Nested structure in parameterized rough reduction

Abstract

In this paper, by strict mathematical reasoning, we discover the relationship between the parameters and the reducts in parameterized rough reduction. This relationship, named the nested reduction, shows that the reducts act as a nested structure with the monotonically increasing parameter. We present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction. Some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning. Based on the nested reduction, we design several quick algorithms to find a different reduct when one reduct is already given. Here ‘different’ refers to the reducts obtained on the different parameters. All these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models. The numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach.

Introduction

Rough set (RS) theory, which was proposed by Pawlak [24], [25], is a mathematical tool to handle uncertainty of indiscernibility. RS is effective in many real applications such as artificial intelligence, data mining and pattern recognition. However, it is limited by its basic definition ‘equivalence relation’. As a result, many generalizations, such as fuzzy rough sets [7], [8], [22], [59], [61], cover rough sets [48], [49], [56], Bayesian rough sets [18], [32], [36], [54] and probabilistic rough sets [42], [43], [51], [55], have been proposed. These generalizations make rough sets feasible to handle many types of practical problems such as the problems with real values, the problems with missing values and the problems with random uncertainty.

One of the important applications of rough set theory is attribute reduction [11], [14], [34], [37], [45], [47], [62], [63]. In recent years, researchers, motivated by a desire to acquire reduction robustly, have proposed many methods to mining valuable and less-sensitive attributes by incorporating parameters into rough set theory [15], [18], [21], [57], [58], [60]. These methods are called parameterized rough reduction. Roughly speaking, parameterized rough reduction is split into two types: reduction on parameterized rough models and parameterized reduction on rough sets. The parameterized rough models share a common characteristic by introducing parameters into the rough approximation operators, whereas parameterized reduction deletes redundant attributes by introducing thresholds into the process of attribute reduction.

Many parameterized rough models are presently being proposed and studied intensively, they are Variable Precision Rough Set Models [2], [3], [50], [53], Robust Fuzzy Rough Set Models [1], [9], [10], [26], [45], Probabilistic Rough Set Models [42], [43], [51], Decision Theoretic Rough Set Models [17], [39], [40], and Bayesian Rough Set Models [18], [32]. The Variable Precision Rough Set Models treat the required parameters as a primitive notion [50]. The interpretation and the process of determination of parameters are based on rather intuitive arguments and left to empirical studies [2], [3], [16], with the lack of theoretical and systematic studies and justifications on the choices of the threshold parameters. K-Nearest Neighbor Fuzzy Rough Sets provide some methods to determine the required parameters [10], whereas the applications of these models focus mainly on making classification predications. Unlike the aforementioned models, the Bayesian Rough Set Models [18], [32] attempt to provide an alternative interpretation of the required parameters. The models are based on Bayes’ Rule that expresses the change from the a priori probability to the a posteriori probability [40], [41]. In these models, the required parameters can be expressed in terms of probabilities. In addition to variable precision analysis and Bayesian analysis, the parameterized approaches have been applied to the theory of rough sets in some other forms, such as decision-theoretic analysis [39], [40] and probabilistic analysis [42], [43], [51]. The probabilistic rough set models and decision-theoretic rough set models provide a unified and comprehensive framework so that many types of parameterized models can be integrated into a whole [38]. These models provide a systematic method for determining the parameters by using more concrete notions of costs and risks.

Instead of introducing parameters into the approximation operators, parameterized reduction on rough sets introduces parameters into the process of attribute reduction. By relaxing the criteria of attribute reduction, some parameters are introduced in the processing (or notions) of attribute reduction, such as the method of fuzzy reductions (Fuzzy-RED) [5], [6] and approximate reductions (Approximate-RED) [28], [29], [30]. The common characteristic of these methods is that the parameters are put into the process of attribute reduction. The main difference between these models is that the distinct measures of discernibility power are used in the process of attribute reduction, such as dependency function, information entropy and monotonic measure. The determination of these parameters is usually adopted in an intuitive way.

Whether reduction on parameterized models or parameterized reduction on rough sets is applied, a difficulty with many existing studies on parameterized rough reductions is that no interpretation or procedure for calculating the required threshold exists. In real applications, the setting of parameters affects the results of the reduction significantly, and different thresholds may lead to different reduction results. Some researchers realize the significance of the parameters, and they have paid attention to the parameter analysis and selection. Intuitively, the parameters are given by the experience of an expert in most of the parameterized rough reductions [11], [12], [52]. Some parameterized rough reduction approaches, based on the assumption that the threshold is in an interval that was determined by the desired level of classification performance, proposed methods for finding an appropriate reduct [2], [3]. Other researchers have tried to find a reasonable threshold based on an assessment of the minimum acceptable upper bound of the misclassification error [33]. Essentially, these approaches utilize the principle of the ‘extent of classification correctness’ to determine an appropriate threshold value under the premise that changing the number of attributes has no effect on the classification results [13]. However, such approaches need to set the classification error beforehand. Instead of setting the parameters intuitively, a systematic method for computing parameters in probabilistic rough sets was provided. Yao et al. introduced risk to probabilistic rough sets and proposed decision theoretic rough sets with Bayesian decision procedures [38], [41].These studies have explicitly noted the parameter effect on the values of approximation operators. Although the aforementioned methods adopt and design different methods of determining and interpreting the parameters, these methods share a common characteristic that all of them seek one optimal or suboptimal value to be the required parameter. These methods omit one possible situation that often occurs in real applications: the required parameter changes frequently. That is, the required parameters are not fixed, and these parameters often change with time or with other conditions in real application.

Some researchers have realized the importance of the parameters and have described various methods for setting parameters. Unfortunately, many researchers are still unaware of the connection between reduction and parameters. The existing methods do not indicate how the parameters affect the performance of reduction. Nobody has mentioned the approaches that can determine the proper reduct based on the given reducts. Discovering the inner relationship between the parameters and the reducts in parameterized rough reduction approaches is now a promising and necessary area of research. In this paper, we propose such a method to quickly find reducts on different parameters in parameterized rough reduction by explicitly showing the connections between the reducts and the parameters.

The remainder of this paper is organized as follows. Section 2 gives some preliminaries, such as rough sets and a parameterized rough reduction. After reviewing many approaches to parameterized rough reduction, we discover the inner connection between the reducts and the parameters in Section 3. By strict mathematical reasoning, we describe the connection, which is called the nested structure. A systematic approach to identify the nested structure is proposed based on a strict mathematical foundation. By using the nested reduction structure, Section 4 proposes some algorithms to find the required reducts quickly. By using numerical experiments, Section 5 clearly demonstrates the effectiveness of these algorithms. Finally, a conclusion is drawn in Section 6.