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

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.