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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Fuzzy Statistical Decision-Making

The classification of decision-making methods can be based on the types of the data in hand. If the data are given as a decision matrix with discrete values, you can use multiple attribute decision-making. If the data are given as unit cost or profit values together with budget or capacity constraints and if you have more than one objective, then you can use multiple objective decision-making in a continuous space. If the data are given as the parameters of certain probability distributions, then you can use statistical decision-making, generally through hypothesis tests. If the data are not exactly known, the fuzzy sets based approaches are incorporated into these decision-making methods. Fuzzy statistical decision-making is one of the most often used methods when insufficient statistical data exist in hand. Fuzzy hypothesis tests, fuzzy variance analysis, and fuzzy design of experiments are the examples of fuzzy statistical decision-making techniques. In this chapter, we survey the literature of fuzzy statistics and fuzzy statistical decision-making and present the results by graphical illustrations.
Cengiz Kahraman, Özgür Kabak

### Fuzzy Probability Theory I: Discrete Case

This chapter introduces the underlying theory of Fuzzy Probability and Statistics related to the differences and similarities between discrete probability and possibility spaces. Fuzzy Probability Theory for Discrete Case starts with the fundamental tools to implement an immigration of crisp probability theory into fuzzy probability theory. Fuzzy random variables are the initial steps to develop this theory. Different models for fuzzy random variables are designated regarding the fuzzy expectation and fuzzy variance. In order to derive the observation related to fuzzy discrete random variables, a brief summary of alpha-cuts is introduced. Furthermore, essential properties of fuzzy probability are derived to present the measurement of fuzzy conditional probability, fuzzy independency and fuzzy Bayes theorem. The fuzzy expectation theory is studied in order to characterize fuzzy probability distributions. Fuzzy discrete distributions; Fuzzy Binomial and Fuzzy Poisson are introduced with different examples. The chapter is concluded with further steps in the discrete case.
I. Burak Parlak, A. Cağrı Tolga

### Fuzzy Probability Theory II: Continuous Case

Continuous probability density functions are widely used in various domains. The characterization of the fuzzy continuous probability theory is similar to the discrete case. However, the possibility space is continuous and the integration between the minimum and the maximum values would set the fuzzy probability through the alpha-cuts. In this chapter, the foundations of fuzzy probability and possibility theory are described for the continuous case. A brief introduction summarized the key concepts in this area with recent applications. The expectation theory is interpreted using the relationship with fuzzy continuous random variables. Fuzzy continuous applications are enriched with different probability density functions. Therefore, fundamental distributions are detailed within their uses and their properties. In this chapter, fuzzy uniform, fuzzy exponential, fuzzy laplace, fuzzy normal and fuzzy lognormal distributions are examined. Several examples are given for the use of these fuzzy distributions regarding the fuzzy interval algebra. Finally, the future suggestions and applications are discussed in the conclusion.
A. Cağrı Tolga, I. Burak Parlak

### On Fuzzy Bayesian Inference

Bayesian inference deals with a-priori information in statistical analysis. However, usually Bayesians assume that all kind of uncertainty can be modeled by probability. Unfortunately, this is not always true due to how uncertainties are defined. The uncertainty of measurement results of continuous quantities differs from probabilistic uncertainty. Individual measurement results also contain another kind of uncertainty, which is called fuzziness. The combination of fuzziness and stochastic uncertainty calls for a generalization of Bayesian inference, i.e. fuzzy Bayesian inference. This chapter explains the generalized Bayes’ theorem in handling fuzzy a-priori information and fuzzy data.
Reinhard Viertl, Owat Sunanta

### Fuzzy Central Tendency Measures

This chapter converts the classical central tendency measures to their fuzzy cases. Fuzzy mean, fuzzy mode and fuzzy median are explained by numerical examples. Fuzzy frequency distribution is another subtitle of this chapter. Classical graphical illustrations are examined under fuzziness. A numerical example for each central tendency measure is given.
Cengiz Kahraman, İrem Uçal Sarı

### Fuzzy Dispersion Measures

Dispersion measures are very useful tools to measure the variability of data. Under uncertainty, the fuzzy set theory can be used to capture the vagueness in the data. This chapter develops the fuzzy versions of classical dispersion measures namely, standard deviation and variance, mean absolute deviation, coefficient of variation, range, and quartiles. Initially, we summarize the classical dispersion measures and then we develop their fuzzy versions for triangular fuzzy data. A numerical example for each fuzzy dispersion measure is given.
İrem Uçal Sarı, Cengiz Kahraman, Özgür Kabak

### Sufficiency, Completeness, and Unbiasedness Based on Fuzzy Sample Space

A new approach is introduced to the estimation of a parameter in the statistical models, based on fuzzy sample space. Two basic concepts of the point estimation theory, i.e. sufficiency and completeness, are extended to the fuzzy data case. Then, the unbiased estimator and the UMVU estimator are defined for such situations. The properties of these estimators are investigated, and some procedures are provided to obtain the UMVU estimators, based on fuzzy data.
Mohsen Arefi, S. Mahmoud Taheri

### Fuzzy Confidence Regions

Confidence regions are usually based on exact data. However, continuous data are always more or less non-precise, also called fuzzy. For fuzzy data the concept of confidence regions has to be generalized. This is possible and the resulting confidence regions are fuzzy subsets of the parameter space. The construction is explained for classical statistics as well as for Bayesian analysis. An example is given in the last section.
Reinhard Viertl, Shohreh Mirzaei Yeganeh

### Fuzzy Extensions of Confidence Intervals: Estimation for µ, σ2, and p

Even though classical point and interval estimations (PIE) are one of the most studied fields in statistics, there are a few numbers of studies covering fuzzy point and interval estimations. In this pursuit, this study focuses on analyzing the works on fuzzy PIE for the years between 1980 and 2015. In the chapter, the literature is reviewed through Scopus database and the review results are given by graphical illustrations. We also use the extensions of fuzzy sets such as interval-valued intuitionistic fuzzy sets (IVIFS) and hesitant fuzzy sets (HFS) to develop the confidence intervals based on these sets. The chapter also includes numerical examples to increase the understandability of the proposed approaches.
Cengiz Kahraman, Irem Otay, Başar Öztayşi

### Testing Fuzzy Hypotheses: A New p-value-based Approach

In this paper, on the basis of Zadeh’s probability measure of fuzzy events, the p-value concept is generalized for testing fuzzy hypotheses. We prove that the introduced p-value has uniform distribution over (0, 1) when the null fuzzy hypothesis is true. Then, based on such a p-value, a procedure is illustrated to test various types of fuzzy hypotheses. Several applied examples are given to show the performance of the method.
Abbas Parchami, S. Mohmoud Taheri, Bahram Sadeghpour Gildeh, Mashaallah Mashinchi

### Fuzzy Regression Analysis: An Actuarial Perspective

The first objective of this paper is to describe from a critical point of view the main types of fuzzy regression methods: those based on minimum fuzziness principle, those that are built up by minimising the squared distance between observations and estimates and models that mix both methodologies. Finally, we revise the actuarial applications of fuzzy regression proposed in the literature and develop in detail two of them: estimating the yield curve and calculating claim reserves.
Jorge de Andrés-Sánchez

### Fuzzy Correlation and Fuzzy Non-linear Regression Analysis

In this chapter, we will deal with fuzzy correlation and fuzzy non-linear regression analyses. Both correlation and regression analyses that are useful and widely employed statistical tools have been redefined in the framework of fuzzy set theory in order to comprehend relation and to model observations of variables collected as either qualitative or approximately known quantities which are no longer being utilized directly in classical sense. When fuzzy correlation and fuzzy non-linear regression are concern, dealing with several computational complexities emerging due to the nature of fuzzy set theory is a challenge. It should be noted that there is no well-established formula or method in order to calculate fuzzy correlation coefficient or to estimate parameters of the fuzzy regression model. Therefore, a rich literature will accompany with the readers. While extension principle based methods are utilized in the computational procedures for fuzzy correlation coefficient, the distance based methods preferred rather than mathematical programming ones are employed in parameter estimation of fuzzy regression models. That extension principle combined with either fuzzy arithmetic or non-linear programming is two different methods proposed in the literature will be examined with small but illustrative examples in detail for fuzzy correlation analysis. Fuzzy non-linear regression has been a relatively new studied method when compared to fuzzy linear regression. However, both employ similar tools. S-curve fuzzy regression and two types of quadratic fuzzy regression models in the literature will be discussed.
Murat Alper Basaran, Biagio Simonetti, Luigi D’Ambra

### Fuzzy Decision Trees

Decision trees are one of the most widely used classification techniques because of their easily understandable representation. In the literature, various methods have been developed to generate useful decision trees. ID3 and SLIQ algorithms are two of the important algorithms generating decision trees. Although they have been applied for various real life problems, they are inadequate to represent ambiguity and vagueness of human thinking and perception. In this study, fuzzy ID3 and fuzzy SLIQ algorithms, which generate fuzzy decision trees, are discussed as well as their enhanced versions. Their performances are also tested using simple training sets from the literature.
Ayca Altay, Didem Cinar

### Fuzzy Shewhart Control Charts

Process Control is the active correction of a process based on the results of process monitoring. Once the process monitoring tools have detected an assignable cause, this cause is removed to bring the process back into control. This chapter presents the process control techniques under fuzziness. Variable and attribute control charts are extended to their fuzzy versions.
Cengiz Kahraman, Murat Gülbay, Eda Boltürk

### Fuzzy EWMA and Fuzzy CUSUM Control Charts

Exponentially Weighted Moving-Averages (EWMA) and Cumulative-Sum (CUSUM) control charts have the ability of detecting small shifts in the process mean. Classical EWMA and CUSUM charts are not capable to capture the uncertainty in case of incomplete data. Fuzzy EWMA and CUSUM control charts are developed in this chapter and numerical illustrations are given.
Nihal Erginel, Sevil Şentürk

### Linear Hypothesis Testing Based on Unbiased Fuzzy Estimators and Fuzzy Significance Level

A wide variety of applied problems of statistical hypothesis testing can be treated under a general setup of the linear models which includes analysis of variance. In this study, a new method is presented to test linear hypothesis using a fuzzy test statistic produced by a set of confidence intervals with non-equal tails. Also, a fuzzy significance level is used to evaluate the linear hypothesis. The method can be used to improve linear hypothesis testing when there is a sensitively in accepting or rejecting the null hypothesis. Also, as a simple case of linear hypothesis testing, one-way analysis of variance based on fuzzy test statistic and fuzzy significance level is investigated. Numerical examples are provided for illustration.
Alireza Jiryaei, Mashaallah Mashinchi

### A Practical Application of Fuzzy Analysis of Variance in Agriculture

For comparing several populations, the fuzzy analysis of variance has been summarized and reviewed where the collected data considered fuzzy rather than crisp numbers. As a practical work based on the real-word data, a case study was carried out to investigate effects of three concentrations (0, 50 and 100 ppm) nanoSiO2 on seedling growth and dry matter weight of fenugreek (Trigonella foenum-graceum L.). All presented data in this study are fuzzy and therefore we need an extended version of analysis of variance to investigate on these fuzzy observations. Although, the presented analysis of variance approach based on vague data can causes to a fuzzy decision, but as an advantage of the proposed approach the vagueness of this fuzzy decision measured.
R. Ivani, S. H. Sanaei Nejad, B. Ghahraman, A. R. Astaraei, H. Feizi

### A Survey of Fuzzy Data Mining Techniques

Data mining is very popular recently due to lots of analysis applications of big data. A well-known algorithm for mining association rules from transactions is the Apriori algorithm. Because transactions may include quantitative values, fuzzy sets which can be used to handle quantitative values are thus utilized to mine fuzzy association rules. Hence in this chapter, some useful fuzzy data mining techniques are introduced. Firstly, with the predefined membership functions, the Apriori-based fuzzy data mining algorithms that provide an easily way to mine fuzzy association rules are described. Since they may be time-consuming when dataset size is large, several tree-based fuzzy data mining methods are then stated to improve the mining efficiency. Besides, how to define appropriate membership functions for fuzzy data mining is important and it can be transferred into an optimization problem. Four types of genetic-fuzzy mining approaches are thus given to find both membership functions and fuzzy association rules. At last, some extended issues are discussed to provide future research directions.
Tzung-Pei Hong, Chun-Hao Chen, Jerry Chun-Wei Lin

### Backmatter

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