Expositing stages of VPRS analysis in an expert system: Application with bank credit ratings

The variable precision rough sets model (VPRS) along with many derivatives of rough set theory (RST) necessitates a number of stages towards the final classification of objects. These include, (i) the identification of subsets of condition attributes (β-reducts in VPRS) which have the same quality of classification as the whole set, (ii) the construction of sets of decision rules associated with the reducts and (iii) the classification of the individual objects by the decision rules. The expert system exposited here offers a decision maker (DM) the opportunity to fully view each of these stages, subsequently empowering an analyst to make choices during the analysis. Its particular innovation is the ability to visually present available β-reducts, from which the DM can make their selection, a consequence of their own reasons or expectations of the analysis undertaken. The practical analysis considered here is applied on a real world application, the credit ratings of large banks and investment companies in Europe and North America. The snapshots of the expert system presented illustrate the variation in results from the ‘asymmetric’ consequences of the choice of β-reducts considered.     

An illustration of variable precision rough sets model: an analysis of the findings of the UK Monopolies and Mergers Commission

This paper introduces a new technique in the investigation of limited-dependent variable models. This paper illustrates that variable precision rough set theory (VPRS), allied with the use of a modern method of classification, or discretisation of data, can out-perform the more standard approaches that are employed in economics, such as a probit model. These approaches and certain inductive decision tree methods are compared (through a Monte Carlo simulation approach) in the analysis of the decisions reached by the UK Monopolies and Mergers Committee. We show that, particularly in small samples, the VPRS model can improve on more traditional models, both in-sample, and particularly in out-of-sample prediction. A similar improvement in out-of-sample prediction over the decision tree methods is also shown.Scope and purposeThe Monopolies and Mergers Commission (MMC) in the UK evaluates whether a given firm, or set of firms is behaving in a manner that is considered to be against the public interest, that is anti-competitive. The interpretation and prediction of the decisions made by the MMC is of importance to firm's possible future investment plans. Through the construction of decision rules using the variable precision rough sets (VPRS) model this interoperation and prediction is able to be undertaken. The importance of the concomitant variables in the decisions made is shown through a ‘leave n out’ Monte Carlo simulation approach. At the technical level this study illustrates one of the first applications of VPRS in an economic environment.     

A transformation algorithm for incompatible information systems based on variable precision values

The variable precision rough set (VPRS) model extends the basic rough set (RS) theory with finite uni- verses and finite evaluative measures. VPRS is concerned with the equivalence and the contained relationship between two sets. In incompatible information systems, the inclusion degree and β upper (lower) approximation of the inconsistent equivalence class to the decision equivalence classes may be affected by the variable precision. The analysis of an example of incompatible decision table shows that there is a critical point in β available-values region. In the new β range limited at the critical point, the incompatible decision table can be converted to the coordination decision table reliably. The method and its algorithm implement are introduced for the critical value search. The examples of the inconsistent equivalence class transformation are exhibited. The results illustrate that this algorithm is rational and precise.     

扩展优势关系下的变精度粗糙集模型

基于优势关系的变精度粗糙集模型将传统粗糙集中的等价关系扩展为优势关系,并结合变精度的思想来定义相关概念,从而可以处理具有偏好关系的信息并具有一定的容错能力。然而,传统优势关系的定义仍然过于严格,只有当一个对象x的每个属性值都优于另一个对象y时,该对象x才优于y。当属性个数较多时,这种优势关系的定义会导致对象的优势集偏小,影响到规则的提取和决策结果。为了解决这一问题,通过引入参数的方法扩展了传统优势关系的定义,并在此基础上进一步给出了扩展后的优势集和近似集的概念,建立了扩展优势关系下的变精度粗糙集模型,采用覆盖率和测试精度作为模型的评估指标。最后给出算例,并在UCI数据集上进行大量的实验将所提模型与传统优势关系下的变精度粗糙集模型进行比较。     

MGRS: A multi-granulation rough set

The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of sets described by a single binary relation on the universe. In the view of granular computing, the classical rough set theory is established through a single granulation. This paper extends Pawlak’s rough set model to a multi-granulation rough set model (MGRS), where the set approximations are defined by using multi equivalence relations on the universe. A number of important properties of MGRS are obtained. It is shown that some of the properties of Pawlak’s rough set theory are special instances of those of MGRS.Moreover, several important measures, such as accuracy measureα, quality of approximationγ and precision of approximationπ, are presented, which are re-interpreted in terms of a classic measure based on sets, the Marczewski-Steinhaus metric and the inclusion degree measure. A concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in MGRS as well. Finally, we discuss how to extract decision rules using MGRS. Unlike the decision rules (“AND” rules) from Pawlak’s rough set model, the form of decision rules in MGRS is “OR”. Several pivotal algorithms are also designed, which are helpful for applying this theory to practical issues. The multi-granulation rough set model provides an effective approach for problem solving in the context of multi granulations.     

A variable precision rough set approach to the remote sensing land use/cover classification

Nowadays the rough set method is receiving increasing attention in remote sensing classification although one of the major drawbacks of the method is that it is too sensitive to the spectral confusion between-class and spectral variation within-class. In this paper, a novel remote sensing classification approach based on variable precision rough sets (VPRS) is proposed by relaxing subset operators through the inclusion error β. The remote sensing classification algorithm based on VPRS includes three steps: (1) spectral and textural information (or other input data) discretization, (2) feature selection, and (3) classification rule extraction. The new method proposed here is tested with Landsat-5 TM data. The experiment shows that admitting various inclusion errors β, can improve classification performance including feature selection and generalization ability. The inclusion of β also prevents the overfitting to the training data. With the inclusion of β, higher classification accuracy is obtained. When β=0 (i.e., the original rough set based classifier), overfitting to the training data occurs, with the overall accuracy=0.6778 and unrecognizable percentage=12%. When β=0.07, the highest classification performance is reached with overall accuracy and unrecognizable percentage up to 0.8873% and 2.6%, respectively.     

基于粗糙集和聚类的纳税属性离散化方法

将粗糙集理论中属性重要度和依赖度的概念与分级聚类离散化算法相结合,提出了一种纳税人连续型属性动态的离散化算法。首先将纳税数据对象的每个连续型属性划分为2类,然后利用粗糙集理论计算每个条件属性对于决策属性的重要度,再通过重要度由大至小排序进行增类运算,最后将保持与原有数据对象集依赖度一致的分类结果输出。该算法能够动态地对数据对象进行类别划分,实现纳税人连续型属性的离散化。通过采用专家分析和关联分析的实验结果,验证了该算法具有较高的纳税人连续型属性离散化精度和性能。     

Optimal βk-stable interval in VPRS-based group decision-making: A further application

This study extends the extant research on use of variable precision rough set (VPRS) for group decision-making (GDM), where the optimal β_k-stable interval is derived for the best group consensus. Firstly, we introduce the basic concepts a VPRS model encompasses, and the approach of VPRS-based GDM. Next, using a mathematical programming approach, we derive the optimal β_k-stable interval for DM_k. Then, an application in petroleum project investment risk management, including risk-based project selection and risk ranking, is investigated. The results suggest that β_k-stable intervals have significant impacts on risk management.     

Dominance relation and rules in an incomplete ordered information system

Rough sets theory has proved to be a useful mathematical tool for classification and prediction. However, as many real‐world problems deal with ordering objects instead of classifying objects, one of the extensions of the classical rough sets approach is the dominance‐based rough sets approach, which is mainly based on substitution of the indiscernibility relation by a dominance relation. In this article, we present a dominance‐based rough sets approach to reasoning in incomplete ordered information systems. The approach shows how to find decision rules directly from an incomplete ordered decision table. We propose a reduction of knowledge that eliminates only that information that is not essential from the point of view of the ordering of objects or decision rules. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 13–27, 2005.     

Probabilistic Decision Tables in the Variable Precision Rough Set Model

The Variable Precision Rough Set Model (VPRS) is an extension of the original rough set model. This extension is directed towards deriving decision table-based predictive models from data with parametrically adjustable degrees of accuracy. The imprecise nature of such models leads to quite significant modification of the classical notion of decision table. This is accomplished by introducing the idea of approximation region-based, or probabilistic decision table which is a tabular specification of three, in general uncertain, disjunctive decision rules corresponding to rough approximation regions: positive, boundary and negative regions. The focus of the paper is on the extraction of such decision tables from data, their relationship to conjunctive rules and probabilistic assessment of decision confidence with such rules.     

Dimensionality reduction based on rough set theory: A review

A rough set theory is a new mathematical tool to deal with uncertainty and vagueness of decision system and it has been applied successfully in all the fields. It is used to identify the reduct set of the set of all attributes of the decision system. The reduct set is used as preprocessing technique for classification of the decision system in order to bring out the potential patterns or association rules or knowledge through data mining techniques. Several researchers have contributed variety of algorithms for computing the reduct sets by considering different cases like inconsistency, missing attribute values and multiple decision attributes of the decision system. This paper focuses on the review of the techniques for dimensionality reduction under rough set theory environment. Further, the rough sets hybridization with fuzzy sets, neural network and metaheuristic algorithms have also been reviewed. The performance analysis of the algorithms has been discussed in connection with the classification.     

变精度粗糙集挖掘技术的应用研究

粗糙集理论是一种处理模糊和不确定知识的一种新型数学工具,在很多领域取得了成功的应用。但是经典粗糙集理论处理的分类必须是完全正确的,在实际应用中,缺乏对噪声数据的适应能力,为了克服这个缺点,提出一种变精度的粗糙集模型,以适应实际应用的需要。对变精度粗糙集理论的数据预处理、属性约简、值约简和规则提取等问题进行了分析和研究,提出属性约简算法和基于求核值属性的归纳值约简算法,并将其运用于医疗系统的手术诊断数据表的数据挖掘分析过程中,所得到的实验结果与专家诊断结果基本吻合,取得了较好的实际应用效果。     

A Comparison of the Rough Sets and Recursive Partitioning Induction Approaches: An Application to Commercial Loans

Credit scoring is the term used to describe methods utilized for classifying applicants for credit into classes of risk. This paper evaluates two induction approaches, rough sets and decision trees, as techniques for classifying credit (business) applicants. Inductive learning methods, like rough sets and decision trees, have a better knowledge representational structure than neural networks or statistical procedures because they can be used to derive production rules. If decision trees have already been used for credit granting, the rough sets approach is rarely utilized in this domain. In this paper, we use production rules obtained on a sample of 1102 business loans in order to compare the classification abilities of the two techniques. We show that decision trees obtain better results with 87.5% of good classifications with a pruned tree, against 76.7% for rough sets. However, decision trees make more type–II errors than rough sets, but fewer type–I errors.     

Dominance-based Rough Set Classifier without Induction of Decision Rules

Rough Sets Theory is often applied to the task of classification and prediction, in which objects are assigned to some pre-defined decision classes. When the classes are preference-ordered, the process of classification is referred to as sorting. To deal with the specificity of sorting problems an extension of the Classic Rough Sets Approach, called the Dominance-based Rough Sets Approach, was introduced. The final result of the analysis is a set of decision rules induced from what is called rough approximations of decision classes. The main role of the induced decision rules is to discover regularities in the analyzed data set, but the same rules, when combined with a particular classification method, may also be used to classify/sort new objects (i.e. to assign the objects to appropriate classes). There exist many different rule induction strategies, including induction of an exhaustive set of rules. This strategy produces the most comprehensive knowledge base on the analyzed data set, but it requires a considerable amount of computing time, as the complexity of the process is exponential. In this paper we present a shortcut that allows classifying new objects without generating the rules. The presented approach bears some resemblance to the idea of lazy learning.     

序决策信息系统中区间决策规则的获取

在序决策信息系统中, 定义区间为支配一个特定的对象同时又被另一个特定的对象所支配的所有对象的集合. 以区间为基本知识颗粒, 建立新的优势关系粗糙集模型, 并由此获取决策值为特定区间范围的区间决策规则. 提出区间的约简的概念, 构造区分函数计算区间的约简, 并由此计算优化区间决策规则. 该方法比初始的优势关系粗糙集方法适应性更强, 且所得区间决策规则可直接应用于序信息系统的分类问题.

     

Interval set clustering

Rough k-means clustering describes uncertainty by assigning some objects to more than one cluster. Rough cluster quality index based on decision theory is applicable to the evaluation of rough clustering. In this paper we analyze rough k-means clustering with respect to the selection of the threshold, the value of risk for assigning an object and uncertainty of objects. According to the analysis, clusters presented as interval sets with lower and upper approximations in rough k-means clustering are not adequate to describe clusters. This paper proposes an interval set clustering based on decision theory. Lower and upper approximations in the proposed algorithm are hierarchical and constructed as outer-level approximations and inner-level ones. Uncertainty of objects in out-level upper approximation is described by the assignment of objects among different clusters. Accordingly, ambiguity of objects in inner-level upper approximation is represented by local uniform factors of objects. In addition, interval set clustering can be improved to obtain a satisfactory clustering result with the optimal number of clusters, as well as optimal values of parameters, by taking advantage of the usefulness of rough cluster quality index in the evaluation of clustering. The experimental results on synthetic and standard data demonstrate how to construct clusters with satisfactory lower and upper approximations in the proposed algorithm. The experiments with a promotional campaign for the retail data illustrates the usefulness of interval set clustering for improving rough k-means clustering results.     

The Model of Fuzzy Variable Precision Rough Sets

The fuzzy rough set (FRS) model has been introduced to handle databases with real values. However, FRS was sensitive to misclassification and perturbation (here misclassification means error or missing values in classification, and perturbation means small changes of numerical data). The variable precision rough sets (VPRSs) model was introduced to handle databases with misclassification. However, it could not effectively handle the real-valued datasets. Now, it is valuable from theoretical and practical aspects to combine FRS and VPRS so that a powerful tool, which not only can handle numerical data but also is less sensitive to misclassification and perturbation, can be developed. In this paper, we set up a model named fuzzy VPRSs (FVPRSs) by combining FRS and VPRS with the goal of making FRS a special case. First, we study the knowledge representation ways of FRS and VPRS, and then, propose the set approximation operators of FVPRS. Second, we employ the discernibility matrix approach to investigate the structure of attribute reductions in FVPRS and develop an algorithm to find all reductions. Third, in order to overcome the NP-complete problem of finding all reductions, we develop some fast heuristic algorithms to obtain one near-optimal attribute reduction. Finally, we compare FVPRS with RS, FRS, and several flexible RS-based approaches with respect to misclassification and perturbation. The experimental comparisons show the feasibility and effectiveness of FVPRS.     

Rough sets for adapting wavelet neural networks as a new classifier system

Classification is an important theme in data mining. Rough sets and neural networks are two techniques applied to data mining problems. Wavelet neural networks have recently attracted great interest because of their advantages over conventional neural networks as they are universal approximations and achieve faster convergence. This paper presents a hybrid system to extract efficiently classification rules from decision table. The neurons of such hybrid network instantiate approximate reasoning knowledge gleaned from input data. The new model uses rough set theory to help in decreasing the computational effort needed for building the network structure by using what is called reduct algorithm and a rules set (knowledge) is generated from the decision table. By applying the wavelets, frequencies analysis, rough sets and dynamic scaling in connection with neural network, novel and reliable classifier architecture is obtained and its effectiveness is verified by the experiments comparing with traditional rough set and neural networks approaches.