Avoiding flatness in factoring ordinal data
Section snippets
Basic factorization problem
A factorization problem, which we consider and which subsumes the well known factorization of Boolean matrices, may be described as follows. Consider an matrix I whose entries , for and , are elements of an ordered scale L; in the basic interpretation, the entry at row i and column j represents a degree to which the object i has the attribute j. In particular, we assume that the degrees form a complete lattice , i.e. a partially ordered set bounded by 0 and 1 in
Flat factors and why they appear
We now present the phenomenon addressed in this paper. Note at the outset that, as shall become apparent, the phenomenon is non-existent in the two-valued Boolean case, i.e. when . In the multiple-valued case, the phenomenon appears on larger data, which is also where we observed it. In particular, we encountered this phenomenon when analyzing data from the British educational system; some of our findings are reported in section 4.
For convenience, we shall visualize matrices with degrees
Avoiding flat factors
In order to avoid flat factors, we propose to retain the basic logic of factorization but change what accounts for the undesirable effects presented in the previous section. We demonstrate below in this section and more thoroughly in section 4 that this new approach results in eliminating flat factors and computation of factors that are natural and have good ability to explain the data.
The observations from the previous section suggest to suppress the role of small values in the matrices ,
Experimental evaluation
In the previous section, we demonstrated using the running example that our approach indeed leads to avoiding flat factors. In this section, we illustrate that the problem addressed in this paper and its solution we proposed are relevant from the viewpoint of existing factorization algorithms. For this purpose, we consider two significant factorization algorithms, namely and , for which we refer to [3], [4], [6] and [4], respectively.
We first show that the current algorithms
Future research
The problem and contributions presented in this paper open way to a diverse set of streams for future research. Some of them are outlined below.
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In a broader context of fuzzy sets, the problem of flat factors presented in this paper may be rephrased in terms of cardinalities of fuzzy sets. In this perspective, our considerations reveal a significant challenge regarding the concept of cardinality that has apparently not yet been addressed.
In more detail, consider the fuzzy relation R between the
CRediT authorship contribution statement
Eduard Bartl: 50%
Radim Belohlavek: 50%
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (20)
- et al.
Discovery of optimal factors in binary data via a novel method of matrix decomposition
J. Comput. Syst. Sci.
(2010) - et al.
Factorization of matrices with grades
Fuzzy Sets Syst.
(2016) - et al.
A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory
Inf. Control
(1972) Cardinality, quantifiers, and the aggregation of fuzzy criteria
Fuzzy Sets Syst.
(1995)- et al.
Toward factor analysis of educational data
Optimal decompositions of matrices with entries from residuated lattices
J. Log. Comput.
(2012)- et al.
Factor analysis of ordinal data via decomposition of matrices with grades
Ann. Math. Artif. Intell.
(2014) - et al.
The discrete basis problem and Asso algorithm for fuzzy attributes
IEEE Trans. Fuzzy Syst.
(2019) - et al.
Factor Analysis of Incidence Data via Novel Decomposition of Matrices
(2009) - et al.
Fuzzy relational matrix factorization and its granular characterization in data description
IEEE Trans. Fuzzy Syst.
(2022)