Feature selection in the data with different types of feature values, i.e., the heterogeneous or mixed data, is especially of practical importance because such types of data sets widely exist in real ...world. The key issue for feature selection in mixed data is how to properly deal with different types of the features or attributes in the data set. Motivated by the fuzzy rough set theory which allows different fuzzy relations to be defined for different types of attributes to measure the similarity between objects and in view of the effectiveness of entropy to measure information uncertainty, we propose in this paper a fuzzy rough set-based information entropy for feature selection in a mixed data set. It is proved that the newly-defined entropy meets the common requirement of monotonicity and can equivalently characterize the existing attribute reductions in the fuzzy rough set theory. Then, a feature selection algorithm is formulated based on the proposed entropy and a filter-wrapper method is suggested to select the best feature subset in terms of classification accuracy. An extensive numerical experiment is further conducted to assess the performance of the feature selection method and the results are satisfactory.
•A novel fuzzy rough set-based information entropy is constructed for mixed data.•The proposed entropy can equivalently characterize the existing attribute reductions in the fuzzy rough set theory.•A feature selection algorithm is formulated based on the proposed entropy.•A filter-wrapper method is suggested to select a best feature subset.
Abstract We investigate a model of becoming—classical sequential growth (CSG)—that has been proposed within the framework of causal sets (causets), with the latter defined as order types of certain ...partial orderings. To investigate how causets grow, we introduce special sequences of causets, which we call “csg-paths”. We prove a number of results concerning relations between csg-paths and causets. These results paint a highly non-trivial picture of csg-paths. There are uncountably many csg-paths, all of them sharing the same beginning, after which they branch. Every infinite csg-path achieves in the limit an infinite causet, and vice versa, every infinite causet is achieved in the limit by an infinite csg-path. However, coalescing csg-paths, i.e., ones that achieve the same causet even after forking off at some point, are ubiquitous.
•Proposing a novel integration of Z numbers and Best Worst Method.•The method results in lower inconsistency.•The uncertainty of the real word decisions is considered in the proposed method.
Best ...Worst Method (BWM) has recently been proposed as a method for Multi Criteria Decision Making (MCDM). Studies show that BWM compared with other methods such as Analytic Hierarchy Process (AHP), leads to lower inconsistency of the results while reducing the number of required pairwise comparisons. MCDM methods such as BWM require accurate information. However, it often happens in practice that a level of uncertainty accompanies the information. The main aim of this paper is to address this problem and provide an integration of BWM and Z-numbers, namely ZBWM. Providing BWM with Z-numbers enables the BWM method to handle the uncertainty of information of a multi-criteria decision. Additionally, the capabilities of the proposed method in the process of utilizing the linguistic information dealing with big data are highlighted. The proposed method is examined to address a supplier development problem. By experimental results, we show that ZBWM results lower inconsistency when compared with BWM. A Z-number contains subjectivity in its fuzzy part, which can be addressed in future applications of ZBWM.
This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27-29, 2015, at Harvard University. It presents a collection of papers ...related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s.The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters.This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas.
Feature selection aims to select a feature subset from an original feature set based on a certain evaluation criterion. Since feature selection can achieve efficient feature reduction, it has become ...a key method for data preprocessing in many data mining tasks. Recently, many feature selection strategies have been developed since in most cases it is infeasible to obtain an optimal/reduced feature subset by using exhaustive search. Among these strategies, fuzzy rough set theory has proved to be an ideal candidate for dealing with uncertain information. This article provides a comprehensive review on the fuzzy rough set theory and two fuzzy rough set theory based feature selection methods, that is, fuzzy rough set based feature selection methods and fuzzy rough neural network based feature selection methods. We review the publications related to the fuzzy rough theory and its applications in feature selection. In addition, the challenges in the two types of feature selection methods are also discussed.
This article is categorized under:
Technologies > Machine Learning
A common framework of fuzzy rough set theory based feature selection approaches.
Attribute reduction is one of the biggest challenges encountered in computational intelligence, data mining, pattern recognition, and machine learning. Effective in feature selection as the rough set ...theory is, it can only handle symbolic attributes. In order to overcome this drawback, the fuzzy rough set model is proposed, which is an extended model of rough sets and is able to deal with imprecision and uncertainty in both symbolic and numerical attributes. The existing attribute selection algorithms based on the fuzzy rough set model mainly take the angle of "attribute set," which means they define the object function representing the predictive ability for an attribute subset with regard to the domain of discourse, rather than following the view of an "object pair." Algorithms from the viewpoint of the object pair can ignore the object pairs that are already discerned by the selected attribute subsets and, thus, need only to deal with part of object pairs instead of the whole object pairs from the discourse, which makes such algorithms more efficient in attribute selection. In this paper, we propose the concept of reduced maximal discernibility pairs, which directly adopts the perspective of the object pair in the framework of the fuzzy rough set model. Then, we develop two attribute selection algorithms, named as reduced maximal discernibility pairs selection and weighted reduced maximal discernibility pair selection, based on the reduced maximal discernibility pairs. Experiment results show that the proposed algorithms are effective and efficient in attribute selection.
By introducing the new concepts of fuzzy β-covering and fuzzy β-neighborhood, we define two new types of fuzzy covering rough set models which can be regarded as bridges linking covering rough set ...theory and fuzzy rough set theory. We show the properties of the two models, and reveal the relationships between the two models and some others. Moreover, we present the matrix representations of the newly defined lower and upper approximation operators so that the calculation of lower and upper approximations of subsets can be converted into operations on matrices. Finally, we generalize the models and their matrix representations to L-fuzzy covering rough sets which are defined over fuzzy lattices.
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