Himpunan Fuzzy dan Rough Sets

The concept of a fuzzy set was introduced by Zadeh in 1965. Fuzzy set is a mathematical model of vague qualitative or quantitative data, frequently generated by means of the natural language. The model is based on the generalization of the classical concepts of set and its characteristic function. Intuitionistic fuzzy sets are sets whose elements have degrees of membership and non-membership. Intuitionistic fuzzy sets have been introduced by Atanassov in 1983 as an extension fuzzy sets. On the other hand, the concept of rough set was proposed by Pawlak 1982. Since then the subject has been investigated in many papers. The overall aim of this paper is to present an introduction to some of main concepts related to fuzzy sets, intuitionistic fuzzy sets and rough sets. We investigate Crisp sets and characteristic functions, fuzzy sets, intuitionistic fuzzy sets, rough sets and probabilistic rough sets

Similarity Measure: An Intuitionistic Fuzzy Rough Set Approach

‎In fuzzy set theory‎, ‎the concept of a non-membership function and the hesitation margin were not considered while these two concepts have been included along with the membership function for intuitionistic fuzzy sets‎. ‎It is also to be noted that the intuitionistic fuzzy set is reflected as an extension of the fuzzy set accommodating both membership and non-membership functions together with a hesitation margin‎. ‎In the intuitionistic fuzzy set theory‎, ‎the sum of the membership function and the non-membership function is a value between 0 and 1‎. ‎In recent times‎, ‎intuitionistic fuzzy rough set theory has emerged as a powerful tool for dealing with imprecision and uncertain information in relational database theory‎. ‎Measures of similarity between fuzzy rough sets as well as intuitionistic fuzzy rough sets provide wide applications in real-life problems and that is why many researchers paid more attention to this concept‎. ‎Intuitionistic fuzzy rough set theory behaves like an excellent tool to tackle impreciseness or uncertainties‎. ‎In this paper‎, ‎we propose a new approach of similarity measure on an intuitionistic fuzzy rough set based on a set-theoretic approach‎. ‎The proposed measure is able to give an exact result‎. ‎In the application part‎, ‎we consider a real-life problem for selecting a fair play award-winning team in a cricket tournament and describe the algorithm‎

Breast Tissue Classification via Interval Type 2 Fuzzy Logic Based Rough Set

BIRADS is a Breast Imaging, Reporting and Data System. A tool to standardize mammogram reports and minimizes ambiguity during mammogram image evaluation. Classification of BIRADS is one of the most challenging tasks to radiologist. An apt treatment can be administered to the patient by the oncologist upon acquiring sufficient information at BIRADS stage. This study aspired to build a model, which classifies BIRADS using mammograms images and reports. Through the implementation of type-2 fuzzy logic as classifier, an automatically generated rules will be applied to the model. Comparison of accuracy, specificity and sensitivity of the modal will be performed vis-à-vis rules given by the experts. The study encompasses a number of steps beginning with collection of the data from Radiology Department of National University of Malaysia Medical Center. The data was initially processed to remove noise and gaps. Then, an algorithm developed by selecting type-2 fuzzy logic using Mamdani model. Three types of membership functions were employed in the study. Among the rules that used by the model were obtained from experts as well as generated automatically by the system using rough set theory. Finally, the model was tested and trained to get the best result. The study shows that triangular membership function based on rough set rules obtains 89% whereas expert driven rules gains about 78% of accuracy rates. The sensitivity using expert rules is 98.24% whereas rough set rules obtained 93.94%. Specificity for using expert rules and rough set rules are 73.33%, 84.34% consecutively. Conclusion: Based on statistical analysis, the model which employed rules generated automatically by rough set theory fared better in comparison to the model using rules given by the experts.

Bipolar quadripartitioned single valued neutrosophic rough set

Here bipolar quadripartitioned single valued neutrosophic rough (BQSVNR) set is introduced. Some basic set theoretic terminologies like constant BQSVNR set, subsethood of two BQSVNR sets are shown. Algebraic operations like union, intersection and complement have also been dened. Dierent types of measure like similarity measure, quasi similarity measure and distance measures between two BQSVNR sets have been discussed with their properties. Again various measures of similarity namely distance based similarity measure, cosine similarity measure, membership function based similarity measure are introduced in this paper. A medical diagonasis problem has been solved using similarity measure at the end

REDUCED ATTRIBUTE ORIENTED HANDLING OF INCONSISTENCY IN DECISION GENERATION

Abstract Due to the discarded attributes, the effectual condition classes of the decision rules are highly different. To provide a unified evaluative measure, the derivation of each rule is depicted by the reduced attributes with a layered manner. Therefore, the inconsistency is divided into two primary categories in terms of the reduced attributes. We introduce the notion of joint membership function wrt. the effectual joint attributes, and a classification method extended from the default decision generation framework is proposed to handle the inconsistency. Keywords: reduced attributes, reduced layer, joint membership function, rough set Introduction Classification in rough set theory [1] is mainly composed of two components: feature extraction and decision synthesis. Many researches focus on the construction of classification algorithm, such as probabilistic method [2], decision trees[3] and parameterized rule inducing method This paper, based on the default rule extracting framewor

Three-way Imbalanced Learning based on Fuzzy Twin SVM

Three-way decision (3WD) is a powerful tool for granular computing to deal with uncertain data, commonly used in information systems, decision-making, and medical care. Three-way decision gets much research in traditional rough set models. However, three-way decision is rarely combined with the currently popular field of machine learning to expand its research. In this paper, three-way decision is connected with SVM, a standard binary classification model in machine learning, for solving imbalanced classification problems that SVM needs to improve. A new three-way fuzzy membership function and a new fuzzy twin support vector machine with three-way membership (TWFTSVM) are proposed. The new three-way fuzzy membership function is defined to increase the certainty of uncertain data in both input space and feature space, which assigns higher fuzzy membership to minority samples compared with majority samples. To evaluate the effectiveness of the proposed model, comparative experiments are designed for forty-seven different datasets with varying imbalance ratios. In addition, datasets with different imbalance ratios are derived from the same dataset to further assess the proposed model's performance. The results show that the proposed model significantly outperforms other traditional SVM-based methods

The Quantification of Perception Based Uncertainty Using R-fuzzy Sets and Grey Analysis

The nature of uncertainty cannot be generically defined as it is domain and context specific. With that being the case, there have been several proposed models, all of which have their own associated benefits and shortcomings. From these models, it was decided that an R-fuzzy approach would provide for the most ideal foundation from which to enhance and expand upon. An R-fuzzy set can be seen as a relatively new model, one which itself is an extension to fuzzy set theory. It makes use of a lower and upper approximation bounding from rough set theory, which allows for the membership function of an R-fuzzy set to be that of a rough set. An R-fuzzy approach provides the means for one to encapsulate uncertain fuzzy membership values, based on a given abstract concept. If using the voting method, any fuzzy membership value contained within the lower approximation can be treated as an absolute truth. The fuzzy membership values which are contained within the upper approximation, may be the result of a singleton, or the vast majority, but absolutely not all. This thesis has brought about the creation of a significance measure, based on a variation of Bayes' theorem. One which enables the quantification of any contained fuzzy membership value within an R-fuzzy set. Such is the pairing of the significance measure and an R-fuzzy set, an intermediary bridge linking to that of a generalised type-2 fuzzy set can be achieved. Simply by inferencing from the returned degrees of significance, one is able to ascertain the true significance of any uncertain fuzzy membership value, relative to other encapsulated uncertain values. As an extension to this enhancement, the thesis has also brought about the novel introduction of grey analysis. By utilising the absolute degree of grey incidence, it provides one with the means to measure and quantify the metric spaces between sequences, generated based on the returned degrees of significance for any given R-fuzzy set. As it will be shown, this framework is ideally suited to domains where perceptions are being modelled, which may also contain several varying clusters of cohorts based on any number of correlations. These clusters can then be compared and contrasted to allow for a more detailed understanding of the abstractions being modelled

Rough Neural Networks Architecture For Improving Generalization In Pattern Recognition

Neural networks are found to be attractive trainable machines for pattern recognition. The capability of these models to accommodate wide variety and variability of conditions, and the ability to imitate brain functions, make them popular research area. This research focuses on developing hybrid rough neural networks. These novel approaches are assumed to provide superior performance with respect to detection and automatic target recognition.In this thesis, hybrid architectures of rough set theory and neural networks have been investigated, developed, and implemented. The first hybrid approach provides novel neural network referred to as Rough Shared weight Neural Networks (RSNN). It uses the concept of approximation based on rough neurons to feature extraction, and experiences the methodology of weight sharing. The network stages are a feature extraction network, and a classification network. The extraction network is composed of rough neurons that accounts for the upper and lower approximations and embeds a membership function to replace ordinary activation functions. The neural network learns the rough set’s upper and lower approximations as feature extractors simultaneously with classification. The RSNN implements a novel approximation transform. The basic design for the network is provided together with the learning rules. The architecture provides a novel method to pattern recognition and is expected to be robust to any pattern recognition problem. The second hybrid approach is a two stand alone subsystems, referred to as Rough Neural Networks (RNN). The extraction network extracts detectors that represent pattern’s classes to be supplied to the classification network. It works as a filter for original distilled features based on equivalence relations and rough set reduction, while the second is responsible for classification of the outputs from the first system. The two approaches were applied to image pattern recognition problems. The RSNN was applied to automatic target recognition problem. The data is Synthetic Aperture Radar (SAR) image scenes of tanks, and background. The RSNN provides a novel methodology for designing nonlinear filters without prior knowledge of the problem domain. The RNN was used to detect patterns present in satellite image. A novel feature extraction algorithm was developed to extract the feature vectors. The algorithm enhances the recognition ability of the system compared to manual extraction and labeling of pattern classes. The performance of the rough backpropagation network is improved compared to backpropagation of the same architecture. The network has been designed to produce detection plane for the desired pattern. The hybrid approaches developed in this thesis provide novel techniques to recognition static and dynamic representation of patterns. In both domains the rough set theory improved generalization of the neural networks paradigms. The methodologies are theoretically robust to any pattern recognition problem, and are proved practically for image environments

Study on Rough Sets and Fuzzy Sets in Constructing Intelligent Information System

Since human being is not an omniscient and omnipotent being, we are actually living in an uncertain world. Uncertainty was involved and connected to every aspect of human life as a quotation from Albert Einstein said: �As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.� The most fundamental aspect of this connection is obviously shown in human communication. Naturally, human communication is built on the perception1-based information instead of measurement-based information in which perceptions play a central role in human cognition [Zadeh, 2000]. For example, it is naturally said in our communication that �My house is far from here.� rather than let say �My house is 12,355 m from here�. Perception-based information is a generalization of measurement-based information, where perception-based information such as �John is excellent.� is hard to represent by measurement-based version. Perceptions express human subjective view. Consequently, they tend to lead up to misunderstanding. Measurements then are needed such as defining units of length, time, etc., to provide objectivity as a means to overcome misunderstanding. Many measurers were invented along with their methods and theories of measurement. Hence, human cannot communicate with measurers including computer as a product of measurement era unless he uses measurement-based information. Perceptions are intrinsic aspect in uncertainty-based information. In this case, information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. 1In psychology, perception is understood as a process of translating sensory stimulation into an organized experience Generally, these various information deficiencies may express different types of uncertainty. It is necessary to construct a computer-based information system called intelligent information system that can process uncertainty-based information. In the future, computers are expected to be able to make communication with human in the level of perception. Many theories were proposed to express and process the types of uncertainty such as probability, possibility, fuzzy sets, rough sets, chaos theory and so on. This book extends and generalizes existing theory of rough set, fuzzy sets and granular computing for the purpose of constructing intelligent information system. The structure of this book is the following: In Chapter 2, types of uncertainty in the relation to fuzziness, probability and evidence theory (belief and plausibility measures) are briefly discussed. Rough set regarded as another generalization of crisp set is considered to represent rough event in the connection to the probability theory. Special attention will be given to formulation of fuzzy conditional probability relation generated by property of conditional probability of fuzzy event. Fuzzy conditional probability relation then is used to represent similarity degree of two fuzzy labels. Generalization of rough set induced by fuzzy conditional probability relation in terms of covering of the universe is given in Chapter 3. In the relation to fuzzy conditional probability relation, it is necessary to consider an interesting mathematical relation called weak fuzzy similarity relation as a generalization of fuzzy similarity relation proposed by Zadeh [1995]. Fuzzy rough set and generalized fuzzy rough set are proposed along with the generalization of rough membership function. Their properties are examined. Some applications of these methods in information system such as α-redundancy of object and dependency of domain attributes are discussed. In addition, multi rough sets based on multi-context of attributes in the presence of multi-contexts information system is defined and proposed in Chapter 4. In the real application, depending on the context, a given object may have different values of attributes. In other words, set of attributes might be represented based on different context, where they may provide different values for a given object. Context can be viewed as background or situation in which somehow it is necessary to group some attributes as a subset of attributes and consider the subset as a context. Finally, Chapter 5 summarizes all discussed in this book and puts forward some future topics of research