Optimal scale selection and attribute reduction in multi-scale decision tables based on three-way decision
Introduction
The three-way decision (3WD) model, which was first proposed by Yao [35], [36], [37], provides a trisecting-and-acting framework for complex problem solving. The core concept of 3WD is to divide a whole into three pair-wise disjoint regions and then devise effective strategies for each region. The three regions, which are referred to as the positive, negative, and boundary regions, can be interpreted as acceptance, rejection, and non-commitment, respectively. Thus far, 3WD has been successfully applied in many fields, such as medical diagnosis [34], email spam filtering [50], malware analysis [19], recommendation systems [42], decision support [14], [41], concept analysis [25], feature fusion [8], classification [32], [40], clustering [1], [39], [17], and uncertainty analysis [46], [44], [48].
The sequential 3WD model [38], which is a multi-step strategy, can transfer objects from the boundary region to the positive or negative regions using newly acquired information. Over the past decade, many sequential 3WD models have been proposed. Savchenko [22] and Li et al. [13] applied sequential 3WD to image recognition. Min et al. [18] proposed a frequent pattern discovery algorithm in which the alphabet was divided into strong, medium, and weak portions. Zhang et al. [43] and Ju et al. [9] studied classification learning problems using sequential 3WD. Zhang et al. [45] proposed a new sequential 3WD model based on penalty function to improve the classification accuracy. Zhang et al. [47] established two sequential 3WD models with intuitionistic fuzzy numbers from the membership degree and nonmembership degree perspectives, respectively. Fang et al. [2] proposed a novel granularity-driven sequential 3WD model, Yang et al. [33] explored dynamic hybrid data using sequential 3WD, and Qian et al. [21] developed a generalized model for sequential 3WD based on multi-granularity.
In many real-world problems, objects are typically measured at different scales under the same attribute [10]. To derive knowledge from data measured at different scales, Wu and Leung [27] first proposed the multi-scale decision table (MDT), where each object can take on different values for the same attribute. This information system was called the Wu-Leung model [11], [12]. Optimal scale selection and attribute reduction are two key issues related to knowledge discovery in MDTs [27], [30], [11]. Wu and Leung [28] studied optimal scale selection under various requirements for MDTs. Gu and Wu [3] discussed rule induction in complete MDTs. Wu et al. [30] investigated optimal scale selection and rule acquisition in incomplete MDTs. Hao et al. [4] and Luo et al. [16] utilized the 3WD model to explore optimal scale selection in dynamic MDTs. Luo et al. [15] designed an incremental method for updating knowledge in hierarchical multi-criteria decision systems, She et al. [23] proposed a local rule induction approach for MDTs, and Xie et al. [31] proposed an optimal scale selection approach for multi-scale formal decision contexts and applied it to a smart city model. Wang et al. [24] discussed optimal granule level selection based on the granule description accuracy. Zhang et al. [49] proposed an optimal scale selection method by integrating uncertainty with cost-sensitive learning. However, in most of the models described above, all attributes are assumed to have the same number of scales.
Recently, Li and Hu [11] proposed a generalized MDT in which different attributes can have different numbers of scales and developed two models, namely a complement model and lattice model, to obtain all optimal scale combinations. Li et al. [12] designed a novel stepwise optimal scale selection method to find a single optimal scale combination. Wu and Leung [29] studied the relationships between various concepts of optimal scale combinations in MDTs. Huang et al. [7] investigated MDTs with multi-scale decision attributes and proposed two algorithms for finding a single optimal scale combination. Huang et al. [6], [5] studied multi-scale intuitionistic fuzzy decision tables.
It is well known that a decision rule with an excessively long description incurs high costs. To acquire concise decision rules from MDTs, knowledge reduction is necessary [30]. To this end, Wu and Leung [27] first proposed the concept of a scale reduct. Most previous studies have performed attribute reduction on optimal scale combinations to obtain optimal scale reducts [3], [30], [23], [6], [5]. In [6], [5], Huang et al. proposed a novel algorithm to quickly obtain a single optimal scale reduct of multi-scale intuitionistic fuzzy decision tables.
However, existing research on MDTs has the following limitations.
- (1)
Previous studies have focused on obtaining a single optimal scale combination. Optimal scale reducts were then obtained by performing attribute reduction of such combinations. From the perspective of practical applications, these optimal scale reducts may not represent the most economical scheme or may be impractical based on various limitations. Therefore, it would be desirable to search for all optimal scale reducts among all scale combinations.
- (2)
A search for all optimal scale reducts may result in a combinatorial explosion and existing approaches are time-consuming [12], [5].
The main goal of this study was to improve the efficiency of searching for all optimal scale reducts. For optimal scale selection of a given MDT, the main search method is to decompose the MDT into several single-scale decision tables and then select an appropriate single-scale decision table based on the expected discrimination ability [27], [11]. Therefore, it is necessary to determine if a given single-scale decision table is consistent. In this study, we attempted to improve the efficiency of searching by reducing the number of consistency checks required for single-scale decision tables and accelerating each check. Therefore, we propose an effective optimal scale selection approach integrating sequential 3WD with simplified MDTs to search for all optimal scale reducts. The main contributions of this paper can be summarized as follows.
- (1)
A new scale combination is defined to perform optimal scale selection and attribute reduction synchronously.
- (2)
A sequential 3WD model of the scale space is proposed to search for all optimal scale reducts quickly. This model can significantly reduce the number of consistency checks required for single-scale decision tables by removing many non-optimal scale reducts from the boundary region. Furthermore, a necessary and sufficient condition under which an MDT has a unique optimal scale reduct is presented.
- (3)
An extended stepwise optimal scale selection method is introduced to search for a single optimal scale reduct in the boundary region quickly. This method further reduces the number of consistency checks required for single-scale decision tables because it is not a traversal method.
- (4)
A simplified MDT is proposed to accelerate consistency checks for single-scale decision tables by deleting duplicate object records and redundant attribute scales.
The remainder of this paper is organized as follows. Section 2 briefly introduces some fundamental concepts related to rough sets and MDTs. Section 3 proposes a sequential 3WD model of the scale space and an extended stepwise optimal scale selection method. Section 4 proposes an optimal scale selection algorithm integrating sequential 3WD with simplified MDTs. Section 5 presents numerical experiments to demonstrate the feasibility and efficiency of the proposed algorithms. Our conclusions are summarized in Section 6.
Section snippets
Preliminaries
In this section, some basic concepts related to rough sets and MDTs are briefly reviewed. It should be noted that we reverse the order of the levels of scales in an MDT (i.e., from the coarse level to the fine level).
Sequential three-way decision model of the scale space
In this section, a new scale combination is defined, a sequential 3WD model of the scale space is established, and its properties are discussed.
Optimal scale selection algorithm integrating sequential three-way decision with simplified multi-scale decision tables
In this section, a simplified MDT is proposed to accelerate consistency checks for single-scale decision tables. Accordingly, an effective optimal scale selection algorithm integrating sequential 3WD with simplified MDTs is presented to search for all optimal scale reducts.
Experiments
In this section, several experiments are presented to demonstrate the effectiveness and efficiency of Algorithm 2. The experimental datasets are the same as those used in the study presented in [12] and were downloaded from the UCI repository of machine learning databases. Descriptions of the datasets are provided in Table 16. The proposed algorithms were coded in MATLAB and were executed on a personal computer with the following specifications: Intel Core(TM) i7-2600 3.40 GHz CPU, 8.0 GB of
Conclusions
Optimal scale selection and attribute reduction are two key issues for knowledge discovery in MDTs. In this paper, an effective optimal scale selection approach integrating sequential 3WD with simplified MDTs was proposed to search for all optimal scale reducts. First, optimal scale selection and attribute reduction were performed synchronously. Second, we proposed a sequential 3WD model of the scale space and an extended stepwise optimal scale selection method that significantly reduce the
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.
CRediT authorship contribution statement
Yunlong Cheng: Methodology, Formal analysis, Validation, Data curation, Visualization, Writing - original draft. Qinghua Zhang: Conceptualization, Methodology, Formal analysis, Writing - review & editing. Guoyin Wang: Methodology, Investigation, Resources, Supervision. Bao Qing Hu: Methodology, Investigation, Resources, Supervision.
Acknowledgement
This work was supported by the National Key Research and Development Program of China (No. 2020YFC2003500), the National Natural Science Foundation of China (No. 61876201, No. 11971365), the Foundation for Innovative Research Groups of Natural Science Foundation of Chongqing (No. cstc2019jcyj-cxttX0002), the Science and Technology Research Project of Chongqing Municipal Education Commission (No. KJQN201800624), and the Doctoral Talent Training Program of Chongqing University of Posts and
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