Abstract
In such areas as knowledge discovery, data mining and logical analysis of data, methodologies to find relations among attributes are considered important. In this paper, given a data set (T, F) of a phenomenon, where T ⊆ |0,1}n denotes a set of positive examples and F ⊆ {0,1}n denotes a set of negative examples, we propose a method to identify decomposable structures among the attributes of the data. Such information will reveal hierarchical structure of the phenomenon under consideration. We first study computational complexity of the problem of finding decomposable Boolean extensions. Since the problem turns out to be intractable (i.e., NP-complete), we propose a heuristic algorithm in the second half of the paper. Our method searches a decomposable partition of the set of all attributes, by using the error sizes of almost-fit decomposable extensions as a guiding measure, and then finds structural relations among the attributes in the obtained partition. The results of numerical experiment on synthetically generated data sets are also reported.
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References
M. Anthony and N. Biggs, Computational Learning Theory (Cambridge University Press, 1992).
E. Boros, V. Gurvich, P.L. Hammer, T. Ibaraki and A. Kogan, Decompositions of partially defined Boolean functions, Discrete Applied Mathematics, 62 (1995) 51–75.
E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of a partially defined Boolean function, Information and Computation, 140 (1998) 254–283.
E. Boros, P. L. Hammer, T. Ibaraki, A. Kogan, E. Mayoraz and I. Muchnik, An implementation of logical analysis of data, RUTCOR Research Report RRR 22-96, Rutgers University, 1996; to appear in IEEE Trans. on Data Engineering.
Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined Boolean functions, Annals of Operations Research, 16 (1988) 299–325.
P. Crescenzi and V. Kann, A compendium of NP optimization problems, http://www.nada.kth.se/viggo/index-en.html.
U. M. Fayyad, G. Piatetsky-Shapiro, P. Smyth, and R. Uthurusamy(eds.), Advances in Knowledge Discovery and Data Mining, 1996, AAAI Press.
M. R. Garey and D. S. Johnson, Computers and Intractability, Freeman, New York, 1979.
N. Garg, V. V. Vazirani and M. Yannakakis, Approximate max-flow min-(multi)cut theorems and their applications, SIAM J. on Computing, 25 (1996) 235–251.
N. Garg, V. V. Vazirani and M. Yannakakis, Multiway cuts in directed and node weighted graphs, ICALP’94, LNCS 820 (1994) 487–498.
K. Makino, K. Yano and T. Ibaraki, Positive and Horn decomposability of partially defined Boolean functions, Discrete Applied Mathematics, 74 (1997) 251–274.
S. Muroga, Threshold Logic and Its Applications, Wiley-Interscience, 1971.
R.C Read and R.E. Tarjan, Bounds on backtrack algorithms for listing cycles, paths, and spanning tress, Networks, 5 (1975) 237–252.
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© 2000 Springer-Verlag Berlin Heidelberg
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Ono, H., Makino, K., Ibaraki, T. (2000). Logical Analysis of Data with Decomposable Structures. In: Du, DZ., Eades, P., Estivill-Castro, V., Lin, X., Sharma, A. (eds) Computing and Combinatorics. COCOON 2000. Lecture Notes in Computer Science, vol 1858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44968-X_39
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DOI: https://doi.org/10.1007/3-540-44968-X_39
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