Rough Entropy Research Articles

Evaluation of geomorphological classification uncertainty using rough set theory: A case study of Shaanxi Province, China

AbstractGeomorphological classification is affected by classification principles, indicators, methods, and data resolution, which can lead to uncertainty in the results. Such uncertainty directly affects the quality and subsequent applications of geomorphological classification. To quantify and control the uncertainty, it is important to select an appropriate and effective method for evaluating the uncertainty of geomorphological classification. This study evaluated the uncertainty of geomorphological classification of Shaanxi Province at the ground‐feature class and image scales, which derived from rough set theory: rough entropy, approximate classification quality, and approximate classification accuracy. The three indicators helped effectively assess the uncertainty of geomorphological classification at multi‐scale and measured the degree to which different factors affected the uncertainty of geomorphological classification. The relative impacts of three factors on the uncertainty of classification decreased in the order of classification methods, data resolution, and classification indicators. This finding is helpful to objectively evaluate and control the uncertainty generated in the process and results of geomorphological classification, and can provide targeted reference and guidance for future geomorphological classification work, which is more conducive to decision‐making and application. At the same time, this study is also a beneficial supplement to the geomorphological research based on digital terrain analysis.

Rough set model of incomplete interval rough number decision systems

The rough set model has been extended to interval rough number decision systems, but the existing studies do not consider interval rough number decision systems with missing values. To this end, a rough set model of incomplete interval rough number decision systems (IIRNDSs) is proposed, and its uncertainty measures are investigated. Firstly, the similarity of two incomplete interval rough numbers (IIRNs) are defined by calculating their optimistic and pessimistic distances of the lower and upper approximation intervals of IIRNs. Then, the rough sets in IIRNDSs are constructed by the induced similarity relation. Next, four uncertainty measures, including approximation accuracy, approximation roughness, conditional entropy, and decision rough entropy are given, which exhibit a monotonic variation with changes in the size of attribute sets, α, and θ. Finally, the experimental results demonstrate the proposed rough set model of IIRNDSs is feasible and effective.

Hybrid similarity relation based mutual information for feature selection in intuitionistic fuzzy rough framework and its applications

Fuzzy rough entropy established in the notion of fuzzy rough set theory, which has been effectively and efficiently applied for feature selection to handle the uncertainty in real-valued datasets. Further, Fuzzy rough mutual information has been presented by integrating information entropy with fuzzy rough set to measure the importance of features. However, none of the methods till date can handle noise, uncertainty and vagueness simultaneously due to both judgement and identification, which lead to degrade the overall performances of the learning algorithms with the increment in the number of mixed valued conditional features. In the current study, these issues are tackled by presenting a novel intuitionistic fuzzy (IF) assisted mutual information concept along with IF granular structure. Initially, a hybrid IF similarity relation is introduced. Based on this relation, an IF granular structure is introduced. Then, IF rough conditional and joint entropies are established. Further, mutual information based on these concepts are discussed. Next, mathematical theorems are proved to demonstrate the validity of the given notions. Thereafter, significance of the features subset is computed by using this mutual information, and corresponding feature selection is suggested to delete the irrelevant and redundant features. The current approach effectively handles noise and subsequent uncertainty in both nominal and mixed data (including both nominal and category variables). Moreover, comprehensive experimental performances are evaluated on real-valued benchmark datasets to demonstrate the practical validation and effectiveness of the addressed technique. Finally, an application of the proposed method is exhibited to improve the prediction of phospholipidosis positive molecules. RF(h2o) produces the most effective results till date based on our proposed methodology with sensitivity, accuracy, specificity, MCC, and AUC of 86.7%, 90.1%, 93.0% , 0.808, and 0.922 respectively.

Uncertainty measurement for incomplete lattice-valued information system

We analyze the properties and characteristics of the information structure in incomplete lattice-valued information system (ILIS), we redefine the information structure and the dependence and information distance between the two information structures. In addition, in order to evaluate the uncertainty of ILIS, the concepts of granular measure and entropy measure are expounded, including information granulation, information quantity, rough entropy and information entropy. Finally, we carry out numerical experiments to verify the feasibility of the method, and carry out effective statistical analysis. These results are conducive to the establishment of granular computing framework in ILIS.

Exploiting fuzzy rough entropy to detect anomalies

Anomaly detection has been used in a wide range of fields. However, most of the current detection methods are only applicable to certain data, ignoring uncertain information such as fuzziness in the data. Fuzzy rough set theory, as an essential mathematical model for granular computing, provides an effective method for processing uncertain data such as fuzziness. Fuzzy rough entropy has been proposed in fuzzy rough set theory and has been employed successfully in data analysis tasks such as feature selection. However, it mainly uses the intersection operation, which may not effectively reflect the similarity between high-dimensional objects. In response to the two challenges mentioned above, distance-based fuzzy rough entropy and its correlation measures are proposed. Further, the proposed fuzzy rough entropy is used to construct the anomaly detection model and the Fuzzy Rough Entropy-based Anomaly Detection (FREAD) algorithm is designed. Finally, the FREAD algorithm is compared and analyzed with some mainstream anomaly detection algorithms (including COF, DIS, INFLO, LDOF, LoOP, MIX, ODIN, SRO, and VarE algorithms) on some publicly available datasets. Experimental results indicate that the FREAD algorithm significantly outperforms other algorithms in terms of performance and flexibility. The code is publicly available online at https://github.com/optimusprimeyy/FREAD.

CDRKD: An improved density peak algorithm based on kernel fuzzy measure in the overlapping community detection

Compared with other traditional community discovery algorithms, density peak clustering algorithm is more efficient in getting network structures through clustering. However, DPC needs to contain the distance information between all nodes as sources, so it cannot directly processing the complex network represented by the adjacency matrix. DPC introduces truncation distance when calculating the local density of nodes, which is usually set as a fixed value according to experience, and lacks self-adaptability for different network structures. A feasible solution to those problems is to combined rough set theory and kernel fuzzy similarity measures. In this work, we present overlapping community detection algorithm based on improved rough entropy fusion density peak. The algorithm applied rough set theory to attribute reduction of massive high-dimensional data. Another algorithm defines the similarity of sample points by the inner product between two vectors on the basis of fuzzy partition matrix. Finally, a community detection algorithm based on rough entropy and kernel fuzzy density peaks clustering (CDRKD) has proposed by combining the two algorithms above, we perform an extensive set of experiments to verify the effectiveness and feasibility of the algorithm.

Uncertainty measure for Z-soft covering based rough graphs with application

Soft graphs are an interesting way to represent specific information. In this paper, a new form of graphs called Z-soft covering based rough graphs using soft adhesion is defined. Some important properties are explored for the newly constructed graphs. The aim of this study is to investigate the uncertainty in Z-soft covering based rough graphs. Uncertainty measures such as information entropy, rough entropy and granularity measures related to Z-soft covering-based rough graphs are discussed. In addition, we develop a novel Multiple Attribute Group Decision-Making (MAGDM) model using Z-soft covering based rough graphs in medical diagnosis to identify the patients at high risk of chronic kidney disease using the collected data from the UCI Machine Learning Repository.

Exploiting fuzzy rough mutual information for feature selection

Feature selection is one of the important applications of rough set theory. Rough entropy proposed in rough set theory has been applied to feature selection. However, rough entropy is based on binary equivalence relation to divide object sets. Therefore, it applies only to nominal attribute data. To this end, this paper extends rough entropy to fuzzy rough set theory, and then proposes the fuzzy rough entropy in fuzzy approximate space. Fuzzy rough entropy decreases monotonically with the increase of the number of features. On this basis, the concepts of fuzzy joint rough entropy, fuzzy conditional rough entropy and fuzzy rough mutual information are defined. In order to measure the importance of features, inner and outer significance functions are constructed by making use of fuzzy rough mutual information. Furthermore, based on the proposed fuzzy rough entropy model, the corresponding feature selection algorithm is designed. It can directly deal with not only nominal data, but also numerical data and even mixed data. The use of inner and outer heuristic significance functions makes the proposed method select features from two complementary perspectives, so that the reduced feature set has better classification performance. In this method, the redundant features are effectively deleted by the backward redundancy elimination strategy. The proposed algorithm is compared with other algorithms on public data. The experimental results show that the proposed method is adaptive and effective.

Design and Research of Economic Management Problem Fusion Method Based on Decision Information System

The construction of the economic management information system ensures the flow and sharing of information, business coordination, and scientific decision-making between economic functions. Therefore, project construction is absolutely necessary and urgent. Project management has achieved significant results in terms of shortening project time and reducing management costs. Information system is an important model information structure in artificial intelligence. This paper focuses on the design of a fusion method for economic management problems based on decision information systems. This paper summarizes the value of the whole project through the research on the project management of the economic management information system. The results of this study show that when the threshold is fixed, the information granularity Gtheta and rough entropy (Er) theta decrease monotonically with increasing attribute subset cardinality. Meanwhile, θ-information granularity Etheta and θ-information entropy Htheta monotonically increase with the increase of attribute subset cardinality, which indicates that when attribute subset cardinality increases, the uncertainty of the information system decreases.

Adaptive granulation Renyi rough entropy image thresholding method with nested optimization

The rough entropy based image thresholding algorithm can effectively deal with the uncertainty information in an image. Nevertheless, the existing rough entropy based thresholding methods have two limitations. One is that there is no generalized rough entropy definition. The other is that the granule size cannot be selected automatically in combination with the image information in the granulation process. In order to address these two problems, an adaptive granulation Renyi rough entropy image thresholding method with nested optimization is proposed in this paper. First, a definition of generalized rough entropy with parameter based on Renyi entropy form is proposed to describe the uncertainty information of complex images. Second, in order to select the granule size in combination with the specific image information, a granule size selection method is proposed by maximizing the uniformity of the segmented image regions. Finally, a nested optimization adaptive granulation Renyi rough entropy thresholding segmentation algorithm is proposed. In the experiments, the Renyi rough entropy is compared with the existing four rough entropy in image thresholding. The new parametric rough entropy can obtain better segmentation results than the other existing rough entropy. The novel adaptive granulation Renyi rough entropy thresholding segmentation algorithm is compared with the state-of-the-art image segmentation methods on several different image data sets, which verifies the effectiveness of the novel algorithm.

Measures of uncertainty for a fuzzy probabilistic approximation space

An approximation space (A-space) is the base of rough set theory and a fuzzy approximation space (FA-space) can be seen as an A-space under the fuzzy environment. A fuzzy probability approximation space (FPA-space) is obtained by putting probability distribution into an FA-space. In this way, it combines three types of uncertainty (i.e., fuzziness, probability and roughness). This article is devoted to measuring the uncertainty for an FPA-space. A fuzzy relation matrix is first proposed by introducing the probability into a given fuzzy relation matrix, and on this basis, it is expanded to an FA-space. Then, granularity measurement for an FPA-space is investigated. Next, information entropy measurement and rough entropy measurement for an FPA-space are proposed. Moreover, information amount in an FPA-space is considered. Finally, a numerical example is given to verify the feasibility of the proposed measures, and the effectiveness analysis is carried out from the point of view of statistics. Since three types of important theories (i.e., fuzzy set theory, probability theory and rough set theory) are clustered in an FPA-space, the obtained results may be useful for dealing with practice problems with a sort of uncertainty.

Greedy Autoaugment for classification of mycobacterium tuberculosis image via generalized deep CNN using mixed pooling based on minimum square rough entropy

Although tuberculosis (TB) is a disease whose cause, epidemiology and treatment are well known, some infected patients in many parts of the world are still not diagnosed by current methods, leading to further transmission in society. Creating an accurate image-based processing system for screening patients can help in the early diagnosis of this disease. We provided a dataset containing1078 confirmed negative and 469 positive Mycobacterium tuberculosis instances. An effective method using an improved and generalized convolutional neural network (CNN) was proposed for classifying TB bacteria in microscopic images. In the preprocessing phase, the insignificant parts of microscopic images are excluded with an efficient algorithm based on the square rough entropy (SRE) thresholding. Top 10 policies of data augmentation were selected with the proposed model based on the Greedy AutoAugment algorithm to resolve the overfitting problem. In order to improve the generalization of CNN, mixed pooling was used instead of baseline one. The results showed that employing generalized pooling, batch normalization, Dropout, and PReLU have improved the classification of Mycobacterium tuberculosis images. The output of classifiers such as Naïve Bayes-LBP, KNN-LBP, GBT-LBP, Naïve Bayes-HOG, KNN-HOG, SVM-HOG, GBT-HOG indicated that proposed CNN has the best results with an accuracy of 93.4%. The improvements of CNN based on the proposed model can yield promising results for diagnosing TB.

Fuzzy Rough Information Measures and their Applications

The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate enough. Some authors use information measure in place of entropy for better understanding which measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough information measures are proposed and their validity is verified. The application of these proposed information measures in decision making problems is studied and also compared with other existing information measures.

New uncertainty measurement for categorical data based on fuzzy information structures: An application in attribute reduction

Categorical data is a significant kind of data in machine learning.Generally, rough set theory (RS-theory) deals with categorical data in the following way.First, an equivalence relation based on the equality of attribute values of categorical data is established.Then, information granules (I-granules) based on equivalence classes are obtained.Finally, information structures (I-structures) consisting of I-granules are formed.However, an equivalence relation is too strict, and there are some limitations in the I-structure of a categorical information system (CIS) that may result in filtering out potentially useful information.This paper investigates fuzzy information structures (FI-structures) and new uncertainty measurements for categorical data from the perspective that “the equality of attribute values is fed back to the attribute set”.First, a fuzzy symmetry relation based on the number of attributes with equal attribute values is established. Then, fuzzy information granules (FI-granules) based on the fuzzy symmetry relation are obtained. Next, FI-structures consisting of FI-granules are formed.Finally, some concepts related to FI-structures in a CIS are given.The set vector is used to denote FI-structures, and the inclusion degree is used to study the dependence between FI-structures.In addition, four new uncertainty measurements based on FI-structures in a CIS are proposed, including fuzzy information granulation (Gf), fuzzy information entropy (Hf), fuzzy rough entropy (Erf) and fuzzy information amount (Ef).Moreover, numerical experiments and statistical tests to evaluate the performance of the proposed new measurements are carried out.The results of the paired t-test show that the performance of the four new measurements based on FI-structures is better than that of the corresponding four measurements based on I-structures.Finally, attribute reduction algorithms based on Gf and Hf are presented, and clustering analysis is conducted on the reduced CIS. The experimental results show that the proposed algorithms are effective and perform well on attribute reduction according to three evaluation indicators of clustering performance.

Measures of uncertainty for a fuzzy probabilistic information system

Fuzzy probabilistic information system (FPIS), a combination of some fuzzy relations on the same universe which satisfies probability distribution, can be regarded as an information system (IS) with fuzzy relations under the probability environment. This means that three types of uncertainty: roughness fuzziness and probability are integrated into an IS. This paper aims to measure the uncertainty of a FPIS. Firstly, information granulation in a FPIS is proposed. Then, information entropy and rough entropy in a FPIS are proposed. Moreover, information amount in a FPIS is also investigated. Finally, the feasibility of the proposed measures are verified by a numerical example, and the statistical analysis is carried out. Those obtained results may be useful for understanding the essence of uncertainty for a FPIS.

Coevolutionary Fuzzy Attribute Order Reduction With Complete Attribute-Value Space Tree

Since big data sets are structurally complex, high-dimensional, and their attributes exhibit some redundant and irrelevant information, the selection, evaluation, and combination of those large-scale attributes pose huge challenges to traditional methods. Fuzzy rough sets have emerged as a powerful vehicle to deal with uncertain and fuzzy attributes in big data problems that involve a very large number of variables to be analyzed in a very short time. In order to further overcome the inefficiency of traditional algorithms in the uncertain and fuzzy big data, in this paper we present a new coevolutionary fuzzy attribute order reduction algorithm (CFAOR) based on a complete attribute-value space tree. A complete attribute-value space tree model of decision table is designed in the attribute space to adaptively prune and optimize the attribute order tree. The fuzzy similarity of multimodality attributes can be extracted to satisfy the needs of users with the better convergence speed and classification performance. Then, the decision rule sets generate a series of rule chains to form an efficient cascade attribute order reduction and classification with a rough entropy threshold. Finally, the performance of CFAOR is assessed with a set of benchmark problems that contain complex high dimensional datasets with noise. The experimental results demonstrate that CFAOR can achieve the higher average computational efficiency and classification accuracy, compared with the state-of-the-art methods. Furthermore, CFAOR is applied to extract different tissues surfaces of dynamical changing infant cerebral cortex and it achieves a satisfying consistency with those of medical experts, which shows its potential significance for the disorder prediction of infant cerebrum.

Decision-making approaches based on color spectrum and D-TOPSIS method under rough environment

A rough set, an extension of a crisp set, is a mathematical tool to understand and model uncertainty without much prior information, additional adjustments or pre-defined membership functions. To manipulate the subjectivity and vagueness of decision-making problems, rough models provide more objective description of given information using upper and lower approximations. In this research paper, we study the absurdity and falsity of existing definition of rough graph. Based on rough relations, we introduce the concepts of rough graphs and rough digraphs and establish certain formulae, lower and upper bounds of color energy of rough graphs. Using D numbers, rough weights and rough entropy weights, we develop rough D-TOPSIS method which incorporates the capability to analyze uncertain and vague information without additional assumptions. We study the importance of rough information for the evaluation of water requirement in agricultural farming, investment analysis in organic and inorganic farming systems and illegal communication networks.

Measures of Uncertainty Based on Gaussian Kernel for Type-2 Fuzzy Information Systems

In data processing, measurement of uncertainty is one of the significant evaluation tools, which can describe the uncertainty essence of data. So far, there are few measurable tools to study the uncertainty of type-2 fuzzy information systems (TFISs-2) (the expanded models of fuzzy information systems). This paper is devoted to looking for effective indicators to describe the uncertainty of TFISs-2. The fuzzy $$T_{\text {cos}}$$ -similarity relations are first introduced, which are generated by TFISs-2 based on Gaussian kernel. Then, the fuzzy information structures are defined on account of this fuzzy $$T_{\text {cos}}$$ -similarity relation. Next, two measures constructed from the upper and lower approximations are given for TFISs-2, that is, $$\delta$$ -accuracy and $$\delta$$ -roughness, which are used to reflect the degree of accuracy and inaccuracy of information by numerical form. Furthermore, by combining roughness and entropy, the $$\delta$$ -rough entropy is investigated. Finally, the practicability of proposed measures is tested by a numerical experiment. The experimental results show that the $$\delta$$ -rough entropy is efficacious and applicable for TFISs-2.

An Intuitionistic Fuzzy Approach With Rough Entropy Measure to Detect Outliers in Two Universal Sets

The process of recognizing patterns, collecting knowledge from massive databases is called data mining. An object which does not obey and deviates from other objects by their characteristics or behavior are known as outliers. Research works carried out so far on outlier detection were focused only on numerical data, categorical data, and in single universal sets. The main goal of this article is to detect outliers significant in two universal sets by applying the intuitionistic fuzzy cut relationship based on membership and non-membership values. The proposed method, weighted density outlier detection, is based on rough entropy, and is employed to detect outliers. Since it is unsupervised, without considering class labels of decision attributes, weighted density values for all conditional attributes and objects are calculated to detect outliers. For experimental analysis, the Iris dataset from the UCI repository is taken to detect outliers, and comparisons have been made with existing algorithms to prove its efficiency.

The uncertainty measures for covering rough set models

Uncertainty measures are important tools for analyzing various data. However, there are relatively few studies on the uncertainty measures for covering rough set models. In this paper, from the viewpoint of the lower and upper approximations, we propose new uncertainty measures, the lower rough entropy and the upper rough entropy, for covering rough set models. Then, we define the concepts of the joint entropy and the conditional entropy in the covering rough set models. Some important properties of these measures are obtained, and their relationships are investigated. Furthermore, we provide a certain characterization of reducible element of a covering by means of the proposed measures, and apply the proposed rough entropy to evaluate the significance of covering granules of a covering. Finally, we apply these rough entropies to measure a dual degree between covering lower and upper approximations. The theoretical analysis and examples show that the proposed uncertainty measures for covering rough set models are reasonable and useful.