Statistical Methods for Fuzzy Data


Book Description

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.




Fundamentals of Statistics with Fuzzy Data


Book Description

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 Statistical Inferences Based on Fuzzy Random Variables


Book Description

This book presents the most commonly used techniques for the most statistical inferences based on fuzzy data. It brings together many of the main ideas used in statistical inferences in one place, based on fuzzy information including fuzzy data. This book covers a much wider range of topics than a typical introductory text on fuzzy statistics. It includes common topics like elementary probability, descriptive statistics, hypothesis tests, one-way ANOVA, control-charts, reliability systems and regression models The reader is assumed to know calculus and a little fuzzy set theory. The conventional knowledge of probability and statistics is required. Key Features: Includes example in Mathematica and MATLAB. Contains theoretical and applied exercises for each section. Presents various popular methods for analyzing fuzzy data. The book is suitable for students and researchers in statistics, social science, engineering, and economics, and it can be used at graduate and P.h.D level. Gholamreza Hesamian is Associate Professor of Statistics at Payame Noor University. His research areas include decision theory, probability theory, fuzzy mathematics, and statistics.




Fuzzy Statistics


Book Description

1. 1 Introduction This book is written in four major divisions. The first part is the introductory chapters consisting of Chapters 1 and 2. In part two, Chapters 3-11, we develop fuzzy estimation. For example, in Chapter 3 we construct a fuzzy estimator for the mean of a normal distribution assuming the variance is known. More details on fuzzy estimation are in Chapter 3 and then after Chapter 3, Chapters 4-11 can be read independently. Part three, Chapters 12- 20, are on fuzzy hypothesis testing. For example, in Chapter 12 we consider the test Ho : /1 = /10 verses HI : /1 f=- /10 where /1 is the mean of a normal distribution with known variance, but we use a fuzzy number (from Chapter 3) estimator of /1 in the test statistic. More details on fuzzy hypothesis testing are in Chapter 12 and then after Chapter 12 Chapters 13-20 may be read independently. Part four, Chapters 21-27, are on fuzzy regression and fuzzy prediction. We start with fuzzy correlation in Chapter 21. Simple linear regression is the topic in Chapters 22-24 and Chapters 25-27 concentrate on multiple linear regression. Part two (fuzzy estimation) is used in Chapters 22 and 25; and part 3 (fuzzy hypothesis testing) is employed in Chapters 24 and 27. Fuzzy prediction is contained in Chapters 23 and 26. A most important part of our models in fuzzy statistics is that we always start with a random sample producing crisp (non-fuzzy) data.




Fuzzy Sets in Decision Analysis, Operations Research and Statistics


Book Description

Fuzzy Sets in Decision Analysis, Operations Research and Statistics includes chapters on fuzzy preference modeling, multiple criteria analysis, ranking and sorting methods, group decision-making and fuzzy game theory. It also presents optimization techniques such as fuzzy linear and non-linear programming, applications to graph problems and fuzzy combinatorial methods such as fuzzy dynamic programming. In addition, the book also accounts for advances in fuzzy data analysis, fuzzy statistics, and applications to reliability analysis. These topics are covered within four parts: Decision Making, Mathematical Programming, Statistics and Data Analysis, and Reliability, Maintenance and Replacement. The scope and content of the book has resulted from multiple interactions between the editor of the volume, the series editors, the series advisory board, and experts in each chapter area. Each chapter was written by a well-known researcher on the topic and reviewed by other experts in the area. These expert reviewers sometimes became co-authors because of the extent of their contribution to the chapter. As a result, twenty-five authors from twelve countries and four continents were involved in the creation of the 13 chapters, which enhances the international character of the project and gives an idea of how carefully the Handbook has been developed.




Federal Statistics, Multiple Data Sources, and Privacy Protection


Book Description

The environment for obtaining information and providing statistical data for policy makers and the public has changed significantly in the past decade, raising questions about the fundamental survey paradigm that underlies federal statistics. New data sources provide opportunities to develop a new paradigm that can improve timeliness, geographic or subpopulation detail, and statistical efficiency. It also has the potential to reduce the costs of producing federal statistics. The panel's first report described federal statistical agencies' current paradigm, which relies heavily on sample surveys for producing national statistics, and challenges agencies are facing; the legal frameworks and mechanisms for protecting the privacy and confidentiality of statistical data and for providing researchers access to data, and challenges to those frameworks and mechanisms; and statistical agencies access to alternative sources of data. The panel recommended a new approach for federal statistical programs that would combine diverse data sources from government and private sector sources and the creation of a new entity that would provide the foundational elements needed for this new approach, including legal authority to access data and protect privacy. This second of the panel's two reports builds on the analysis, conclusions, and recommendations in the first one. This report assesses alternative methods for implementing a new approach that would combine diverse data sources from government and private sector sources, including describing statistical models for combining data from multiple sources; examining statistical and computer science approaches that foster privacy protections; evaluating frameworks for assessing the quality and utility of alternative data sources; and various models for implementing the recommended new entity. Together, the two reports offer ideas and recommendations to help federal statistical agencies examine and evaluate data from alternative sources and then combine them as appropriate to provide the country with more timely, actionable, and useful information for policy makers, businesses, and individuals.




Statistical Methods for Ranking Data


Book Description

This book introduces advanced undergraduate, graduate students and practitioners to statistical methods for ranking data. An important aspect of nonparametric statistics is oriented towards the use of ranking data. Rank correlation is defined through the notion of distance functions and the notion of compatibility is introduced to deal with incomplete data. Ranking data are also modeled using a variety of modern tools such as CART, MCMC, EM algorithm and factor analysis. This book deals with statistical methods used for analyzing such data and provides a novel and unifying approach for hypotheses testing. The techniques described in the book are illustrated with examples and the statistical software is provided on the authors’ website.




Fuzzy Probability and Statistics


Book Description

This book combines material from our previous books FP (Fuzzy Probabilities: New Approach and Applications,Physica-Verlag, 2003) and FS (Fuzzy Statistics, Springer, 2004), plus has about one third new results. From FP we have material on basic fuzzy probability, discrete (fuzzy Poisson,binomial) and continuous (uniform, normal, exponential) fuzzy random variables. From FS we included chapters on fuzzy estimation and fuzzy hypothesis testing related to means, variances, proportions, correlation and regression. New material includes fuzzy estimators for arrival and service rates, and the uniform distribution, with applications in fuzzy queuing theory. Also, new to this book, is three chapters on fuzzy maximum entropy (imprecise side conditions) estimators producing fuzzy distributions and crisp discrete/continuous distributions. Other new results are: (1) two chapters on fuzzy ANOVA (one-way and two-way); (2) random fuzzy numbers with applications to fuzzy Monte Carlo studies; and (3) a fuzzy nonparametric estimator for the median.




Fuzzy Data Analysis


Book Description

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.




Soft Methods in Probability, Statistics and Data Analysis


Book Description

Classical probability theory and mathematical statistics appear sometimes too rigid for real life problems, especially while dealing with vague data or imprecise requirements. These problems have motivated many researchers to "soften" the classical theory. Some "softening" approaches utilize concepts and techniques developed in theories such as fuzzy sets theory, rough sets, possibility theory, theory of belief functions and imprecise probabilities, etc. Since interesting mathematical models and methods have been proposed in the frameworks of various theories, this text brings together experts representing different approaches used in soft probability, statistics and data analysis.