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As the oxygen redox ability shows great effects on the catalytic performances of ceria-zirconia based materials, many strategies have been utilized to improve the oxygen storage capacity. Here in this study, we report a simple and facile approach to prepare a SO42–-modified La, Y-doped ceria-zirconia material (SO/CZLY-f) with high oxygen storage capacity. Due to the additional redox process between SO42− and S2−, oxygen storage capacity of SO/CZLY-f (745.3 μmol O2/g) is about 1.6 times higher than that of La, Y-doped ceria-zirconia material without SO42– modification. Moreover, the catalytic activities and stability of the corresponding Pd-only three-way catalyst were measured. Compared to that of Pd@CZLY-f, the operation window of CO, full conversion temperature of HC and NO over Pd@SO/CZLY-f are obviously widened and lowered, respectively. After aging treatment at 1100 °C for 4 h, the superiority of aged Pd-loading composite is still maintained.

Formal concept analysis, rough sets, and three-way decisions are prominent theories and methods for data representation and analysis. They have been applied to data mining, machine learning, artificial intelligence as well as many other areas. This special issue contains thirty high quality state-of-art research that addresses theoretic and applicational aspects of formal concept analysis, rough sets, and three-way decisions.

In double-sided lapping and polishing processes, the friction coefficients between the workpiece and platens or pads are important parameters for understanding or improving the lapping and polishing processes. However, the friction coefficients have not been investigated in detail owing to insufficient measurement methods for the torques acting on the upper and lower platens, which are necessary to evaluate the friction coefficients. Thus, a novel measurement method for the torque acting on the upper platen in a three-way double-sided machine was proposed in this study. In the method, the torque is measured based on the forces acting on the holders supporting the upper platen rotation. The torques were measured with the method in double-sided lapping experiments, and they were confirmed to agree with the theoretical torque variation. Furthermore, the friction coefficient between the workpiece and upper pad identified with the proposed method was also verified by comparing with the friction coefficient measured in the single-sided lapping experiment. In addition, the estimation method of the lower platen torque based on the motor torque was investigated by utilizing the proposed method, and we found the lower platen torque was estimated accurately by taking a sufficient warm-up time or by modeling the torque loss in transmission path. Therefore, accurate measurement of the torques acting on the upper and lower platens, which is an effective tool for understanding or improving the process, was realized.

The work proposes a new method to detect influential news in online communities. Influential news are articles that induce shifts in users’ opinions or, in general, lead to a polarization of opinions or change like-mindedness of users. The method aims at supporting online platform managers and editors in understanding the impact that social content and news can have on the dynamics of opinions. The influential news detection is conduced by using the Three-Way Decisions approach based on Probabilistic Rough Sets to perform a tri-partitioning of online users. The three parts are then mapped onto a structure, namely Hexagons of Opposition, allowing to reason on opinions, related to a given set of news, of specific communities. More in detail, several hexagons of opposition are constructed along the timeline, as recent news are considered, and compared to detect which news contribute to change opinions of the considered communities. Moreover, two indicators have been introduced to measure the impact of the news. The proposed method has been experimented on real data derived from existing datasets and the promising results have been discussed by using a qualitative approach.

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.

A three-way decision (3WD) with group consensus using intuitionistic fuzzy sets (IFSs) involves two pivotal decision steps: achieving a consensus of loss functions and determining the threshold pair in 3WD. We focus on these decision steps and propose a convex combination-based approach to a three-way intuitionistic fuzzy group decision (3WIFGD) considering a group consensus. First, a similarity measure between IFSs is introduced to define a group consensus index (GCI) for an expert group based on loss functions. Then, an automated algorithm is designed with the GCI-based convex combination strategy to improve the group consensus of loss functions. Moreover, we theoretically prove that the GCI in this algorithm is improved and even converges linearly to “1” as the iteration number increases. Second, based on the aggregated collective consensus loss functions, we construct the optimization model pair by extending the existing models and prove its unique solution , leading to the thresholds. Third, a two-decision-steps-based method for 3WIFGD is developed to capture the rules underlying a group consensus. Finally, an illustrative example and its related comparisons are demonstrated to show the validity of our method.

A nanometric Rh overlayer on a Fe − Cr − Al metal foil prepared by pulsed arc-plasma deposition showed simultaneous removal of NO, CO and C3H6 in a simulated three-way catalyst reaction under stoichiometric conditions, whereas the conversions of NO and C3H6 decreased under lean conditions. The decrease of C3H6 conversion despite of excess O2 was caused by the preferential adsorption of O2 on the surface of Rh overlayer, inhibiting C3H6 adsorption and oxidation. Very small amount of Pt deposited on Rh overlayer can act as adsorption sites of C3H6, and the activity of Rh overlayer for C3H6 oxidation under lean conditions was successfully improved.

Decision-theoretic three-way approximation of a fuzzy set F exploits a three-element set { 0 , 0.5 , 1 } in order to approximate F. By relying on an optimum pair of thresholds ( α , β ) , it changes elements’ membership grade μ F ( x ) to 0 , 0.5 and 1 if μ F ( x ) α , α ⩽ μ F ( x ) ⩽ β and μ F ( x ) > β respectively. A general three-element system, { n , m , p } , 0 ⩽ n m p ≤ 1 , has been proposed in the literature. However, the main issue is to determine appropriate n ≠ 0 , m ≠ 0.5 and p ≠ 1 . A recent advancement has determined m ( 0 m 1 ) . However, n = 0 and p = 1 are still imposed by the model. This restriction on the values of n and p lacks general adaptation for different types of membership distribution. In this paper, a novel way of determining appropriate values of n , m , and p is given without the aforesaid restriction. We consider an alternative { n , m , p } formula for computing the optimum pair ( α , β ) in cost-sensitive three-way approximation context. We use synthetic fuzzy sets and some datasets from UCI Machine Learning repository to demonstrate the suitability of the { n , m , p } system in minimizing approximation error and producing well-guided approximation regions.

Three-way catalytic converters are widely used to regulate emissions from gasoline-powered vehicles. Although significant effort over the past 40 years has resulted in the discovery of several metal additives that improve the thermal stability of three-way catalysts (TWCs), their effects on the actual catalytic process have not been studied systematically. The present work examines the roles of the typical basic metal additives La, Ba, and Sr in Pd-based TWC systems, using various spectroscopic and kinetic studies . Metallic Pd 0 species on Sr/Al2O3 and Ba/Al2O3 supports were found to be more electron-rich than those on pristine Al2O3, whereas those on La/Al2O3 were more electron-deficient. Consequently, Pd/La/Al2O3 showed a lessened CO poisoning effect during NO reduction reactions. Evaluations were performed using powdered catalysts as well as monolithic honeycomb catalysts under conditions simulating actual use. Pd/La/Al2O3 was observed to promote the catalytic reduction of NO most efficiently, while Pd/Ba/Al2O3 exhibited the highest activity for the oxidations of CO and C3H6. The present data suggest that the optimal metal additive for a Pd-based TWC will be determined by the specific application. The selection of such metals should take into account not only the stability but also the promotional effect during the exhaust purification process.

As a typical methodology to deal with uncertain issues, the three-way decision (3WD) has been developed rapidly in nearly ten years, both in theories and applications. Three-way classification is one of the important research fields of 3WD, which utilizes the idea of 3WD to solve classification problems. In this paper, we focus on investigating the effectiveness of three-way classification through two evaluation indicators: the classification quality ( Precision , Recall , Accuracy and F 1 ) and the decision cost. The comparisons between two-way classification and three-way classification are concretely analyzed, some mathematical properties , judging conditions and decision criteria of these two classification methods are also discussed in detail. Finally, the experimental results on eight UCI data sets validate the mathematical analysis, which reveal the effectiveness of three-way classification.

Triangular fuzzy numbers (TFNs) can not only provide the range of fuzzy points, but contain the three most representative fuzzy points, which play an essential role in describing fuzzy information. Combining with conflict situations, the agents' caution in the decision-making process can be well depicted by TFNs. However, little effort has been paid to conflict analysis on triangular fuzzy information systems (TFISs). In this paper, we study three-way conflict analysis based on TFISs. First, we propose the concept of TFISs and then establish a TFIS for conflict analysis. Further, relative area ΔS is defined to describe concrete attitudes of agents about issues. Second, since the importance of the same issue may vary for agents, we attach fuzzy weights to issues according to triangular fuzzy symmetric judgment matrices (TFSJMs) of agents. Third, we determine the total attitude of each agent to all issues. Then we illustrate how to compute thresholds α and β using the theory of triangular fuzzy decision-theoretic rough sets. Finally, a tri-partition of agents is obtained based on the thresholds.

In decision making processes, an expert with hesitant attitudes may experience difficulties when evaluating alternatives via a single assessment value. By allowing the membership degree of an element to a set represented by several possible values, hesitant fuzzy sets (HFSs) are usually needed to address this situation. Thus, it is meaningful to put forward a kind of multi-attribute decision making (MADM) method in the hesitant fuzzy (HF) environment. In addition, three-way decision (3WD) is a decision making method by introducing the idea of non-commitments, and it can effectively reduce decision risks. In this paper, the 3WD method-based MADM with HF information is proposed. According to the information table, the membership function of an objective HFS is given. The effectiveness of the proposed method is verified by solving an infectious disease diagnosis problem. Finally, we give a comparative analysis between the proposed method and the existing HF MADM methods. Further, three experimental results show that the proposed method is reasonable and effective.

The notion of fuzzy concept is proposed to deal with object-attribute data with L-values (where L is a truth-value structure). One disadvantage of fuzzy concept is that a fuzzy context contains a considerable number of fuzzy concepts. This makes it very time-consuming to generate a fuzzy concept lattice, and it is very difficult to find important concepts. In addition, the fuzzy concept shows great strictness when applying to crisp sets. To overcome these problems, we propose several new kinds of variable-precision concepts within L-contexts in this paper. First, we present two kinds of variable-precision two-way (VP2W) concepts: α-positive concept and β-negative concept. The family of each kind of VP2W concept forms a complete lattice. Next, considering both the positive and negative parts, we investigate two kinds of variable-precision three-way (VP3W) concepts: ( α , β ) -object-induced three-way concept and ( α , β ) -attribute-induced three-way concept. The family of each kind of VP3W concept forms a complete lattice. Then, we study the relationships between VP2W concepts and VP3W concepts. The results show that VP3W concept lattices can be directly generated by VP2W concept lattices. Finally, the experiments are preformed to verify the effectiveness of our model.

Circulating tumor DNA (ctDNA) refers to a class of acellular nucleic acids carrying genetic features of primary tumor, which can be regarded as a promising noninvasive biomarker for cancer diagnosis. The development of ctDNA assay is an important component of liquid biopsy. In this study, we have fabricated a novel electrochemical strategy for ultrasensitive detection of ctDNA combining the merits of strand displacement amplification and DNA nanostructures. Stable DNA triangular prism is firstly self-assembled and modified on the electrode surface. After target initiated strand displacement polymerization reaction, the generated DNA product helps the formation of three-way junction nanostructure on triangular prism, which localizes electrochemical species. By carefully investigating the electrochemical responses, the limit of detection (LOD) for ctDNA assay as low as 48 amol/L is achieved. This proposed electrochemical biosensor shows great potential for clinical applications.

We presented a low-abundance mutation detection method with lambda exonuclease and DNA three-way junction structure. The assistant strand in the DNA three-way junction structure could regulate the reaction system from the kinetics and thermodynamics aspects. The optimization of the assistant strand helps to improve the selectivity of the mutant-type DNA to the wild-type DNA about 35 times. Moreover, the cost of the optimization process could be saved by about 90%. The method was applied to the detection of a human ovarian cancer-related gene mutation BRCA1 (rs1799949, c.2082C > T). The limit of detection to the mutation abundance in the DNA three-way junction structure system (0.2%) was one order lower compared with that in the double-stranded DNA structure system (2%). The mutation abundance in different standard samples was quantitively measured, and the results were consistent with the initial abundance in the standard samples.

The theory of three-way decision plays an important role in decision making and knowledge reasoning. However, little attention has been paid to the problem of learning from partially labeled data with three-way decision. In this paper, we propose a three-way co-decision model for partially labeled data. More specifically, the problem of attribute reduction for partially labeled data is first investigated, and two semi-supervised attribute reduction algorithms based on novel confidence discernibility matrix are proposed. Then, a three-way co-decision model is introduced to classify unlabeled data into useful, useless, and uncertain data, and the model is iteratively retrained on the carefully selected useful data to improve its performance. Moreover, we theoretically analyze the effectiveness of the proposed model. The experimental results conducted on UCI data sets demonstrate that the proposed model is promising, and even compares favourably with the single supervised classifier trained on all training data with true labels.

Using modern information theory to classify and identify high-risk disease groups is one of the research concerns in medical decision-making. The early diagnosis of gout is missing a single indicator, and relying on artificial labeling of disease characteristics is not only costly for decision-making, but also has a high misdiagnosis rate. Aiming at incomplete and attribute-related random large sample data, we propose a three-way clustering algorithm based on neighborhood rough sets, which is used to initially label the data, reduce the rate of misdiagnosis, and improve decision-making efficiency. Firstly, a neighborhood rough set theory in a heterogeneous information system is established. Secondly, the Best-Worst method-based neighborhood rough set attribute reduction model considering attribute correlation is constructed. Thirdly, a neighborhood rough set-based three-way clustering method for heterogeneous information system is proposed. Finally, we use 2,683 random samples and the proposed model to identify and classify potential gout patients in the samples. The results show that the proposed model can be used to mark and cluster potential gout groups in random samples without prior probability and with fuzzy decision rules, which is helpful for clinical decision-making.

By reformulating and extending the properties of three-way operators, this paper investigates the relationship between different kinds of three-way concept lattices. Three-way operators are defined through eight kinds of two-way operators which are connected by the complement operation. To examine the interrelations systematically, we study (a) the relationship between two-way operators, (b) the relationship between two-way concepts, (c) the relationship between three-way operators, and (d) the relationship between three-way concepts. The results show that the four kinds of object-induced three-way concept lattices are order isomorphic to each other and the four kinds of attribute-induced three-way concept lattices are also order isomorphic to each other.

Decision conflict is a crucial issue of three-way decisions of multi-class. To overcome this limitation, there are mainly two ways. One is conflict resolution after decision-making, and the other is conflict resolution before decision-making. Considering insufficient information is a main cause of decision conflict, we propose a sequential approach to solve this problem by adding more detailed information step by step, which is conflict resolution during decision-making. Combining sequential approach with three methods to handle multi-class decision, which are converting m-class decision to m two-class decisions, based on pair-wise comparisons of m decision classes and directly making three-way decisions for each decision class of m-class simultaneously, we propose three sequential multi-class three-way decision models. At each level of three models, we define conflict region. In the sequential process of decision-making, for each model, only the objects belonging to one positive region with certain decision are assigned into corresponding decision classes, the objects in conflict region will be processed at the next level by adding more information sequentially. Finally, we compare the performance of three conflict resolutions and the performance of the proposed three models. Experimental results validate the effectiveness of the proposed three sequential multi-class three-way decision models.

This paper proposes a sequential three-way decision-making approach based on BWM (best-worst method) and MULTIMOORA for multiple levels of granularity to deal with the multi-attribute group decision-making problems under uncertainty. First, using BWM to preprocess the attribute indicators of the decision problem, the relationship between the criteria important for decision problem is determined. Then, according to the importance of the determined level of granularity, from rough to clear, three-way decisions are made in sequence at each level. In this way, we obtain a lattice based three-way decision-making process. Also, in the process at each granularity, we consider the costs of both the decision process and the decision result. Thus, this paper builds a multigranulation sequential three-way decisions model with cost-sensitive. Next, based on the above-mentioned model, we propose a decision procedure and an algorithm for multi-attribute group decision making. It is worth noting that according to the research goals of the decision problem, MULTIMOORA is used to sort the classified alternatives. And a ranking result of the decision problem is given. Finally, the effectiveness and validity of the proposed model is verified by a multi-attribute group decision problem. The problem is the case of emergency diagnosis and treatment after a highway accident. And put forward reasonable suggestions for further development direction.

Recently, with the trisecting and acting models of human cognitive behaviors, three-way decisions are introduced to deal with uncertain decisions in many applications. To define and conduct the three-way decisions on the possible or necessary loser or co-loser and/or winner or co-winner, we first define the score-based common voting rules, where a score-based function is introduced to evaluate certain utility, including positional scoring rules, maximin, Bucklin, and so on. Then on one hand, it has been proved that under certain conditions three-way decisions on the possible loser or co-loser and/or winner or co-winner are NP-complete with positional scoring rules, maximin, and Bucklin, respectively. On the other hand, we have presented algorithms whose time complexity is polynomial to conduct three-way decisions on the necessary winner or co-winner and/or necessary loser or co-loser with positional scoring rules, maximin, and Bucklin in time O ( n m 2 ) , O ( n m 3 ) , and O ( n m 2 ) , respectively, which has been validated experimentally.

Hierarchical classification (HC) is effective when categories are organized hierarchically. However, the blocking problem makes the effect of hierarchical classification greatly reduced. Blocking means that samples are easily getting misclassified in high-level classifiers so that the samples are blocked at the high-level of the hierarchy. This issue is caused by the inconsistency between the artificially defined hierarchy and the actual hierarchy of the raw data. Another issue is that it is flippant to strictly process data following the hierarchy. Therefore, special treatment is required for some uncertain data. To address the first issue, we learn category relationships and modify the hierarchy. To address the second issue, we introduce three-way decisions (3WD) to targetedly deal with the ambiguous data. We extend original studies and propose two HC models based on 3WD, collectively referred to as TriHC, for carefully modifying the hierarchy to alleviate the blocking problem. The proposed TriHC model learns new category hierarchies by the following three steps: (1) mining category relations; (2) modifying category hierarchies according to the latent category relations; and (3) using 3WD to divide observed objects into three regions: positive region, boundary region, and negative region, and making decisions based on different strategies. Specifically, based on different category relation mining methods, there are two versions of TriHC, cross-level blocking priori knowledge based TriHC (CLPK-TriHC) and expert classifier based TriHC (EC-TriHC). The CLPK-TriHC model defines a cross-level blocking distribution matrix to mine the category relations between the higher and lower levels. To better exploit category hierarchical relations, the EC-TriHC model builds expert classifiers using topic model to learn latent category topics. Experimental results validate that the proposed methods can simultaneously reduce the blocking and improve the classification accuracy.

Each mixing valve is characterized by its kvs-value and the valve characteristics. In mounted condition, the valve characteristics are deformed and the degree of the deformation is determined by the valve authority. Considering already installed mixing valves, neither the valve authority nor the course of the undeformed valve characteristics are exactly known. The lack of information poses a challenge when physically modelling the relationship between the inlet temperatures, the control signal and the outlet temperature. Based on Bezier curves, a universal non-linear grey-box model to describe the thermal valve behaviour is proposed. It is compared with a state space as well as a Hammerstein model and shows the advantages over black-box approaches. The trained model is very robust, able to predict measurement data very accurately and can be used in control design. The parameter estimation is performed using the Open Source platform JModelica.org.

Three-way concept analysis provides a novel model for three-way decisions, and also extends the classical formal concept analysis, which has attracted many attentions in recent years. In order to construct three-way concept lattices, original formal context and its complement context need to be considered simultaneously. In this paper, a new method of constructing attribute-induced three-way (AE) and object-induced three-way (OE) concept lattices is proposed. Firstly, we introduce an AE-oriented composite operator and an OE-oriented composite operator that combine a pair of formal concepts coming from the concept lattices of original formal context and its complement context. Secondly, we define the candidate AE concepts and redundant AE concepts based on the AE-oriented composite operator, and define the candidate OE concepts and redundant OE concepts based on the OE-oriented composite operator, respectively. Finally, we propose an AE lattice construction algorithm and an OE lattice construction algorithm. Theoretical analysis and experimental results demonstrate that the proposed method is simple and effective, and it is a useful supplement to current three-way concept lattice construction methods.

In this paper, we address ambiguity, intended as a characteristic of any data expression for which a unique meaning cannot be associated by the computational agent for either lack of information or multiple interpretations of the same configuration. In particular, we will propose and discuss ways in which a decision-support classifier can accept ambiguous data and make some (informative) value out of them for the decision maker. Towards this goal we propose a set of learning algorithms within what we call the three-way-in and three-way-out approach, that is, respectively, learning from partially labeled data and learning classifiers that can abstain. This approach is based on orthopartitions, as a common representation framework, and on three-way decisions and evidence theory, as tools to enable uncertain and approximate reasoning and inference. For both the above learning settings, we provide experimental results and comparisons with standard Machine Learning techniques, and show the advantages and promising performances of the proposed approaches on a collection of benchmarks, including a real-world medical dataset.

Rough set models on two universes are important generalizations of Pawlak’s classical model. In most of the two-universe models, the upper and lower approximations are constructed based on inclusion relation. We generalize inclusion relation to the general evaluation function and define the models of three-way decision on two universes. As an important class of evaluation functions, subsethood measures are considered. We compare our models with other five existing two-universe models in rough set theory and point out that the model of three-way decision on two-universe unifies the five two-universe models. Besides, properties of the two-universe model of three-way decision are also given. More importantly, we propose an approach to computing the pair of thresholds α and β. Our approach is based on the maximum value of the accuracy measure with respect to tri-partitions of a universe induced by all thresholds.

The trisecting-acting-outcome (TAO) model of three-way decisions comprises three components, i.e., constructing a trisection, strategies for action, and measuring the outcome. Existing research seldom involves the third component, i.e., measuring the effectiveness of the outcome. A previous study provides a strategy select method by analyzing benefits or costs in the movement-based three-way decision. However, we often choose a strategy depend on the probability or distribution of three regions instead of the benefits or costs. In this paper, we propose a probabilistic movement model of three-way decision and give a strategy selection mechanism based on information entropy. Conditional entropy and cross entropy are used to select strategies by measuring these probabilities according to the user's preferences and structural features of the three partitions. It selects the action or strategy which has the maximum probability of desirable movements based on the minimization of the conditional entropy. It also selects the action or strategy which has the least difference between actual probabilistic distributions and expected probabilistic distribution based on the minimization of the cross entropy. We give two examples to illustrate the practicality and effectiveness of the strategy selection method in the probabilistic movement model of three-way decision.

In order to use three-way decision (TWD) to solve multiple attribute decision making (MADM) problems, in this article, a new TWD model with intuitionistic fuzzy numbers (IFNs) is proposed. First of all, we define the relative loss functions to demonstrate some features of loss functions in TWDs, which is the basis for future research. Then, based on the correlation between the loss functions and the IFNs, we get the relative loss functions based on IFNs. At the same time, the classification rules of the TWDs are discussed from different viewpoints, including the thresholds and their properties. Aiming at MADM problems with unreasonable values, a new integrated method of relative loss functions is established to obtain a fairer loss integration result of alternatives. In addition, considering that there are no decision attributes and only condition attributes in MADM, we use grey relational degree to calculate the condition probability. In the end, a novel TWD model is proposed to solve MADM problems with IFNs, and a practical example on selecting suppliers is used to demonstrate its effectiveness and practicability.

The methods of mining incomplete data based on characteristic sets or characteristic relation have been intensively studied in recent years. With the development of related research, many modifications of the definition of characteristic relation have been proposed. However, few of them can be used for decision rule induction due to the so-called definability problem. In this paper, by using the wide sense of the three-way decision theory, we extend the notions of characteristic relation and characteristic set to the systems with four types of characteristic relations and characteristic sets, respectively. Then we study the probabilistic approximations based on the extended characteristic set system. Moreover, we extend the maximal characteristic neighborhood system into four types of maximal characteristic neighborhood systems, and investigate the probabilistic approximations based on them. Finally, we generalize the definition of local definability for incomplete data processing. In particular, several existing methods based on characteristic sets are integrated into our model, and the new types of characteristic relations proposed by us seem more practical than the classical one in some aspects. In this way, our model illustrates the effectiveness and comprehensiveness of thinking in threes.

Conflict analysis, which plays an important role in our society, has attracted more and more attentions. However, the existing conflict analysis models are not effective enough to evaluate the inconsistency degree of cliques that have more than two individuals. Besides, for each clique, its allied, conflict and neutral attributes have not been explicitly defined and thoroughly explored. Therefore, due to the lack of enough explicit clues, we cannot further formulate some strategies manipulating the conflict degrees of related cliques to realize the specific objectives. Last but not least, one-vote veto is seldom considered, although it plays a vital role in many fields, such as venture capital and United Nations Security Council resolutions. In order to solve these problems, we resort to three-way concept analysis and describe a clique by using the intent of a specific concept. On the basis of the obtained specific concepts, we derive the allied, conflict and neutral attributes of cliques, and further make decisions and explore the maximal coalitions and minimum conflict sets. Finally, in order to cater dynamic environment, we also describe an incremental algorithm for conflict analysis.

Three-way concept lattices can supply much more information than classical concept lattices since they contain the positive information and negative information between objects and attributes simultaneously. Taking advantage of this, the rules acquisition for formal decision contexts from the perspective of three-way concept lattices are discussed, and the results are compared with the common decision rules based on classical concept lattices. Firstly, the definition of object-induced three-way consistence of a formal decision context is presented. Then, positive decision rules and negative decision rules are proposed for an object-induced three-way consistent formal decision context. Furthermore, we give semantic explanation for these rules, and compare them with common rules obtained from a strongly consistent lattice . In parallel, the similar issues are investigated from the perspective of attributes, including the definition of attribute-induced three-way consistence of a formal decision context, rules acquisition based on an attribute-induced three-way consistent formal decision context, and rules comparison. Finally, the relationships among the three types of consistence, namely, object-induced three-way consistence, attribute-induced three-way consistence, and existing strongly consistence are discussed to make the contents compact. Some theoretical examples are given to illustrate the main results of this paper.

The conventional clustering approaches are not very effective in dealing with clusters having overlapping regions. The three-way clustering (3WC) is an effective and promising approach in this regards. A key issue in 3WC is the determination of thresholds which plays a crucial and important role in accurate estimation of the overlapping region. In this article, we propose different variance based criteria for determining the thresholds. In particular, we examine the variance or spread in evaluation function values of objects contained in the three regions obtained with 3WC of objects. An algorithm called 3WC-OR is introduced that considers the optimization of the proposed criteria for determining effective thresholds by incorporating approaches such as genetic algorithms and game-theoretic rough sets. Experimental results on five UCI datasets indicate that the proposed algorithm significantly improves results on datasets with overlapping clusters and provide comparable results on datasets with non-overlapping clusters.

In this paper, we study convex systems under the frame of three-way decision theory. Firstly, we give the notion of three-way convex systems. We obtain an equivalence characterization of three-way convex systems. Then, we propose the concept of three-way fuzzy convex systems in term of L-convex systems. Some related examples are presented. Finally, we establish a one-to-one correspondence between three-way convex systems and three-way fuzzy convex systems.

Sequential three-way decision (S3WD) is a multi-stage decision-making progress that emulates human cognition. S3WD employs coarse-to-fine information granularity and uses various types of costs to determine thresholds to make appropriate decisions. To date, however, few existing models discuss the construction of granules or consider costs in the decision process. This paper describes efforts to develop a granularity-driven sequential three-way decision model to address these two issues, and handle dual-constraint satisfaction problems considering both decision process and decision result costs. A key feature of the model is the incorporation of information granularity into the decision-making process. The model also incorporates a cost structure that accounts for the costs of both the decision process and the decision result. Using this model, we design two algorithms to minimize the cost of the decision process or the cost of the decision result. Our experimental results validate the effectiveness of the algorithms and the viability of the new model.

Credit scoring is a crucial task within risk management for any company in the financial sector. On the one hand, it is in the self-interest of banks to avoid approving credits to customers who probably default. On the other hand, regulators require strict risk management systems from banks to protect their customers and, from “too big to fail institutions”, to avoid bankruptcy with negative impacts on an economy as a whole. However, credit scoring is also expensive and time-consuming. So, any possible method, like three-way decisions, to further increase its efficiency, is worth a try. We propose a two-step approach based on three-way decisions. Customers whose credit applications can be approved or rejected right away are decided in a first step. For the remaining credit applications, additional information is gathered in a second step. Hence, these decisions are more expensive than the ones in the first step. In our paper, we present a methodology to apply three-way decisions with probabilistic rough sets for credit scoring and an extensive case study with more than 7000 credit applications from Chilean micro-enterprises.

In the study, non-deterministic information systems-Apriori-based (NIS-Apriori-based) rule generation from table data sets with incomplete information, SQL implementation, and the unique characteristics of the new framework are presented. Additionally, a few unsolved new research topics are proposed based on the framework. We follow the framework of NISs and propose certain rules and possible rules based on possible world semantics. Although each rule τ depends on a large number of possible tables, we prove that each rule τ is determined by examining only two τ-dependent possible tables. The NIS-Apriori algorithm is an adjusted Apriori algorithm that can handle such tables. Furthermore, it is logically sound and complete with regard to the rules. Subsequently, the implementation of the NIS-Apriori algorithm in SQL is described and a few new topics induced by effects of NIS-Apriori-based rule generation are confirmed. One of the topics that are considered is the possibility of estimating missing values via the obtained certain rules. The proposed methodology and the environment yielded by NIS-Apriori-based rule generation in SQL are useful for table data analysis with three-way decisions.

A model of decision-theoretic three-way approximations of fuzzy sets that can calculate a pair of interpretable thresholds (α, β) by combining shadowed sets with three-way decisions was established by Deng. In that model, the significant contribution is that the value 0.5, which denotes a membership grade with the highest uncertainty, is used to replace the uncertain region with the interval [0,1] in shadowed sets. From the principle of the minimum decision cost, although the lowest overall cost can be achieved precisely in some data distributions based on the value 0.5, there is a difference between the obtained overall cost and the least cost in some cases. Therefore, in this paper, based on Deng’s model, the concept of a general three-way approximation of a fuzzy set is proposed to replace 0.5 with a variable value c(0

Neighborhood Covering (NC) is the union of homogeneous neighborhoods and provides a set-level approximation of data distribution. Because of the nonparametric property and the robustness to complex data, neighborhood covering has been widely used for data classification . Most existing methods directly classify data samples according to the nearest neighborhoods. However, the certain classification methods strictly classify the uncertain data and may lead to serious classification mistakes. To tackle this problem, we extend traditional neighborhood coverings to fuzzy ones and thereby propose a Three-Way Classification method with Fuzzy Neighborhood Covering (3WC-FNC). Fuzzy neighborhood covering consists of membership functions and forms an approximate distribution of neighborhood belongingness. Based on the soft partition induced by the memberships of fuzzy neighborhood coverings of different classes, data samples are classified into Positive (certainly belonging to a class), Negative (certainly beyond classes) and Uncertain cases. Experiments verify that the proposed three-way classification method is effective to handle the uncertain data and in the meantime reduce the classification risk.

Three-way granular computing is a paradigm of thinking and information processing in three granules. We introduce a framework of three-way granular computing based on a hexagon induced by a trisection of a set. This enables us to unify rough-set concept analysis and formal concept analysis. For an object granule (i.e., a subset of a universal set of objects), we construct three attribute granules. Conversely, for an attribute granule (i.e., a subset of a universal set of attributes), we construct three object granules. Rough-set concept analysis uses one kind of trisections to study four types of disjunctive formal concepts and the associated concept lattices. Formal concept analysis uses another kind of trisections to study four types of conjunctive formal concepts and the associated concept lattices. The eight concept lattices can be grouped into two classes such that four lattices in each class are isomorphic.

In recent years, multi-view clustering algorithms have shown promising performance by combining multiple sources or views of datasets. A problem that has not been addressed satisfactorily is the uncertain relationship between an object and a cluster. Thus, this paper investigates an active three-way clustering method via low-rank matrices that can improve clustering accuracy as clustering proceeds for the multi-view data of high dimensionality. We adopt a three-way clustering representation to reflect the three types of relationships between an object and a cluster, namely, belong-to definitely, uncertain and not belong-to definitely. We construct the consensus low-rank matrix from each weighted low-rank matrix by taking account of the diversity of views, and give the method to solve the optimization problem of objective function based on the improved augmented Lagrangian multiplier algorithm. We suggest an active learning strategy to learn important informative pairwise constraints after measuring the uncertainty of an object based on the entropy concept. The experimental results conducted on real-world datasets have validated the effectiveness of the proposed method.

Autoencoder network is an efficient representation learning method. In general, a finer feature set obtained from autoencoder leads to a lower error rate and lower total misclassification cost. However, the network is usually trained for a long time to obtain a finer feature set, leading to a high time cost and total cost. To address this issue, a Sequential Three-Way Decision (S3WD) model is developed to balance the misclassification cost and the time cost in autoencoder based classifications and decisions. To implement the tradeoff strategy, it is necessary to extract a multi-granular feature set. In the network, the associated discriminative information in the extracted features increases with training epochs, which constructs a multi-granular feature structure. An autoencoder-based multi-granular feature description definition is presented. Based on the definition, an autoencoder composed of restricted Boltzmann machines is adopted to extract the multi-granular features. Then, a new cost-sensitive S3WD model is proposed, which aims to find the optimal granule level with the lowest total cost. Finally, the experiments demonstrate the effectiveness of the proposed approaches.

Three-way decisions provide a trisecting-and-acting framework for complex problem solving. For a cost-sensitive decision-making problem under multiple levels of granularity, sequential three-way decisions have come into being. Within this framework, how to act upon the three pair-wise disjoint regions is the most important issue. To this end, we propose a generalized model of sequential three-way decisions via multi-granularity in this paper. Subsequently, we adopt the typical aggregation strategies to implement the following five kinds of multigranulation sequential three-way decisions—the weighted arithmetic mean multigranulation sequential three-way decisions, the optimistic multigranulation sequential three-way decisions, the pessimistic multigranulation sequential three-way decisions, the pessimistic-optimistic multigranulation sequential three-way decisions and the optimistic-pessimistic multigranulation sequential three-way decisions. Furthermore, we discuss the rightness and rationality of the five kinds of multigranulation sequential three-way decisions and also analyze the relationships and differences between them. Finally, the experimental results demonstrate that the first four different multigranulation sequential three-way decisions are effective. These models will accelerate and enrich the development of multigranulation three-way decisions.

Three-way decisions (3WD) is an important methodology in solving problems with uncertainty. A systematic analysis on three-way based uncertainty measures is conducive to the promotion of three-way decisions. Meanwhile, confusion matrix, with multifaceted views, serves as a fundamental role in evaluating classification performance. In this paper, confusion matrix is endowed with semantics of three-way decisions. A collection of measures are thus deduced and summarized into seven measure modes. We further investigate the formulation of three-way regions from a measure driven view. To satisfy the preferences of stakeholder, two different objective functions are formulated, and each of them can include different combinations of measures. To demonstrate the effectiveness, we generate probabilistic three-way decisions for a wealth of datasets. Compared with Gini coefficient based and Shannon entropy based objective functions, our model can deduce more satisfying three-way regions.

Cluster ensemble has emerged as a powerful technique for combining multiple clustering results. To address the problem of clustering on large-scale data, this paper presents an efficient three-way cluster ensemble approach based on Spark, which has the ability to deal with both hard clustering and soft clustering. First, this paper proposes the framework of three-way cluster ensemble based on Spark inspired by the theory of three-way decisions, and develops a distributed three-way k-means clustering algorithm. Then, we introduce the concept of cluster unit, which reflects the minimal granularity distribution structure agreed by all the ensemble members. We also introduce quantitative measures for calculating the relationships between units and between clusters. Finally, we propose a consensus clustering algorithm based on cluster units, and we devise various three-way decision strategies to assign small cluster units and no-unit objects. The experimental results using 19 real-world data sets validate the effectiveness of the proposed approach from different indices such as ARI, ACC, NMI and F1-Measure. The experimental results show that the proposed approach can effectively deal with large-scale data, and the proposed consensus clustering algorithm has a lower time cost and does not sacrifice the clustering quality.

This note gives a specific model of Yao's three elements of granular structures: granules, levels, and hierarchies [Three-way decision and granular computing, Int. J. Approx. Reas. 103 (2018) 107–123]. Besides, we point out the double-granular structure of three-way fuzzy matroids.

Irony detection in social networks has drawn much attention in recent years. In this work, we propose a three-way decisions based feature fusion method for Chinese irony detection in microblog. First, we build a set of discriminating features for Chinese irony detection by considering the characteristics of both Chinese language and social networks, including lexical feature, homophonic pun, consecutive punctuation, length of microblog, passive verbs, and affective imbalance. Second, to tackle the problem of extreme imbalance between different kinds of features, we propose a two-stage classification method for Chinese irony detection by considering feature fusion. In the first stage, a three-way decisions based classifier is trained to predict the ironic tendency based on the lexical feature only. This procedure can reduce the thousands-of-dimensions based lexical feature into a one-dimensional feature. In the second stage, the constructed new feature is merged with other five features into a new feature vector for learning. Experimental results show that our proposed two-stage classification method can get a significant improvement than one-stage classification method.

Three-way approximations of fuzzy sets aim at abstracting fuzzy sets into three pair-wise disjoint categories which facilitate semantic-oriented interpretations and reduce computing burden. Shadowed sets are a schema of three-way approximations of fuzzy sets which are formed based on a specific optimization mechanism. Among different principles guiding the construction of shadowed sets, the criterion of minimum distance offers a new insight within the framework of three-way decision theory. In this paper, the essential mathematical properties of the objective function used as a criterion to construct three-way approximations of fuzzy sets based on the principle of minimal distance, as well the characteristics of the optimal solutions, are analyzed. It is demonstrated that this optimization objective function is continuous but nonconvex with respect to the optimized variables. The nonconvex property makes the solution difficult and different approximate region partitions are obtainable even under the same optimization model. Therefore, further criteria are required to select final partition thresholds and make the construction process well-defined. To address this limitation, the notion of constrained three-way approximations of fuzzy sets is proposed from the perspective of minimal distance. Moreover, a constructive algorithm is provided to obtain the proposed constrained three-way approximations rather than using a direct enumeration method, and its performance is illustrated by considering some typical fuzzy sets along with some data from UCI repository.

This paper introduces a model of three-way conflict analysis by reformulating and extending the original model by Pawlak. The proposed model is based on the philosophy, methodology, and mechanism of thinking and processing in threes of a theory of three-way decision. We examine a trisection of agents and a trisection of pairs of agents, as well as their relationships. We adopt a hexagon to represent geometrically a trisection, which gives an intuitive and effective tool for studying three-way decision. According to the two trisections, we suggest three levels of conflict consisting of strong conflict, weak conflict, and non-conflict. The results are described geometrically by a triangle of conflict. The new model of three-way conflict analysis (a) clarifies important conceptual and semantics interpretations of conflict, (b) removes some inconsistencies in the Pawlak model, and (c) gives a more general conception of conflict analysis with three levels of conflict by using many-valued ratings.