Incremental updating reduction for relation decision systems with dynamic conditional relation sets
Introduction
Rough set theory put forward by Pawlak [25], [26] is an effective mathematical tool for mining hidden knowledge from uncertain data. Being one of the most important applications of rough sets, attribute reduction plays a vital role in machine learning and pattern recognition, and provides a theoretical framework for feature selection of decision systems with symbols and discrete data. With the progress of study and application of rough set [1], [22], [33], [45], more attention has been paid to attribute reduction methods for decision systems [7], [10], [14], [27], [32], [38], [39], [41], [49].
To deal with various new decision systems such as set-valued decision systems, different extension models of rough sets have been proposed resulting in different types of attribute reduction methods including distribution reduction [42], variable precision reduction [21], and covering reductions [4], [36]. In [13], Kryszkiewicz investigated attribute reduction of consistent and inconsistent incomplete decision systems based on similarity relations. Yang et al. [47] developed attribute reduction algorithm for incomplete ordered information systems based on the dominance-based rough set. More and more attention has been paid to incomplete decision systems [15]. It is worth noting that the above-mentioned attribute reduction methods were all proposed for a certain type of decision system. In [37], Wang et al. proposed the concept of relation decision systems, and conducted systematical study on the reduction of relation decision systems where the conditional relations (conditional attributes) are general relations. Although the reduction algorithms for such relation decision systems perform well in various types of decision systems such as complete, incomplete, numerical and covering ones, they show weakness in decision systems with missing decision values where the decision relations are no longer equivalence relations.
Recently, Liu et al. [16], [17], [18] extended Wang's research by redefining the relation decision systems where the decision relations are no longer limited to equivalence relations, and established a general discernibility matrix-based reduction algorithm which can find all reductions of a relation decision system. However, this reduction algorithm ignores the computationally complexity. When dealing with the reductions of relation decision systems with large-scale data or dynamic relation decision systems, it may not be suitable or invalid. It should be pointed out that among the reduction algorithms which can be divided into three categories: positive region-based ones [20], [24], [28], discernibility matrix-based ones [30], [43], and entropy-based ones [15], [31], the positive region-based algorithms [29] can indeed improve the efficiency of reduction. Thus, the positive region is employed in the attribute reduction algorithms proposed by this paper.
In real-world applications, a decision system usually varies with time in attribute set [9], object set [35] or attribute values [40]. Such a decision system is called a dynamic decision system. In the face of dynamic decision systems, how to get its reduction becomes the primary problem. For this problem, one way to get the reduction after variation is to do from scratch with ignoring the reduction obtained before variation. We call such a reduction algorithm non-incremental reduction algorithm. It can be easily seen that a non-incremental reduction algorithm tends to be time-consuming or even infeasible [48]. Incremental learning, being an efficient method for dealing with dynamic decision systems and mining dynamic data, can avoid some recalculations by using previous data [8]. Hu et al. [11] proposed an incremental approach for updating attribute reduction based on elementary sets, which can be only applied to complete information systems. Based on the positive region reduction method, Shu et al. [34] proposed an incremental updating mechanism for tolerance classes and a dynamic attribute reduction algorithm for incomplete decision systems. For decision systems with dynamic object sets, Chen et al. [5] proposed an incremental method to solve the updating problem of the approximate set based on equivalence relation.
Although the studies on incremental reduction research of dynamic decision systems are fairly progressive [6], [23], [44], [46], the focus of the existing incremental methods are dynamic decision systems whose decision relations are equivalence. When facing with dynamic decision systems whose decision relations are not equivalence ones, the existing reduction methods may be inapplicable or invalid. In fact, these problems can be attributed to the reduction problem of dynamic relation decision systems. In real application, the dynamic decision systems whose changes are caused by the change of attribute set occurs in most cases [12], [50]. Therefore, this paper concentrates on the incremental updating reduction algorithms for relation decision systems with dynamic conditional relation sets.
The main contributions of this study are summarized as follows: (1) an incremental updating reduction algorithm is proposed for relation decision systems whose conditional relation sets have increased; (2) an incremental updating reduction algorithm is proposed for relation decision systems whose conditional relation sets have decreased; (3) the proposed algorithms unify the reduction of some dynamic decision systems including those whose decision relations are not equivalence relations; (4) experiments carried out on different data sets show the effectiveness of the proposed algorithms.
The rest of this paper is organized as follows. In Section 2, some important concepts about relation decision systems are reviewed. Based on the positive region-based reduction method, Section 3 presents the incremental updating mechanisms for positive region and right neighbor in relation decision systems with dynamic conditional relation sets. In Section 4, the proposed updating mechanisms of positive region and right neighbor are employed to update the reductions of relation decision systems. We propose the incremental updating reduction algorithms for relation decision systems with dynamic conditional relation sets. In Section 5, numerical experiments are conducted to verify our proposed algorithms. Section 6 gives a conclusion of the paper and discusses the future work.
Section snippets
Preliminaries
In this section, we recall some basic definitions and properties of binary relations and relation decision systems [16], [37]. Suppose that is a finite set of objects called the universal set and is the power set of . Let be a binary relation on . The left and right neighborhood of with respect to are respectively defined as follows.
and .
Based on the left and right neighborhoods, several special binary relations are given as follows.
(1) is
Updating positive region when conditional relation set has varied
The algorithm given in [16] has greater uniformity and generality than the other existing algorithms. In fact, it is a reduction method based on the positive region of a relation decision systems, and can be applied to attribute reduction of all kinds of decision systems with missing decision value, such as complete or incomplete decision systems, set-valued decision systems, and covering decision systems. Obviously, it is more general to various decision systems. However, for a relation decision
The non-incremental and incremental updating reduction algorithms for dynamic relation decision systems
Based on the positive region incremental updating mechanism given in Section 3, the non-incremental and incremental updating reduction algorithms for a relation decision system with a dynamic conditional relation set are developed in this section.
Firstly, we design a non-incremental updating reduction algorithm for relation decision system whose conditional relation sets have varied, named Algorithm 1.
Algorithm 1 A non-incremental updating reduction algorithm for a relation decision system with
Experimental studies
In this section, to demonstrate the effectiveness of our proposed incremental updating algorithm, some comparative experiments are designed on nine datasets with missing values. The datasets are downloaded from UCI's Machine Learning Database Repository (https://www.ics.uci.edu/mlearn/MLRepository.html), which are listed in Table 1. All experiments are performed on computer with Intel Xeon Gold 6136 CPU 3.0GHZ, 256.0 GB of memory, running Win 10, and Algorithms are coded in MATLAB 2020.
Being
Conclusions
In many tasks in the real world, decision systems may be dynamic, which can be regarded as dynamic relation decision systems. In dynamic relation decision systems, incremental update of reduction is helpful to improve the efficiency of knowledge discovery. In this paper, the incremental updating reduction algorithms are developed in relation decision systems with dynamic conditional relation sets. Firstly, the positive region incremental updating mechanism is established by the update method of
CRediT authorship contribution statement
Lirun Su: Methodology, Writing – original draft. Fusheng Yu: Conceptualization, Supervision, Writing – review & editing. Jinjin Li: Conceptualization, Writing – review & editing. Xubo Du: Visualization. Hanliang Huang: Conceptualization, Writing – review & editing.
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.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11971065, No. 11571001, No. 11871259, No. 12271191), the Natural Science Foundation of Fujian Province (No. 2020J02043, No. 2022J01306, No. 2022J05169).
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