Indiscernibility Relation Research Articles

Rough Set Extension under Incomplete Information System with “?” Values

Classical rough set theory (RST) can't process incomplete information system (IIS) because it is based on an indiscernibility relation which is a kind of equivalent relation. In the literature a non-symmetric similarity relation based rough set model (NS-RSM) has been introduced as an extended model under IIS with ``?" values directly. Unfortunately, in this model objects in the same similarity class are not necessarily similar to each other and may belong to different target classes. In this paper, a new inequivalent relation called Maximal Limited Consistent block relation (MLC) is proposed. The proposed MLC relation improves the lower approximation accuracy by finding the maximal limited blocks of indiscernible objects in IIS with ``?" values. Maximal Limited Similarity rough set model (MLS) is introduced as an integration between our proposed relation (MLC) and NS-RSM. The resulted MLS model works efficiently under IIS with ``?" values. Finally, an illustrative example is given to validate MLS model. Furthermore, approximation accuracy comparisons have been conducted among NS-RSM and MLS. The results of this work demonstrate that the MLS model outperform NS-RSM.

Planetary gearbox fault diagnosis based on data-driven valued characteristic multigranulation model with incomplete diagnostic information

There are many uncertain factors that may result in incomplete diagnostic information of planetary gearboxes, such as sensor malfunctions, communication lags, and data discretization, etc. Therefore, incomplete diagnostic information of planetary gearboxes may simultaneously contain two categories of unknown attribute values. However, existing fault diagnosis methods of planetary gearboxes are hard to realize fault diagnosis using incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. To overcome this issue, a fault diagnosis method of planetary gearboxes based on data-driven valued characteristic multigranulation model with incomplete diagnostic information is proposed. First, a calculation method of characteristic similarity degrees among cases is introduced, and a data-driven valued characteristic relation is defined. The data-driven valued characteristic relation is used to analyze and process incomplete diagnostic information that simultaneously contains two categories of unknown attribute values. Then, a data-driven valued characteristic multigranulation model is defined according to multigranulation model. An attribute reduction algorithm based on pessimistic data-driven valued characteristic multigranulation model is employed to extract fault diagnosis decision rules. Finally, naive Bayesian classifier is constructed to identify planetary gearbox conditions. The effectiveness of this method is validated and the advantages are investigated using a fault diagnosis experiment of planetary gearbox. Experimental results demonstrate that this method can accurately determine indiscernibility relation among cases, reduce computational complexity, and enhance fault diagnosis accuracy.

Rough Set Based Ontology Matching

Ontologies enriches the knowledge and add meaning to the data residing in semantic web. Ontology matching identifies concepts for matching in source ontologies to the target ontologies to eliminate heterogeneities. Despite using similarity measures for identifying similar concepts, the ontology matching systems fails to handle uncertainty. This paper proposes a rough set based ontology matching system to handle uncertainty which aims (i)to optimize concepts considered for matching by using concept type classification(ii) to use rough set concepts using indiscernibility relations and reducts (iii) to apply criterion of realism – a decision making under uncertainty criteria. The experiments conducted in the OAEI benchmark data sets, RSOM system yielded an increase of 8% in precision. The combined approach of using reduct and indiscenibilty relations reduces the number of concepts considered for matching among uncertain entities to about 70% in comparison to the existing systems and increases the accuracy of results by using criterion of realism.

Granular computing on information tables: Families of subsets and operators

In this work we use the granular computing paradigm to study specific types of families of subsets, operators and families of ordered pairs of sets of attributes which are naturally induced by information tables. In an unifying perspective, by means of some representation results, we connect the study of finite closure systems, matroids and finite lattice theory in the scope of the more general notion of attribute dependency based on information tables. For a fixed finite set Ω and for a corresponding information table J having attribute set Ω, the fundamental tool we use to proceed in our investigation is the equivalence relation ≈J on the power set P(Ω) which identifies two any sets of attributes inducing the same indiscernibility relation on the object set of J. We interpret the attribute dependency as a preorder ≥J on P(Ω) whose induced equivalence relation coincides with ≈J. Then we investigate in detail the links between the preorder ≥J, a closure system and an abstract simplicial complex on Ω naturally induced by ≈J and specific families of ordered pairs of sets of attributes on Ω.

THE MODEL OF TERITORIAL SYSTEM IN NATURAL DISASTER CONDITIONS

Context. The spatial model of territorial system in natural emergency conditions dedicated to decision support tasks solving and represented as the model of complex dynamic system is described in the paper. The literature data analysis has shown that applying the established approaches to the class of complex dynamic systems under consideration doesn’t provide the required calculation speed andefficiency of decision support system. This determines the timeliness of the further searching of nontraditional models and methods of decision support, which provide the fulfillment of calculation speed and efficiency requirements applicable to such systems. Objective. The goal of research is decreasing the damage from the natural emergency by means of improving the quality and timeless of forecasting the territorial system dynamics in the natural emergency conditions Method. The concept of territorial system in natural emergency conditions is represented in the form of overlaying static and dynamic topological spaces induced by indiscernibility relation, each of which allows representing geographical and attributive information about nature conditions, value objects demanding protection against natural emergency, as well as about natural emergency dynamics. The model of natural emergency dynamics is based on the discretization of the space by the grid of square cells of equal area and represented in the form of fuzzy dynamic topological space in the set of cells. Results. The web-oriented decision support system is created on the base of developed concept and model. The experiments have been conducted, which have shown that the proposed natural emergency model can provide reasonable characteristics in terms of accuracy and speed providing that the space is discretized with the size of cell being from 8 m to 18 m. Conclusions. The concept of territorial system representation in the natural emergency conditions is first developed in the form of overlaying static and dynamic topological spaces. The formal model of the natural emergency dynamics in the form of moving the bounding region of fuzzy-rough set, which has allowed reducing computational complexity and provided adaptation to the conditions of incomplete and inaccurate information is first created. The software for realization of suggested concept and method is developed. The software has allowed performing the practical task of decision support in natural emergency conditions.

Redefining core preliminary concepts of classic Rough Set Theory for feature selection

Data is growing at an exponential pace. To cope with this data explosion, we need effective data processing and analysis techniques. Feature selection is selecting a subset of features from a dataset that still provides most of the useful information. Various tools are available as underlying framework for this process however, Rough Set Theory is the most prominent tool due to its analysis friendly nature. Majority of Rough Set based feature selection algorithms use positive region based dependency measure as the sole criteria to select feature subset. Calculating positive region requires calculation of lower approximation which consequently involves indiscernibility relation. In this paper, new definitions of two Rough Set preliminaries i.e. lower and upper rough set approximation are proposed. New definitions of approximations are computationally less expensive as compared to the conventional. The proposed redefinitions showed 42.78% decrease in execution time for redefined lower approximation and 43.06% decrease in case of redefined upper approximation, for five publicly available datasets while maintaining 100% accuracy. Finally based on these redefined approximations we proposed a feature selection algorithm, which when compared with state of the art techniques showed significant increase in performance without the affecting the accuracy.

Statistical Inference in Rough Set Theory Based on Kolmogorov–Smirnov Goodness-of-Fit Test

Dependence degree (DD) and importance degree (ID) of patterns are crucial for pattern appraisement and model reconstruction. In rough set data analysis (RSDA), DD and ID lack robustness because the lower approximation set is terribly unstable under the indiscernibility relation perturbation. Statistical inference is a good way to deal with this instability. However, the fixed-value hypothesis testing and interval estimation of DD and ID were only discussed by the $\chi ^2$ test, which merely suits for two contingency tables (CT) with the same number of elements, and their nonzero elements must exist at the same positions. These requirements are too strict for practical applications. In this paper, the Kolmogorov–Smirnov (K–S) goodness-of-fit test is introduced to generalize the statistical inferences of DD and ID. As the bridge between data and corresponding measures, CT lies at the core of RSDA. By transforming CT to a random sample of a hypothesized random variable, the elementary algorithm for the goodness-of-fit test of contingency tables and the advanced algorithm for the goodness-of-fit test of contingency tables are proposed based on K–S statistic to implement goodness-of-fit tests of CTs. Better than $\chi ^2$ test, all CTs, even with different numbers and different positions of nonzero elements, are permitted. Subsequently, by generating a CT with expected DD value, the fixed-value hypothesis of DD is transformed to the goodness-of fit test between the original and expected CTs. Three algorithms, i.e., hypothesis test of dependence degree based on the K–S test, region estimation of the dependence degree, and significance test and region estimation of importance degree of attribute set, are proposed as fixed-value hypothesis tests and region estimations of DD and ID. These algorithms can be used to verify the importance of attributes and choose the attribute subset with the highest likelihood of maintaining the original discrimination ability. Experiments verify that the discrimination ability, disturbance tolerance ability, and stability under varying discretization strategies of DD and ID are significantly enhanced by the proposed algorithms.

Knowledge pairing systems in granular computing

In Pawlak’s theory of information systems, one has a collection of objects and knows the values of any object with respect a certain class of properties, usually called attributes. An implicit assumption of this theory is that the difference between the nature of objects and attributes is well outlined. In our paper we generalize the concept of information system by analyzing the case in which there is no a priori distinction in the nature of objects and attributes and, hence, both the interpretations are admissible. We call the structure arising in the previous context knowledge pairing system. We study the indiscernibility relations induced in both the admissible interpretations by means of up-down operators, in such a way to have a direct analogy with the extent and intent operators used in Formal Context Analysis. In particular, we investigate three models of knowledge pairing systems arising from real contexts and modeled respectively by graphs, digraphs and hypergraphs. We show the real convenience to use the notion of knowledge pairing system focusing on interpretation of this structure and discussing the two admissible perspectives obtained by avoiding the difference between the roles of objects and attributes.

Quick attribute reduction with generalized indiscernibility models

The efficiency of attribute reduction is one of the important challenges being faced in the field of Big Data processing. Although many quick attribute reduction algorithms have been proposed, they are tightly coupled with their corresponding indiscernibility relations, and it is difficult to extend specific acceleration policies to other reduction models. In this paper, we propose a generalized indiscernibility reduction model(GIRM) and a concept of the granular structure in GIRM, which is a quantitative measurement induced from multiple indiscernibility relations and which can be used to represent the computation cost of varied models. Then, we prove that our GIRM is compatible with three typical reduction models. Based on the proposed GIRM, we present a generalized attribute reduction algorithm and a generalized positive region computing algorithm. We perform a quantitative analysis of the computation complexities of two algorithms using the granular structure. For the generalized attribute reduction, we present systematic acceleration policies that can reduce the computational domain and optimize the computation of the positive region. Based on the granular structure, we propose acceleration policies for the computation of the generalized positive region, and we also propose fast positive region computation approaches for three typical reduction models. Experimental results for various datasets prove the efficiency of our acceleration policies in those three typical reduction models.

Multigranulation fuzzy rough set over two universes and its application to decision making

The original Pawlak’s rough set approach based on indiscernibility relation (single granularity) has been extended to multigranulation rough set structure in the recent years. Multigranulation rough set approach has become a flouring research direction in rough set theory. This paper considers rough approximation of a fuzzy concept under the framework of multigranulation over two different universes of discourse, i.e., multigranulation fuzzy rough set models over two universes. We present three types of multigranulation fuzzy rough set over two universes by the constructive approach, respectively. Some interesting properties of the proposed models are discussed and also the interrelationships between the proposed models and the existing rough set models are given. We then propose a new approach to a kind of multiple criteria group decision making problem based on multigranulation fuzzy rough set model over two universes. The decision rules and algorithm of the proposed method are given and an example of handling multiple criteria group decision making problem of clothes ranking illustrates this approach. The main contribution of this paper is twofold. One is to establish the multigranulation fuzzy rough set theory over two universes. Another is to try presenting a new approach to multiple criteria group decision making based on multigranulation fuzzy rough set over two universes. The proposed models not only enrich the theory of multigranulation rough set but also make a tentative to provide a new perspective for multiple criteria group decision making with uncertainty.

Dynamic variable precision rough set approach for probabilistic set-valued information systems

Set-valued information systems are important type of data tables in many real applications, where the attribute values are described by sets to characterize uncertain and incomplete information. However, in some real situations, set-values may be depicted by probability distributions, which results in that the traditional tolerance relation based on intersection operation could not reasonably describe the indiscernibility relation of objects. To address this issue, we introduce the concept of probabilistic set-valued information systems (PSvIS), and present the extended variable precision rough set model (VPRS) based on the λ-tolerance relation in terms of Bhattacharyya distance. Considering the features of information systems will evolve over time in a dynamic data environment, it will lead to the change of information granulation and approximation structures. A matrix representation of rough approximation is presented based on two matrix operators and two vector functions in PSvIS. Then incremental mechanisms by the utilization of previously learned approximation results and region relation matrices for updating rough approximations are proposed, and the corresponding algorithms are developed and analyzed. Experimental results show that the proposed algorithms outperform the static algorithms and related incremental algorithms while inserting into or removing from attributes in PSvIS.

Developing the model of ecosystem in natural disasters conditions

The spatial model of ecosystem in natural emergency conditions dedicated to decision support tasks solving is described in the paper. The goal of research is decreasing the damage from the natural emergency by means of improving the quality and timeless of forecasting the territorial system dynamics in the natural emergency conditions. The methods of topology, fuzzy sets theories, as well as geoinformation systems and web-technologies were used when performing research. The concept of territorial system in natural emergency conditions in the form of overlaying static and dynamic topological spaces induced by indiscernibility relation is described. Each of the topological spaces allows representing geographical and attributive information about nature conditions, value objects demanding protection against natural emergency, as well as about natural emergency dynamics. The model of natural disaster dynamics in the form of fuzzy dynamic topological space is also described in the paper. This representation of natural disaster model has allowed to provide adaptability to incomplete and inaccurate information. The web-oriented decision support system is created on the base of developed concept and model. The experiments have been conducted, which have shown that the proposed natural emergency model can provide reasonable characteristics in terms of accuracy and speed providing that the space is discretized with the size of cell from 8 m to 18 m.

Micro and macro models of granular computing induced by the indiscernibility relation

In rough set theory (RST), and more generally in granular computing on information tables (GRC-IT), a central tool is the Pawlak’s indiscernibility relation between objects of a universe set with respect to a fixed attribute subset. Let us observe that Pawlak’s relation induces in a natural way an equivalence relation ≈ on the attribute power set that identifies two attribute subsets yielding the same indiscernibility partition. We call indistinguishability relation of a given information table I the equivalence relation ≈, that can be considered as a kind of global indiscernibility. In this paper we investigate the mathematical foundations of indistinguishability relation through the introduction of two new structures that are, respectively, a complete lattice and an abstract simplicial complex. We show that these structures can be studied at both a micro granular and a macro granular level and that are naturally related to the core and the reducts of I. We first discuss the role of these structures in GrC-IT by providing some interpretations, then we prove several mathematical results concerning the fundamental properties of such structures.

An Empirical Analysis of Rough Set Categorical Clustering Techniques.

Clustering a set of objects into homogeneous groups is a fundamental operation in data mining. Recently, many attentions have been put on categorical data clustering, where data objects are made up of non-numerical attributes. For categorical data clustering the rough set based approaches such as Maximum Dependency Attribute (MDA) and Maximum Significance Attribute (MSA) has outperformed their predecessor approaches like Bi-Clustering (BC), Total Roughness (TR) and Min-Min Roughness(MMR). This paper presents the limitations and issues of MDA and MSA techniques on special type of data sets where both techniques fails to select or faces difficulty in selecting their best clustering attribute. Therefore, this analysis motivates the need to come up with better and more generalize rough set theory approach that can cope the issues with MDA and MSA. Hence, an alternative technique named Maximum Indiscernible Attribute (MIA) for clustering categorical data using rough set indiscernible relations is proposed. The novelty of the proposed approach is that, unlike other rough set theory techniques, it uses the domain knowledge of the data set. It is based on the concept of indiscernibility relation combined with a number of clusters. To show the significance of proposed approach, the effect of number of clusters on rough accuracy, purity and entropy are described in the form of propositions. Moreover, ten different data sets from previously utilized research cases and UCI repository are used for experiments. The results produced in tabular and graphical forms shows that the proposed MIA technique provides better performance in selecting the clustering attribute in terms of purity, entropy, iterations, time, accuracy and rough accuracy.

Incremental update of rough set approximation under the grade indiscernibility relation

The incremental updating of lower and upper approximations under the variation of information systems is an important issue in rough set theory. Many incremental updating approaches with respect to different kinds of indiscernibility relations have been proposed. The grade indiscernibility relation is a fuzzification of classical Pawlak’s indiscernibility relation which can characterize the similarity between objects more precisely. Based on fuzzy rough set model, this paper discusses the approaches for dynamically acquiring of the upper and lower approximations with respect to the grade indiscernibility relation when adding and removing an attribute or an object, and changing the attribute value of the object, respectively. Since the approaches are used in succession, they make the approximations can be updated correctly and effectively when any kind of possible change in the information system. Finally, extensive experiments on data sets from University of California, Irvine (UCI) show that the incremental methods effectively reduce the computing time in comparison with the traditional non-incremental method.

A Variable Precision Fuzzy Rough Set Approach to a Fuzzy-Rough Decision Table

In a real world, non-empty boundaries between classes may be both rough and fuzzy. In order to make decision in fuzzy approximation space, a fuzzy VPRS (variable precision rough set) approach is proposed based on substitution of the indiscernibility relation by a fuzzy indiscernibility relation in the rough approximation of decision classes, which can obtain probabilistic rules from fuzzy decision tables. Some set theoretic properties of the proposed approach are discussed.

The adjacency matrix of a graph as a data table: a geometric perspective

In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph G in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph G by assuming a generic vertex subset W as reference system with respect to which examine the symmetry of all vertex subsets of G. The change of perspective from G without reference system to the pair (G, W) is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system (G, W) as particular equivalence classes of vertices in G, and we study the geometric properties of all reference systems (G, W), when W runs over all vertex subsets of G. We also introduce three hypergraph models and a vertex set partition lattice associated to G, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph.

An Algebraic Approach to Unital Quantities and their Measurement

Abstract The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system’s ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity’s value into its measuring instrument’s pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument’s indiscernibility relation, manifested by not being able to distinguish the measured system’s topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment.

A Novel Soft Set Approach for Feature Selection

Feature selection is an important preprocess for data mining. The soft set theory is a new mathematical tool to deal with uncertainties. Merging the soft set theory into the feature selection process based on rough sets facilitates the computation with equivalence classes and improves the efficiency. We propose a paired relation soft set model based on equivalence classes of the information system. Then we use it to present the lower approximate set in the form of soft sets and calculate the degree of dependency between relations. Furthermore, we give a new mapping to obtain equivalence classes of indiscernibility relations and propose a feature selection algorithm based on the paired relation soft set model. Compared to the algorithm based NSS, this algorithm shows 18.17% improvement on an average. Meantime, both of the algorithms show a good scalability.

A modified Fuzzy k-Partition based on indiscernibility relation for categorical data clustering

Categorical data clustering has been adopted by many scientific communities to classify objects from large databases. In order to classify the objects, Fuzzy k-Partition approach has been proposed for categorical data clustering. However, existing Fuzzy k-Partition approaches suffer from high computational time and low clustering accuracy. Moreover, the parameter maximize of the classification likelihood function in Fuzzy k-Partition approach will always have the same categories, hence producing the same results. To overcome these issues, we propose a modified Fuzzy k-Partition based on indiscernibility relation. The indiscernibility relation induces an approximation space which is constructed by equivalence classes of indiscernible objects, thus it can be applied to classify categorical data. The novelty of the proposed approach is that unlike previous approach that use the likelihood function of multivariate multinomial distributions, the proposed approach is based on indescernibility relation. We performed an extensive theoretical analysis of the proposed approach to show its effectiveness in achieving lower computational complexity. Further, we compared the proposed approach with Fuzzy Centroid and Fuzzy k-Partition approaches in terms of response time and clustering accuracy on several UCI benchmark and real world datasets. The results show that the proposed approach achieves lower response time and higher clustering accuracy as compared to other Fuzzy k-based approaches.