Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Complexity and Systems Science pp 4283–4305Cite as

Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach

  • Reference work entry

Definition of the Subject

This article describes the dominance‐based rough set approach (DRSA) to granular computing and data mining. DRSA was first introduced asa generalization of the rough set approach for dealing with multicriteria decision analysis, where preference order is important. The ordering isalso important, however, in many other problems of data analysis. Even when the ordering seems absent, the presence or the absence of a property canbe represented in ordinal terms, because if two properties are related, the presence, rather than the absence, of one property should make more (or less)probable the presence of the other property. This is even more apparent when the presence or the absence of a property is graded or fuzzy, because inthis case, the more credible the presence of a property, the more (or less) probable the presence of the other property. Since the presence ofproperties, possibly fuzzy, is the basis of any granulation, DRSA can be seen as a general basis for...

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   3,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   549.99
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

Learn about institutional subscriptions

Abbreviations

Case-based reasoning:

Case-based reasoning is a paradigm in machine learning whose idea is that a new problem can be solved by noticing its similarity to a set of problems previously solved. Case-based reasoning regards the inference of some proper conclusions related to a new situation by the analysis of similar cases from a memory of previous cases. Very often similarity between two objects is expressed on a graded scale and this justifies application of fuzzy sets in this context. Fuzzy case-based reasoning is a popular approach in this domain.

Decision rule:

Decision rule is a logical statement of the type “if…, then…”, where the premise (condition part) specifies values assumed by one or more condition attributes and the conclusion (decision part) specifies an overall judgment.

Dominance‐based rough set approach (DRSA):

DRSA permits approximation of a set in universe U based on available ordinal information about objects of U. Also the decision rules induced within DRSA are based on ordinal properties of the elementary conditions in the premise and in the conclusion, such as “if property \( { f_{i1} } \) is present in degree at least \( { \alpha_{i1} } \) and … property f ip is present in degree at least α ip , then property f iq is present in degree at least α iq ”.

Fuzzy sets:

Differently from ordinary sets in which an object belongs or does not belong to a given set, in a fuzzy set an object belongs to a set in some degree. Formally, in universe U a fuzzy set X is characterized by its membership function \( { \mu_X\colon U \rightarrow [0,1] } \), such that for any \( { y \in U } \), y certainly does not belong to set X if \( { \mu_X(y)=0 } \), y certainly belongs to X if \( { \mu_X(y)=1 } \), and y belongs to X with a given degree of certainty represented by the value of \( { \mu_X(y } \)) in all other cases.

Granular computing:

Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory.

Ordinal properties and monotonicity:

Ordinal properties in description of objects are related to graduality of the presence or absence of a property. In this context, it is meaningful to say that a property is more present in one object than in another object. It is important that the ordinal descriptions are handled properly, which means, without introducing any operation, such as sum, averages, or fuzzy operators, like t-norm or t‑conorm of Łukasiewicz, taking into account cardinal properties of data not present in the considered descriptions, which would, therefore, give not meaningful results. Monotonicity is strongly related to ordinal properties. It regards relationships between degrees of presence or absence of properties in the objects, like “the more present is property f i , the more present is property f j ”, or “the more present is property f i , the more absent is property f j ”. The graded presence or absence of a property can be meaningfully represented using fuzzy sets. More precisely, the degree of presence of property f i in object \( { y \in U } \) is the value given to y by the membership function of the set of objects having property f i .

Rough set:

A rough set in universe U is an approximation of a set based on available information about objects of U. The rough approximation is composed of two ordinary sets called lower and upper approximation. Lower approximation is a maximal subset of objects which, according to the available information, certainly belong to the approximated set, and upper approximation is a minimal subset of objects which, according to the available information, possibly belong to the approximated set. The difference between upper and lower approximation is called boundary.

Bibliography

  1. Cattaneo G, Ciucci D (2004) Algebraic structures for rough sets. In: Transactionon rough sets II. LNCS, vol 3135. Springer, Berlin, pp 208–252

    Google Scholar 

  2. Cattaneo G, Giuntini R, Pilla R (1999) BZMVdM algebras and stonian MV‐algebras, (applications to fuzzy sets and rough approximations). Fuzzy Sets Syst108:201–222

    MathSciNet  MATH  Google Scholar 

  3. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J GeneralSyst 17:191–209

    MATH  Google Scholar 

  4. Dubois D, Prade H (1992) Gradual inference rules in approximate reasoning. InfSci 61:103–122

    MathSciNet  MATH  Google Scholar 

  5. Dubois D, Prade H (1992) Putting rough sets and fuzzy sets together. In:Słowiński R (ed) Intelligent decision support – Handbook of applications and advances of the rough sets theory. Kluwer, Dordrechtpp 203–232

    Google Scholar 

  6. Dubois D, Prade H, Esteva F, Garcia P, Godo L, Lopez de Mantara R (1998) Fuzzyset modelling in case-based reasoning. Int J Intell Syst 13:345–373

    MATH  Google Scholar 

  7. Dubois D, Grzymala‐Busse J, Inuiguchi M, Polkowski L (eds) (2004)Transations on rough sets II: Rough sets and fuzzy sets. LNCS, vol 3135. Springer, Berlin

    Google Scholar 

  8. Dyer J (2005) MAUT – Multiattribute utility theory, In:Figueira J, Greco S, Ehrgott M (eds) Multiple criteria decision analysis: State of the art surveys. Springer, Berlin,pp 266–294

    Google Scholar 

  9. Figueira J, Greco S, Ehrgott M (eds) (2005) Multiple criteria decision analysis:State of the art surveys. Springer, Berlin

    Google Scholar 

  10. Fodor J, Roubens M (1994) Fuzzy preference modelling and multicriteriadecision support. Kluwer, Dordrecht

    MATH  Google Scholar 

  11. Fortemps P, Greco S, Słowiński R (2008) Multicriteria decisionsupport using rules that represent rough‐graded preference relations. Eur J Operational Res 188:206–223

    Google Scholar 

  12. Gilboa I, Schmeidler D (2001) A theory of case-based decisions.Cambridge University Press, Cabmridge

    MATH  Google Scholar 

  13. Ginsburg S, Hull R (1983) Order dependency in the relational model. TheorComput Sci 26:149–195

    MathSciNet  MATH  Google Scholar 

  14. Greco S, Inuiguchi M, Słowiński R (2002) Dominance‐based roughset approach using possibility and necessity measures. In: Alpigini JJ, Peters JF, Skowron A, Zhong N (eds) Rough sets and current trends incomputing. LNAI, vol 2475. Springer, Berlin, pp 85–92

    Google Scholar 

  15. Greco S, Inuiguchi M, Słowiński R (2004) A new proposal forrough fuzzy approximations and decision rule representation. In: Dubois D, Grzymala‐Busse J, Inuiguchi M, Polkowski L (eds) Transations on roughsets II: Rough sets and fuzzy sets. LNCS, vol 3135. Springer, Berlin, pp 156–164

    Google Scholar 

  16. Greco S, Inuiguchi M, Słowiński R (2006) Fuzzy rough sets andmultiple‐premise gradual decision rules. Int J Approx Reason 41:179–211

    Google Scholar 

  17. Greco S, Matarazzo B, Słowiński R (1999) The use of rough sets andfuzzy sets in MCDM. In: Gal T, Stewart T, Hanne T (eds) Advances in multiple criteria decision making. Kluwer, Boston,pp 14.1–14.59

    Google Scholar 

  18. Greco S, Matarazzo B, Słowiński R (2000) Rough set processing ofvague information using fuzzy similarity relations. In: Calude C, Paun G (eds) From finite to infinite. Springer, Berlin,pp 149–173

    Google Scholar 

  19. Greco S, Matarazzo B, Słowiński R (2000) A fuzzy extension ofthe rough set approach to multicriteria and multiattribute sorting. In: Fodor J, De Baets B, Perny P (eds) Preferences and decisions under incompleteinformation. Physica, Heidelberg, pp 131–154

    Google Scholar 

  20. Greco S, Matarazzo B, Słowiński R (2001) Rough sets theory formulticriteria decision analysis. Eur J Operational Res 129:1–47

    Google Scholar 

  21. Greco S, Matarazzo B, Słowiński R (2001) Rough set approach todecisions under risk. In: Ziarko W, Yao Y (eds) Rough sets and current trends in computing. LNAI, vol 2005. Springer, Berlin,pp 160–169

    Google Scholar 

  22. Greco S, Matarazzo B, Słowiński R (2002) Preference representationby means of conjoint measurement and decision rule model. In: Bouyssou D, Jacquet‐Lagréze E, Perny P, Słowiński R, Vanderpooten D,Vincke P (eds) Aiding decisions with multiple criteria – Essays in Honor of Bernard Roy. Kluwer, Dordrecht,pp 263–313

    Google Scholar 

  23. Greco S, Matarazzo B, Słowiński R (2004) Axiomatic characterizationof a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules. Eur J Operational Res158:271–292

    Google Scholar 

  24. Greco S, Matarazzo B, Słowiński R (2004) Dominance‐based roughset approach to knowledge discovery (I) – General perspective. In: Zhong N, Liu J (eds) Intelligent technologies for information analysis.Springer, Berlin, pp 513–552

    Google Scholar 

  25. Greco S, Matarazzo B, Słowiński R (2004) Dominance‐based roughset approach to knowledge discovery (II) – Extensions and applications. In: Zhong N, Liu J (eds) Intelligent technologies for informationanalysis. Springer, Berlin, pp 553–612

    Google Scholar 

  26. Greco S, Matarazzo B, Słowiński R (2005) Decision rule approach. In:Figueira J, Greco S, Ehrgott M (eds) Multiple criteria decision analysis: State of the art surveys. Springer, Berlin,pp 507–563

    Google Scholar 

  27. Greco S, Matarazzo B, Słowiński R (2005) Generalizing rough settheory through dominance‐based rough set approach. In: Slezak D, Yao J, Peters J, Ziarko W, Hu X (eds) Rough sets, fuzzy sets, data mining, andgranular computing. LNAI, vol 3642. Springer, Berlin, pp 1–11

    Google Scholar 

  28. Greco S, Matarazzo B, Słowiński R (2006) Dominance‐based roughset approach to case-based reasoning. In: Torra V, Narukawa Y, Valls A, Domingo‐Ferrer J (eds) Modelling decisions for artificial intelligence.LNAI, vol 3885. Springer, Berlin, pp 7–18

    Google Scholar 

  29. Greco S, Matarazzo B, Słowiński R (2007) Dominance‐based roughset approach as a proper way of handling graduality in rough set theory. In: Transactions on rough sets VII. LNAI, vol 4400. Springer, Berlin,pp 36–52

    Google Scholar 

  30. Greco S, Matarazzo B, Słowiński R (2008) An algebraic structure fordominance‐based rough set approach. In: Proc. 3rd Int Conference on rough sets and knowledge technology (RSKT 2008), LNAI. Springer, Berlin,pp 252–259

    Google Scholar 

  31. Greco S, Matarazzo B, Słowiński R, Stefanowski J (2001) Variableconsistency model of dominance‐based rough set approach, In: Ziarko W, Yao Y (eds) Rough sets and current trends in computing. LNAI, vol 2005.Springer, Berlin, pp 170–181

    Google Scholar 

  32. Greco S, Predki B, Słowiński R (2002) Searching for an equivalencebetween decision rules and concordance‐discordance preference model in multicriteria choice problems. Control Cybern31:921–935

    Google Scholar 

  33. Hume D (1748) An enquiry concerning human understanding. Oxford, ClarendonPress

    Google Scholar 

  34. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer,Dordrecht

    MATH  Google Scholar 

  35. Kolodner J (1993) Case-based reasoning. Morgan Kaufmann, SanMateo

    Google Scholar 

  36. Leake DB (1996) CBR in context: the present and future. In: Leake D (ed)Case-based reasoning: Experiences, lessons, and future directions. AAAI Press/MIT Press, Menlo Park, pp 1–30

    Google Scholar 

  37. Lin TY (1988) Neighborhood systems and relational databases. In: Proceedingsof the ACM Conference on Computer Science, Atlanta, p 725

    Google Scholar 

  38. Lin TY (1989) Neighborhood systems and approximation in database and knowledgebase systems. In: Proceedings of the Fourth International Symposium on Methodologies of Intelligent Systems, Poster Session, October 12–15,pp 75–86

    Google Scholar 

  39. Lin TY (1992) Topological and fuzzy rough sets. In: Slowinski R (ed)Intelligent decision support – Handbook of application and advances of the rough sets theory. Kluwer, Dordrecht,pp 287–304

    Google Scholar 

  40. Lin TY (1997) Granular computing. Announcement of the BISC Special InterestGroup on Granular Computing

    Google Scholar 

  41. Lin TY (1998) Granular computing on binary relations I: Data mining andneighborhood systems. In: Skowron A, Polkowski L (eds) Rough sets in knowledge discovery. Physica, Heidelberg,pp 107–121

    Google Scholar 

  42. Lin TY (1998) Granular computing on binary relations II: Rough setrepresentations and belief functions. In: Skowron A, Polkowski L (eds) Rough sets in knowledge discovery. Physica, Heidelberg,pp 121–140

    Google Scholar 

  43. Loemker L (ed and trans), Leibniz GW (1969) Philosophical papers and letters,2nd edn. Reidel, Dordrecht

    Google Scholar 

  44. Nakamura A, Gao JM (1991) A logic for fuzzy data analysis. Fuzzy SetsSyst 39:127–132

    MathSciNet  MATH  Google Scholar 

  45. Pal SK, Skowron A (eds) (1999) Rough-fuzzy hybridization: A new trends indecision making. Springer, Singapore

    Google Scholar 

  46. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci11:341–356

    MathSciNet  MATH  Google Scholar 

  47. Pawlak Z (1991) Rough sets. Kluwer, Dordrecht

    MATH  Google Scholar 

  48. Pawlak Z (2001) Rough set theory. Künstliche Intelligenz 3:38–39

    Google Scholar 

  49. Peters JF, Skowron A, Dubois D, Grzymala‐Busse J, Inuiguchi M, PolkowskiL (eds) (2005) Rough sets and fuzzy sets, transaction on rough sets II. Springer, Berlin

    Google Scholar 

  50. Polkowski L (2002) Rough set: mathematical foundations. Physica,Heidelberg

    MATH  Google Scholar 

  51. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy roughsets. Fuzzy Sets Syst 126:137–155

    MathSciNet  MATH  Google Scholar 

  52. Słowiński R, Greco S, Matarazzo B (2002) Axiomatization of utility,outranking and decision‐rule preference models for multiple‐criteria classification problems under partial inconsistency with the dominanceprinciple. Control Cybern 31:1005–1035

    Google Scholar 

  53. Słowiński R, Greco S, Matarazzo B (2002) Mining decision‐rulepreference model from rough approximation of preference relation. In: Proc. 26th IEEE Annual Int. Conference on Computer Software & Applications(COMPSAC 2002), Oxford, pp 1129–1134

    Google Scholar 

  54. Słowiński R, Greco S, Matarazzo B (2005) Rough set based decisionsupport. In: Burke EK, Kendall G (eds) Search methodologies: Introductory tutorials in optimization and decision support techniques. Springer, New York,pp 475–527

    Google Scholar 

  55. Stewart T (2005) Dealing with uncertainties in MCDA. In: Figueira J, Greco S,Ehrgott M (eds) Multiple criteria decision analysis: State of the art surveys. Springer, Berlin, pp 445–470

    Google Scholar 

  56. Zadeh LA (1965) Fuzzy sets. Inf Control8:338–353

    MathSciNet  MATH  Google Scholar 

  57. Zadeh LA (1979) Fuzzy sets and information granularity. In: Gupta M, RagadeRK, Yager RR (eds) Advances in fuzzy set theory and applications. North‐Holland, Amsterdam, pp 3–18

    Google Scholar 

  58. Zadeh LA (1996) Key roles of information granulation and fuzzy logic in humanreasoning, concept formulation and computing with words. In: Proceedings of the 5th IEEE International Conference on Fuzzy Systems, New Orleans,p 1

    Google Scholar 

  59. Zadeh LA (1997) Towards a theory of fuzzy information granulation and itscentrality in human reasoning and fuzzy logic. Fuzzy Sets Syst 90:111–127

    MathSciNet  MATH  Google Scholar 

  60. Zadeh LA (1999) From computing with numbers to computing withwords – from manipulation of measurements to manipulation of perception. IEEE Trans Circuits Syst – I: Fundament Theor Appl45:105–119

    Google Scholar 

  61. Ziarko W (1993) Variable precision rough sets model. J Comput Syst Sci46:39–59

    MathSciNet  MATH  Google Scholar 

  62. Ziarko W (1998) Rough sets as a methodology for data mining. In:Polkowski L, Skowron A (eds) Rough Sets in Knowledge Discovery, vol 1. Physica, Heidelberg, pp 554–576

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Economics, University of Catania, Catania, Italy

    Salvatore Greco & Benedetto Matarazzo

  2. Poznań University of Technology, Institute of Computing Science, Poznan, Poland

    Roman Słowiński

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Roman Słowiński

Authors
  1. Salvatore Greco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benedetto Matarazzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roman Słowiński
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Greco, S., Matarazzo, B., Słowiński, R. (2009). Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_251

Download citation

Publish with us

Policies and ethics

Search

Navigation