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

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 which T (resp., F ) f0; 1g n denotes a set of positive (resp., negative) examples for the phenomenon under consideration. In this paper, we consider the case in which extracted knowledge has a decomposable structure; i.e., f is described as a form f(x) = g(x[S 0 ]; h(x[S 1 ])) for some S 0 ; S 1 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 jT j and jF j must be su ciently large. In this paper, we provide an index for such indispensable number of examples, based on probabilistic analysis. Using p = jT j=(jT j+ jF j) and q = jF j=(jT j+ jF j), we claim that there exist many deceptive decomposable structures of (T;F ) if jT j+ jF j p 2 n 1 =pq. The computational results on synthetically generated data sets show that the above index gives a good lower bound on the indispensable data size.