Probabilistic Rough Set Research Articles

Breast cancer pre-diagnosis based on incomplete picture fuzzy multi-granularity three-way decisions

PurposeThree-way decision (3WD) and probabilistic rough sets (PRSs) are theoretical tools capable of simulating humans' multi-level and multi-perspective thinking modes in the field of decision-making. They are proposed to assist decision-makers in better managing incomplete or imprecise information under conditions of uncertainty or fuzziness. However, it is easy to cause decision losses and the personal thresholds of decision-makers cannot be taken into account. To solve this problem, this paper combines picture fuzzy (PF) multi-granularity (MG) with 3WD and establishes the notion of PF MG 3WD.Design/methodology/approachAn effective incomplete model based on PF MG 3WD is designed in this paper. First, the form of PF MG incomplete information systems (IISs) is established to reasonably record the uncertain information. On this basis, the PF conditional probability is established by using PF similarity relations, and the concept of adjustable PF MG PRSs is proposed by using the PF conditional probability to fuse data. Then, a comprehensive PF multi-attribute group decision-making (MAGDM) scheme is formed by the adjustable PF MG PRSs and the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method. Finally, an actual breast cancer data set is used to reveal the validity of the constructed method.FindingsThe experimental results confirm the effectiveness of PF MG 3WD in predicting breast cancer. Compared with existing models, PF MG 3WD has better robustness and generalization performance. This is mainly due to the incomplete PF MG 3WD proposed in this paper, which effectively reduces the influence of unreasonable outliers and threshold settings.Originality/valueThe model employs the VIKOR method for optimal granularity selections, which takes into account both group utility maximization and individual regret minimization, while incorporating decision-makers' subjective preferences as well. This ensures that the experiment maintains higher exclusion stability and reliability, enhancing the robustness of the decision results.

Three-way decisions with dual hesitant fuzzy covering-based rough set and their applications in medical diagnosis

As an extension of partition, fuzzy β-covering can provide a more realistic and accurate description of incomplete information. In this paper, we mainly integrate the idea of fuzzy β-covering with dual hesitant fuzzy (DHF) information and construct some novel three-way decision (3WD) models with DHF covering-based rough set. Firstly, we propose the notions of DHF β-covering, DHF β-minimal description and DHF β-maximal description, and then construct three types of DHF neighborhood operators. Meanwhile, we introduce DHF conditional probability by using DHF neighborhood operator. Secondly, in terms of DHF conditional probability, DHF covering-based probabilistic rough set, DHF covering-based decision-theoretic rough set and their 3WD models are established. In light of DHF neighborhood operator, we propose two pairs of optimistic and pessimistic DHF decision evaluation functions and build two 3WD models through them. Lastly, a numerical example is employed to elaborate the application of the above models, which is effective and credible to medicine screening for Alzheimer’s disease.

Multigranulation fuzzy probabilistic rough sets induced by overlap functions and their applications

As we all know, t-norms can be used to construct fuzzy probabilistic rough set (FPRS). Meanwhile, overlap functions (OFs), as a sort of novel aggregation functions different from t-norms, have shown a flourishing situation in terms of applications and theory, especially for the study involving combination of OFs with rough sets. In this paper, we propose a novel OFs-based FPRS named as OFPRS. Specifically, first, we provide a pair of approximation operators of OFPRS via the conditional probability based on OFs. Meanwhile, we present a new OFs-based multigranulation fuzzy probabilistic rough set named as OMGFPRS. Then, we study elementary properties of OFPRS and OMGFPRS. Furthermore, we list practical examples to illustrate the feasibility as well as effectiveness of OFPRS and OMGFPRS, and give a short comparison of the proposed models with existing corresponding FPRS models. Lastly, we develop numerical experiments where OFPRS and OMGFPRS have better classification performance than t-norms-based FPRS.

Three-way decisions with evaluative linguistic expressions

The theory of three-way decisions (3WD) requires dividing a finite, non-empty universe into three disjoint sets called positive, negative, and boundary regions. Three types of decisions are then made on the objects in each region: acceptance, rejection, and abstention (or non-commitment), respectively. Until today, a large number of 3WD extensions and applications have been proposed; some of the most recent ones also include aspects of linguistics. In this article, we first propose an innovative linguistic interpretation of three-way decisions, where the positive, negative, and boundary regions are constructed by means of the so-called evaluative linguistic expressions. These are expressions of natural language, such as small, medium, very short, quite roughly strong, extremely good, etc., and they are described within a logical theory based on the formal system of higher-order fuzzy logic. Furthermore, in line with our linguistic 3WD approach, we introduce the novel notion of linguistic rough sets, thus contributing to the development of Rough Set Theory. Finally, we connect the theory of linguistic three-way decisions with the standard 3WD model based on probabilistic rough sets, establishing conditions under which the two approaches coincide. Our results highlight connections between two different research areas: three-way decisions and the theory of evaluative linguistic expressions.

Fuzzy Intelligence Learning Based on Bounded Rationality in IoMT Systems: A Case Study in Parkinson’s Disease

As a cause of interfering with routine activities, freezing of gait (FOG) is a severe syndrome of Parkinson’s disease (PD) and usually performs as an abrupt and momentary inability to effective stepping forward. Advanced wearable acceleration sensors based on socially implemented Internet of medical things (IoMT) devices can remotely provide a platform for recognizing FOG. However, due to the diverse data acquisition modes that appear in classic IoMT devices, the obtained data may contain imprecise, hesitant, and incomplete ones. Meanwhile, the bounded rationality owned by neurologists usually has a big impact on using wearable acceleration sensors to predict illnesses. Therefore, the objective of this article lies in exploring a fuzzy intelligence learning approach based on bounded rationality in IoMT systems and providing a valid scheme for biomedical data analysis. Specifically, a brand-new three-way group decision-making approach by means of TODIM (an acronym in Portuguese for interactive multicriteria decision-making) with incomplete dual hesitant fuzzy (DHF) information and its applications in detecting FOG in PD using IoMT devices are systematically explored. First, taking advantage of DHF sets (DHFSs) when depicting realistic group decision information, the concept of multigranulation (MG) incomplete DHF information systems is built. Second, adjustable MG DHF probabilistic rough sets (PRSs) are further put forward via DHF similarity relations. Third, a three-way group decision-making approach is constructed by virtue of adjustable MG DHF PRSs and TODIM. Finally, the validity, effectiveness, and practicality of the constructed three-way group decision-making approach are investigated by a University of California, Irvine (UCI) dataset with several experimental analyses in the background of FOG detection in PD using IoMT devices. The experimental result indicates that the developed fuzzy intelligence learning approach achieves reasonable diagnostic conclusions for FOG detection in PD from the perspective of uncertain information processing abilities, decision risks, and bounded rationality.

A regret theory-based multi-granularity three-way decision model with incomplete T-spherical fuzzy information and its application in forest fire management

Forest fires are an abrupt and highly destructive meteorological disaster that can occur in all regions of the world, resulting in significant ecological, economic and social losses. Moreover, the causes of forest fire disasters are usually complex, involving several uncertain factors such as temperature, relative humidity, wind speed and rainfall. All of those pose the greatest challenge to the study of forest fire management (FRM). In order to efficiently explore FRM via valid intelligent decision-making techniques, a novel model of regret theory (RT)-based multi-granularity (MG) three-way decisions (TWD) in incomplete T-spherical fuzzy (T-SF) environments has been constructed, where incomplete T-spherical fuzzy sets (T-SFSs) have been employed to describe diverse types of uncertain information in FRM, and RT-based MG TWD is conducive to analyzing multi-source T-SF information via reducing decision risks and modeling bounded rationality owned by decision-makers (DMs). Specifically, the concept of MG T-SF incomplete information systems (IISs) has been first constructed for information depictions of FRM. Then, MG T-SF IISs have been processed via the presented T-SF similarity principles for developing adjustable MG T-SF probabilistic rough sets (PRSs). Afterwards, an RT-based MG TWD approach has been built with the support of adjustable MG T-SF PRSs. Finally, a real-world FRM case analysis has been performed by using the built RT-based MG TWD approach, and extensive comparative and experimental analyses have been performed to validate the practicability of the presented methodology. To sum up, the presented methodology has simultaneously incorporated MG T-SF IISs, MG TWD and RT to model various uncertainties, valid information fusion processes and bounded rationality for FRM, which serves as a valid intelligent decision-making technique in processing incomplete and imprecise multi-source information with plentiful decision risks and regret emotions.

Valued outranking relation-based heterogeneous multi-decision multigranulation probabilistic rough set and its use in medical decision-making

Regarding to group decision-making, decision makers may express different decision preferences for the same target. Since existing decision-making methods only consider a single decision preference at most, it is difficult to integrate the multi-decision preferences of multiple decision makers effectively. In this paper, we explore a class of heterogeneous multi-attribute group decision-making (MAGDM) problems with multiple decisions. Over the framework of granular computing, we present a new MAGDM method with the aid of the valued outranking relations in the ELECTRE III method. Considering the non-compensability among attributes, we first define a special valued outranking relation on a heterogeneous attribute multi-decision information system, and then present the notion of valued outranking classes. Subsequently, given the fact that the fault tolerance of probability approximations can reduce the vulnerability of decision-making methods, we construct multi-decision multigranulation probabilistic rough sets (MD-MGPRSs) form different perspectives, which extend existing rough set models to a more general situation. After that, we establish a novel MAGDM method, and apply the method to a real medical case to illustrate its applicability. Lastly, the experimental results on seven datasets demonstrate the effectiveness, robustness, and superiority of the method. Our study not only enriches the theoretical research of rough sets, but also extends the application scope of decision-making theory.

Incomplete intuitionistic fuzzy behavioral group decision-making based on multigranulation probabilistic rough sets and MULTIMOORA for water quality inspection

Water quality inspection (WQI) is one of the primary ways to ensure the safe utilization of water resources, and complicated data modeling, fusion and analysis play a significant role in seeking the resource with the best water quality. Nevertheless, the challenges of missing data, relatively large differences in decision results and bounded rationality owned by decision-makers (DMs) in terms of WQI still exist nowadays. Thus, from the aspect of stable and behavioral decision-making in multi-granularity incomplete intuitionistic fuzzy information systems (MG-IIFISs), the paper investigates a comprehensive multi-attribute group decision-making (MAGDM) approach for the application of WQI. First, the concept of MG-IIFISs is built by modeling MAGDM problems with intuitionistic fuzzy numbers (IFNs), then a new transformation scheme is constructed for transforming MG-IIFISs into multi-granularity intuitionistic fuzzy information systems (MG-IFISs) based on the similarity principle. Second, three types of multigranulation intuitionistic fuzzy probabilistic rough sets (MG IF PRSs) are developed by referring to the MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the full MULTIplicative form) method. Afterwards, attribute weights are objectively calculated based on the best-worst method (BWM), and a new stable and behavioral MAGDM approach is constructed by means of the TODIM (an acronym in Portuguese for interactive multi-criteria decision-making) method. At last, a case study in the setting of WQI is conducted with the support of a UCI data set, and sensitivity analysis, comparative analysis and experimental analysis are investigated to display the validity of the proposed approach. In general, the proposed approach improves the stability of decision results via MULTIMOORA and BWM, and also fully considers the bounded rationality of DMs’ psychological behaviors from the aspect of the TODIM method, which has certain advantages in the community of MAGDM studies.

Three-way decisions with fuzzy probabilistic covering-based rough sets and their applications in credit evaluation

As a generalization of equivalent class, fuzzy covering makes descriptions of the objective world more realistic, practical and accurate in some cases. In this paper, we establish four kinds of fuzzy probabilistic covering-based rough set, which combine the fuzzy covering-based neighborhood operator and the fuzzy probabilistic rough set, and then obtain the result of their three-way decision. The differences among the above models lie in the selection of thresholds, which further leads to different classification results. In addition, in order to interpret the thresholds rationally under different models, Bayesian decision procedure is adopted to construct the fuzzy probabilistic covering-based upper and lower approximation operators. Finally, a numerical example of the credit evaluation is applied to illustrate the validity of the proposed model.

An artificial bee colony-based framework for multi-objective optimization of three-way decisions with probabilistic rough sets

The cogent area, Probabilistic rough sets, offers methods that are used to trisect the data into positive, negative and boundary regions for optimum (α, β) pairs. These basic methods generate three regions based on a single quality, including cost, entropy, impurity, correlation and variance, thereby the best (α, β) pair is generated. The optimization of multiple qualities has significance in real-life applications; however, experiments rarely discussed the optimization of different criteria together in probabilistic rough sets. This probe conducts multi-objective optimization of uncertainty, impurity and correlation, to determine a trisection at optimal (α, β) pairs. For that, this work proposes a hybrid method that involves Weighted Sum and Artificial Bee Colony Algorithm to optimize the thresholds. The results are compared with the Information-theoretic rough sets and Game-theoretic rough sets. The proposed method outperforms regarding optimal qualities, multiple optimum thresholds, minimal size of boundary regions, and better evaluation results. By attesting the study on experimental data sets, optimal (α, β) pairs are obtained at which the uncertainty and impurity are minima. Moreover, the correlation at this threshold is reasonable. From the application viewpoint, it reduces the cost of further analysis by generating the minimum delayed decision and maximizes the benefit with optimal decisions by considering multiple optimized qualities simultaneously.

Data analysis using Maximum Probabilistic Rough Set in R Environment

Data analysis using Maximum Probabilistic Rough Set in R Environment

Reduction Algorithm of Interval-Valued Intuitionistic Fuzzy Probability Rough Set Under Dominant Relation

Attribute reduction is a part of the most basic and significant research contents in rough set theory. The so-called attribute reduction is the smallest independent subset that keeps certain properties of information table unchanged. In this paper, four interval-valued intuitionistic fuzzy probabilistic rough set models and their natures are given on the basis of the dominant-, inferior- and interval-valued intuitionistic fuzzy probabilistic rough set models. At the same time, the interval-valued intuitionistic fuzzy numbers are transformed by fuzzy degree, and the approximate accuracy and approximate classification quality under the dominance relation are used for reduction, at last, the feasibility is verified by an example.

Three-Way Group Decisions with Incomplete Spherical Fuzzy Information for Treating Parkinson’s Disease Using IoMT Devices

As Internet of Things (IoT) is extensively employed in diverse realistic areas, it is vital to effectively and timely analyze the data collected by IoT devices. To cope with this problem, by integrating adjustable MG SF probabilistic rough sets (PRSs) with the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method, this paper explores a three-way multi-attribute group decision-making (MAGDM) approach in the context of multigranulation (MG) spherical fuzzy (SF) incomplete information systems (IISs) and further applies the presented method to the analysis of leg muscle data obtained from Internet of Medical Things (IoMT) devices for Parkinson’s patients. First, the concept of MG SF IISs is established, and the completion method is provided. Then, adjustable MG SF PRSs are proposed for information fusion. Afterwards, considering the bounded rationality of decision-makers (DMs), a new three-way MAGDM method is designed by fusing adjustable MG SF PRSs with the TODIM method. Finally, in the context of IoMT-based detecting abnormal knee joints in Parkinson’s patients, the applicability and validity of the presented method are eventually verified.

Incomplete three-way multi-attribute group decision making based on adjustable multigranulation Pythagorean fuzzy probabilistic rough sets

Different from the traditional two-way decision paradigm, three-way decisions (3WD) have been extensively explored in addressing realistic multi-attribute decision making (MADM) by virtue of its powerful performances in minimizing decision risks with the added non-commitment option. Since existing 3WD models can not efficiently handle MADM issues in the group setting with incomplete high-order fuzzy information systems (ISs), the current paper focuses on exploring viable three-way multi-attribute group decision making (MAGDM) methods in light of the multi-granularity 3WD paradigm under incomplete Pythagorean fuzzy (PF) environments. First, the notion of Multigranulation (MG) incomplete PF ISs is constructed. Then, optimistic, pessimistic and adjustable MG PF probabilistic rough sets (PRSs) are proposed after patching lost values. Afterwards, the decision-making-driven 3WD method is extended via adjustable MG PF PRSs, and an improved technique for order preference by similarity to ideal solution (TOPSIS) method with entropy weights is utilized for optimal granularity selections. According to adjustable MG PF PRSs and the improved TOPSIS method, a new three-way MAGDM method is eventually established. At last, two case studies are investigated in the background of mine ventilator fault diagnosis (MVFD) and air quality evaluations (AQE), and several comparative experiments and sensitivity analysis are performed for presenting the effectiveness of the constructed theoretical methodology.

Interval prediction approach to crude oil price based on three-way clustering and decomposition ensemble learning

Prediction methods have become a hot topic in intelligent decision making. Most of the existing prediction methods focus on the prediction accuracy and stability. As a second choice, accurate interval prediction can provide a relatively reliable reference in the sense of probability and provide help for assisting decision management. Therefore, we propose a novel interval prediction approach. Firstly, the decomposition method based on ensemble empirical mode decomposition (EEMD) is utilized to alleviate the complexity of the original time series, thereby generating a series of relatively smooth subseries. Secondly, a three-way clustering (TWC) algorithm is established by integrating sample entropy into probabilistic rough set, enriching the three-way clustering theory from the perspective of entropy. Thirdly, aiming at determining the optimal input dimensions of different neural networks, the feature selection technique based on phase space reconstruction (PSR) is constructed. Furthermore, an interval prediction system based on TWC is proposed to provide a new data-driven prediction method. Finally, the proposed approach is applied to predict the interval price of crude oil. On the one hand, the practicability of the constructed prediction approach is verified; on the other hand, it provides a new theoretical method for interval prediction of crude oil price. The experiment results show the proposed prediction approach can assist the decision-makers to make scientific and reasonable decisions.

A unified incremental updating framework of attribute reduction for two-dimensionally time-evolving data

In the open-world environment, the incremental updating approaches to attribute reduction based on rough sets are efficient and effective to evaluate and search an optimal subset of attributes from two-dimensionally time-evolving data, which can be interpreted as the complex changes of dynamic data, i.e., four types of combinations induced by the insertion/deletion of objects and the addition/remove of attributes. To avoid the time-consuming and repetitive computation from scratch in such dynamic data, this paper mainly focuses on constructing a unified incremental framework to attribute reduction by the matrix-based accelerated updating strategies. We systematically discuss and present a series of incremental updating mechanisms and algorithms of approximation quality in the neighborhood-based probabilistic rough sets. Besides, a unified framework of dynamic attribute reduction in four situations of changes is proposed to develop the performance of updating reduct. Finally, we report the comparative experiments between the non-incremental and incremental algorithms of reduct to demonstrate the feasibility and efficiency of proposed approaches.

Multigranulation fuzzy probabilistic rough set model on two universes

Fuzzy probabilistic rough set is an important mathematical model in rough set theory and has been widely used in applications. Although many studies have been taken on this topic, the existing fuzzy probabilistic rough set models are insufficient for dealing with data noise and data discretization. This motivates us to develop a better method to handle complex data. In this paper, we propose a novel fuzzy probabilistic rough set model on two universes by using λ-fuzzy sets. To further extend application field, we define upper and lower approximation operators of fuzzy probabilistic rough set in a multigranulation context and propose a novel multigranulation fuzzy probabilistic rough set model on two universes. Moreover, some interesting properties of the models are studied. The effectiveness and superiority of them are illustrated by numerical examples and experiments compared with existing models. Experimental results show that the proposed models have better performance in classification and have potential applications in decision making problems.

Label-based Approach for Dynamic Updating Approximations in Incomplete Fuzzy Probabilistic Rough Sets over Two Universes

Label-based Approach for Dynamic Updating Approximations in Incomplete Fuzzy Probabilistic Rough Sets over Two Universes

A novel method to attribute reduction based on weighted neighborhood probabilistic rough sets

Attribute reduction is an important application of rough set theory. Most existing rough set models do not consider the weight information of attributes in information systems. In this paper, we first study the weight of each attribute in information systems by using data binning method and information entropy theory. Then, we propose a new rough set model of weighted neighborhood probabilistic rough sets (WNPRSs) and investigate its basic properties. Meanwhile, the dependency degree formula of an attribute relative to an attribute subset is defined based on WNPRSs. Subsequently, we design a novel attribute reduction method by using WNPRSs and the corresponding algorithm is also given. Finally, to evaluate the performance of the proposed algorithm, we conduct a data experiment and compare it with other existing attribute reduction algorithms on eight public datasets. Experimental result demonstrates that the proposed attribute reduction algorithm is effective and performs better than some of the existing algorithms.

Multiple attribute group decision making based on multigranulation probabilistic models, MULTIMOORA and TPOP in incomplete q-rung orthopair fuzzy information systems

The concept of q-rung orthopair fuzzy sets (q-ROFSs) severs as an extended form of orthopair fuzzy sets, which excels in flexibly depicting imprecise information existed in multiple attribute group decision making (MAGDM) via allowing a greater space in terms of membership degrees (MD) and non-membership degrees (ND). Moreover, incomplete information systems play a significant role in depicting incomplete information existed in MAGDM. In this paper, for the sake of exploring MAGDM with imprecise and incomplete information, a new MAGDM method based on multigranulation probabilistic models, MULTIMOORA (Multi-Objective Optimization by Ratio Analysis plus the full MULTIplicative form) and the technique of precise order preference (TPOP) method in incomplete q-rung orthopair fuzzy (q-ROF) information systems is systematically investigated. First, the concept of multigranulation (MG) incomplete q-ROF information systems is established. Then, a completion technique is put forward to obtain MG q-ROF information systems via patching lost values. Moreover, three versions of MG q-ROF probabilistic rough sets (PRSs) are put forward in light of MG q-ROF information systems. Afterwards, a new MAGDM method is constructed by means of MG q-ROF PRSs, MULTIMOORA and the TPOP method. At last, two experimental studies from open datasets are performed to present the applicability of the constructed MAGDM method, and corresponding sensitivity analysis and comparative analysis are further conducted to demonstrate the validity of the presented methodology.