Elsevier

Information Sciences

Volume 629, June 2023, Pages 471-487
Information Sciences

Avoiding flatness in factoring ordinal data

https://doi.org/10.1016/j.ins.2023.02.002Get rights and content

Abstract

Factorization of classical, two-valued Boolean data became a widely studied topic in the past decade due to its role in analyzing relational data as well as its significance for other fields. Recently, various extensions to factorization of ordinal data, or data with graded (fuzzy) attributes, have been proposed. We identify and describe a fundamental problem regarding quality of factors, which is non-existent in the Boolean case, but naturally appears in the more general setting of ordinal data. As we demonstrate, the problem gets more significant with growing size of the factorized data. We analyze the problem, propose a method to alleviate it, and evaluate experimentally our solution to the problem. We also provide a discussion regarding ramifications of our findings for the concept of cardinality of fuzzy sets.

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 n×m matrix I whose entries Iij, for 1in and 1jm, are elements of an ordered scale L; in the basic interpretation, the entry Iij 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 L,,0,1, 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 L={0,1}. 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 IAB,

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 GreConDL and AssoL, 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.

  • 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.

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