General relation-based variable precision rough fuzzy set

A comparative study of variable precision fuzzy rough sets based on residuated lattices

The variable precision rough set (VPRS) model generalizes the Pawlak rough set model with variable parameters. The Bayesian rough set (BRS) model improves the VPRS model with non-parametric modification by using the prior probability as a reference. This paper presents two research results related to rough set model and rough vague sets. One is a modification to the Bayesian rough set model and the other is a modification to rough vague sets. First, the Bayesian rough set model is analyzed and discussed. Second, a modification to this model is proposed and verified. Finally, a modification to rough vague sets is presented and its related properties are discussed..

Rough sets and fuzzy sets are both theories for handling uncertain problems, and the viewpoints between them are different. They are complementary than competitive. Much seminal research has been carried out into combining them both; the results are rough fuzzy sets and fuzzy rough sets. In this paper, we introduce the idea of fuzzy rough sets in research of decision tables with quantitative data. Thus, we can handle the decision table directly without discretization. Combining the idea of variable precision rough sets, we make the form of the upper and lower approximation uniform.

Rough set theory for the interval-valued fuzzy information systems

The aim of feature selection (FS) is to select a small subset of most important and discriminative features. Many FS approaches based on rough set theory up to now, have employed reduct analysis using feature dependency measures. However the critical shortcoming for such approaches is that they are not able to manage useful information that may be destroyed by noise elements. Therefore several extensions to the original theory have been proposed. Three notable extensions are fuzzy rough set (FRS), variable precision rough set (VPRS), and tolerance rough set model (TRSM). Although successful, each of the extensions exhibits a critical shortcoming which makes that extension inapplicable in most of scenarios. For example, FRS is able to describe the existing dependencies between different attributes accurately, but its high run-times makes it inapplicable to larger datasets. As another example, VPR is very fast, but requires more information than contained within the data itself, which is inaccessible for most of the applications. This paper examines a rough set FS technique which uses a noise resistant dependency measure to quantify information that may be hidden due to the noise elements. Experimental results demonstrate that the use of this measure can result more discriminative reducts than those obtained using other RSFS approaches. Moreover, the proposed measure is as fast as VPRS and as accurate as FRS and TRSM, while it need no additional information other than contained within the data.

This paper is devoted to propose generalized L-fuzzy variable precision rough sets, based on generalized residuated lattices with conjunctions not necessarily commutative over two universes, as a generalization of the notion of L-fuzzy variable precision rough sets based on residuated lattices with commutative conjunctions over one universe only. Then we define and investigate several classes of generalized L-fuzzy variable precision rough sets. The topological properties and granular representation of generalized L-fuzzy variable precision rough sets are also given. Our main concern is to propose a real number valued function for each approximation operator in generalized L-fuzzy variable precision rough sets and to measure its approximating ability. We observe that the functions of lower and upper approximation operators are natural generalizations of belief and plausibility functions respectively as given in the evidence theory. Using these functions, accuracy measure and degree of roughness is defined for generalized L-fuzzy variable precision rough sets. As an application of generalized L-fuzzy variable precision rough sets, I-fuzzy variable precision rough sets are proposed on the unit interval I = [0, 1], based on generalized residuated lattices induced by left-continuous pseudo-t-norms.

In this paper some properties of the variable precision fuzzy rough sets model will be considered. A new way of determining the positive region of classification will be proposed, which is useful in evaluation of approximation quality in variable precision fuzzy or crisp rough sets applications. The notions of the fuzzy rough weighted mean u-lower and l-upper approximation will be discussed. Fuzzy rough approximations will be evaluated basing on selected R-implicators.

Grade and precision are two important confidence levels. In this paper, the connection and combination between them are studied. The connection between graded approximations and variable precision approximations is studied, and the important formula of conversion is achieved. Approximation of intersection of grade and precision is defined based on logical ”and” operation of grade and precision, and its construction is achieved, which becomes important theory base for decreasing computing complexity and computing quantity of application algorithm of graded rough sets and variable precision rough sets. Graded rough sets and variable precision rough sets are united and included by the approximation of intersection of grade and precision. The study on combination and unity of graded rough sets and variable precision rough sets in this paper, has important value on both theory development and practice application.KeywordsArtificial IntelligenceRough SetsGraded Rough SetsVariable Precision Rough SetsConfidence LevelsGradePrecision

Uncertainty about future developments constitutes the most important and most difficult challenge for mankind. Despite this fact, uncertainty is not a part of general science. General science assumes that the future development follows cause-effect relations which can be described by mathematical functions, where the argument represents the cause and the image represents the effect. Scientific theories have exactly this form and it is widely believed that these functions represent “truth”. Of course, this is nonsense as all the scientific theories are with certainty wrong and cannot describe the real evolution correctly. The inappropriate handling of uncertainty in science has produced a strange variety of “uncertainty theories” that causes confusion and helplessness. A friend of mine has expressed his confusion by the following words: ‘Crisp sets’, ‘fuzzy sets’, ‘rough sets’, ‘grey sets’, ‘fuzzy rough sets’, ‘rough fuzzy sets’, ‘fuzzy grey sets’, ‘grey fuzzy sets’, ‘rough grey sets’, ‘grey rough sets’, and now ‘affinity sets’. My goodness! Is there anybody around who can enlighten me, i.e., help me to see a clear pattern in this set of sets, allegedly providing powerful tools to model various kinds of uncertainty? This paper examines the role and the handling of uncertainty in quality control. How is uncertainty quantified in quality control for making decisions aiming at maintaining or improving quality of processes and products.KeywordsRandomnessIgnoranceKnowledgeFuzzynessQuantificationProbabilityCredibilityProbability spaceUncertainty spaceBernoulli space

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

This paper researched and constructed residuation implication operator of intuitionistic fuzzy T-norms.Based on this,the paper deducted the intuitionistic fuzzy rough sets model,and proved that the Pawlak rough sets,intuitionistic fuzzy sets,fuzzy rough sets,rough fuzzy sets and fuzzy T-norms rough sets were special cases of intuitionistic fuzzy rough sets.At last,some properties of intuitionistic fuzzy sets were presented and proved.

The definition of basic rough sets [3] depends upon either a single equivalence relation defined on a universe or several equivalence relations defined over the universe, taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. Extending this notion, a rough set model based on multi-granulations (MGRS) was introduced in [5]. In this approach, approximations of sets were defined through multiple equivalence relations on the universe and their properties were investigated. Using hybridization of fuzzy set [13] with rough set the concept of rough fuzzy set was introduced by Dubois and Prade [1]. Recently, a Rough Fuzzy Set Model was introduced and studied by Wu and Kou [12], which is based on Multiple Granulation. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [10]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [11]. In this paper we introduce incomplete multigranulation on rough fuzzy sets, study their basic properties and extend the topological properties in [11] to this context. Our findings are true for both complete and incomplete fuzzy rough set models based upon multi granulation.KeywordsRough SetsFuzzy rough setsequivalence relationstolerance relationstype of rough setsmulti granular fuzzy rough sets

Fuzzy rough sets (FRSs) play a significant role in the field of data analysis, and one of the common methods for constructing FRSs is the use of the fuzzy logic operators. To further extend FRSs theory to more diverse information backgrounds, this article proposes a covering variable precision fuzzy rough set model based on overlap functions and fuzzy β-neighbourhood operators (OCVPFRS). Some necessary properties of OCVPFRS have also been studied in this work. Furthermore, multi-label classification is a prevalent task in the realm of machine learning. Each object (sample or instance) in multi-label data is associated with various labels (classes), and there are numerous features or attributes that need to be taken into account within the attribute space. To enhance various performance metrics in the multi-label classification task, attribute reduction is an essential pre-processing step. Therefore, according to overlap functions and fuzzy rough sets’ excellent work on applications: such as image processing and multi-criteria decision-making, we establish an attribute reduction method suitable for multi-label data based on OCVPFRS. Through a series of experiments and comparative analysis with existing multi-label attribute reduction methods, the effectiveness and superiority of the proposed method have been verified.

By the combination of feature and concept hierarchy model and the definition of innovation about concept, the method of structured processing and knowledge hierarchical representation for injection mould repair schemes is put forward under the condition of non-fuzzy or fuzzy data. Rule sets can be provided by knowledge induction for injection mould repairs based on basic rough set, but the rule sets are large-scale and the applicability is poor in practice. Aiming at achieving the rule sets more efficiently and applicably, a new method of knowledge hierarchical induction based on variable precision rough set is proposed. For injection mould repair schemes based on fuzzy data, by the abstraction of innovation about concept from the feature decision tables, feature fuzzy similar matrix is constructed and the feature decision table is divided into some fuzzy equivalent subsets by introducing confidence level vector. Finally, the algorithm of feature reduction and knowledge induction based on fuzzy rough set is put forward by defining single or multi-fuzzy feature equivalence relations. The feasibility and effectiveness of two methods for knowledge hierarchical induction under different data environments are both analyzed.

In e-learning environment, optimal learning patterns are discovered for realising and understanding the effective learning styles. The value of uncertain and imprecise knowledge collected has to be categorised into classes known as membership grades. Rough set theory is potential in categorising data into equivalent classes and fuzzy logic may be applied through soft thresholds for refining equivalence relation that quantifies correlation between each class of elucidated data. In this paper, probabilistic variable precision fuzzy rough set technique (PVPFRST) is proposed for deriving robust approximations and generalisations that handles the types of uncertainty namely stochastic, imprecision and noise in membership functions. The result infers that the degree of accuracy of PVPFRST is 21% superior to benchmark techniques. Result proves that PVPFRST improves effectiveness and efficiency in identifying e-learners styles and increases the performance by 27%, 22% and 25% in terms of discrimination rate, precision and recall value than the benchmark approaches.