### Abstract

Logical analysis of data (LAD) is one of the methodologies for extracting knowledge in the form of 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) ⊆{0,1} ^{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 f has a decomposable structure; 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, however, it is considered that the sizes |T| and |F| must be sufficiently large. In this paper, based on probabilistic analysis, we provide an index for such indispensable number of examples to detect decomposability; we claim that there exist many deceptive decomposable structures of (T,F) if |T||F|≤2^{n-1}. The computational results on synthetically generated data sets and real-world data sets show that the above index gives a good lower bound on the indispensable data size.

Original language | English |
---|---|

Pages (from-to) | 165-180 |

Number of pages | 16 |

Journal | Discrete Applied Mathematics |

Volume | 142 |

Issue number | 1-3 SPEC. ISS. |

DOIs | |

Publication status | Published - Aug 15 2004 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*142*(1-3 SPEC. ISS.), 165-180. https://doi.org/10.1016/j.dam.2004.02.001

**A decomposability index in logical analysis of data.** / Ono, Hirotaka; Yagiura, Mutsunori; Ibaraki, Toshihide.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 142, no. 1-3 SPEC. ISS., pp. 165-180. https://doi.org/10.1016/j.dam.2004.02.001

}

TY - JOUR

T1 - A decomposability index in logical analysis of data

AU - Ono, Hirotaka

AU - Yagiura, Mutsunori

AU - Ibaraki, Toshihide

PY - 2004/8/15

Y1 - 2004/8/15

N2 - Logical analysis of data (LAD) is one of the methodologies for extracting knowledge in the form of 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) ⊆{0,1} 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 f has a decomposable structure; 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, however, it is considered that the sizes |T| and |F| must be sufficiently large. In this paper, based on probabilistic analysis, we provide an index for such indispensable number of examples to detect decomposability; we claim that there exist many deceptive decomposable structures of (T,F) if |T||F|≤2n-1. The computational results on synthetically generated data sets and real-world data sets show that the above index gives a good lower bound on the indispensable data size.

AB - Logical analysis of data (LAD) is one of the methodologies for extracting knowledge in the form of 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) ⊆{0,1} 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 f has a decomposable structure; 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, however, it is considered that the sizes |T| and |F| must be sufficiently large. In this paper, based on probabilistic analysis, we provide an index for such indispensable number of examples to detect decomposability; we claim that there exist many deceptive decomposable structures of (T,F) if |T||F|≤2n-1. The computational results on synthetically generated data sets and real-world data sets show that the above index gives a good lower bound on the indispensable data size.

UR - http://www.scopus.com/inward/record.url?scp=3142528914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3142528914&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2004.02.001

DO - 10.1016/j.dam.2004.02.001

M3 - Article

AN - SCOPUS:3142528914

VL - 142

SP - 165

EP - 180

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 1-3 SPEC. ISS.

ER -