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Fuzzy rough sets (FRSs) are considered to be a powerful model for analyzing uncertainty in data. This model encapsulates two types of uncertainty: 1) fuzziness coming from the vagueness in human concept formation and 2) roughness rooted in the granulation coming with human cognition. The rough set theory has been widely applied to feature selection, attribute reduction, and classification. However, it is reported that the classical FRS model is sensitive to noisy information. To address this problem, several robust models have been developed in recent years. Nevertheless, these models do not consider a statistical distribution of data, which is an important type of uncertainty. Data distribution serves as crucial information for designing an optimal classification or regression model. Thus, we propose a data-distribution-aware FRS model that considers distribution information and incorporates it in computing lower and upper fuzzy approximations. The proposed model considers not only the similarity between samples, but also the probability density of classes. In order to demonstrate the effectiveness of the proposed model, we design a new sample evaluation index for prototype-based classification based on the model, and a prototype selection algorithm is developed using this index. Furthermore, a robust classification algorithm is constructed with prototype covering and nearest neighbor classification. Experimental results confirm the robustness and effectiveness of the proposed model.

Support vector machines (SVMs), which were originally designed for binary classifications, are an excellent tool for machine learning. For the multiclass classifications, they are usually converted into binary ones before they can be used to classify the examples. In the one-against-one algorithm with SVMs, there exists an unclassifiable region where the data samples cannot be classified by its decision function. This paper extends the one-against-one algorithm to handle this problem. We also give the convergence and computational complexity analysis of the proposed method. Finally, one-against-one, fuzzy decision function (FDF), and decision-directed acyclic graph (DDAG) algorithms and our proposed method are compared using five University of California at Irvine (UCI) data sets. The results report that the proposed method can handle the unclassifiable region better than others.

The support vector machine (SVM) has been demonstrated to be a very effective classifier in many applications, but its performance is still limited as the data distribution information is underutilized in determining the decision hyperplane. Most of the existing kernels employed in nonlinear SVMs measure the similarity between a pair of pattern images based on the Euclidean inner product or the Euclidean distance of corresponding input patterns, which ignores data distribution tendency and makes the SVM essentially a "local" classifier. In this paper, we provide a step toward a paradigm of kernels by incorporating data specific knowledge into existing kernels. We first find the data structure for each class adaptively in the input space via agglomerative hierarchical clustering (AHC), and then construct the weighted Mahalanobis distance (WMD) kernels using the detected data distribution information. In WMD kernels, the similarity between two pattern images is determined not only by the Mahalanobis distance (MD) between their corresponding input patterns but also by the sizes of the clusters they reside in. Although WMD kernels are not guaranteed to be positive definite (pd) or conditionally positive definite (cpd), satisfactory classification results can still be achieved because regularizers in SVMs with WMD kernels are empirically positive in pseudo-Euclidean (pE) spaces. Experimental results on both synthetic and real-world data sets show the effectiveness of "plugging" data structure into existing kernels.

The generalization error bounds found by current error models using the number of effective parameters of a classifier and the number of training samples are usually very loose. These bounds are intended for the entire input space. However, support vector machine (SVM), radial basis function neural network (RBFNN), and multilayer perceptron neural network (MLPNN) are local learning machines for solving problems and treat unseen samples near the training samples to be more important. In this paper, we propose a localized generalization error model which bounds from above the generalization error within a neighborhood of the training samples using stochastic sensitivity measure. It is then used to develop an architecture selection technique for a classifier with maximal coverage of unseen samples by specifying a generalization error threshold. Experiments using 17 University of California at Irvine (UCI) data sets show that, in comparison with cross validation (CV), sequential learning, and two other ad hoc methods, our technique consistently yields the best testing classification accuracy with fewer hidden neurons and less training time.

The one-class classification problem aims to distinguish a target class from outliers. The spherical one-class classifier (SOCC) solves this problem by finding a hypersphere with minimum volume that contains the target data while keeping outlier samples outside. SOCC achieves satisfactory performance only when the target samples have the same distribution tendency in all orientations. Therefore, the performance of the SOCC is limited in the way that many superfluous outliers might be mistakenly enclosed. The authors propose to exploit target data structures obtained via unsupervised methods such as agglomerative hierarchical clustering and use them in calculating a set of hyperellipsoidal separating boundaries. This method is named the structured one-class classifier (TOCC). The optimization problem in TOCC can be formulated as a series of second-order cone programming problems that can be solved with acceptable efficiency by primal-dual interior-point methods. The experimental results on artificially generated data sets and benchmark data sets demonstrate the advantages of TOCC.

When fuzzy production rules are used to approximate reasoning, interaction exists among rules that have the same consequent. Due to this interaction, the weighted average model frequently used in approximate reasoning does not work well in many real-world problems. In order to model and handle this interaction, this paper proposes to use a nonadditive nonnegative set function to replace the weights assigned to rules having the same consequent, and to draw the reasoning conclusion based on an integral with respect to the nonadditive nonnegative set function, rather than on the weighted average model. Handling interaction in fuzzy production rule reasoning in this way can lead to a good understanding of the rules base and an improvement of reasoning accuracy. This paper also investigates how to determine from data the nonadditive set function that cannot be specified by a domain expert.

Fuzzy production rules (FPRs) have been used for years to capture and represent fuzzy, vague, imprecise and uncertain domain knowledge in many fuzzy systems. There have been a lot of researches on how to generate or obtain FPRs. There exist two methods to obtain FPRs. One is by painstakingly, repeatedly and time-consuming interviewing domain experts to extract the domain knowledge. The other is by using some machine learning techniques to generate and extract FPRs from some training samples. These extracted rules, however, are found to be nonoptimal and sometimes redundant. Furthermore, these generated rules suffer from the problem of low accuracy of classifying or recognizing unseen examples. The reasons for having these problems are 1) the FPRs generated are not powerful enough to represent the domain knowledge, 2) the techniques used to generate FPRs are pre-matured, ad-hoc or may not be suitable for the problem, and 3) further refinement of the extracted rules has not been done. In this paper we look into the solutions of the above problems by 1) enhancing the representation power of FPRs by including local and global weights, 2) developing a fuzzy neural network (FNN) with enhanced learning algorithm, and 3) using this FNN to refine the local and global weights of FPRs. By experimenting our method with some existing benchmark examples, the proposed method is found to have high accuracy in classifying unseen samples without increasing the number of the FPRs extracted and the time required to consult with domain experts is greatly reduced.

Approximate reasoning in a fuzzy system is concerned with inferring an approximate conclusion from fuzzy and vague inputs. There are many ways in which different forms of conclusions can be drawn. Fuzzy sets are usually represented by fuzzy membership functions. These membership functions are assumed to have a clearly defined base. For other fuzzy sets such as intelligent, smart, or beautiful, etc., it would be difficult to define clearly its base because its base may consist of several other fuzzy sets or unclear nonfuzzy bases. A method to handle this kind of fuzzy set is proposed. A fuzzy neural network (FNN) is also proposed to tune knowledge representation parameters (KRPs). The contributions are that we are able to handle a broader range of fuzzy sets and build more powerful fuzzy systems so that the conclusions drawn are more meaningful, reliable, and accurate. An experiment is presented to demonstrate how our method works.

Fuzzy decision tree induction is an important way of learning from examples with fuzzy representation. Since the construction of optimal fuzzy decision tree is NP-hard, the research on heuristic algorithms is necessary. In this paper, three heuristic algorithms for generating fuzzy decision trees are analyzed and compared. One of them is proposed by the authors. The comparisons are twofold. One is the analytic comparison based on expanded attribute selection and reasoning mechanism; the other is the experimental comparison based on the size of generated trees and learning accuracy. The purpose of this study is to explore comparative strengths and weaknesses of the three heuristics and to show some useful guidelines on how to choose an appropriate heuristic for a particular problem.

If the given fact for an antecedent in a fuzzy production rule (FPR) does not match exactly with the antecedent of the rule, the consequent can still be drawn by technique such as fuzzy reasoning. Many existing fuzzy reasoning methods are based on Zadeh's Compositional Rule of Inference (CRI) which requires setting up a fuzzy relation between the antecedent and the consequent part. There are some other fuzzy reasoning methods which do not use Zadeh's CRI. Among them, the similarity-based fuzzy reasoning methods, which make use of the degree of similarity between a given fact and the antecedent of the rule to draw the conclusion, are well known. In this paper, six similarity-based fuzzy reasoning methods are compared and analyzed. Two of them are newly proposed by the authors. The comparisons are two-fold. One is to compare the six reasoning methods in drawing appropriate conclusions for a given set of FPRs. The other is to compare them based on five issues: 1) types of FPR handled by these methods; 2) the complexity of the methods; 3) the accuracy of the conclusion drawn; 4) the accuracy of the similarity measure; and 5) the multi-level reasoning capability. The results have shed some lights on how to select an appropriate fuzzy reasoning method under different environments.