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Factor Analysis of Incidence Data via Novel Decomposition of Matrices

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Formal Concept Analysis (ICFCA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5548))

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Abstract

Matrix decomposition methods provide representations of an object-variable data matrix by a product of two different matrices, one describing relationship between objects and hidden variables or factors, and the other describing relationship between the factors and the original variables. We present a novel approach to decomposition and factor analysis of matrices with incidence data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red or a person performs well in a test. We assume that the grades belong to a scale bounded by 0 and 1 which is equipped with certain aggregation operators and forms a complete residuated lattice. We present an approximation algorithm for the problem of decomposition of such matrices with grades into products of two matrices with grades with the number of factors as small as possible. Decomposition of binary matrices into Boolean products of binary matrices is a special case of this problem in which 0 and 1 are the only grades. Our algorithm is based on a geometric insight provided by a theorem identifying particular rectangular-shaped submatrices as optimal factors for the decompositions. These factors correspond to formal concepts of the input data and allow for an easy interpretation of the decomposition. We present the problem formulation, basic geometric insight, algorithm, illustrative example, experimental evaluation.

Supported by research plan MSM 6198959214.

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Author information

Authors and Affiliations

  1. State University of New York at Binghamton, PO Box 6000, Binghamton, NY 13902–6000, USA

    Radim Belohlavek & Vilem Vychodil

  2. Palacky University, Olomouc, Tomkova 40, CZ-779 00, Olomouc, Czech Republic

    Radim Belohlavek & Vilem Vychodil

Authors
  1. Radim Belohlavek
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  2. Vilem Vychodil
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Editor information

Editors and Affiliations

  1. Irisa/Ifsic, Université de Rennes 1, Campus Universitaire de Beaulieu, 35042, Rennes cedex, France

    Sébastien Ferré

  2. Institut AIFB, Universität Karlsruhe (TH), 76128, Karlsruhe, Germany

    Sebastian Rudolph

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Belohlavek, R., Vychodil, V. (2009). Factor Analysis of Incidence Data via Novel Decomposition of Matrices. In: Ferré, S., Rudolph, S. (eds) Formal Concept Analysis. ICFCA 2009. Lecture Notes in Computer Science(), vol 5548. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01815-2_8

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  • DOI: https://doi.org/10.1007/978-3-642-01815-2_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-01814-5

  • Online ISBN: 978-3-642-01815-2

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