Local and global approximations for incomplete data

  • Authors:

  • Jerzy W. Grzymała-Busse;Wojciech Rząsa

  • Affiliations:

  • Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS and Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland;Department of Computer Science, University of Rzeszow, Rzeszow, Poland

  • Venue:
  • Transactions on rough sets VIII
  • Year:
  • 2008

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Abstract

For completely specified decision tables lower and upper approximations are unique, the lower approximation is the largest definable set contained in the approximated set X and the upper approximation of X is the smallest definable set containing X. For incomplete decision tables the existing definitions of upper approximations provide sets that, in general, are not minimal definable sets. The same is true for generalizations of approximations based on relations that are not equivalence relations. In this paper we introduce two definitions of approximations, local and global, such that the corresponding upper approximations are minimal. Local approximations are more precise than global approximations. Global lower approximations may be determined by a polynomial algorithm. However, algorithms to find both local approximations and global upper approximations are NP-hard. Additionally, we show that for decision tables with all missing attribute values being lost, local and global approximations are equal to one another and that they are unique.