Low Approximation Research Articles

Approximations of semiring-valued fuzzy sets with applications in new fuzzy structures

Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic or fuzzy soft sets, can be transformed into one common type of fuzzy sets, called (R,R⁎)-fuzzy sets, with values in a set R that is a common underlying set of complete commutative idempotent semirings R and R⁎. For (R,R⁎)-fuzzy sets, the theory of lower and upper approximations by (R,R⁎)-relations is defined and properties of these approximations are investigated. The transformation of new types of fuzzy sets into (R,R⁎)-fuzzy sets is used to show the possibility of introducing approximation operators in new types of fuzzy sets without having to define these operators separately for each new type of fuzzy set.

CP decomposition for tensors via alternating least squares with QR decomposition

AbstractThe CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low‐rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill‐conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP‐ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP‐ALS subproblems efficiently, have the same complexity as the standard CP‐ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill‐conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

The axiomatic characterization on fuzzy variable precision rough sets based on residuated lattice

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.

Attribute reduction algorithms for hesitant fuzzy decision systems based on hypergraphs

The rough set has been proven to be a powerful tool for dealing with uncertainty and vagueness in different fields. The hesitant fuzzy rough set is a generalization of rough sets to solve more complex problems. Since the existing hesitant fuzzy rough sets don’t satisfy the inclusive property, a novel hesitant fuzzy rough set based on the dual score functions is proposed. Then four generalized hesitant fuzzy rough set about the novel one are presented. Next, the lower approximation distribution reductions can be obtained by the discernibility matrix. Moreover, it is discovered that finding the lower approximation distribution reductions of a hesitant fuzzy decision system is equivalent to finding the minimal transversals of its hypergraph. An improved algorithm for hesitant fuzzy decision systems based on hypergraphs is presented to accelerate the reduction process. Finally, The hybrid data of Hepatitis C Virus from UCI is used as an instance background for data analyses to reflect the feasibility of the proposed algorithm.

Novel Graph Neighborhoods Emerging from Ideals

Rough set theory is a mathematical approach that deals with the problems of uncertainty and ambiguity in knowledge. Neighborhood systems are the most effective instruments for researching rough set theory in general. Investigations on boundary regions and accuracy measures primarily rely on two approximations, namely lower and upper approximations, by using these systems. The concept of the ideal, which is one of the most successful and effective mathematical tools, is used to obtain a better accuracy measure and to decrease the boundary region. Recently, a generalization of Pawlak’s rough set concept has been represented by neighborhood systems of graphs based on rough sets. In this research article, we propose a new method by using the concepts of the ideal and different neighborhoods from graph vertices. We examine important aspects of these techniques and produce accuracy measures that exceed those previously = reported in the literature. Finally, we show that our method yields better results than previous techniques utilized in chemistry.

A further study on generalized neighborhood systems-based pessimistic rough sets

The generalized neighborhood system-based rough set is an important extension of Pawlak’s rough set. The rough sets based on generalized neighborhood systems include two basic models: optimistic and pessimistic rough sets. In this paper, we give a further study on pessimistic rough sets. At first, to regain some properties of Pawlak’s rough sets that are lost in pessimistic rough sets, we introduce the mediate, transitive, positive (negative) alliance conditions for generalized neighborhood systems. At second, some approximation operators generated by special generalized neighborhood systems are characterized, which include serial, reflexive, symmetric, mediate, transitive, and negative alliance generalized neighborhood systems and their combinations (e.g. reflexive and transitive). At third, we discuss the topologies generated by the upper and lower approximation operators of the pessimistic rough sets. Finally, combining practical examples, we apply pessimistic rough sets to rule extraction of incomplete information systems. Particularly, we prove that different decision rules can be obtained when different neighborhood systems are chosen. This enables decision makers to choose decisions based on personal preferences.

Approximation spaces inspired by subset rough neighborhoods with applications

Abstract In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of u u and ⟨ u ⟩ \langle u\rangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.

Three-way conflict analysis and resolution based on q-rung orthopair fuzzy information

In solving some complex conflict problems, compared with intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs), q-rung orthopair fuzzy sets (q-ROFSs) can better express and model imprecise information. In addition, three-way decisions (TWDs), an emerging theory of simulating human decision-making behavior, play an increasingly important role in the study of conflict problems. In order to solve conflict problems under q-rung orthopair fuzzy information (q-ROFI), we propose a three-way conflict analysis and resolution model based on TWDs and decision-theoretic rough sets (DTRSs) theory, which provides another answer for Deja's fundamental questions about Pawlak's conflict model, namely, “what are the inherent causes of conflict?” and “how can a viable strategy be found?” different from the existing ones. To be specific, firstly, we define new conflict distances and conflict function for q-ROFI system using aggregation functions and weight determination method, and prove some of their properties. On this basis, the concepts of three-level of contention set and agent set, a trisection of conflict set under individual contention and contention set, and maximum doughty coalition of q-ROFI are introduced. Secondly, after introducing the rough approximation threshold calculation method on the basis of standard evaluation set, to find the optimal viable strategy under the given q-ROFI, we calculate and sort the score function by the upper and lower approximation operators in the proposed q-rung orthopair fuzzy decision-theoretic rough sets (q-ROFDTRSs) model. Thirdly, we use the proposed model to study a land-use conflict case in a metropolitan area to verify the feasibility. Finally, from sensitivity and comparative analysis, we verify the stability and validity of the q-ROFDTRSs model.

Nano SC-Open Sets in Nano Topological Spaces

The objective of this paper is to define and introduce a new type of nano semi-open set which called nano -open set as a strong form of nano semi-open set which is related to nano closed sets in nano topological spaces. In this paper, we find all forms of the family of nano -open sets in term of upper and lower approximations of sets and we can easily find nano -open sets and they are a gate to more study. Several types of nano open sets are known, so we study relationship between the nano -open sets with the other known types of nano open sets in nano topological spaces. The Operators such as nano -interior and nano -closure are the part of this paper.

Axiomatic characterizations of (𝔾, 𝕆)-fuzzy rough approximation operators via overlap and grouping functions on a complete lattice

Recently, Jiang, H. B., and B. Q. Hu. [2022. “On (O,G)-Fuzzy Rough Sets Based on Overlap and Grouping Functions Over Complete Lattices.” International Journal of Approximate Reasoning 144: 18–50. doi:10.1016/j.ijar.2022.01.012] constructed a -fuzzy rough set model with the logical connectives–a grouping function and an overlap function on a complete lattice, which provided a new constructive approach to fuzzy rough set theory. The axiomatic approach is as important as the constructive approach in rough set theory. In this paper, we continue to study axiomatic characterizations of -fuzzy rough set. Traditionally, the associativity of the logical connectives plays a vital role in the axiomatic research of existing fuzzy rough set models. However, a grouping function and an overlap function lack the associativity. So we explore a novel axiomatic approach to -upper and -lower fuzzy rough approximation operators without associativity. Further, we provide single axioms to characterize -upper and -lower fuzzy rough approximation operators instead of sets of axioms. Finally, we use single axioms to characterize fuzzy rough approximation operators generated by various kinds of fuzzy relations including serial, reflexive, symmetric, -transitive, -transitive fuzzy relations as well as their compositions.

Feature selection in a discrete feature space based on fuzzy conditional information entropy iterative model and matrix operation

A discrete feature space consists of the set of samples, the set of categorical attributes (features) describing the samples and a decision attribute. Generally, many feature selection algorithms for a discrete feature space are based on the lower approximation or information entropy. However, the calculation of the lower approximation is more cumbersome, and information entropy may result based on equivalence class in a poor selection of features. This paper proposes feature selection algorithm for a discrete feature space by using fuzzy conditional information entropy iterative strategy and matrix operation. Firstly, the fuzzy symmetry relation induced by a discrete feature space is defined. Then, fuzzy conditional and joint information entropy for a discrete feature space are presented, and some properties are obtained. Subsequently, fuzzy conditional information entropy iterative model (FCIEI-model) is proposed. Moreover, difference, block diagonal, and decision block diagonal matrices are introduced. Next, a feature selection algorithm (denoted as FDM-algorithm) on account of FCIEI-model and matrix operations is designed and its time complexity is analyzed. Finally, the performance of the algorithm is evaluated through a series of experiments. The results show that the given algorithm is better than the existing algorithms.

Quantum mixed state compiling

The task of learning a quantum circuit to prepare a given mixed state is a fundamental quantum subroutine. We present a variational quantum algorithm (VQA) to learn mixed states which is suitable for near-term hardware. Our algorithm represents a generalization of previous VQAs that aimed at learning preparation circuits for pure states. We consider two different ansätze for compiling the target state; the first is based on learning a purification of the state and the second on representing it as a convex combination of pure states. In both cases, the resources required to store and manipulate the compiled state grow with the rank of the approximation. Thus, by learning a lower rank approximation of the target state, our algorithm provides a means of compressing a state for more efficient processing. As a byproduct of our algorithm, one effectively learns the principal components of the target state, and hence our algorithm further provides a new method for principal component analysis. We investigate the efficacy of our algorithm through extensive numerical implementations, showing that typical random states and thermal states of many body systems may be learnt this way. Additionally, we demonstrate on quantum hardware how our algorithm can be used to study hardware noise-induced states.

Two new fuzzy covering-based rough approximation operators via inclusion degree

Approximate accuracy is an important concept in rough set theory, which is defined by upper and lower approximations. Generally speaking, the higher precision means the better application performance. The approximation accuracy can be improved by minimizing the upper approximation and maximizing the lower approximation. Recently, Zhou [52] introduced two types of fuzzy-covering based rough set models by using inclusion relation between fuzzy sets. In this paper, by replacing inclusion relation with implication degree, we investigate two new fuzzy covering-based rough set models. Compared with inclusion relationship, the inclusion degree can describe the contained relation between fuzzy sets in more detail. This makes our upper approximation smaller than Zhou’s upper approximation, while the lower approximation is larger than Zhou’s. Therefore, the approximate accuracy of our model is higher than that of Zhou. Furthermore, we apply the new model to the study of multi-attribute decision-making (MADM). Combined with the car buying problem, we demonstrate the effectiveness of our model and compare it with other methods. The results show that we can get the same optimal choice as other methods. However, according to Zhou’s model, we cannot get the optimal choice.

Generalization Approximation Spaces Using Combined Edges Systems of a finite number of undirected graphs

In this paper we will study approximate coefficients. New based on a finite family of Lower approximation and Upper approximation and we present a generalization of some concepts and definitions and boundary and we will also study the accuracy factor for this family

Another view on tolerance based multigranulation q-rung orthopair fuzzy rough sets with applications

The q-rung orthopair fuzzy sets accommodate more uncertainties than the Pythagorean fuzzy sets and hence their applications are much extensive. Under the q-rung orthopair fuzzy set, the objective of this paper is to develop new types of q-rung orthopair fuzzy lower and upper approximations by applying the tolerance degree on the similarity between two objects. After employing tolerance degree based q-rung orthopair fuzzy rough set approach to it any times, we can get only the six different sets at most. That is to say, every rough set in a universe can be approximated by only six sets, where the lower and upper approximations of each set in the six sets are still lying among these six sets. The relationships among these six sets are established. Furthermore, we propose tolerance degree based multi granulation optimistic/pessimistic q-rung orthopair fuzzy rough sets and investigate some of their properties. Another main contribution of this paper is to disclose the ideas of different kinds of approximations called approximate precision, rough degree, approximate quality and their mutual relationship. Finally a technique is devloped to rank the alternatives in a q-rung orthopair fuzzy information system based on similarity relation. We find that the proposed method/technique is more efficient when compared with other existing techniques.

(O, G)-granular variable precision fuzzy rough sets based on overlap and grouping functions

Since Bustince et al. introduced the concepts of overlap and grouping functions, these two types of aggregation functions have attracted a lot of interest in both theory and applications. In this paper, the depiction of (O, G)-granular variable precision fuzzy rough sets ((O, G)-GVPFRSs for short) is first given based on overlap and grouping functions. Meanwhile, to work out the approximation operators efficiently, we give another expression of upper and lower approximation operators by means of fuzzy implications and co-implications. Furthermore, starting from the perspective of construction methods, (O, G)-GVPFRSs are represented under diverse fuzzy relations. Finally, some conclusions on the granular variable precision fuzzy rough sets (GVPFRSs for short) are extended to (O, G)-GVPFRSs under some additional conditions.

On three types of L-fuzzy β-covering-based rough sets

In this paper, we mainly establish three types of L-fuzzy β-covering-based rough set models and study the axiomatic characterizations, matrix representations and interdependency of their rough approximation operators. Firstly, we propose three pairs of L-fuzzy β-covering-based rough approximation operators by constructing the notions such as β-subsethood degree and β-degree of intersection. And then, three pairs of the axiomatic characterizations about L-fuzzy β-covering-based rough approximation operators are investigated, respectively. In the meantime, we verify the independence of each axiom set. Thirdly, we give the matrix representations of three pairs of L-fuzzy β-covering-based rough approximation operators for efficient calculation of the lower and upper approximation operators through operations on matrices. Finally, the interdependency of three pairs of rough approximation operators based on L-fuzzy β-covering is explored in light of reducible elements and independent elements. Meanwhile, we present the necessary and sufficient conditions under which two L-fuzzy β-coverings can generate the same lower and upper rough approximation operations.

Analysis of a nonuniform guiding structure by the adaptive finite‐difference and singular value decomposition methods

AbstractThis paper presents a flexible finite‐difference technique for analyzing the nonuniform guiding structures. Because the voltage and current variations along the nonuniform structure differ for each segment, this work considers the adaptable discretization steps. This technique increases the accuracy of the final response. Moreover, by applying the singular value decomposition and discarding the nonprincipal singular values, an optimal lower rank approximation of the discretization matrix is obtained. The computational cost of the introduced method is significantly reduced using the optimal discretization matrix. Also, the proposed method can be extended to the nonuniform waveguides. The technique is verified by analyzing several practical transmission lines and waveguides with nonuniform profiles.

Notes on the improvement of concept-cognitive learning accuracy

The concept-cognitive learning (CCL) process is the specific implementation step of simulating the human brain to learn concepts, and the CCL model is its core carrier. Different CCL models constructed by different cognitive minds will produce different concept learning results. The existing CCL model based on sufficient and necessary granule approximations regards the human's approximation idea in the face of inconsistent information as a logical criterion, and aims at finding the closest concept pair of the clue as concept learning results with a certain learning accuracy. However, the existing CCL method based on sufficient and necessary granule approximations cannot guarantee that the clue must be between its lower approximation and upper approximation, which causes the fact that the learning accuracy may not effectively measure the consistency of concept learning results. What is more, although the computational process obeys logical cognitive condition, the concept learning results may not conform to the actual situation, such as the case of merely generating full concepts and empty concepts. For the first problem, we improve the learning accuracy of the existing CCL method, propose a new CCL method with learning accuracy under hybrid lattice structure, and develop CCL algorithms for the cases of objects and attributes as clues. Moreover, experiments show the effectiveness of the proposed CCL method with learning accuracy under hybrid lattice structure. For the second problem, we put forward a CCL method based on non-logical associative mechanism to handle the unreasonable situation where the concept learning results are full concepts and empty concepts. Finally, two associative CCL algorithms are explored, and experiments are conducted to show their effectiveness.

A dynamic framework for updating approximations with increasing or decreasing objects in multi-granulation rough sets

The data we need to deal with are getting bigger and bigger in recent years, and the same happens to multi-granulation rough sets, so updated schemes have been proposed with the variation of attributes or attribute values in multi-granulation rough sets. This paper puts forward a dynamic mechanism to update the approximations in multi-granulation rough sets when increasing or decreasing objects. Firstly, the relationships between the original approximations and updated approximations are explored when adding or deleting objects and the dynamic processes of updating the lower and upper approximations in optimistic and pessimistic approximations are proposed. Secondly, two corresponding dynamic algorithms and their time complexity are given. Finally, the experimental evaluations show the effectiveness of the proposed dynamic updating algorithms compared with the static algorithm.