Sampling aspects of rough set theory

Rough Set Theory (RST) originated as an approach to approximating a given set, but has found its main applications in the statistical domain of classification problems. It generates classification rules, and can be seen in general terms as a technique for rule induction. Expositions of RST often stress that it is robust in requiring no (explicit) assumptions of a statistical nature. The argument here, however, is that this apparent strength is also a weakness which prevents establishment of general statistical properties and comparison with other methods. A sampling theory is developed for the first time, using both the original RST model and its probabilistic extension, Variable Precision Rough Sets. This is applied in the context of examples, one of which involves Fishers Iris data.Bruce Curry: The author is grateful to two anonymous referees for various helpful suggestions     

A heuristic approach to classifying labeled/unlabeled data sets

A classification method, which comprises Fuzzy C-Means method, a modified form of the Huang-index function and Variable Precision Rough Set (VPRS) theory, is proposed for classifying labeled/unlabeled data sets in this study. This proposed method, designated as the MVPRS-index method, is used to partition the values of per conditional attribute within the data set and to achieve both the optimal number of clusters and the optimal accuracy of VPRS classification. The validity of the proposed approach is confirmed by comparing the classification results obtained from the MVPRS-index method for UCI data sets and a typical stock market data set with those obtained from the supervised neural networks classification method. Overall, the results show that the MVPRS-index method could be applied to data sets not only with labeled information but also with unlabeled information, and therefore provides a more reliable basis for the extraction of decision-making rules of labeled/unlabeled datasets.     

Value reducts and bireducts: A comparative study

In Rough Set Theory, the notion of bireduct allows to simultaneously reduce the sets of objects and attributes contained in a dataset. In addition, value reducts are used to remove some unnecessary values of certain attributes for a specific object. Therefore, the combination of both notions provides a higher reduction of unnecessary data. This paper is focused on the study of bireducts and value reducts of information and decision tables. We present theoretical results capturing different aspects about the relationship between bireducts and reducts, offering new insights at a conceptual level. We also analyze the relationship between bireducts and value reducts. The studied connections among these notions provide important profits for the efficient information analysis, as well as for the detection of unnecessary or redundant information.     

Rough sets for pattern classification using pairwise-comparison-based tables

Rough set theory is a useful mathematical tool to deal with vagueness and uncertainty in available information. The results of a rough set approach are usually presented in the form of a set of decision rules derived from a decision table. Because using the original decision table is not the only way to implement a rough set approach, it could be interesting to investigate possible improvement in classification performance by replacing the original table with an alternative table obtained by pairwise comparisons among patterns. In this paper, a decision table based on pairwise comparisons is generated using the preference relation as in the Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE) methods, to gauges the intensity of preference for one pattern over another pattern on each criterion before classification. The rough-set-based rule classifier (RSRC) provided by the well-known library for the Rough Set Exploration System (RSES) running under Windows as been successfully used to generate decision rules by using the pairwise-comparisons-based tables. Specifically, parameters related to the preference function on each criterion have been determined using a genetic-algorithm-based approach. Computer simulations involving several real-world data sets have revealed that of the proposed classification method performs well compared to other well-known classification methods and to RSRC using the original tables.     

Classification systems based on rough sets under the belief function framework

In this paper, we present two classification approaches based on Rough Sets (RS) that are able to learn decision rules from uncertain data. We assume that the uncertainty exists only in the decision attribute values of the Decision Table (DT) and is represented by the belief functions. The first technique, named Belief Rough Set Classifier (BRSC), is based only on the basic concepts of the Rough Sets (RS). The second, called Belief Rough Set Classifier, is more sophisticated. It is based on Generalization Distribution Table (BRSC-GDT), which is a hybridization of the Generalization Distribution Table and the Rough Sets (GDT-RS). The two classifiers aim at simplifying the Uncertain Decision Table (UDT) in order to generate significant decision rules for classification process. Furthermore, to improve the time complexity of the construction procedure of the two classifiers, we apply a heuristic method of attribute selection based on rough sets. To evaluate the performance of each classification approach, we carry experiments on a number of standard real-world databases by artificially introducing uncertainty in the decision attribute values. In addition, we test our classifiers on a naturally uncertain web usage database. We compare our belief rough set classifiers with traditional classification methods only for the certain case. Besides, we compare the results relative to the uncertain case with those given by another similar classifier, called the Belief Decision Tree (BDT), which also deals with uncertain decision attribute values.     

Monotonic Variable Consistency Rough Set Approaches

We consider probabilistic rough set approaches based on different versions of the definition of rough approximation of a set. In these versions, consistency measures are used to control assignment of objects to lower and upper approximations. Inspired by some basic properties of rough sets, we find it reasonable to require from these measures several properties of monotonicity. We consider three types of monotonicity properties: monotonicity with respect to the set of attributes, monotonicity with respect to the set of objects, and monotonicity with respect to the dominance relation. We show that consistency measures used so far in the definition of rough approximation lack some of these monotonicity properties. This observation led us to propose new measures within two kinds of rough set approaches: Variable Consistency Indiscernibility-based Rough Set Approaches (VC-IRSA) and Variable Consistency Dominance-based Rough Set Approaches (VC-DRSA). We investigate properties of these approaches and compare them to previously proposed Variable Precision Rough Set (VPRS) model, Rough Bayesian (RB) model, and previous versions of VC-DRSA.     

Multi-criteria classification – A new scheme for application of dominance-based decision rules

We are considering the problem of multi-criteria classification. In this problem, a set of “if … then …” decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes. Given a sample of classification examples, called learning data set, the rules are induced from dominance-based rough approximations of preference-ordered decision classes, according to the Variable Consistency Dominance-based Rough Set Approach (VC-DRSA). The main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by (i) no rule, (ii) exactly one rule, (iii) several rules. The proposed classification scheme can be applied to both, learning data set (to restore the classification known from examples) and testing data set (to predict classification of new objects). A hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals.     

Rough approximation of a preference relation by a multi-attribute stochastic dominance for determinist and stochastic evaluation problems

Let A be a set of actions evaluated by a set of attributes. Two kinds of evaluations will be considered in this paper: determinist or stochastic in relation to each attribute. The multi-attribute stochastic dominance (MSD_r) for a reduced number of attributes will be suggested to model the preferences in this kind of problem. The case of mixed data, where we have the attributes of different natures is not well known in the literature, although it is essential from a practical point of view. To apply the MSD_r the subset R of attributes from which approximation of the global preference is valid should be known. The theory of Rough Sets gives us an answer on this issue allowing us to determine a minimal subset of attributes that enables the same classification of objects as the whole set of attributes. In our approach these objects are pairs of actions. In order to represent preferential information we shall use a pairwise comparison table. This table is built for subset B⊂A described by stochastic dominance (SD) relations for particular attributes and a total order for the decision attribute given by the decision maker (DM). Using a Rough Set approach to the analysis of the subset of preference relations, a set of decision rules is obtained, and these are applied to a set A⧹B of potential actions. The Rough Set approach of looking for the reduction of the set of attributes gives us the possibility of operating with MSD_r.     

Rough sets,decision algorithms and Bayes' theorem

Rough set-based data analysis starts from a data table, called an information system. The information system contains data about objects of interest characterized in terms of some attributes. Often we distinguish in the information system condition and decision attributes. Such information system is called a decision table. The decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table. With every decision table a set of decision rules, called a decision algorithm, can be associated. It is shown that every decision algorithm reveals some well-known probabilistic properties, in particular it satisfies the total probability theorem and Bayes' theorem. These properties give a new method of drawing conclusions from data, without referring to prior and posterior probabilities, inherently associated with Bayesian reasoning.     

Fisher information measure of temporal fluctuations in satellite advanced very high resolution radiometer (AVHRR) thermal signals recorded in the volcanic area of Etna (Italy)

The time dynamics of long-term time series of satellite thermal signal, measured at Mount Etna, has been investigated. The signal has been analyzed by means of a recently proposed multi-temporal and robust technique (RST), which has already shown to be better capable to detect and monitor volcanic hotspots, compared to traditional satellite approaches. The temporal fluctuations of the thermal signal detected by RST over a long series (1995–2006) of advanced very high resolution radiometer (AVHRR) satellite data, have been investigated by means of the Fisher information measure, which is a powerful tool to investigate complex and nonstationary signals. The preliminary obtained results indicate that the proposed nonlinear approach can be used to dynamically characterize the volcanic phenomena and to recognize possible pre-eruptive temporal patterns.     

Rough set approach to multiple criteria classification with imprecise evaluations and assignments

Dominance-based Rough Set Approach (DRSA) has been introduced to deal with multiple criteria classification (also called multiple criteria sorting, or ordinal classification with monotonicity constraints), where assignments of objects may be inconsistent with respect to dominance principle. In this paper, we consider an extension of DRSA to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes. The imprecisions are given in the form of intervals of possible values. In order to solve the problem, we reformulate the dominance principle and introduce second-order rough approximations. The presented methodology preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and precisiation. Moreover, the meaning of the precisiation property is extended to the considered case. The paper presents also a way to reduce decision tables and to induce decision rules from rough approximations.     

不完备信息下的灰色粗糙集成决策模型侯亚林

In this paper, for multiple attribute decision-making problem in which attribute values are interval grey numbers and some of them are null values, a decision model based on grey rough sets integration with incomplete information is proposed. We put forward incidence degree coefficient formula for grey interval, by information entropy theory and analysis technique, the method and principle is presented to fill up null values. We also establish the method of grey interval incidence cluster. Because grey system theory and Rough set theory are complementary each other, decision table with preference information is obtained by the result of grey incidence cluster. An algorithm for inducing decision rules based on rough set theory and the dominance relationship is presented. In some extent, this algorithm can deal with decision-making problem in which the attribute values are interval grey numbers and some of them are null values. Contrasted with classical model of cluster decision-making, the algorithm has an advantage of flexibility and compatibility to new information.     

粗糙集的矩阵关系

将粗糙集中的集合转化为矩阵刻画,通过引入矩阵算子、类矩阵算子,借助截矩阵和关系矩阵,讨论了Paw lak粗糙集和变精度粗糙集中集合关系的矩阵计算及其所具有的一些基本性质.     

A modified rough c-means clustering algorithm based on hybrid imbalanced measure of distance and density

Traditional c-means clustering partitions a group of objects into a number of non-overlapping sets. Rough sets provide more flexible and objective representation than classical sets with hard partition and fuzzy sets with subjective membership function for a given dataset. Rough c-means clustering and its extensions were introduced and successfully applied in many real life applications in recent years. Each cluster is represented by a reasonable pair of lower and upper approximations. However, the most available algorithms pay no attention to the influence of the imbalanced spatial distribution within a cluster. The limitation of the mean iterative calculation function, with the same weight for all the data objects in a lower or upper approximation, is analyzed. A hybrid imbalanced measure of distance and density for the rough c-means clustering is defined, and a modified rough c-means clustering algorithm is presented in this paper. To evaluate the proposed algorithm, it has been applied to several real world data sets from UCI. The validity of this algorithm is demonstrated by the results of comparative experiments.     

Rough set theory fuzzy belief functions related to statistical confidence:application & evaluation for golf course closing

Rough set theory is developed in terms of fuzzy sets that will allow a decision maker to determine certain and possible rules. In the application presented these rules are determined for the decision to close a golf course when the number of rounds of golf to be played is anticipated to be low. Attributes having an effect on the number of rounds of golf to be played are statistically analyzed and those negative factors are selected for generating rules for closing. Furthermore, the degree of belief of the decision maker in these generated rules is determined to be directly related to statistical confidence intervals.     

Using decision rules to achieve mass customization of airline services

This paper uses the Dominance-based Rough Set Approach (DRSA) to formulate airline service strategies by generating decision rules that model passenger preference for airline service quality. DRSA could help airlines eliminate some services associated with dispensable attributes without affecting passenger perception of service quality. DRSA could also help airlines achieve mass customization of airline services and generate additional revenues by active or passive targeting of quality services to passengers.     

Granular computing applied to ontologies

Granular Computing is an emerging conceptual and computing paradigm of information-processing. A central notion is an information-processing pyramid with different levels of clarifications. Each level is usually represented by ‘chunks’ of data or granules, also known as information granules. Rough Set Theory is one of the most widely used methodologies for handling or defining granules.Ontologies are used to represent the knowledge of a domain for specific applications. A challenge is to define semantic knowledge at different levels of human-depending detail.In this paper we propose four operations in order to have several granular perspectives for a specific ontological commitment. Then these operations are used to have various views of an ontology built with a rough-set approach. In particular, a rough methodology is introduced to construct a specific granular view of an ontology.     

Information sets in abelian codes: defining sets and Groebner basis

In Bernal and Simón (IEEE Trans Inf Theory 57(12):7990–7999, 2011) we introduced a technique to construct information sets for every semisimple abelian code by means of its defining set. This construction is a non trivial generalization of that given by Imai (Inf Control 34:1–21, 1977) in the case of binary two-dimensional cyclic (TDC) codes. On the other hand, Sakata (IEEE Trans Inf Theory IT-27(5):556–565, 1981) showed a method for constructing information sets for binary TDC codes based on the computation of Groebner basis which agrees with the information set obtained by Imai. Later, Chabanne (IEEE Trans Inf Theory 38(6):1826–1829, 1992) presents a generalization of the permutation decoding algorithm for binary abelian codes by using Groebner basis, and as a part of his method he constructs an information set following the same ideas introduced by Sakata. In this paper we show that, in the general case of q-ary multidimensional abelian codes, both methods, that based on Groebner basis and that defined in terms of the defining sets, also yield the same information set.     

基于区间AHP与粗糙集的决策系统的排序

针对决策系统中属性值以区间数形式给出的多指标决策问题,根据区间层次分析法(IAHP)与粗糙集的特点,提出了一种合理的排序方法.在保持分类能力不变的前提下,利用粗糙集理论中的知识约简方法,删除其中不相关或不重要的知识,结合区间层次分析法计算权重进行有效的排序.最后,给出算例说明了该方法的可行性和有效性.     

A simple method for estimating the fractal dimension from digital images: The compression dimension

The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.