Measures and approximations using empirical structures

Abstract

Suppose that \({\mathscr {N}}\) is a set of subsets of a finite set Q. \({\mathscr {N}}\) may be interpreted, for example, as a set of observed states as a result of an experiment or as a model of some theory. In a further step, the elements of \({\mathscr {N}}\) can be weighted by some set function \(f: {\mathscr {N}}\rightarrow {\mathbb {R}}\) such as various kinds of uncertainty measures or probabilities. In this paper, we demonstrate the process of analysing data based on observed granules according to the model proposed by G. Gigerenzer, and argue the necessity of an error theory when passing from a theoretical model to empirical data. We also investigate changes that occur when using granule structures weaker than a Boolean algebra, and under which conditions results based on a granule set can be extended to the whole power set algebra. Our main examples are results of a test taken by a population of students, and their interpretation by various probability functions such as basic probability assignments and belief functions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

Notes

  1. 1.

    bpas are often called mass functions and their semantics have been subject to controversial discussions. By introducing them as normalized weights, a very common tool in data analysis and decision support, we hope to avoid such controversies.

  2. 2.

    For the statistical background we invite the reader to consult Hand (2008).

References

  1. Berger V, Zhang J (2005) Structural zeros. In: Everitt B, Howell D (eds) Encyclopedia of statistics in behavioral science, vol 4. Wiley, Chichester, pp 1958–1959

    Google Scholar 

  2. Berthold M, Hand D (eds) (2007) Intelligent data analysis, 2nd edn. Springer, Berlin

    Google Scholar 

  3. Bilgiç T, Türkşen IB (2008) Measurement and elicitation of membership functions. In: Pedrycz W, Skowron A, Kreinovich V (eds) Handbook of granular computing. Wiley, Chicester, pp 141–151

    Google Scholar 

  4. Birkhoff G (1948) Lattice theory, American-mathematical-society-colloquium, vol 25, 2nd edn. AMS, Providence

    Google Scholar 

  5. Dempster A (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339

    MathSciNet  MATH  Article  Google Scholar 

  6. Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York

    Google Scholar 

  7. Dubois D, Prade H (2016a) Bridging gaps between several forms of granular computing. Granul Comput 1:115–126

    Article  Google Scholar 

  8. Dubois D, Prade H (2016b) Practical methods for constructing possibility distributions. Int J Intell Syst 31:215–239

    Article  Google Scholar 

  9. Düntsch I, Gediga G (2008) Probabilistic granule analysis. In: Chan CC, Grzymala-Busse JW, Ziarko WP (eds) Proceedings of the Sixth International Conference on Rough Sets and Current Trends in Computing (RSCTC 2008), Springer Verlag, Lecture Notes in Computer Science, vol 5306, pp 223–231

  10. Düntsch I, Gediga G (2015) PRE and variable precision models in rough set data analysis. In: Peters J, Skowron A (eds) Transactions on Rough Sets, Lecture Notes in Computer Science, vol XIX, Springer Verlag, Heidelberg, pp 17–37, MR3618228

  11. D’Urso P (2017) Exploratory multivariate analysis for empirical information affected by uncertainty and modeled in a fuzzy manner: a review. Granular Comput 2:225–247

    Article  Google Scholar 

  12. Efron B, Tibshirani RJ (1993) An Introduction to the bootstrap. Chapman & Hall, New York

    Google Scholar 

  13. Falmagne JC, Doignon JP (2011) Learning spaces. Springer, Heidelberg

    Google Scholar 

  14. Falmagne JC, Koppen M, Villano M, Doignon JP, Johannesen J (1990) Introduction to knowledge spaces: How to build, test and search them. Psychol Rev 97(2):201–224

    Article  Google Scholar 

  15. Gediga G, Düntsch I (2003) On model evaluation, indices of importance, and interaction values in rough set analysis. In: Pal S, Polkowski L, Skowron A (eds) Rough-neural computing: techniques for computing with words. Physica Verlag, Heidelberg, pp 251–276

    Google Scholar 

  16. Gediga G, Düntsch I (2014) Standard errors of indices in rough set data analysis. In: Peters J, Skowron A (eds) Transactions on rough sets, vol 17. Lecture Notes in Computer Science, vol 8375. Springer, Heidelberg, pp 33–47

    Google Scholar 

  17. Gigerenzer G (1981) Messung und Modellbildung in der Psychologie. Birkhäuser, Basel

    Google Scholar 

  18. Grabisch M (2004) The Möbius transform on symmetric ordered structures and its application to capacities on finite sets. Discret Math 287:17–34

    MATH  Article  Google Scholar 

  19. Grabisch M (2009) Belief functions on lattices. Int J Intell Syst 24(1):76–95

    MATH  Article  Google Scholar 

  20. Grabisch M (2016) Set functions, games and capacities in decision making, theory and decision library c, vol 46. Springer, Berlin

    Google Scholar 

  21. Guttman L (1944) A basis for scaling qualitative data. Am Soc Rev 9:139–150

    Article  Google Scholar 

  22. Haertel E (1989) Using restricted latent class models to map the skill structure of achievement items. J Educ Meas 26:301–324

    Article  Google Scholar 

  23. Hand D (2008) Statistics: a very short introduction. Oxford University Press, Oxford

    Google Scholar 

  24. Pawlak Z, Skowron A (1994) Rough membership functions. Advances in the dempster-shafer theory of evidence. Wiley, Hoboken, pp 251–271

    Google Scholar 

  25. Pedrycz W (2000) Granular computing: an introduction. In: Kasabov N (ed) Future directions for intelligent systems and information sciences. Springer, Berlin, pp 309–328

    Google Scholar 

  26. Pedrycz W (2018) Granular computing for data analytics: a manifesto of human-centric computing. IEEE/CAA J Autom Sin 5(6):1025–1034

    MathSciNet  Article  Google Scholar 

  27. Rota G (1964) On the foundations of combinatorial theory I. Theory of Möbius functions. Z Wahrscheinlichkeitstheorie 2:340–368

    MathSciNet  MATH  Article  Google Scholar 

  28. Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton

    Google Scholar 

  29. Smets P (1988) Belief functions. In: Smets P, Mandani A, Dubois D, Prade H (eds) Non-standard logics for automated reasoning. Academic Press, London

    Google Scholar 

  30. Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66(2):191–234

    MathSciNet  MATH  Article  Google Scholar 

  31. Syau Y, Skowron A, Lin E (2017) Inclusion degree with variable-precision model in analyzing inconsistent decision tables. Granul Comput 2:65–72

    Article  Google Scholar 

  32. Wang H (2003) Contextual probability. J Telecommun Inf Technol 3:92–97

    Google Scholar 

  33. Wang G (2017) DGCC: data-driven granular cognitive computing. Granul Comput 2:343–355

    Article  Google Scholar 

  34. Wang H, Murtagh F (2008) A study of the neighborhood counting similarity. IEEE Trans Knowl Data Eng 20(4):449–461

    Article  Google Scholar 

  35. Wilson N (1993) Decision making with belief functions and pignistic probabilities. In: Clarke M, Kruse R, Moral S (eds) Symbolic and Quantitative Approaches to Reasoning Under Uncertainty, Springer Verlag, Lecture Notes in Computer Science, vol 747, pp 364–371, European Conference ECSQARU ’93

  36. Yager R, Liu L (eds) (2008) Classic works of the dempster-shafer theory of belief functions, studies in fuzziness and soft computing, vol 219. Springer, Berlin

    Google Scholar 

  37. Yao Y (1998) On generalizing Pawlak approximation operators. In: Polkowski L, Skowron A (eds) Proceedings of the 1st International Conference on Rough Sets and Current Trends in Computing (RSCTC-98), Springer Verlag, Berlin, LNAI, vol 1424, pp 298–307

  38. Yao Y (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353

    MathSciNet  Article  Google Scholar 

  39. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    MATH  Article  Google Scholar 

  40. Zhou C (2013) Belief functions on distributive lattices. Artif Intell 201:1–31

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

We should like to thank the anonymous referees for careful reading and helpful suggestions. I. Düntsch and H. Wang gratefully acknowledge support by the National Natural Science Foundation of China, Grant No.61976053.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ivo Düntsch.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The ordering of authors is alphabetical and equal authorship is implied.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Düntsch, I., Gediga, G. & Wang, H. Measures and approximations using empirical structures. Granul. Comput. 6, 47–58 (2021). https://doi.org/10.1007/s41066-019-00198-y

Download citation

Keywords

  • Granular structures
  • Empirical knowledge structures
  • Rough sets
  • Evidence theory
  • Error theory