Three-way Regions Research Articles

Matrix-based approaches for updating three-way regions in incomplete information systems with the variation of attributes

As a commonly used framework for uncertainty reasoning, tolerance rough set has achieved remarkable success in handling incomplete information systems with missing values. Three-way regions generated from tolerance rough set model play an increasingly crucial role in decision making and intelligent data analysis. Nevertheless, the dynamic change of attributes often exists in incomplete information systems. With this dynamic characteristic, three-way regions must be effectively updated for potential decision-making processes. Therefore, we develop incremental algorithms for maintenance of three-way regions in incomplete information systems when adding or deleting attributes, accelerating the calculation by making use of prior information. First, we put forward an effective matrix-based approach to calculate three-way regions in incomplete data. With the dynamic change of attributes, we further investigate the updating strategies of related matrices for constructing three-way regions. Accordingly, matrix-based algorithms for incrementally updating three-way regions are developed and discussed while the attributes vary over time. In addition, the complexity comparisons of non-incremental and incremental algorithms are illustrated. Finally, empirical experiments are performed to reveal the efficiency of the incremental algorithms compared with matrix-based non-incremental and related incremental algorithms.

Three-way neighborhood based stream computing for incomplete hybrid information system

Stream computing paradigm focuses on efficiently knowledge updating when immigration and emigration occur simultaneously. Existing stream computing for reasoning with three-way decisions aims at symbolic and complete data. In contrast, the studies on more realistic cases of numerical and symbolic types with potentially incomplete data are preliminary. To efficiently cope with the concept uncertainty, a dynamic neighborhood stream three-way decisions algorithm for incomplete hybrid information systems (DNS3WD) is proposed. Following the setting of flowing into one object and flowing out of another, we dynamically generate all neighborhood classes and decision equivalence classes in the data streaming window. Finally, we deduce the principles of variations of conditional probabilities and update three-way regions accordingly. Theoretically, the time complexity of the non-incremental three-way decisions algorithm (NIH3WD) is about N times that of DNS3WD, where N is the size of the data streaming window. In addition, extensive comparisons against NIH3WD demonstrate that DNS3WD can efficiently leverage the previously acquired knowledge.

Hierarchical Fuzzy Rough Approximations With Three-Way Multigranularity Learning

The approximation learning of fuzzy concepts associated with fuzzy rough sets and three-way decisions is a useful technology for the representation, learning, and transformation of uncertain knowledge. However, how to integrate the advantages of fuzzy rough approximations and three-way approximations has rarely been carefully investigated so far. In this article, we focus on exploring the connection and interplay of these two methods, and further propose the hierarchical fuzzy rough approximations in the dynamic fuzzy open-world environment. We utilize the temporal-spatial perspectives of three-way decisions to construct multigranularity structures and implement multigranularity learning. The temporality of data and the spatiality of model parameters are both considered in such frameworks. Subsequently, we discuss the interpretation and representation of fuzzy three-way regions in fuzzy rough sets with some definitions and properties. The max, double, and aggregated evaluation-based models of three-way approximations are proposed to gradually transform a fuzzy concept to a crisp concept by the time-evolving attributes. Finally, the comparative experimental results between static and dynamic approaches demonstrate the effectiveness of our proposed hierarchical approximation learning models.

Correction to: Incremental maintenance of three-way regions with variations of objects and values in hybrid incomplete decision systems

Correction to: Incremental maintenance of three-way regions with variations of objects and values in hybrid incomplete decision systems

Incremental maintenance of three-way regions with variations of objects and values in hybrid incomplete decision systems

Incremental maintenance of three-way regions with variations of objects and values in hybrid incomplete decision systems

Dynamic three-way neighborhood decision model for multi-dimensional variation of incomplete hybrid data

The theory of three-way decisions, as a powerful methodology of granular computing, has been widely used in making decision under uncertainty environments. Decision tasks in incomplete hybrid data including heterogeneous and missing features are of abundance in realistic situations. To deal with these tasks, some work based on three-way decisions has been investigated. However, the losses used for evaluating objects are precise real numbers, which makes these decision models have some limitations in applications when there exist missing values in incomplete hybrid data. Thus, this paper constructs a generalized three-way neighborhood decision model by assigning the interval-valued loss function to each object and further adopting an average strategy to integrate the interval-valued loss functions of objects in each data-driven neighborhood class. Moreover, considering that the objects and attributes of incomplete hybrid data will simultaneously change over time, this paper also provides an efficient framework to dynamically maintain three-way regions of the proposed model. An approach based on matrix to compute the three-way regions is first presented by introducing the matrix operations and the matrix forms of related concepts. Then, with the simultaneous variation of objects and attributes, the matrix-based incremental mechanism and algorithm are proposed for updating the three-way regions, respectively. Experimental results on nine datasets indicate that the proposed incremental algorithm can effectively improve the computational performance for evolving data in comparison with the static algorithm.

Dynamic maintenance of variable precision fuzzy neighborhood three-way regions in interval-valued fuzzy decision system

Dynamic maintenance of variable precision fuzzy neighborhood three-way regions in interval-valued fuzzy decision system

Incremental updating three-way regions with variations of objects and attributes in incomplete neighborhood systems

The data collected from the real world are diverse and include categorical data, numerical data, incomplete data and noisy data. In addition, many real data sets may dynamically vary, and dynamic data display characteristics with multi-dimensional variations. However, for mixed incomplete data systems, most of the existing incremental methods only work well with single-dimensional dynamic data sets and are not suitable for processing specific multi-dimensional variations of objects and attributes. In this paper, we focus on researching dynamic approaches to efficiently update three-way regions based on the simultaneous variations of the object set and the attribute set in an incomplete neighborhood decision system (INDS). First, considering the complexity of data, we utilize matrix approaches to calculate three-way regions of the INDS based on a proposed neighborhood tolerance relation. Then, under the simultaneous addition of the object set and the attribute set in the INDS, we research incremental mechanisms based on the matrix to obtain three-way regions from previous knowledge. Subsequently, an incremental algorithm for updating three-way regions is proposed when the object set and the attribute set are simultaneously added to the INDS. Finally, the results of a series of experiments and comparisons based on UCI data sets show that the performance of the proposed incremental algorithm is much better than that of the traditional static algorithm, the integrated single-dimensional incremental algorithm and the single-level combined incremental algorithm.

Incremental three-way neighborhood approach for dynamic incomplete hybrid data

In practical applications, there generally exist incomplete hybrid data with heterogeneous and missing features. The complex structures and the fast update of incomplete hybrid data bring a series of challenges for decision making in dynamic data environments. Three-way decisions, as an important cognitive method for analyzing uncertain problems, have been extensively applied into various fields. However, the existing studies rarely focus on exploring three-way decisions with incomplete hybrid information. To tackle this issue, we propose a Three-Way Neighborhood Decision Model (TWNDM) based on the data-driven neighborhood relation in terms of two pseudo-distance functions only satisfying the reflexivity. Considering that the addition and deletion of objects will result in the variation of information granules and decision structures, this paper presents a matrix-based dynamic framework for updating three-way regions (positive, boundary and negative regions) in TWNDM. A novel relation matrix is first constructed by using a pair of values to replace single value in the classical relation matrix. Then, the matrix-based approach for computing the three-way regions is established in the light of the new relation matrix, the decision matrix and the related induced matrices. Moreover, the matrix-based incremental mechanisms and algorithms for the maintenance of the three-way regions are presented when adding and removing objects, respectively. The results of comparative experiments demonstrate that the proposed incremental algorithms can improve the computational performance for maintaining three-way regions in TWNDM compared with the static algorithm.

Incremental updating probabilistic neighborhood three-way regions with time-evolving attributes

Neighborhood decision systems (NDSs) are commonly-used systems in real-world applications such as credit scoring. Probabilistic neighborhood rough sets (PNRSs) model, which is a generalized rough set model by combining neighborhood rough sets (NRSs) with probabilistic theory, can efficiently handle numerical data with noise values and allow tolerance of errors. Approximations of a target concept are fundamental and vital notions of the PNRSs, which can induce the certain and uncertain decision rules. Nevertheless, data evolves over time due to dynamic characteristics, especially, the addition of new attributes and the deletion of redundant or irrelevant attributes. Therefore, the potential meaningful knowledge may alter over time accordingly. To improve the efficiency of acquiring decision rules, the three-way regions (i.e., the positive, negative, and boundary regions) of the PNRSs need to be updated in an incremental fashion. To address this issue, two incremental algorithms for updating the three-way regions of each decision class of the PNRSs are investigated from the matrix perspective with time-evolving attributes (namely, adding attributes and deleting attributes). First, a matrix-based characterization of the three-way regions of each decision class of the PNRSs is constructed. Subsequently, considering the variation of the attributes, matrix-based updating mechanisms for relevant matrices are developed by means of the previously calculated information. Furthermore, the incremental algorithms are developed by taking advantage of the matrix-based updating mechanisms. Comprehensive experiments are performed on benchmark UCI data sets, and comparative results demonstrate our proposed algorithms consistently outperform the non-incremental algorithm in terms of computational efficiency.

Three-way confusion matrix for classification: A measure driven view

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.

Evolutionary history of the human multigene families reveals widespread gene duplications throughout the history of animals

BackgroundThe hypothesis that vertebrates have experienced two ancient, whole genome duplications (WGDs) is of central interest to evolutionary biology and has been implicated in evolution of developmental complexity. Three-way and Four-way paralogy regions in human and other vertebrate genomes are considered as vital evidence to support this hypothesis. Alternatively, it has been proposed that such paralogy regions are created by small-scale duplications that occurred at different intervals over the evolution of life.ResultsTo address this debate, the present study investigates the evolutionary history of multigene families with at least three-fold representation on human chromosomes 1, 2, 8 and 20. Phylogenetic analysis and the tree topology comparisons classified the members of 36 multigene families into four distinct co-duplicated groups. Gene families falling within the same co-duplicated group might have duplicated together, whereas genes belong to different co-duplicated groups might have distinct evolutionary origins.ConclusionTaken together with previous investigations, the current study yielded no proof in favor of WGDs hypothesis. Rather, it appears that the vertebrate genome evolved as a result of small-scale duplication events, that cover the entire span of the animals’ history.

Three-way class-specific attribute reducts from the information viewpoint

By virtue of granular computing, attribute reduction of rough sets can effectively perform information processing. Regarding the decision table with granular structures, the traditional classification-based attribute reducts at the macro-top already have the algebra and information interpretations, while the new class-specific attribute reducts at the meso-middle currently have only the algebra explanation. Aiming at the class-specific reducts, this paper establishes three-way types from the information viewpoint, and it further compares the information and algebra types. At first, the three-way information class-specific reducts (including the likelihood, prior, and posterior types) are systematically proposed by using the three-way weighted entropies with uncertainty measurement. Then, optimization preservation conditions of the positive region and weighted entropy are deeply mined to describe reduction targets of the algebra and information types, and relevant conditions related to knowledge coarsening are effectively represented by the granular merging and three-way regions. Furthermore, all four types of class-specific reducts gain their relationships, including the strong-weak relationship based on optimization preservation conditions, the degeneration relationship based on consistency and inconsistency, and the information relationship based on systematic equalities. Finally, algebra and information class-specific reducts and their relationships are verified by a consistency example and an inconsistency example. As a conclusion, the four types of class-specific reducts adopt distinctive algebra or information perspectives to present the difference and emphasis, especially in the inconsistency case, and they have in-depth mutual relationships. This study provides novel insight into attribute reduction based on granular computing, mainly via the information theory and three-way decisions.

Dynamic Online learning Algorithm For Three-way decision

Three-way decision is an important theory for solving uncertain problems. Online computing is a new dynamic Stream computing form. How to execute three-way decision quickly in online computing is a challenging topic. In this paper, Online computing process is divided into incremental computing portion and decreasing computing portion. And a three-way decision dynamic incremental and decreasing learning algorithm for online computing is proposed. Firstly, the dynamic incremental and decreasing learning models is studied for stream computing based on probabilistic rough set . Then, the logical reasoning for three-way decision regions changing are discussed based on the dynamic incremental and decreasing learning models. And a novel dynamic online learning algorithm for three-way decision online computing is proposed based on the above theory. Finally, the experiment by UCI data set show that the proposed algorithms are superior than classical static three-way decision method in time efficiency.

A unified framework of dynamic three-way probabilistic rough sets

The incremental learning technology has been widely applied in efficient and effective data mining with big data based on granular computing, rough sets and three-way approaches. In real-life applications, the information systems will evolve over time with four levels of variational situations, which can be described by the combination of the variations of attributes, objects, condition attributes values and decision attributes values. Considering updating knowledge with multilevel variations of data, this paper proposes a unified dynamic framework of decision-theoretic rough sets for incrementally updating three-way probabilistic regions, namely, positive region, boundary region and negative region. Through improving the representation of three-way regions based on the well-established Bayesian decision procedure, a novel matrix approach is introduced by the construction of Boolean matrix and specific definition of matrix operation. Subsequently, at the variations of level-1, the fundamental updating propositions, which can induce the corresponding propositions with the variations of level-2, level-3, level-4, respectively, are presented by the matrix updating strategies. Finally, experiments with four incremental algorithms developed for the verification of feasibility and efficiency under multilevel variations of data are conducted by comparison with non-incremental algorithm.

A three-way decisions model with probabilistic rough sets for stream computing

Stream computing paradigm, with the characteristics of real-time arrival and departure, has been admitted as a major computing paradigm in big data. Relevant theories are flourishing recently with the surge development of stream computing platforms such as Storm, Kafka and Spark. Rough set theory is an effective tool to extract knowledge with imperfect information, however, related discussions on synchronous immigration and emigration of objects have not been investigated. In this paper, stream computing learning method is proposed on the basis of existing incremental learning studies. This method aims at solving challenges resulted from simultaneous addition and deletion of objects. Based on novel learning method, a stream computing algorithm called single-object stream-computing-based three-way decisions (SS3WD) is developed. In this algorithm, the probabilistic rough set model is applied to approximate the dynamic variation of concepts. Three-way regions can be determined without multiple scans of existing information granular. Extensive experiments not only demonstrate better efficiency and robustness of SS3WD in the presence of objects streaming variation, but also illustrate that stream computing learning method is an effective computing strategy for big data.

Gini objective functions for three-way classifications

The three-way classifications aim to divide the universe of objects into three disjoint regions, i.e., acceptance, rejection, and non-commitment regions. We can induce different types of classification rules from these regions. There exist different measures to evaluate the quality of regions. The partition of the three regions based on certain measures such as Gini coefficient is one of the challenges in three-way classifications. When using Gini coefficients to evaluate the impurities of three-way regions, there may exist contradiction on changing of various regions towards the preferred measure levels. The impurity of one region decreases at the expense of the increase of other regions' impurities when regions change. It is impossible to decrease one region's impurity without increasing the other regions' impurities. In this paper, we formulate Gini objective functions to balance the contradictions among the impurities of three-way regions. Three Gini objective functions, i.e., minimizing the overall impurity of three regions, minimizing impurities of immediate and non-commitment decision regions simultaneously, and minimizing impurities of acceptance, rejection and non-commitment regions simultaneously, are discussed in detail. These Gini objective functions express different preferred situations of three-way regions. The balanced three-way regions representing the trade-off among impurities can be obtained by finding the solutions to these Gini objective functions. An example shows how and what three-way regions are obtained by tuning impurities of these regions to satisfy certain Gini objective functions. It is suggested that with the proposed Gini objective functions more efficient and applicable three-way regions may be induced.