Numerical analysis forms a cornerstone of numeric computing and optimization, in particular recently, interval numerical computations play an important role in these topics. The interest of researchers in computations involving uncertain data, namely interval data opens new avenues in coping with real-world problems and deliver innovative and efficient solutions. This book provides the basic theoretical foundations of numerical methods, discusses key technique classes, explains improvements and improvements, and provides insights into recent developments and challenges. The theoretical parts of numerical methods, including the concept of interval approximation theory, are introduced and explained in detail. In general, the key features of the book include an up-to-date and focused treatise on error analysis in calculations, in particular the comprehensive and systematic treatment of error propagation mechanisms, considerations on the quality of data involved in numerical calculations, and a thorough discussion of interval approximation theory. Moreover, this book focuses on approximation theory and its development from the perspective of linear algebra, and new and regular representations of numerical integration and their solutions are enhanced by error analysis as well. The book is unique in the sense that its content and organization will cater to several audiences, in particular graduate students, researchers, and practitioners.
As an extension of multiplicative preference relations (MPRs), intuitionistic MPRs (IMPRs) reflect experts' hesitant quantitative judgments. This paper presents an intuitionistic multiplicative preference information-based group analytic hierarchy process (AHP) and develops an intuitionistic multiplicative group AHP (IMGAHP), which addresses multicriteria group decision-making (MCGDM) that uses IMPRs to capture experts' preference judgments. First, we introduce a consistency index to gauge the consistency of IMPRs and describe the concept of acceptably consistent IMPRs. Second, we propose an algorithm for repairing an inconsistent IMPR to an acceptable level. Third, we propose an aggregation operator to integrate acceptably consistent IMPRs into a collective IMPR with acceptable consistency. We also propose an approach to derive an intuitionistic multiplicative priority weight vector from an acceptably consistent IMPR. An IMGAHP method is then described as a means of solving an MCGDM process with IMPRs. Finally, a practical example and comparative analysis are presented.
Holmes is a software product line tool that supports all core activities of software product line analysis and development. Holmes integrates its tools using a blackboard architecture based on a Linda tuple space. A novel feature is the use of a critiquing system to provide semantic support. This is demonstrated with an example.
Kernel machines and rough sets are two classes of commonly exploited learning techniques. Kernel machines enhance traditional learning algorithms by bringing opportunities to deal with nonlinear classification problems, rough sets introduce a human-focused way to deal with uncertainty in learning problems. Granulation and approximation play a pivotal role in rough sets-based learning and reasoning. However, a way how to effectively generate fuzzy granules from data has not been fully studied so far. In this study, we integrate kernel functions with fuzzy rough set models and propose two types of kernelized fuzzy rough sets. Kernel functions are employed to compute the fuzzy T-equivalence relations between samples, thus generating fuzzy information granules in the approximation space. Subsequently fuzzy granules are used to approximate the classification based on the concepts of fuzzy lower and upper approximations. Based on the models of kernelized fuzzy rough sets, we extend the measures existing in classical rough sets to evaluate the approximation quality and approximation abilities of the attributes. We discuss the relationship between these measures and feature evaluation function ReliefF, and augment the ReliefF algorithm to enhance the robustness of these proposed measures. Finally, we apply these measures to evaluate and select features for classification problems. The experimental results help quantify the performance of the KFRS.