Rule Extraction Using Soft-Computing Techniques with Fuzzy Linguistic Representation

This study presents a rough-fuzzy hybridization method to generate fuzzy if-then rules automatically from a medical diagnosis dataset with quantitative data values, based on fuzzy set and rough set theory. The proposed method consists of four stages: preprocessing inputs with fuzzy linguistic representation; rough set theory in finding notable reducts; candidate fuzzy if-then rules generation by data summarization, and truth evaluation the effectiveness of fuzzy if-then rules. The main contributions of the proposed method are the capability of fuzzy linguistic representation of the fuzzy if-then rules, finding concise fuzzy if-then rules from medical diagnosis dataset, and tolerance of imprecise data.

Soft rough fuzzy sets and soft fuzzy rough sets

Uncertainty about future developments constitutes the most important and most difficult challenge for mankind. Despite this fact, uncertainty is not a part of general science. General science assumes that the future development follows cause-effect relations which can be described by mathematical functions, where the argument represents the cause and the image represents the effect. Scientific theories have exactly this form and it is widely believed that these functions represent “truth”. Of course, this is nonsense as all the scientific theories are with certainty wrong and cannot describe the real evolution correctly. The inappropriate handling of uncertainty in science has produced a strange variety of “uncertainty theories” that causes confusion and helplessness. A friend of mine has expressed his confusion by the following words: ‘Crisp sets’, ‘fuzzy sets’, ‘rough sets’, ‘grey sets’, ‘fuzzy rough sets’, ‘rough fuzzy sets’, ‘fuzzy grey sets’, ‘grey fuzzy sets’, ‘rough grey sets’, ‘grey rough sets’, and now ‘affinity sets’. My goodness! Is there anybody around who can enlighten me, i.e., help me to see a clear pattern in this set of sets, allegedly providing powerful tools to model various kinds of uncertainty? This paper examines the role and the handling of uncertainty in quality control. How is uncertainty quantified in quality control for making decisions aiming at maintaining or improving quality of processes and products.KeywordsRandomnessIgnoranceKnowledgeFuzzynessQuantificationProbabilityCredibilityProbability spaceUncertainty spaceBernoulli space

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

Being classic K-means++ clustering algorithm only for static data, dynamically incremental K-means++ clustering algorithm (DK-Means++) is presented based on fuzzy rough set theory in this paper. Firstly, in DK-Means++ clustering algorithm, the formula of similar degree is improved by weights computed by using of the important degree of attributes which are reduced on the basis of rough fuzzy set theory. Secondly, new data only need match granular which was clustered by K-means++ algorithm or seldom new data is clustered by classic K-means++ algorithm in global data. In this way, that all data is re-clustered each time in dynamic data set is avoided, so the efficiency of clustering is improved. Throughout our experiments showing, DK-Means++ algorithm can objectively and efficiently deal with clustering problem of dynamically incremental data.

Rough set theory for the interval-valued fuzzy information systems

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

In order to effectively handle the real-valued data sets in practice, it is valuable from theoretical and practical aspects to combine fuzzy rough set and variable precision rough set so that a powerful tool can be developed. That is, the model of fuzzy variable precision rough set, which not only can handle numerical data but also is less sensitive to misclassification and perturbation,In this paper, we propose a new variable precision rough fuzzy set by introducing the variable precision parameter to generalized rough fuzzy set, i.e., the variable precision rough fuzzy set based on general relation. We, respectively, define the variable precision rough lower and upper approximations of any fuzzy set and it level set with variable precision parameter by constructive approach. Also, we present the properties of the proposed model in detail. Meanwhile, we establish the relationship between the variable precision rough approximation of a fuzzy set and the rough approximation of the level set for a fuzzy set. Furthermore, we give a new approach to uncertainty measure for variable precision rough fuzzy set established in this paper in order to overcome the limitations of the traditional methods. Finally, some numerical example are used to illuminate the validity of the conclusions given in this paper.

Extracting useful patterns is an important theme in data mining. Fuzzy logic and Rough sets are the most common techniques applied in data mining problems where rule sets are used to classify new cases. In order to extract minimum rules with high data coverage and fast processing time based on rule induction. This paper presents a hybrid method called, Updating Induction of Decision Rules based on Fuzzy Rough Set (UIDR-FRS). The UIDR-FRS method is based on the integration of fuzzy logic and rough sets and hence constructs a novel rule extraction model. The attribute reduction algorithm using fuzzy logic and rules extraction method based on rough sets are proposed. The attributes of decision table are analyzed using fuzzy logic and subsequently a rule extraction model based on these attributes is built through integration of rough sets and fuzzy logic model. Experiments on real-life data sets are conducted to test and verify the validity of the proposed measure. Applications of the proposed rule induction in rule extraction and accuracy are also studied with experiments. Extensive experimental results verify the effectiveness and efficiency of the attribute reduction algorithms for a compacted decision table.

This paper introduces a novel framework that uses fuzzy IF-THEN rules in an ensemble system. Our model tackles several drawbacks. First, IF-THEN rules approaches have problems with high dimensional data since computational cost is exponential. In our framework, rules are operated on outputs of base classifiers which frequently have lower dimensionality than the original data. Moreover, outputs of base classifiers are scaled within the range [0, 1] so it is convenient to apply fuzzy rules directly instead of requiring data transformation and normalization before generating fuzzy rules. The performance of this model was evaluated through experiments on 6 commonly used datasets from UCI Machine Learning Repository and compared with several state-of-art combining classifiers algorithms and fuzzy IF-THEN rules approaches. The results show that our framework can improve the classification accuracy.KeywordEnsemble methodCombining multiple classifiersClassifier fusionFuzzy IF-THEN rules

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

This paper presents an artificial intelligence approach towards classification of persons based on verbal descriptions of their facial features. Frame knowledge representation, fuzzy sets, fuzzy IF-THEN rules, and fuzzy granulation are employed. Features of face elements (nose, eyes, etc.) are extracted by use of existing detection techniques, such as measurements of horizontal and vertical sizes. Linguistic variables that correspond to fuzzy sets, representing selected facial features, are applied in the frames and fuzzy rules. Linguistic values defined by the fuzzy sets conform the terminology applied by law enforcement to create an eyewitness verbal description. Classification results are illustrated in three cases of the system’s input: facial composites (sketches) created by an artist, images (digital pictures) from a face database, and verbal descriptions.

The definition of basic rough sets [3] depends upon either a single equivalence relation defined on a universe or several equivalence relations defined over the universe, taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. Extending this notion, a rough set model based on multi-granulations (MGRS) was introduced in [5]. In this approach, approximations of sets were defined through multiple equivalence relations on the universe and their properties were investigated. Using hybridization of fuzzy set [13] with rough set the concept of rough fuzzy set was introduced by Dubois and Prade [1]. Recently, a Rough Fuzzy Set Model was introduced and studied by Wu and Kou [12], which is based on Multiple Granulation. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [10]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [11]. In this paper we introduce incomplete multigranulation on rough fuzzy sets, study their basic properties and extend the topological properties in [11] to this context. Our findings are true for both complete and incomplete fuzzy rough set models based upon multi granulation.KeywordsRough SetsFuzzy rough setsequivalence relationstolerance relationstype of rough setsmulti granular fuzzy rough sets

This paper researched and constructed residuation implication operator of intuitionistic fuzzy T-norms.Based on this,the paper deducted the intuitionistic fuzzy rough sets model,and proved that the Pawlak rough sets,intuitionistic fuzzy sets,fuzzy rough sets,rough fuzzy sets and fuzzy T-norms rough sets were special cases of intuitionistic fuzzy rough sets.At last,some properties of intuitionistic fuzzy sets were presented and proved.

Rule extraction from data is one of the key technologies for solving the bottlenecks in artificial intelligence. Artificial neural networks are well suited for representing any knowledge in given data. Extraction of logical/fuzzy rules from the trained artificial neural network is of great importance to researchers in the fields of artificial intelligence and soft computing. Fuzzy rule sets are capable of approximating any nonlinear mapping relationships. Extraction of rules from data has been discussed in terms of fuzzy modeling, fuzzy clustering, and classification with fuzzy rule sets. This special issue entitled"Rule Extraction from Data" is aimed at providing the readers with good insights into the advanced studies in the field of rule extraction from data using neural networks/fuzzy rule sets. I invited seven research papers best suited for the theme of this special issue. All the papers were reviewed rigorously by two reviewers each. The first paper proposes an interesting rule extraction method from data using neural networks. Ishikawa presents a combination of learning with an immediate critic and a structural learning with forgetting. This method is capable of generating skeletal networks for logical rule extraction from data with correct and wrong answers. The proposed method is applied to rule extraction from lense data. The second paper presents a new methodology for logical rule extraction based on transformation of MLP (multilayered perceptron) to a logical network. Duck et al. applied their C-MLP2LN to the Iris benchmark classification problem as well as real-world medical data with very good results. In the third paper, Geczy and Usui propose fuzzy rule extraction from trained artificial neural networks. The proposed algorithm is implied from their theoretical study, not from heuristics. Their study enables to initially consider derivation of crisp rules from trained artificial neural network, and in case of conflict, application of fuzzy rules. The proposed algorithm is experimentally demonstrated with the Iris benchmark classification problem. The fourth paper presents a new framework for fuzzy modeling using genetic algorithm. The authors have broken new ground of fuzzy rule extraction from neural networks. For the fuzzy modeling, they have proposed a particular type of neural networks containing nodes representing membership functions. In this fourth paper, the authors discuss input variable selection for the fuzzy modeling under multiple criteria with different importance. A target system with a strong nonlinearity is used for demonstrating the proposed method. Kasabov, et al. present, in the fifth paper, a method for extraction of fuzzy rules that have different level of abstraction depending on several modifiable thresholds. Explanation quality becomes better with higher threshold values. They apply the proposed method to the Iris benchmark classification problem and to a real world problem. J. Yen and W. Gillespie address interpretability issue of Takagi-Sugeno-Kang model, one of the most popular fuzzy mdoels, in the fifth paper. They propose a new approach of fuzzy modeling that ensures not only a high approximation of the input-output relationship in the data, but also good insights about the local behavior of the model. The proposed method is applied to fuzzy modeling of sinc function and Mackey-Glass chaotic time series data. The last paper discusses fuzzy rule extraction from numerical data for high-dimensional classification problems. H.Ishibuchi, et al. have been pioneering methods for classification of data using fuzzy rules and genetic algorithm. In this last paper, they introduced a new criterion, simplicity of each rule, together with the conventional ones, compactness of rule base and classification ability, for high-dimensional problem. The Iris data is used for demonstrating their new classification method. They applied it also to wine data and credit data. I hope that the readers will be encouraged to explore the frontier to establish a new paradigm in the field of knowledge representation and rule extraction.