Interval-valued Decision Research Articles

A novel attribute reduction approach using coverage-credibility-based rough decision entropy for interval-valued data

Attribute reduction is an important method in data analysis and machine learning, and it usually relies on algebraic and informational measures. However, few existing informational measures have considered the relative information of decision class cardinality, and the fusion application of algebraic and informational measures is also limited, especially in attribute reductions for interval-valued data. In interval-valued decision systems, this paper presents a coverage-credibility-based condition entropy and an improved rough decision entropy, further establishes corresponding attribute reduction algorithms for optimization and applicability. Firstly, the concepts of interval credibility, coverage and coverage-credibility are proposed, and thus, an improved condition entropy is defined by virtue of the integrated coverage-credibility. Secondly, the fused rough decision entropy is constructed by the fusion of improved condition entropy and roughness degree. By introducing the coverage-credibility, the proposed uncertainty measurements enhance the relative information of decision classes. In addition, the nonmonotonicity of the improved condition entropy and rough decision entropy is validated by theoretical proofs and experimental counterexamples, with respect to attribute subsets and thresholds. Then, the two rough decision entropies drive monotonic and nonmonotonic attribute reductions, and the corresponding reduction algorithms are designed for heuristic searches. Finally, data experiments not only verify the effectiveness and improvements of the proposed uncertainty measurements, but also illustrate the reduction algorithms optimization through better classification accuracy than four comparative algorithms.

Attribute reductions based on δ-fusion condition entropy and harmonic similarity degree in interval-valued decision systems

This paper defines an improved similarity degree based on inclusion degree as well as advanced information system based on interval coverage and credibility, and thus an attribute reduction framework embodying 4×2 = 8 reduct algorithms is systematically constructed for application and optimization in interval-valued decision systems. Firstly, a harmonic similarity degree is constructed by introducing interval inclusion degree and harmonic average mechanism, which has better semantic interpretation and robustness. Secondly, interval credibility degree and coverage degree are defined for information fusion, and they are combined to propose a δ-fusion condition entropy. The improved condition entropy achieves the information reinforcement and integrity by dual quantization fusion of credibility and coverage, and it obtains measure development from granularity monotonicity to non-monotonicity. In addition, information and joint entropies are also constructed to obtain system equations. Furthermore, 8 reduct algorithms are designed by using attribute significance for heuristic searches. Finally, data experiments show that our five novel reduct algorithms are superior to the three contrast algorithms on classification performance, which also further verify the effectiveness of proposed similarity degree, information measures and attribute reductions.

Two-dimensional improved attribute reductions based on distance granulation and condition entropy in incomplete interval-valued decision systems

Attribute reductions rely on knowledge granulation and information measurement. Aiming at incomplete interval-valued decision systems (IIVDSs), an attribute reduction (with the FSR-AR/HS-AR algorithm) emerges by combining distance granulation and condition entropies; however, its distance measurement has defects for maximum and minimum fillings, while its condition entropy has promotion space on information enrichment. This paper utilizes FSR-AR/HS-AR to make the two-dimensional improvements of distance granulation and condition entropy to improve IIVDS-driven attribute reductions. Concretely, distance measures and similarity degrees are corrected via range completion and statistical enhancement, and the condition entropy is deepened by replacing credibility with coverage-credibility, so 2×2=4 reduction algorithms extend and improve FSR-AR/HS-AR. First, new maximal and minimal distances are defined via complete search and statistical optimization of missing values, so an amended fuzzy α-similarity relation induces knowledge granulation. Then, improved informational, conditional, joint entropies, and mutual information are established via coverage-credibility, and they obtain system equations and granulation nonmonotonicity. Furthermore, the two-dimensional improvements generate 2×2=4 attribute reductions, and the corresponding heuristic algorithms include FSR-AR/HS-AR and three improved algorithms. Finally, improvements in uncertainty measures and reduction algorithms are validated via table examples and data experiments, and the three novel algorithms outperform FSR-AR/HS-AR and contrastive algorithms for classification performances.

A novel information fusion method using improved entropy measure in multi-source incomplete interval-valued datasets

Multi-source data is a comprehensive data type that combines multiple sources of information or datasets. Compared to point-valued data, interval-valued data provides a more accurate representation of the uncertainty and variability associated with objects. In practical situations, data obtained from multiple sources may contain missing values for various reasons. Therefore, it is essential to develop multi-source information fusion technology in order to achieve information fusion or information extraction from multi-source incomplete data. This paper aims to explore the information fusion problem of multi-source incomplete interval-valued datasets. The primary contributions of this study involve utilizing the principle of statistical distribution and KL divergence to establish a metric for measuring the similarity between intervals. Firstly, this approach helps to reduce the problem of disregarding internal information within interval values, which can result in the loss of valuable information. Secondly, we establish an interval fuzzy similarity relation based on the mentioned concept of similarity among interval values. Moreover, we investigate the uncertainty measurement of incomplete interval-valued decision datasets and design an emerging information entropy fusion method. Finally, we comprehensively evaluate the effectiveness of the proposed method. Experimental results indicate that the proposed approach has advantage over the maximum, minimum, mean, and information entropy fusion method based on tolerance relationship. In addition, the distance metric used in this article can improve the fusion classification effect compared to several common interval-valued distance measures.

A Novel Interval-Valued Decision Theoretic Rough Set Model with Intuitionistic Fuzzy Numbers Based on Power Aggregation Operators and Their Application in Medical Diagnosis

Intuitionistic fuzzy information is a potent tool for medical diagnosis applications as it can represent imprecise and uncertain data. However, making decisions based on this information can be challenging due to its inherent ambiguity. To overcome this, power aggregation operators can effectively combine various sources of information, including expert opinions and patient data, to arrive at a more accurate diagnosis. The timely and accurate diagnosis of medical conditions is crucial for determining the appropriate treatment plans and improving patient outcomes. In this paper, we developed a novel approach for the three-way decision model by utilizing decision-theoretic rough sets and power aggregation operators. The decision-theoretic rough set approach is essential in medical diagnosis as it can manage vague and uncertain data. The redesign of the model using interval-valued classes for intuitionistic fuzzy information further improved the accuracy of the diagnoses. The intuitionistic fuzzy power weighted average (IFPWA) and intuitionistic fuzzy power weighted geometric (IFPWG) aggregation operators are used to aggregate the attribute values of the information system. The established operators are used to combine information within the intuitionistic fuzzy information system. The outcomes of various alternatives are then transformed into interval-valued classes through discretization. Bayesian decision rules, incorporating expected loss factors, are subsequently generated based on this foundation. This approach helps in effectively combining various sources of information to arrive at more accurate diagnoses. The proposed approach is validated through a medical case study where the participants are classified into three different regions based on their symptoms. In conclusion, the decision-theoretic rough set approach, along with power aggregation operators, can effectively manage vague and uncertain information in medical diagnosis applications. The proposed approach can lead to timely and accurate diagnoses, thereby improving patient outcomes.

Feature selections based on three improved condition entropies and one new similarity degree in interval-valued decision systems

Feature selections facilitate classification learning in various data environments. Aiming at interval-valued decision systems (IVDSs), feature selections rely on information measures and similarity degrees, whereas current selection algorithms on credibility-based condition entropy and classical similarity degree are accompanied with some measurement limitations and advancement space. In this paper based on IVDSs, three coverage-credibility-based condition entropies and one geometry-probabilistic similarity degree are proposed across two dimensions of informationization and granulation, and they improve the existing condition entropy and similarity degree; accordingly, 4 × 2 feature selections emerge for optimization and applicability, and they systematically contain one initial selection algorithm and seven new/robuster algorithms. At first, three-way granular measures (i.e., credibility, coverage, and integrated coverage-credibility) are formulated in IVDSs, and three novel condition entropies are established by implementing three information structures on coverage-credibility. These condition entropies acquire in-depth improvements, hierarchical algorithms, size relationships, maximum/minimum conditions, and granulation non-monotonicity. Then, the probabilistic similarity degree is defined by a six-piecewise function with quadratic factors, and this new measure gains the geometry-probability mechanism and high-quality improvement. Furthermore, feature selections are determined by preserving condition entropies and by mining feature significances, so eight selection algorithms are obtained by combining condition entropies and similarity degrees. Finally, data experiments are performed to validate relevant uncertainty measures and feature selections, and seven constructional selection algorithms outperform three contrastive algorithms to achieve better classification performances.

Interval-valued fuzzy discernibility pair approach for attribute reduction in incomplete interval-valued information systems

Interval-valued information systems provide rich semantic interpretation and greater flexibility compared to real-valued information systems. Meanwhile, incomplete interval-valued information systems containing missing values exist widely in reality. One of the key issues in handling incomplete interval-valued information systems is attribute reduction. Unfortunately, there are few studies on attribute reduction for incomplete interval-valued information systems. Moreover, most of the existing studies use fuzzy similarity relations to measure the similarity of objects. Due to the complexity of incomplete interval-valued information systems, traditional fuzzy similarity relations cannot adequately represent the information contained within incomplete interval-valued information systems. To address this issue, this study proposes interval-valued fuzzy min-max similarity relations for incomplete interval-valued information systems, along with two attribute reduction algorithms based on interval-valued fuzzy discernibility pairs model for incomplete interval-valued information systems. Firstly, this study defines interval-valued fuzzy min-max similarity relations using distance-based extended restricted equivalence functions. Secondly, this study presents the interval-valued fuzzy discernibility pairs and discerning ability measure for attribute sets. Furthermore, an attribute reduction algorithm is proposed for incomplete interval-valued information systems. Thirdly, taking into account decision attributes, the relative interval-valued fuzzy discernibility pairs and an attribute reduction algorithm for incomplete interval-valued decision information systems are proposed. Finally, the effectiveness of the proposed method is compared to four representative attribute reduction algorithms in experiments. The experimental results illustrate that our method is effective and has advantages over the compared algorithms.

An Interval-Valued Three-Way Decision Model Based on Cumulative Prospect Theory

In interval-valued three-way decision, the reflection of decision-makers’ preference under the full consideration of interval-valued characteristics is particularly important. In this paper, we propose an interval-valued three-way decision model based on the cumulative prospect theory. First, by means of the interval distance measurement method, the loss function and the gain function are constructed to reflect the differences of interval radius and expectation simultaneously. Second, combined with the reference point, the prospect value function is utilized to reflect decision-makers’ different risk preferences for gains and losses. Third, the calculation method of cumulative prospect value for taking action is given through the transformation of the prospect value function and cumulative weight function. Then, the new decision rules are deduced based on the principle of maximizing the cumulative prospect value. Finally, in order to verify the effectiveness and feasibility of the algorithm, the prospect value for decision-making and threshold changes are analyzed under different risk attitudes and different radii of the interval-valued decision model. In addition, compared with the interval-valued decision rough set model, our method in this paper has better decision prospects.

Feature selection for interval-valued data via FRIC-model

Feature selection is one basic technology for data mining. This paper investigates feature selection for interval-valued data via fuzzy rough iterative computation model (FRIC-model). To depict the similarity between samples in an interval-valued decision information system (IVDIS), the fuzzy symmetry relation in an IVDIS is first introduced from the perspective of “The similarity between information values is fed back to the feature set”. After that, several attribute evaluation functions, such as fuzzy positive regions, dependency functions and attribute importance functions are defined. Subsequently, FRIC-model for interval-valued data is established by using the iterations of these functions. Next, An feature selection algorithm in an IVDIS based on this model is presented. Lastly, numerical experiments and statistics tests are carried out to estimate the performance of the presented algorithm. The experimental results illustrate that the presented algorithm maintains high classification accuracy, and does not occupy too much memory. These findings will provide new perspective for feature selection in an IVDIS.

Feature selection using a weighted method in interval-valued decision information systems

Feature selection using a weighted method in interval-valued decision information systems

Incremental fuzzy rough sets based feature subset selection using fuzzy min-max neural network preprocessing

Fuzzy rough sets (FRS) provides effective ways for selecting the compact/relevant feature subset for hybrid decision systems. However, the underlying complexity of the existing FRS methods through batch processing is often costly or intractable on large data and also suffer from continuous model adaptation on dynamic data. This paper proposes a FRS based incremental feature subset selection (IvFMFRS) framework using fuzzy min-max neural network (FMNN) as a preprocessor step in aiding to deal with data dynamically without sacrificing classification performance. FMNN is a single epoch learning algorithm employed to construct fuzzy hyperboxes (information granules) of pattern spaces very fast. Fuzzy hyperboxes facilitate the formation of interval-valued decision system (IDS) from the numerical decision system of much smaller size. In IvFMFRS, on each sample subset arrival, an incremental mechanism for updating fuzzy discernibility matrix (FDM) based on constructed IDS is first formulated and then update feature subset by adding and deleting features based on updated FDM. A comparative analysis has been conducted comprehensively to assess the performance of the proposed algorithm with the existing FRS methods on numerical datasets. And, the results show that the IvFMFRS obtained the relevant feature subsets with similar classification accuracy with significantly less computational time than existing FRS methods.

A fuzzy [formula omitted]-similarity relation-based attribute reduction approach in incomplete interval-valued information systems

As generalizations of single-valued information systems, interval-valued information systems (IVISs) can better express the real data with uncertainty in some applications. Attribute reduction methods for complete IVISs or complete interval-valued decision systems (IVDSs) have been developed. However, there are few researches on attribute reduction for incomplete interval-valued information systems (IIVISs). The paper aims to investigate the attribute reduction issue in IIVISs. Firstly, the maximal and minimal distances, which characterize the difference between two interval values, are defined, and the maximal and minimal similarity degrees are given. Secondly, the fuzzy α-similarity relation is defined based on similarity between interval values, and the concept of α-equivalence relation is raised. Thirdly, entropy measures are investigated for IIVISs in view of α-equivalence relations. Fourthly, a new attribute reduction approach for IIVISs is proposed by using conditional entropy, and its corresponding algorithm is given. Finally, experiments to verify the effectiveness and feasibility of the newly proposed approach for attribute reduction in IIVISs are presented. These results will be helpful to perfect the uncertainty measurement model, and provide an approach for attribute reduction in IIVISs.

Three-level and three-way uncertainty measurements for interval-valued decision systems

Uncertainty measurements underlie the system interaction and data learning. Their relevant studies are extensive for the single-valued decision systems, but become relatively less for the interval-valued decision systems. Thus, three-level and three-way uncertainty measurements of the interval-valued decision systems are proposed, mainly by systematically constructing vertical-horizontal weighted entropies. Firstly, the interval-valued decision systems are endowed with three-level structures, including Micro-Bottom, Meso-Middle, and Macro-Top. Secondly, a three-level decomposition is hierarchically made for the existing conditional entropy. Thirdly, three-way weighted entropies are systematically and hierarchically constructed at the three levels, and they achieve their hierarchy, systematicness, algorithm, boundedness, and granulation monotonicity/non-monotonicity. The three-level and three-way weighted entropies deepen and extend the conditional entropy, and they realize the ingenious criss-cross informatization for the interval-valued decision systems. Their effectiveness of uncertainty measurements is ultimately verified by table examples and data experiments.

Attribute Reduction in an Incomplete Interval-Valued Decision Information System

An incomplete interval-valued decision information system (IIVDIS) is a significant type of data decision table, which is ubiquitous in real life. Interval value is a form of knowledge representation, and it seems to be an embodiment of the uncertainty of research objects. In this paper, we focus on attribute reduction on the basis of a parameterized tolerance-based rough set model in an IIVDIS. Firstly, we give the similarity degree between information values on each attribute in an IIVDIS by considering incomplete information. Then, we present tolerance relations on the object set of an IIVDIS based on this similarity degree. Next, we define the rough approximations by means of the presented tolerance relation. Based on Kryszkiewicz’s ideal, we introduce $\alpha $ -generalized decision and consider attribute reduction in an IIVDIS by means of this decision. Furthermore, we put forward the notions of $\alpha $ -information entropy, $\alpha $ -conditional information entropy and $\alpha $ -joint information entropy in an IIVDIS. And we prove that $\alpha $ -positive region reduction theorem, $\alpha $ -conditional entropy reduction theorem, $\alpha $ -dependency reduction theorem and $\alpha $ -generalized decision reduction theorem are equivalent to each other. Finally, we propose two attribute reduction methods in an IIVDIS by using entropy measurement and the rough approximations, and design the relevant algorithms. We carry out a series of numerical experiments to verify the effectiveness of the proposed algorithms. The experimental results show that proposed algorithms often choose fewer attributes and improve classification accuracies in most cases.

Partial-overall dominance three-way decision models in interval-valued decision systems

Three-way decisions are a generalization of classical decision theory and receive increasing attentions from various fields to handle decision-making problems, especially when involving in incomplete information. An interval is a typical notion of information representation with incompleteness and uncertainty. To measure the dominance degree of one interval dominating or being dominated by another is a hot issue. In this paper, a novel dominance measure is constituted by considering both lengths and locations of intervals. The proposed dominance measure distinguishes two overlapped or coincided intervals, and is able to quantize the separation degree of disjoint intervals. A notion of variable precision overall dominance relation is introduced by integrating both the attribute-wise evaluation information and overall dominance degree of objects. Based on the constituted dominance relation, two three-way decision models are presented in interval-valued decision systems with categorical and interval-valued decision attributes, respectively. Numerical examples of three-way decisions and simulated experiments on classification over two synthetic and four benchmark datasets are presented. Experimental results indicate that the proposed model can produce lower classification errors in comparison with existing methods.

Scalable Fuzzy Rough Set Reduct Computation Using Fuzzy Min–Max Neural Network Preprocessing

A fuzzy rough set (FRS) is a hybridization of rough sets and fuzzy sets and provides a framework for reduct (feature subset selection) computation for hybrid decision systems. However, the existing FRS-based feature selection approaches are intractable for large decision systems due to the space complexity of the FRS methodology. We propose a novel fuzzy min–max neural network (FMNN)-FRS reduct computation approach utilizing the FMNN to enhance the scalability of FRS approaches. The FMNN provides a single pass epoch learning of arriving at granules of objects in the form of fuzzy hyperboxes for multiple decision classes. In the proposed approach, the FMNN model is used to reconstruct the object-based decision system into a fuzzy hyperbox-based interval-valued decision system. Then, a novel way of constructing the fuzzy discernibility matrix (FDM) from the interval-valued decision system is introduced. A fuzzy rough approximate reduct computation algorithm is developed with the induced FDM. The FMNN-FRS approach reduces the space complexity of FRS reduct computation significantly and results in enhanced scalability. Comparative experimental analysis has been done with the existing FRS reduct approaches on benchmark hybrid decision systems and established the relevance of the FMNN-FRS approach. The FMNN-FRS approach obtained the exact reduct in most of the datasets in much lesser computational time than existing FRS approaches while preserving similar classification accuracy. The FMNN-FRS method achieved enhanced scalability to such large decision systems, at which it is not possible to obtain reduct by existing FRS approaches.

Dynamically updating approximations based on multi-threshold tolerance relation in incomplete interval-valued decision information systems

With the development of society, data noise and other factors will cause the incompleteness of information systems. Objects may increase or decrease over time in information systems. The classical information system can be extended to the incomplete interval-valued decision information system (IIDIS) that is the researching object of this paper. Incremental learning technique is a significant method for solving approximate sets under dynamic data. This article defines a multi-threshold tolerance relation based on the set pair analysis theory and establishes a rough set model in IIDIS. Then, several methods and algorithms for statically/dynamically solving approximate sets are shown. Finally, comparative experiments from six UCI data sets show both dynamic algorithms take less time than the static algorithm to calculate the approximate sets no matter how object set changes.

A four-way decision-making approach using interval-valued fuzzy sets, rough set and granular computing: a new approach in data classification and decision-making

In this paper, a novel method is presented to solve the problems of classification and decision-making by employing the interval-valued fuzzy set, rough set and granular computing (GrC) concepts. The proposed method can classify the objects available in a system, called an interval-valued fuzzy decision table into the four distinct regions as interval-valued fuzzy-positive region, interval-valued fuzzy-negative region, interval-valued completely fuzzy region and interval-valued gray fuzzy region. These four regions constitute a new space for decision-making, which is termed as a four-way interval-valued decision space (4WIVDS). Based on the classified objects, various decision rules are generated from the distinct regions of the 4WIVDS. The study shows that the interval-valued gray fuzzy region has included most prominent decision rules. For taking precise level of decisions based on this particular region, this study utilizes the GrC to extract granular level of decision rules which are prominent by nature. The proposed 4WIVDS method is verified and validated with various benchmark datasets. Experimental results include statistical and comparison analyses which signify the efficiency of the proposed method in classifying the objects and generating the decision rules.

Rough Set Model for Cognitive Expectation Embedded Interval‐Valued Decision Systems

Interval-valued information system is a kind of knowledge representation model for uncertain information. An interval-valued attribute has an expectation by experience or background knowledge, called cognitive expectation. There are few studies aiming at intervalvalued attributes with cognitive expectations. We propose the concept of Interval-valued decision system with expectations (IDSE). A new dominance relation based on the distances between expectations and interval values is constructed. Based on the constructed dominance relation, a rough set model for IDSE is investigated. Attribute reduction in IDSE is also examined by using discernibility matrices and discernibility functions.

Multi-Granulation Rough Set for Incomplete Interval-Valued Decision Information Systems Based on Multi-Threshold Tolerance Relation

A relation is viewed as a granularity from a granular computing perspective. A classic rough set contains only one granularity. A multi-granulation rough set contains multiple granularities, which promotes the applications of classical rough set. Firstly, this paper uses the incomplete interval-valued decision information system (IIVDIS) as research object and constructs two rough set models in the light of single granularity rough set model for applying the rough set theory to real life more widely, which are optimistic multi-granulation rough set (OMGRS) model and pessimistic multi-granulation rough set (PMGRS) model in the IIVDIS. Secondly, we design two algorithms to compute the roughness and the degree of dependence that are two tools for measuring uncertainty of rough set. Finally, several experiments are performed on six UCI data sets to verify the validity of the proposed theorems.