade, R. and Yager, R. (Eds.), North-Holland, Amsterdam, 1979, pp. 3-18.
[43] L. A. Zadeh, “Towards a Theory of Fuzzy Information Granulation and its Centrality in Human Reasoning and Fuzzy Logic,” Fuzzy Sets and Systems, Vol. 19, 1997, pp. 111-127. doi:10.1016/S0165-0114(97)00077-8
[44] L. A. Zadeh, “Generalized Theory of Uncertainty (GTU)—Principal Concepts and Ideas,” Computational Statistics & Data Analysis, Vol. 51, No. 1, 2006, pp. 15-46. doi:10.1016/j.csda.2006.04.029
[45] L. A. Zadeh, “Toward a Perception-Based Theory of Probabilistic Reasoning with Imprecise Probabilities,” Journal of Statistical Planning and Inference, Vol. 105, No. 1, 2002, pp. 233-264. doi:10.1016/S0378-3758(01)00212-9

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