Shadowed Sets Research Articles

Shadowed-set-based three-way clustering methods: An investigation of new optimization-based principles

Shadowed set approximation form a cornerstone for the explainable decision advice provided by shadowed C means (SCM) clustering in unsupervised learning. Due to its advantage of managing uncertainty in fuzzy clustering, it has been used in data classification. Existing SCM clustering method requires the overall amount of uncertainty associated with a fuzzy cluster ck to be preserved in its boundary region. This requirement may suffer serious risk of having high number of unclassified patterns, especially when the uncertainty in ck is very high. Consequently, the ensuing clustering may not adequately maximize the inter-cluster separation necessary for achieving optimum cluster validity results. To tackle this problem, this paper considers new SCM clustering methods arising from (i) trade-off between uncertain and certain regions, which is necessary for refraining from making uncertain classification as much as possible, (ii) measure of sharpness balance, which helps to leverage on the location of a pattern from borderline and identify included or excluded patterns by means of their location from borderline, (iii) measure of gradualness balance, which exploits the degree of transition of a pattern out of or into ck. Each method comes with some advantages. For instance, the first and third methods may minimize the overall amount of unclassified patterns. To provide an overall evaluation of the performance of the proposed methods, a comparative study with some other shadowed set-based optimization methods are involved by considering some data sets from UCI Machine Learning repository. Friedman testing followed by Holm-Bonferroni testing are also exploited to provide statistical analysis on the performance significance of the compared methods.

The Shadowed Set, Intuitionistic Fuzzy Set and their Three-way Decision

The theory of three-way decision provides an effective tool for decision-making under uncertainty and incomplete information when a two-way decision is difficult to make. In this paper, we try to transform an intuitionistic fuzzy set into a pessimistic shadowed set or an optimistic shadowed set by using two intuitionistic fuzzy parameters and the classical shadowed set. Accordingly, the pessimistic three-way decision or the optimistic three-way decision of intuitionistic fuzzy set are investigated by means of the proposed shadowed sets. In addition, a method to calculate the thresholds by the minimum cost principle is proposed and some examples are given for illustrating the validity of the method.

Granular rough sets and granular shadowed sets: Three-way approximations in Pawlak approximation spaces

A Pawlak approximation space is a pair of a ground set/space and a quotient set/space of the ground set induced by an equivalence relation on the ground set. The quotient space is a simple granulation of the ground space such that an equivalence class is a granule of objects in the ground space and, at the same time, a single granular object in the quotient space. The new two-space view leads to more insights into and a deeper understanding of rough set theory. In this paper, we revisit results from rough sets from the two-space perspective and introduce the notions of granular rough sets and probabilistic granular rough sets in the quotient space, as three-way approximations of sets in the ground space. We propose a concept of granular shadowed sets in the quotient space, as three-way approximations of fuzzy sets in the ground space. We formulate a cost-sensitive method to construct a granular shadowed set from a fuzzy set. We show that, when the costs satisfy some conditions, the three granular approximations become the same for the special case where a fuzzy set is in fact a set.

A three-way decision based construction of shadowed sets from Atanassov intuitionistic fuzzy sets

By combining ideas from three-way decision and Pedrycz shadowed sets, this paper investigates two possible solutions to the problem of constructing a shadowed set from an Atanassov intuitionistic fuzzy set. The first solution treats the pair of membership and nonmembership functions as two evaluations and directly applies the two-evaluation based model of three-way decision. The second solution transforms the pair of membership and nonmembership functions into an evaluation and applies a single-evaluation based model of three-way decision. To address a fundamental issue of the determination of a pair of thresholds on evaluations, we present a general optimization based formulation and use Chebyshev distance to calculate the required thresholds. The results are useful for three-way decision with fuzzy sets, Atanassov intuitionistic fuzzy sets, and shadowed sets.

Anti-unwinding sliding mode attitude control via two modified Rodrigues parameter sets for spacecraft

In this paper, the attitude maneuver control without unwinding for rigid spacecraft is addressed based on the modified Rodrigues parameters (MRPs). First of all, an attitude control system model for rigid spacecraft is constructed by designing a selection logic for the MRP set and its shadow set. To guarantee the asymptotic stability in the large and unwinding-free performance of the closed-loop system on the sliding phase, a sliding function is presented. Then, a sliding mode control law is developed to ensure that the system states are driven to the sliding surface. Moreover, a dynamic parameter is designed in the proposed control scheme, such that the unwinding phenomenon is avoided when the system states are outside the sliding surface. Further, a boundary layer is introduced for the constructed controller to avoid the chattering phenomenon. It is also proven that the convergence property and anti-unwinding performance are ensured when the system states are within the boundary layer. Numerical simulations are performed to demonstrate the unwinding-free performance of the closed-loop system under the proposed control scheme.

Fuzzy-Entropy-Based Game Theoretic Shadowed Sets: A Novel Game Perspective From Uncertainty

As a three-way approximation of fuzzy sets, shadowed sets have attracted extensive attention in recent years. A fundamental issue in the process of constructing shadowed sets is the interpretation and determination of the threshold pair <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({\alpha, \beta } )$</tex-math></inline-formula> , and the uncertainty consistency, that is, the consistency of fuzzy entropy. However, there may be a large fuzzy entropy loss between a fuzzy set and its corresponding game theoretic shadowed sets (GTSS), and the GTSS model is also accompanied by a large time cost when the precision of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({\alpha, \beta } )$</tex-math></inline-formula> is improved. Therefore, the fuzzy-entropy-based GTSS (FeGTSS) is proposed in this article from the perspective of fuzzy entropy loss. First, based on the compromise principle of game theory, the fuzzy entropy loss of shadowed sets is analyzed in this article. Second, in the process of calculating <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$({\alpha, \beta } )$</tex-math></inline-formula> , the optimal game strategy is searched based on the dichotomy algorithm. Third, the FeGTSS model is extended and discussed based on the analysis of different data distribution types. Finally, the rationality and validity of the FeGTSS model are illustrated through instances and experimental analysis.

A Self-Adaptive Weighted Fuzzy c-Means for Mixed-Type Data

The influence of features on each cluster is not the same in a mixed-type dataset. Based on the rough set and shadow set theories, the fuzzy distribution centroid was defined to represent the clustering center of the discrete feature so that the fuzzy c-means algorithm (FCM) could be extended to cluster the data with both continuous and discrete features. Then, considering the different contributions of the features to each cluster, a new weighted objective function was constructed in accordance with the principles of fuzzy compactness and separation. Because the learning feature weight is the key step in feature-weighted FCM, this paper regarded the feature weight as a variable optimized in the clustering process and put forward a self-adaptive mixed-type weighted FCM. The experimental results showed that the algorithm could be effectively applied to a heterogeneous mixed-type dataset.

(t, k, n) XOR-based visual cryptography scheme with essential shadows

Visual cryptography scheme (VCS) shares a binary secret image into multiple shadows, only qualified set of shadows can reveal the secret image by stacking operation. However, VCS suffers the problems of low visual quality of the revealed image and large shadow size. A (t, k, n) XOR-based visual cryptography scheme (XVCS) shares the secret image into n shadows including t essentials and n-t non-essentials. A qualified set of shadows contains any k shadows including t essentials. The revealing process is implemented by XOR operation on the involved shadows. In this paper, we propose a construction method for (t, k, n)-XVCS with essential shadows. The secret image can be revealed perfectly, and the shadow size is small compared with VCS. Theoretical analysis and experimental results show the security and effectiveness of the proposed scheme.

An Extended VIKOR Method Based on q-Rung Orthopair Shadowed Set and Its Application to Multi-Attribute Decision Making

In the multi-attribute decision making (MADM) process, the attribute values are sometimes provided by experts or the public in the form of words. To model the linguistic evaluation more accurately, this paper proposes the q-rung orthopair shadowed set (q-ROSS) to represent attribute values and extends the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method to solve MADM problems in the q-ROSS context. First, we propose the q-ROSS to express evaluation information. Some basic operation rules and distance measures are investigated accordingly. When the amount of data is large, the left and right endpoints of the collected interval numbers will obey symmetric normal distribution. Secondly, based on the normal distribution assumption, the collected data intervals are mapped to shadowed sets through a data processing approach. Furthermore, we extend the VIKOR model to tackle the MADM problem where the evaluation values are expressed by q-rung orthopair shadowed numbers. A location selection problem verifies the practicability of our method, and the effectiveness and superiority of the presented approach are reflected through comparative analysis.

Entropy‐based shadowed set approximation of intuitionistic fuzzy sets

Entropy‐based shadowed set approximation of intuitionistic fuzzy sets

Artificial immune network clustering based on a cultural algorithm

Data mining technology has been applied in many fields. Prototype-based cluster analysis is an important data mining method, but its ability to discover knowledge is limited because of the need to know the number of target data categories and cluster prototypes in advance. Artificial immune evolutionary network clustering is a clustering method based on network structure. Compared with prototype-based cluster analysis, it has the advantage of realizing unsupervised learning and clustering without any prior knowledge of data. However, artificial immune evolutionary network clustering also has problems such as a lack of guidance in the clustering process, fuzzy boundary sensitivity, and difficulty in determining parameters. To solve these problems, an artificial immune network clustering algorithm based on a cultural algorithm is proposed. First, three kinds of knowledge are constructed: normative knowledge is used to regulate the spatial range of population initialization to avoid blindness; state knowledge is used to distinguish the type of antigen, and immune defense measures are taken to prevent the network structure caused by noise and boundaries from being unclear; and topology knowledge is used to guide the antigen for optimal antibody search. Second, topology knowledge in the cultural algorithm is used to characterize the distribution of antigens and antibodies in space, and elite learning is used to improve the traditional clone mutation operator. Based on the shadow set theory, a method for adaptively determining the compression threshold is proposed. Finally, the results of simulation experiments show that the proposed algorithm can effectively overcome the above problems, and the clustering performances on a synthetic dataset and an actual dataset are satisfactory.

A shadowed set-based TODIM method and its application to large-scale group decision making

As an effective tool to describe the fuzziness of qualitative information, the shadowed sets have received increasing attention. By establishing reasonable compromise between qualitative information and quantitative membership grades, the shadowed sets provide a new way to model linguistic variables. In this paper, we propose an extended TODIM method based on shadowed sets to solve large-scale group decision making problems with linguistic information. First, considering the linguistic terms cannot be directly computed, a codebook used to model linguistic terms is constructed with shadowed sets by the data-driven and the percentile method. Next, to improve the decision efficiency, a shadowed set-based clustering model is proposed to cluster the decision makers on the basis of new similarity measure and the Louvain algorithm. Then, we propose a new decision-making method by taking into account the decision makers’ psychological behavior which has a vital influence on the decision results. Finally, we utilize a case study about assembly factory site selection to illustrate the feasibility of the proposed method, meanwhile comparison, parameter discussion and sensitivity analysis are also conducted to verify the superiority and the efficiency.

A Maze Matrix-Based Secret Image Sharing Scheme with Cheater Detection.

Secret image sharing is a technique for sharing a secret message in such a fashion that stego image shadows are generated and distributed to individual participants. Without the complete set of shadows shared among all participants, the secret could not be deciphered. This technique may serve as a crucial means for protecting private data in massive Internet of things applications. This can be realized by distributing the stego image shadows to different devices on the Internet so that only the ones who are authorized to access these devices can extract the secret message. In this paper, we proposed a secret image sharing scheme based on a novel maze matrix. A pair of image shadows were produced by hiding secret data into two distinct cover images under the guidance of the maze matrix. A two-layered cheat detection mechanism was devised based on the special characteristics of the proposed maze matrix. In addition to the conventional joint cheating detection, the proposed scheme was able to identify the tampered shadow presented by a cheater without the information from other shadows. Furthermore, in order to improve time efficiency, we derived a pair of Lagrange polynomials to compute the exact pixel values of the shadow images instead of resorting to time-consuming and computationally expensive conventional searching strategies. Experimental results demonstrated the effectiveness and efficiency of the proposed secret sharing scheme and cheat detection mechanism.

A Parametric Model for the Shapes of Black Hole Shadows in Non-Kerr Spacetimes

Abstract The Event Horizon Telescope (EHT) is taking the first images of black holes resolved at horizon scales to measure their shadows and probe accretion physics. A promising avenue for testing the hypothesis that astrophysical black holes are described by the Kerr solution to Einstein’s equations is to compare the size and shape of the shadow a black hole casts on the surrounding emission to the predictions of the Kerr metric. We develop here an efficient parametric framework to perform this test. We carry out ray-tracing simulations for several parameterized non-Kerr metrics to create a large data set of non-Kerr shadows that probe the allowed parameter space for the free parameters of each metric. We then perform principal components analysis (PCA) on this set of shadows and show that only a small number of components are needed to accurately reconstruct all shadows within the set. We further show that the amplitude of the PCA components are smoothly related to the free parameters in the metrics and, therefore, that these PCA components can be fit to EHT observations in order to place constraints on the free parameters of these metrics that will help quantify any potential deviations from the Kerr solution.

Shadowed sets with higher approximation regions

This paper mainly discusses three points involving shadowed set approximation of a given fuzzy set. Firstly, a principle of uncertainty balance, which guarantees that preservation of uncertainty in the induced shadowed set is studied. Secondly, an alternative formulation for determining the optimum partition thresholds of shadowed sets is suggested. This formulation helps us study principle of uncertainty balance in shadowed sets with higher approximation regions. Thirdly, five-region shadowed set, which effectively deals with the issue of uncertainty balance, is introduced. We provide a closed-form formula for determining its optimum partition thresholds and generalize it to $$n (\ge 5)$$ -region shadowed sets. Finally, some examples from synthetic and real dataset are provided to demonstrate the feasibility of the suggested methods.

Shadowed Neighborhoods Based on Fuzzy Rough Transformation for Three-Way Classification

Neighborhoods form a set-level approximation of data distribution for learning tasks. Due to the advantages of data generalization and nonparametric property, neighborhood models have been widely used for data classification. However, the existing neighborhood-based classification methods rigidly assign a certain class label to each data instance and lack the strategies to handle the uncertain instances. The far-fetched certain classification of uncertain instances may suffer serious risks. To tackle this problem, in this article, we propose a novel shadowed set to construct shadowed neighborhoods for uncertain data classification. For the fuzzy–rough transformation in the proposed shadowed set, a step function is utilized to map fuzzy neighborhood memberships to the set of triple typical values $\lbrace 0, 1, 0.5\rbrace$ and thereby partition a neighborhood into certain regions and uncertain boundary (neighborhood shadow). The threshold parameter in the step function for constructing shadowed neighborhoods is optimized through minimizing the membership loss in the mapping of shadowed sets. Based on the constructed shadowed neighborhoods, we implement a three-way classification algorithm to distinguish data instances into certain classes and uncertain case. Experiments validate the proposed three-way classification method with shadowed neighborhoods is effective in handling uncertain data and reducing the classification risk.

Estymacja kosztów eksploatacyjnych rurociągu dla U-kształtnej funkcji intensywności uszkodzeń przy nieprecyzyjnym podejściu

In this paper, we discuss imprecise settings for an evaluation of the maintenance costs of a water distribution system (WDS). Moments of failures of pipes are modelled using a newly proposed three-piece convex hazard rate function (HRF) for which number of previous failures is taken into account, too. Both fuzzy sets and shadowed sets are used to model the impreciseness of important parameters of this HRF and the costs of maintenance services. Contrary to more classical and widely-used approaches to cost analysis (i.e. a constant yield or nominal value of money), a strictly stochastic process (i.e. the one-factor Vasicek model) of an interest rate is assumed in the analysis of maintenance costs. This approach models future behaviour of the interest rate (i.e. the future value of money) in a more realistic way. Respective algorithms together with exemplary results of numerical simulations for two setups, which are related to fuzzy and shadowed sets, are also provided.

Three-way decisions of rough vague sets from the perspective of fuzziness

Vague set, as well as intuitionistic fuzzy set, is an extended model of fuzzy sets. On the basis of fuzzy sets, vague sets describe the membership degree of a vague concept by using an interval value instead of a single value. To a certain degree, vague sets have a more powerful ability to process fuzzy information than fuzzy sets. Thus, when characterizing a target concept by vague sets, identifying methods to make scientific and reasonable decisions has become an essential issue. However, existing decision methods always focus on the decisions based on fuzzy concepts, and research on how to make three-way decisions based on vague concepts is still lacking. Therefore, in this paper, the concept of rough vague sets is proposed to construct a rough approximation framework of vague concepts. Then, the fuzziness of the existing approximation approaches is analyzed. Next, improved step-vague set model which is a better approximation approach than existing approaches and the algorithm used to search for a improved step-vague set are proposed. Furthermore, based on the improved step-vague sets, probabilistic rough vague sets and a three-way approximation model with shadowed sets are introduced. Finally, several illustrative examples and relative experiment are listed to verify the effectiveness and significance of the proposed models.

Mean-entropy-based shadowed sets: A novel three-way approximation of fuzzy sets

Shadowed set provide a three-way approximation scheme for transforming a fuzzy set into three disjoint areas (elevated, reduced, and shadow areas). A fundamental issue in the construction of shadowed sets is the interpretation and determination of a pair of thresholds (α,β). Several extended shadowed set models have been proposed to calculate (α,β). However, the construction of a few of these models may have a large fuzzy entropy loss, and the determination of (α,β) involves artificial subjective parameters. Therefore, in this study, a novel shadowed set model is proposed, namely, mean-entropy-based shadowed sets (MESS). At first, based on the principle of uncertainty invariance, a novel framework of three-way approximations of fuzzy sets is proposed based on the mean of fuzzy entropy. Secondly, new decision rules are generated based on the fuzzy entropy loss, and (α,β) is obtained. Thirdly, the MESS model is optimized more reasonably using an iterative method, and thus the fuzzy entropy loss of the MESS model can be minimized. Finally, the validity and rationality of the proposed model are verified by instances and experimental analysis.

Set-theoretic models of three-way decision

The theory of three-way decision is about a philosophy of thinking in threes, a methodology of working with threes, and a mechanism of processing in threes. We approach a whole through three parts, in terms of three units, or from three perspectives. A trisecting–acting–outcome (TAO) model of three-way decision involves trisecting a whole into three parts and acting on the three parts, in order to produce an optimal outcome. In this paper, we further explore the TAO model in a set-theoretic setting and make three new contributions. The first contribution is an examination of three-way decision with nonstandard sets for representing concepts under the two kinds of objective/ontic and subjective/epistemic uncertainty. The second contribution is an introduction of an evaluation-based framework of three-way decision. We present a classification of trisections and investigate the notion of an evaluation space. The third contribution is, within the proposed framework, a systematical study of three-way decision with rough sets, interval sets, fuzzy sets, shadowed sets, rough fuzzy sets, interval fuzzy sets (or equivalently, vague sets, interval-valued fuzzy sets, intuitionistic fuzzy sets), and soft sets.