Skip to main content
SpringerLink
Log in

Similarity and Granulation

  • Chapter
  • First Online:
Granular Computing in Decision Approximation

Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 77))

Abstract

This chapter introduces into topics of similarity and granulation. We define similarity relations as tolerance and weak tolerance relations and give basic information on their structure. In preparation for theme of granulation, we extend notions of tolerance to graded tolerance and weak tolerance. We outline approaches to granulation and the notion of a granule.

You can please some of the people all of the time, you can please all of the people some of the time, but you can’t please all of the people all of the time

[John Lydgate]

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Albatineh, A.N., Niewiadomska-Bugaj, M., Mihalko, D.: On similarity indices and correction for chance agreement. J. Classif. 23, 301–313 (2006)

    Article  MathSciNet  Google Scholar 

  2. Bocheński, I.M.: Die Zeitg\(\ddot{{\rm o}}\)nossischen Denkmethoden. A. Francke AG, Bern (1954)

    Google Scholar 

  3. Casati, R., Varzi, A.C.: Parts and Places: The Structures of Spatial Representations. MIT Press, Cambridge (1999)

    Google Scholar 

  4. Gomolińska, A., Wolski, M.: Rough inclusion functions and similarity indices. In: Proceedings of CSP 2013 (22nd International Workshop on Concurrency, Specification and Programming), Warsaw University, September 2013. Białystok University of Technology, pp. 145–156 (2013). ISBN 978-83-62582-42-6

    Google Scholar 

  5. Grzymala-Busse, J.W.: Data with missing attribute values: generalization of indiscernibility relation and rule induction. Transactions on Rough Sets I. Lecture Notes in Computer Science, vol. 3100, pp. 78–95. Springer, Berlin (2004)

    Google Scholar 

  6. Grzymala-Busse, J.W., Ming, H.: A comparison of several approaches to missing attribute values in data mining. Lecture Notes in Artificial Intelligence, vol. 2005, pp. 378–385. Springer, Berlin (2004)

    Google Scholar 

  7. Kuratowski, C., Mostowski, A.: Set Theory. Polish Scientific Publishers, Warszawa (1968)

    MATH  Google Scholar 

  8. Lin, T.Y.: Neighborhood systems and relational database. Abstract. In: Proceedings of CSC’88, p. 725 (1988)

    Google Scholar 

  9. Lin, T.Y.: Neighborhood systems and approximation in database and knowledge based systems. In: Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems (ISMIS), pp. 75–86 (1989)

    Google Scholar 

  10. Lin, T.Y.: Topological and fuzzy rough sets. In: Słowiński, R. (ed.) Intelligent Decision Support. Handbook of Applications and Advances of the Rough Sets Theory, pp. 287–304. Kluwer, Dordrecht (1992)

    Google Scholar 

  11. Lin, T.Y.: From rough sets and neighborhood systems to information granulation and computing with words. In: Proceedings of the European Congress on Intelligent Techniques and Soft Computing, pp. 1602–1606 (1994)

    Google Scholar 

  12. Lin, T.Y.: Granular computing: fuzzy logic and rough sets. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with Words in Information/Intelligent Systems, vol. 1, pp. 183–200. Physica Verlag, Heidelberg (1999)

    Chapter  Google Scholar 

  13. Lin, T.Y.: Granular Computing. Lecture Notes in Computer Science, vol. 2639, pp. 16–24. Springer, Berlin (2003)

    Google Scholar 

  14. Lin, T.Y.: Granular computing: examples, intuitions, and modeling. In: Proceedings of IEEE 2005 Conference on Granular Computing GrC05, pp. 40–44. IEEE Press, Beijing (2005)

    Google Scholar 

  15. Lin, T.Y.: A roadmap from rough set theory to granular computing. In: Proceedings RSKT 2006, 1st International Conference on Rough Sets and Knowledge Technology, Chongqing, China. Lecture Notes in Artificial Intelligence, vol. 4062, pp. 33–41. Springer, Berlin (2006)

    Google Scholar 

  16. Łukasiewicz, J.: Die Logischen Grundlagen der Wahrscheinlichtkeitsrechnung. Cracow (1913)

    Google Scholar 

  17. Menger, K.: Statistical metrics. Proc. Natl. Acad. Sci. USA 28, 535–537 (1942)

    Google Scholar 

  18. Nguyen, S.H.: Regularity analysis and its applications in data mining. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Rough Set Methods and Applications. New Developments in Knowledge Discovery in Information Systems, pp. 289–378. Physica Verlag, Heidelberg (2000)

    Google Scholar 

  19. Nieminen, J.: Rough tolerance equality and tolerance black boxes. Fundamenta Informaticae 11, 289–296 (1988)

    MATH  MathSciNet  Google Scholar 

  20. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer, Dordrecht (1991)

    Book  MATH  Google Scholar 

  21. Poincaré, H.: La Science et l’Hypothèse. Flammarion, Paris (1902)

    Google Scholar 

  22. Polkowski, L., Skowron, A, Żytkow, J.: Tolerance based rough sets. In: Lin, T.Y., Wildberger, M. (eds.) Soft Computing: Rough Sets, Fuzzy Logic, Neural Networks, Uncertainty Management, Knowledge Discovery, pp. 55–58. Simulation Councils Inc., San Diego (1994)

    Google Scholar 

  23. Polkowski, L.: A rough set paradigm for unifying rough set theory and fuzzy set theory. Fundamenta Informaticae 54, 67–88; also. In: Proceedings RSFDGrC03, Chongqing, China, 2003. Lecture Notes in Artificial Intelligence, vol. 2639, pp. 70–78. Springer, Berlin (2003)

    Google Scholar 

  24. Polkowski, L.: A note on 3-valued rough logic accepting decision rules. Fundamenta Informaticae 61, 37–45 (2004)

    MATH  MathSciNet  Google Scholar 

  25. Polkowski, L.: Toward rough set foundations. Mereological approach. In: Proceedings RSCTC04, Uppsala, Sweden. Lecture Notes in Artificial Intelligence, vol. 3066, pp. 8–25. Springer, Berlin (2004)

    Google Scholar 

  26. Polkowski, L.: A rough-neural computation model based on rough mereology. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough-Neural Computing. Techniques for Computing with Words, pp. 85–108. Springer, Berlin (2004)

    Google Scholar 

  27. Polkowski, L.: Formal granular calculi based on rough inclusions. In: Proceedings of IEEE 2005 Conference on Granular Computing GrC05, pp. 57–62. IEEE Press, Beijing (2005)

    Google Scholar 

  28. Polkowski, L.: Rough-fuzzy-neurocomputing based on rough mereological calculus of granules. Int. J. Hybrid Intell. Syst. 2, 91–108 (2005)

    MATH  Google Scholar 

  29. Polkowski, L.: A model of granular computing with applications. In: Proceedings of IEEE 2006 Conference on Granular Computing GrC06, pp. 9–16. IEEE Press, Atlanta (2006)

    Google Scholar 

  30. Polkowski, L.: Granulation of knowledge in decision systems: the approach based on rough inclusions. The method and its applications. In: Proceedings RSEiSP’07 (in memory of Z. Pawlak). Lecture Notes in Artificial Intelligence, vol. 4585, pp. 69–79 (2007)

    Google Scholar 

  31. Polkowski, L.: The paradigm of granular rough computing. In: Proceedings ICCI’07. 6th IEEE International Conference on Cognitive Informatics, pp. 145–163. IEEE Computer Society, Los Alamitos (2007)

    Google Scholar 

  32. Polkowski, L.: Rough mereology in analysis of vagueness. In: Proceedings RSKT 2008. Lecture Notes in Artificial Intelligence, vol. 5009, pp. 197–205. Springer, Berlin (2008)

    Google Scholar 

  33. Polkowski, L.: A unified approach to granulation of knowledge and granular computing based on rough mereology: a survey. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of Granular Computing, pp. 375–400. Wiley, Chichester (2008)

    Google Scholar 

  34. Polkowski, L.: Granulation of knowledge: similarity based approach in information and decision systems. In: Meyers, R.A. (ed.) Springer Encyclopedia of Complexity and System Sciences. Springer, Berlin, article 00 788 (2009)

    Google Scholar 

  35. Polkowski, L.: Approximate Reasoning by Parts. An Introduction to Rough Mereology. ISRL, vol. 20. Springer, Berlin (2011)

    Google Scholar 

  36. Ruspini, E.H.: On the semantics of fuzzy logic. Int. J. Approx. Reason. 5, 45–88 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  37. Schreider, Yu.: Equality, Resemblance, and Order. Mir Publishers, Moscow (1975)

    Google Scholar 

  38. Tarski, A.: Zur Grundlegung der Booleschen Algebra. I. Fundamenta Mathematicae 24, 177–198 (1935)

    Google Scholar 

  39. Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered Sets, pp. 445–470. D. Reidel, Dordrecht (1982)

    Google Scholar 

  40. Yao, Y.Y.: Granular computing: basic issues and possible solutions. In: Proceedings of the 5th Joint Conference on Information Sciences, vol. 1, pp. 186–189. Association for Intelligent Machinery, Atlantic (2000)

    Google Scholar 

  41. Yao, Y.Y.: Perspectives of granular computing. In: Proceedings of IEEE 2005 Conference on Granular Computing GrC05, pp. 85–90. IEEE Press, Beijing (2005)

    Google Scholar 

  42. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  43. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  44. Zadeh, L.A.: Fuzzy sets and information granularity. In: Gupta, M., Ragade, R., Yager, R.R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North-Holland, Amsterdam (1979)

    Google Scholar 

  45. Zeeman, E.C.: The topology of the brain and the visual perception. In: Fort, K.M. (ed.) Topology of 3-manifolds and Selected Topics, pp. 240–256. Prentice Hall, Englewood Cliffs (1965)

    Google Scholar 

Further Reading

  1. Bargiela, A., Pedrycz, W.: Granular Computing: An Introduction. Kluwer Academic Publishers, Dordrecht (2003)

    Book  Google Scholar 

  2. Kacprzyk, J., Nurmi, H., Zadrożny, S.: Towards a comprehensive similarity analysis of voting procedures using rough sets and similarity measures. In: Skowron, A., Suraj, Z. (eds.) Rough Sets and Intelligent Systems—Professor Zdzisław Pawlak in Memoriam, vol. 1. Springer, Heidelberg, ISRL vol. 42, pp. 359–380 (2013)

    Google Scholar 

  3. Pal, S.K., Polkowski, L., Skowron, A. (eds.): Rough-Neural Computing: Techniques for Computing with Words. Springer, Heidelberg (2004)

    Google Scholar 

  4. Pedrycz, W. (ed.): Granular Computing: An Emerging Paradigm. Physica Verlag, Heidelberg (2001)

    Google Scholar 

  5. Pedrycz, W., Skowron, A., Kreinovich, V. (eds.): Handbook of Granular Computing. Wiley, Chichester (2008)

    Google Scholar 

  6. Pawlak, Z.: Elementary rough set granules: Toward a rough set processor. In: Pal, S.K., Polkowski, L., Skowron, A. (eds.) Rough Neuro-Computing: Techniques for Computing with Words, pp. 5–14. Springer, Heidelberg (2013)

    Google Scholar 

  7. Polkowski, L., Skowron, A.: Grammar systems for distributed synthesis of of approximate solutions extracted from experience. In: Paun, Gh., Salomaa, A. (eds.) Grammatical Models of Multi-Agent Systems, pp. 316–333. Gordon and Breach, Amsterdam (1999)

    Google Scholar 

  8. Polkowski, L., Skowron, A.: Towards adaptive calculus of granules. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with Words in Information/Intelligent Systems. Foundations, vol. 1. Physica Verlag, Heidelberg (1999)

    Google Scholar 

  9. Polkowski, L., Skowron, A.: Rough mereological calculi of granules: a rough set approach to computation. Comput. Intell. Int. J. 17(3), 472–492 (2001)

    MathSciNet  Google Scholar 

  10. Polkowski, L., Semeniuk-Polkowska, M.: On rough set logics based on similarity relations. Fundamenta Informaticae 64, 379–390 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Skowron, A., Stepaniuk, J.: Information granules: towards foundations of granular computing. Int. J. Intell. Syst. 16(1), 57–86 (2001)

    Article  MATH  Google Scholar 

  12. Skowron, A., Suraj, Z. (eds.): Rough Sets and Intelligent Systems. Professor Zdzisław Pawlak in Memoriam, vol. 1. Springer, ISRL, vol. 42. Heidelberg (2013)

    Google Scholar 

  13. Stepaniuk, J. (ed.): Rough-Granular Computing in Knowledge Discovery and Data Mining. Springer, Heidelberg (2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Polish-Japanese Institute of Information Technology, Warsaw, Poland

    Lech Polkowski

  2. Department of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland

    Lech Polkowski & Piotr Artiemjew

Authors
  1. Lech Polkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr Artiemjew
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lech Polkowski .

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Polkowski, L., Artiemjew, P. (2015). Similarity and Granulation. In: Granular Computing in Decision Approximation. Intelligent Systems Reference Library, vol 77. Springer, Cham. https://doi.org/10.1007/978-3-319-12880-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-12880-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12879-5

  • Online ISBN: 978-3-319-12880-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation