Rule Reduction

Fuzzy decision tree induction algorithms require the fuzzy quantization of the input variables. This paper demonstrates that supervised fuzzy clustering combined with similarity-based rule-simplification algorithms is an effective tool to obtain the fuzzy quantization of the input variables, so the synergistic combination of supervised fuzzy clustering and fuzzy decision tree induction can be effectively used to build compact and accurate fuzzy classifiers.

A novel evolving semi-supervised classifier, namely Parsimonious Classifier? (pClass?), is proposed in this paper. pClass? enhances a recently developed classifier, namely pClass, for a semi-supervised learning scenario. As with its predecessor, pClass? is capable of initiating its learning process from scratch with an empty rule base and adopts an open network structure, where fuzzy rules are evolved, pruned, and recalled automatically on demands. The novelty of pClass? lies in an online active learning technique, which decreases operator's annotation efforts and expedites its training process. pClass? is also equipped with a new parameter identification strategy to cope with the class overlapping situation. The efficacy of pClass? has been experimentally validated with numerous synthetic and real-world study cases, confirmed by thorough statistical tests and comparisons against state-of-the art classifiers, where pClass? outperforms its counterparts in achieving the best trade-off between accuracy and complexity.

Complex fuzzy theory has strong practical background in many important applications, especially in decision-making support systems. Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been introduced as an effective tool for handling events that are not restricted to only values of a given time point but also include all values within certain time intervals (i.e., the phase term). In such decision-making problems, the complex fuzzy theory allows us to observe both the amplitude and phase values of an event, thus resulting in better performance. However, one of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset. In order to handle the problem, we propose a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing called M-CFIS-R. Several fuzzy similarity measures such as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), a...

This paper describes links between computational intelligence (CI), data mining and knowledge discovery. The generating elements of soft computing based data mining algorithms are defined where the extracted knowledge is represented by fuzzy rule-based expert systems. It is recognized that both model performance and interpretability are of major importance, and effort has to make to keep the resulting rule bases small and comprehensible. Therefore, the model is derived from data by CI techniques developed for feature selection, feature extraction, model optimization and model reduction (rule base simplification). Applications of these generating elements to the Wine data classification problem are also shown.

Complex fuzzy theory has strong practical background in many important applications, especially in decision-making support systems. Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been introduced as an effective tool for handling events that are not restricted to only values of a given time point but also include all values within certain time intervals (i.e., the phase term). In such decision-making problems, the complex fuzzy theory allows us to observe both the amplitude and phase values of an event, thus resulting in better performance. However, one of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset. In order to handle the problem, we propose a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing called M-CFIS-R. Several fuzzy similarity measures such as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), a...

Complex fuzzy theory has strong practical background in many important applications, especially in decision-making support systems. Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been introduced as an effective tool for handling events that are not restricted to only values of a given time point but also include all values within certain time intervals (i.e., the phase term). In such decision-making problems, the complex fuzzy theory allows us to observe both the amplitude and phase values of an event, thus resulting in better performance. However, one of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset. In order to handle the problem, we propose a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing called M-CFIS-R. Several fuzzy similarity measures such as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), a...

Complex fuzzy theory has strong practical background in many important applications, especially in decision-making support systems. Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been introduced as an effective tool for handling events that are not restricted to only values of a given time point but also include all values within certain time intervals (i.e., the phase term). In such decision-making problems, the complex fuzzy theory allows us to observe both the amplitude and phase values of an event, thus resulting in better performance. However, one of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset. In order to handle the problem, we propose a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing called M-CFIS-R. Several fuzzy similarity measures such as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), a...

The synchronous Boolean network model is a simple and powerful tool in describing, analyzing and simulating cellular biological networks. This paper seeks a complete understanding of the dynamics of such a model by utilizing conventional matrix methods, rather than scalar methods, or matrix methods employing the non-conventional semi-tensor products (STP) of matrices. The paper starts by relating the network transition matrix to its function matrix via a self-inverse (involutary) state matrix, which has a simple recursive expression, provided a recursive ordering is employed for the underlying basis vector. Once the network transition matrix is obtained, it can be used to generate a wealth of information including its powers, characteristic equation, minimal equation, 1-eigenvectors, and 0-eigenvectors. These might be used to correctly predict both the transient behavior and (more importantly) the cyclic behavior of the network. In a short-cut partial variant of the proposed approac...

In order to assess the performance indexes of some practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems, we propose various types of techniques for analyzing a capacitated network. These include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method.

A multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karn...

This paper presents a new approach aimed to design a fuzzy face recognition system. Face feature lines, new features proposed in the paper, are incorporated in the feature vector used to design the patter recognition system. Face feature lines are considered as new features based on previous studies related to face recognition tasks on newborns. Besides the face feature lines the feature vector incorporates eigenvectors of the face image obtained with the Karhunen-Loeve transformation. The fuzzy face recognition system is based on the Gath-Gheva fuzzy clustering method and the Abonyi and Szeifert classification scheme. The performance of the face recognition system turned out to be 90% of correct classification using the ORL data base.

Robustness is defined as a system's ability to withstand under disturbances. In real-life applications, where problem parameters are often uncertain, incorporating robustness in decision making is important. In this study, we propose a robust decision making (RDM) approach using intuitionistic trapezoidal fuzzy number (ITrFN). Fuzzy linguistic quantifier (FLQ) is used in the proposed approach to compute the uncertain optimism degree of the decision maker. Initially, decision maker expresses his/her opinion using linguistic terms, which are presented numerically using ITrFNs. The aggregated ITrFN for each of the alternatives is evaluated using intuitionistic trapezoidal fuzzy ordered weighted averaging operator (ITrFOWA). Then, we find out the expected value and variance of the aggregated ITrFN for each alternative, which are subsequently used for robust decision making. A collective measure of these two values of each alternative is considered to find an interval of the corresponding alternative, known as optimal interval. Alternative with maximum optimal interval is selected as the robust solution. Applicability of the proposed approach has been demonstrated on a site selection problem of nuclear power plant. Site selection for installing a nuclear power plant has become a crucial problem throughout the world, especially after the Fukushima (2011) and Chernobyl (1986) nuclear disasters. Keywords Multi-criteria decision making Á Intuitionistic trapezoidal fuzzy number Á Intuitionistic trapezoidal fuzzy ordered weighted averaging operator Á Robust decision making Á Nuclear power plant

Belief  rule-based  inference  system  introduces  a  belief  distribution  structure  into the conventional  rule-based  system,  which  can  effectively  synthesize  incomplete  and  fuzzy information.  In  order  to  optimize  reasoning  efficiency  and  reduce  redundant  rules,  this  paper proposes a rule reduction method based on regularization. This method controls the distribution of  rules  by  setting  corresponding  regularization  penalties  in  different  learning  steps  and reduces redundant rules. This paper first proposes the use of the Gaussian membership function to  optimize  the  structure  and  activation  process  of the  belief  rule  base,  and  the  corresponding regularization penalty construction method. Then, a step-by-step training method is used to set a  different  objective  function  for  each  step  to  control  the  distribution  of  belief  rules,  and  a reduction  threshold  is  set  according  to  the  distribution  information  of  the  belief  rule  base  to perform rule reduction. Two experiments will be conducted based on the synthetic classification data  set  and  the  benchmark  classification  data  set  to  verify  the  performance  of  the  reduced belief rule base.

In order to assess the performance indexes of some practical systems having fixed channel capacities, such as telecommunication networks, power transmission systems or commodity pipeline systems, we propose various types of techniques for analyzing a capacitated network. These include Karnaugh maps, capacity-preserving network reduction rules associated with delta-star transformations, and a generalization of the max-flow min-cut theorem. All methods rely on recognizing the network capacity function as a random pseudo-Boolean function of link successes; a fact that allows the expected value of this function to be easily obtainable from its sum-of-products expression. This network capacity has certain advantages for representation of nonbinary discrete random functions, mostly employed in the analysis of flow networks. Five tutorial examples demonstrate the afore-mentioned methods and illustrate their computational advantages over the exhaustive state enumeration method.

This paper proposes an approach to deriving a fuzzy classifier based on evolutionary supervised clustering, which identifies the optimal clusters necessary to classify classes. The clusters are formed by multi-dimensional weighted Euclidean distance, which allows clusters of varying shapes and sizes. A cluster induces a Gaussian fuzzy antecedent set with unique variance in each dimension, which reflects the tightness of the cluster. The fuzzy classifier is composed of as many classification rules as classes. The clusters identified for each class constitute fuzzy sets, which are joined by an "and" connective in the antecedent part of the corresponding rule. The approach is evaluated using six data sets. The comparative results with different classifiers are given.

A novel fuzzy clustering method is proposed here for separating the breast cancer data, which operates with reasonable accuracy, allows flexibility in dataset & is modestly time consuming. This method can be applied to any type of cancer data set with some initial labels to obtain high accuracy result in the classification of unlabeled samples. Further, the curse of dimensionality is not an issue for the proposed scheme as it can be applied to data having any number of dimensions or attributes. The DifFUZZY unsupervised clustering algorithm is applied at the initial stage, giving an accuracy of 96.28% over Wisconsin Breast Cancer Dataset (WBCD); the result is further improved to 98.14% by using the proposed Back-Retreat algorithm. The formed clusters are estimated using three internal cluster validation indices and the performance of the method is evaluated using receiver operating characteristic (ROC) curves. The clustering algorithm is compared with Fuzzy C-Means (FCM) algorithm and the results are compared with different classifiers and clustering techniques.

Fuzzy decision tree induction algorithms require the fuzzy quantization of the input variables. This paper demonstrates that supervised fuzzy clustering combined with similarity-based rule-simplification algorithms is an effective tool to obtain the fuzzy quan- tization of the input variables, so the synergistic combination of supervised fuzzy clustering and fuzzy decision tree induction can be effectively used to build compact and accurate fuzzy classifiers.

A novel approach to nonlinear classification is presented. In the training phase of the classifier, the training data is first clustered in an unsupervised way by fuzzy c c c-means or a similar algorithm. The class labels are not used in this step. Then, a fuzzy relation between the clusters and the class identifiers is computed. This approach allows the number of prototypes to be independent of the number of actual classes. For the classification of unseen patterns, the membership degrees of the feature vector in the clusters are first computed by using the distance measure of the clustering algorithm. Then, the output fuzzy set is obtained by relational composition. This fuzzy set contains the membership degrees of the pattern in the given classes. A crisp decision is obtained by defuzzification, which gives either a single class or a "reject" decision, when a unique class cannot be selected based on the available information. The principle of the proposed method is demonstrated on an artificial data set and the applicability of the method is shown on the identification of live-stock from recorded sound sequences. The obtained results are compared with two other classifiers.

The classical fuzzy classifier consists of rules each one describing one of the classes. In this paper a new fuzzy model structure is proposed where each rule can represent more than one classes with different probabilities. The obtained classifier can be considered as an extension of the quadratic Bayes classifier that utilizes mixture of models for estimating the class conditional densities. A supervised clustering algorithm has been worked out for the identification of this fuzzy model. The relevant input variables of the fuzzy classifier have been selected based on the analysis of the clusters by Fisher’s interclass separability criteria. This new approach is applied to the well-known wine and Wisconsin breast cancer classification problems.

Classification is one of the most popular data mining techniques applied to many scientific and industrial problems. The efficiency of a classification model is evaluated by two parameters, namely the accuracy and the interpretability of the model. While most of the existing methods claim their accurate superiority over others, their models are usually complex and hardly understandable for the users. In this paper, we propose a novel classification model that is based on easily interpretable fuzzy association rules and fulfils both efficiency criteria. Since the accuracy of a classification model can be largely affected by the partitioning of numerical attributes, this paper discusses several fuzzy and crisp partitioning techniques. The proposed classification method is compared to 15 previously published association rule-based classifiers by testing them on five benchmark data sets. The results show that the fuzzy association rule-based classifier presented in this paper, offers a compact, understandable and accurate classification model.

This paper presents a new approach aimed to design a fuzzy face recognition system. Face feature lines, new features proposed in the paper, are incorporated in the feature vector used to design the patter recognition system. Face feature lines are considered as new features based on previous studies related to face recognition tasks on newborns. Besides the face feature lines the feature vector incorporates eigenvectors of the face image obtained with the Karhunen-Loeve transformation. The fuzzy face recognition system is based on the Gath-Gheva fuzzy clustering method and the Abonyi and Szeifert classification scheme. The performance of the face recognition system turned out to be 90% of correct classification using the ORL data base.