Fundamentals Of Statistics With Fuzzy Data

This book presents basic aspects for a theory of statistics with fuzzy data, together with a set of practical applications. Theories of fuzzy logic and of random closed sets are used as basic ingredients in building statistical concepts and procedures in the context of imprecise data, including coarse data analysis. The book aims at motivating statisticians to examine fuzzy statistics to enlarge the domain of applicability of statistics in general.

Statistical data are not always precise numbers, or vectors, or categories. Real data are frequently what is called fuzzy. Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively. Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information. Key Features: Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data. Describes methods of increasing importance with applications in areas such as environmental statistics and social science. Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples. Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data. This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.

This book offers a comprehensive reference guide to fuzzy statistics and fuzzy decision-making techniques. It provides readers with all the necessary tools for making statistical inference in the case of incomplete information or insufficient data, where classical statistics cannot be applied. The respective chapters, written by prominent researchers, explain a wealth of both basic and advanced concepts including: fuzzy probability distributions, fuzzy frequency distributions, fuzzy Bayesian inference, fuzzy mean, mode and median, fuzzy dispersion, fuzzy p-value, and many others. To foster a better understanding, all the chapters include relevant numerical examples or case studies. Taken together, they form an excellent reference guide for researchers, lecturers and postgraduate students pursuing research on fuzzy statistics. Moreover, by extending all the main aspects of classical statistical decision-making to its fuzzy counterpart, the book presents a dynamic snapshot of the field that is expected to stimulate new directions, ideas and developments.

This book presents basic aspects for a theory of statistics with fuzzy data, together with a set of practical applications. Theories of fuzzy logic and of random closed sets are used as basic ingredients in building statistical concepts and procedures in the context of imprecise data, including coarse data analysis. The book aims at motivating statisticians to examine fuzzy statistics to enlarge the domain of applicability of statistics in general.

Fuzzy data such as marks, scores, verbal evaluations, imprecise observations, experts' opinions and grey tone pictures, are quite common. In Fuzzy Data Analysis the authors collect their recent results providing the reader with ideas, approaches and methods for processing such data when looking for sub-structures in knowledge bases for an evaluation of functional relationship, e.g. in order to specify diagnostic or control systems. The modelling presented uses ideas from fuzzy set theory and the suggested methods solve problems usually tackled by data analysis if the data are real numbers. Fuzzy Data Analysis is self-contained and is addressed to mathematicians oriented towards applications and to practitioners in any field of application who have some background in mathematics and statistics.

Mainly focusing on processing uncertainty, this book presents state-of-the-art techniques and demonstrates their use in applications to econometrics and other areas. Processing uncertainty is essential, considering that computers – which help us understand real-life processes and make better decisions based on that understanding – get their information from measurements or from expert estimates, neither of which is ever 100% accurate. Measurement uncertainty is usually described using probabilistic techniques, while uncertainty in expert estimates is often described using fuzzy techniques. Therefore, it is important to master both techniques for processing data. This book is highly recommended for researchers and students interested in the latest results and challenges in uncertainty, as well as practitioners who want to learn how to use the corresponding state-of-the-art techniques.

The current volume “New Advances in Intelligent Signal Processing” contains extended works based on a careful selection of papers presented originally at the jubilee sixth IEEE International Symposium on Intelligent Signal Processing (WISP’2009), held in Budapest Hungary, August 26-28, 2009 - celebrating the 10 years anniversary of the WISP event series. The present book does not intent to be an overall survey on the fields of interest of the area, but tries to find topics which represent new, hot, and challenging problems. The book begins with papers investigating selected problems of Modeling, Identification, and Clustering such as fuzzy random variables, evolutionary multi-objective neural network models, a structural learning model of neural networks within a Boltzmann machine, a robust DNA-based clustering techniques, and the advances of combining multi-criteria analysis of signals and pattern recognition using machine learning principles. In the second part of the book Image Processing is treated. The carefully edited chapters deal with fuzzy relation based image enhancement, image contrast control technique based on the application of Łukasiewicz algebra operators, low complexity situational models of image quality improvement, flexible representation of map images to quantum computers, and object recognition in images. The last chapter presents an image processing application for elderly care, performing real-time 3D tracking based on a new evaluative multi-modal algorithm.