Approximations from Anywhere and General Rough Sets

Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the \emph{inverse problem}. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical perspective, a few (as opposed to one) theoretical frameworks may be suitable for formulating the problem itself. \emph{Granular operator spaces} have been recently introduced and investigated by the present author in her recent work in the context of antichain based and dialectical semantics for general rough sets. The nature of the inverse problem is examined from number-theoretic and combinatorial perspectives in a higher order variant of granular operator spaces and some necessary conditions are proved. The results and the novel approach would be useful in a number of unsupervised and semi supervised learning contexts and algorithms.Comment: 20 Pages. Scheduled to appear in IJCRS'2017 LNCS Proceedings, Springe

Knowledge structure, knowledge granulation and knowledge distance in a knowledge base

AbstractOne of the strengths of rough set theory is the fact that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. Knowledge structures in knowledge bases have two categories: complete and incomplete. In this paper, through uniformly expressing these two kinds of knowledge structures, we first address four operators on a knowledge base, which are adequate for generating new knowledge structures through using known knowledge structures. Then, an axiom definition of knowledge granulation in knowledge bases is presented, under which some existing knowledge granulations become its special forms. Finally, we introduce the concept of a knowledge distance for calculating the difference between two knowledge structures in the same knowledge base. Noting that the knowledge distance satisfies the three properties of a distance space on all knowledge structures induced by a given universe. These results will be very helpful for knowledge discovery from knowledge bases and significant for establishing a framework of granular computing in knowledge bases

Determinants of Long-term Economic Development: An Empirical Cross-country Study Involving Rough Sets Theory and Rule Induction

Empirical findings on determinants of long-term economic growth are numerous, sometimes inconsistent, highly exciting and still incomplete. The empirical analysis was almost exclusively carried out by standard econometrics. This study compares results gained by cross-country regressions as reported in the literature with those gained by the rough sets theory and rule induction. The main advantages of using rough sets are being able to classify classes and to discretize. Thus, we do not have to deal with distributional, independence, (log-)linearity, and many other assumptions, but can keep the data as they are. The main difference between regression results and rough sets is that most education and human capital indicators can be labeled as robust attributes. In addition, we find that political indicators enter in a non-linear fashion with respect to growth.Economic growth, Rough sets, Rule induction

Variable Precision Rough Set Model for Incomplete Information Systems and Its Beta-Reducts

As the original rough set model is quite sensitive to noisy data, Ziarko proposed the variable precision rough set (VPRS) model to deal with noisy data and uncertain information. This model allowed for some degree of uncertainty and misclassification in the mining process. In this paper, the variable precision rough set model for an incomplete information system is proposed by combining the VPRS model and incomplete information system, and the beta-lower and beta-upper approximations are defined. Considering that classical VPRS model lacks a feasible method to determine the precision parameter beta when calculating the beta-reducts, we present an approach to determine the parameter beta. Then, by calculating discernibility matrix and discernibility functions based on beta-lower approximation, the beta-reducts and the generalized decision rules are obtained. Finally, a concrete example is given to explain the validity and practicability of beta-reducts which is proposed in this paper

A Comparison of the Quality of Rule Induction from Inconsistent Data Sets and Incomplete Data Sets

In data mining, decision rules induced from known examples are used to classify unseen cases. There are various rule induction algorithms, such as LEM1 (Learning from Examples Module version 1), LEM2 (Learning from Examples Module version 2) and MLEM2 (Modified Learning from Examples Module version 2). In the real world, many data sets are imperfect, either inconsistent or incomplete. The idea of lower and upper approximations, or more generally, the probabilistic approximation, provides an effective way to induce rules from inconsistent data sets and incomplete data sets. But the accuracies of rule sets induced from imperfect data sets are expected to be lower. The objective of this project is to investigate which kind of imperfect data sets (inconsistent or incomplete) is worse in terms of the quality of rule induction. In this project, experiments were conducted on eight inconsistent data sets and eight incomplete data sets with lost values. We implemented the MLEM2 algorithm to induce certain and possible rules from inconsistent data sets, and implemented the local probabilistic version of MLEM2 algorithm to induce certain and possible rules from incomplete data sets. A program called Rule Checker was also developed to classify unseen cases with induced rules and measure the classification error rate. Ten-fold cross validation was carried out and the average error rate was used as the criterion for comparison. The Mann-Whitney nonparametric tests were performed to compare, separately for certain and possible rules, incompleteness with inconsistency. The results show that there is no significant difference between inconsistent and incomplete data sets in terms of the quality of rule induction