Factorizing Three-Way Ordinal Data Using Triadic Formal Concepts
The paper presents a new approach to factor analysis of three-way ordinal data, i.e. data described by a 3-dimensional matrix I with values in an ordered scale. The matrix describes a relationship between objects, attributes, and conditions. The problem consists in finding factors for I, i.e. finding a decomposition of I into three matrices, an object-factor matrix A, an attribute-factor matrix B, and a condition-factor matrix C, with the number of factors as small as possible. The difference from the decomposition-based methods of analysis of three-way data consists in the composition operator and the constraint on A, B, and C to be matrices with values in an ordered scale. We prove that optimal decompositions are achieved by using triadic concepts of I, developed within formal concept analysis, and provide results on natural transformations between the space of attributes and conditions and the space of factors. We present an illustrative example demonstrating the usefulness of finding factors and a greedy algorithm for computing decompositions.
Unable to display preview. Download preview PDF.
- 1.Belohlavek, R.: Optimal decomposition of matrices with entries from residuated lattices. J. Logic and Computation (to appear, preliminary version appeared in Proc. IEEE Intelligent Systems, pp. 15-2–15-7 (2008))Google Scholar
- 2.Belohlavek, R., Glodeanu, C.V., Vychodil, V.: Optimal factorization of three-way binary data using triadic concepts. (submitted, preliminary version appeared in Proc. IEEE GrC 2010, pp. 61–66 (2010))Google Scholar
- 3.Belohlavek, R., Osicka, P.: Triadic concept analysis of data with fuzzy attributes. In: Proc. 2010 IEEE International Conference on Granular Computing, San Jose, California, August 14–16, pp. 661–665 (2010)Google Scholar
- 10.Jäschke, R., Hotho, A., Schmitz, C., Ganter, B., Stumme, G.: TRIAS – An Algorithm for Mining Iceberg Tri-Lattices. In: ICDM 2006, pp. 907–911 (2006)Google Scholar
- 16.Mickey, M.R., Mundle, P., Engelman, L.: Boolean factor analysis. In: Dixon, W.J. (ed.) BMDP Statistical Software Manual, vol. 2, pp. 849–860. University of California Press, Berkeley (1990)Google Scholar
- 19.Outrata, J.: Preprocessing input data for machine learning by FCA. In: Kryszkiewicz, M., Obiedkov, S. (eds.) Proc. CLA 2010, vol. 672, pp. 187–198. University of Sevilla, CEUR WS (2010)Google Scholar
- 22.Stockmeyer, L.J.: The set basis problem is NP-complete. IBM Research Report RC5431, Yorktown Heights, NY (1975)Google Scholar
- 23.Tang, F., Tao, H.: Binary principal component analysis. In: Proc. British Machine Vision Conference 2006, pp. 377–386 (2006)Google Scholar
- 24.Tatti, N., Mielikäinen, T., Gionis, A., Mannila, H.: What is the dimension of your binary data? In: ICDM 2006, pp. 603–612 (2006)Google Scholar