Advertisement

An Index for the Data Size to Extract Decomposable Structures in LAD

  • Hirotaka Ono
  • Mut unori Yagiura
  • Toshihide Ibaraki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2223)

Abstract

Logical analysis of data (LAD)is one of the methodologies for extracting knowledge as a Boolean function f from a given pair of data sets (T,F)on attributes set S of size n in whch T (resp.,F)0 , 1n denotes a set of positive (resp.,negative)examples for the phenomenon under cons deration.In this paper,we consider the case n which extracted knowledge has a decomposable structure;i.e.,f is described as aform f (x)=g(x[S0],h x[S1]))for some S0,S1 .S and Boolean functions g and h where x[I]denotes the projection of vector x on I In order to detect meaningful decomposable structures,it is expected that the sizes ∣T∣and ∣F∣ must be sufficiently large.In this paper,we provide an index for such indispensable number of examples,based on probabilistic analysis.Using p = ∣T ∣/ ∣T ∣+ ∣F ∣)and q = ∣F ∣/ ∣T ∣+ ∣F ∣),we claim that there exist many deceptive decomposable structures of (T,F) if ∣T + ∣F ∣≤√p n - 1 /pq The computat onal results on synthetically generated data sets show that the above index gives a good lower bound on the indispensable data size.

Keywords

logical analysis of data Boolean functions decomposable functions computational learning theory random graphs probabilistic analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alon and J.H. Spencer,The Probablistic Method, Second Edition (John Wiley & Sons,2000).Google Scholar
  2. 2.
    M. Anthony and N. Biggs,Computational Learning Theory, (Cambridge Univer-sity Press,1992).Google Scholar
  3. 3.
    E. Boros, V. Gurvich, P.L. Hammer, T. Ibarak and A. Kogan,Decompositions of partially defined Boolean functions,Discrete Applied Mathematics 62 (1995) 51–5.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    E. Boros, P.L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchn k,An implementat on of logical analysis of data,IEEE Trans. on Knowledge and Data Engineering 12 (2000)292–306.CrossRefGoogle Scholar
  5. 5.
    E. Boros, P.L. Hammer, T. Ibarak and A. Kogan,Logical analysis of numerical data,Mathematical Programming 79 (1997)163–190.MathSciNetMATHGoogle Scholar
  6. 6.
    Y. Crama, P.L. Hammer and T. Ibaraki,Cause-effect relationships and partially defined Boolean functions,Annals of Operations Research 16 (1988)299–325.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Erdös and A. Rényi,On the evolution of random graphs,Publications of the Mathematical Institute of the Hungarian Academy of Sciences 5 (1960)17–61.MathSciNetMATHGoogle Scholar
  8. 8.
    U.M. Fayyad, G. Piatetsky-Shapiro, P. Smyth,and R. Uthurusamy,Advances in Knowledge Discovery and Data Mining, (AAAI Press,1996).Google Scholar
  9. 9.
    J. Kivinen and H. Mannila,The power of sampling in knowledge discovery,Pro-ceedings of the 1994 ACM SIGACT-SIGMOD-SIGACT Symposium on Principles of Database Theory (PODS’94),(1994)77–85.Google Scholar
  10. 10.
    K. Makino, K. Yano and T. Ibaraki,Positive and Horn decomposability of par-tially defined Boolean functions,Discrete Applied Mathematics, 74 (1997)251–274.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    S. Mii,Feature Determination Algorithms in the Analysis of Data,Master Thesis, Department of Applied Mathematics and Phys cs,Graduate School of Informat-ics,Kyoto University,March 2001.Google Scholar
  12. 12.
    H. Ono, K. Makino and T. Ibarak,Logical Analysis of Data with Decomposable Structures,COCOON2000 Lecture Notes in Computer Science 1858,(2000)396–406.Google Scholar
  13. 13.
    H. Toivonen,Sampling Large Databases for Association Rules,Proceedings of 22th International Conference on Very Large Data Bases (VLDB’96),(1996)134–145.Google Scholar
  14. 14.
    M. Yagiura, T. Ibarak and F. Glover,An Ejection Chain Approach for the Gen-eralized Assignment Problem,Technical Report #99013,Department of Applied Mathematics and Physics,Graduate School of Informatics,Kyoto University, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Hirotaka Ono
    • 1
  • Mut unori Yagiura
    • 1
  • Toshihide Ibaraki
    • 1
  1. 1.Department of Applied Mathematics and PhysicsGraduate School of Informatics, Kyoto UniversityKyotoJapan

Personalised recommendations