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Z-numbers contain fuzzy restrictions, credibility measures, and probability distributions to effectively represent uncertain information. Converting Z-numbers to fuzzy numbers facilitates extensive applications (such as multi-attribute decision-making (MADM)), thus becoming valuable for research purposes. Regarding Z-number conversions, the original method never considers the association probability, while probabilistic strategies offer better informatization. Recently, a probability-driven conversion starts with a linear transformation of the centroid difference between the fuzzy restriction and probabilistic distribution. However, it has the invalidation weakness of edge information due to underlying non-normalization. To improve this probability-linear conversion, a Z-number conversion is proposed by using underlying probability-exponential descriptions, and this new method is further applied to MADM. At first, the current probability-linear conversion is analyzed based on the initial non-probabilistic conversion, and its intrinsic weakness and correctional improvement are revealed. Then, the novel probability-exponential conversion resorts to an exponential characterization of centroid difference between the restriction and distribution, and it gains information enrichment due to underlying normalization. The refined method preserves the inherent characteristics of Z-numbers more effectively, facilitating their application in subsequent engineering practices. This is especially pertinent in decision-making systems based on expert input and initial value problems. The proposed method for converting Z-numbers aims to minimize information loss in transitions between Z-numbers and classical fuzzy numbers. This approach will be further explored in future research. Furthermore, the probability-exponential conversion induces an ExpTODIM algorithm for MADM, called PE-ExpTODIM. Three Z-number conversions (i.e., the non-probabilistic, probability-linear, and probability-exponential types) and three decision algorithms (i.e., ExpTODIM, EDAS, MOORA) are combined to establish a 3 × 3 framework of Z-number-driven MADM. Finally, the systematical 9 algorithms are applied to the problem of site selection of carbon storage. They are validated by criss-cross contrast analyses and statistical significance tests. Thus, PE-ExpTODIM exhibits the desired optimization. The last technology of statistical testing is original, ingenious, and valuable for MADM. [ABSTRACT FROM AUTHOR]

A fuzzy set extension known as the hesitant fuzzy set (HFS) has increased in popularity for decision making in recent years, especially when experts have had trouble evaluating several alternatives by employing a single value for assessment when working in a fuzzy environment. However, it has a significant problem in its uses, i.e., considerable data loss. The probabilistic hesitant fuzzy set (PHFS) has been proposed to improve the HFS. It provides probability values to the HFS and has the ability to retain more information than the HFS. Previously, fuzzy regression models such as the fuzzy linear regression model (FLRM) and hesitant fuzzy linear regression model were used for decision making; however, these models do not provide information about the distribution. To address this issue, we proposed a probabilistic hesitant fuzzy linear regression model (PHFLRM) that incorporates distribution information to account for multi-criteria decision-making (MCDM) problems. The PHFLRM observes the input–output (IPOP) variables as probabilistic hesitant fuzzy elements (PHFEs) and uses a linear programming model (LPM) to estimate the parameters. A case study is used to illustrate the proposed methodology. Additionally, an MCDM technique called the technique for order preference by similarity to ideal solution (TOPSIS) is employed to compare the PHFLRM findings with those obtained using TOPSIS. Lastly, Spearman's rank correlation test assesses the statistical significance of two rankings sets. [ABSTRACT FROM AUTHOR]

This paper mainly discusses the system integration mode of industrialized residential collaborative design, and proposes a system integration model based on Building Information Modeling (BIM) and Integrated Project Delivery (IPD) industrialized residential collaborative design, which solves the collaborative design problem of industrialized residential life cycle. This system integration mode will industrialize residential coordination. The key elements of the design are integrated to form a complete system methodology. Through the analysis of variance, the average and standard deviation of the respondents' scores were tested for significance. Then I used Fuzzy Set Theory to analyze the membership degree of key elements to verify my argument. [ABSTRACT FROM AUTHOR]

Three-way approximations of fuzzy sets are an important scheme of granular computing, by abstracting a fuzzy set to its discrete three option-alternatives which adhere to human cognitive behaviors and reduce the computational burden. The key point of such three-way approximations of fuzzy sets is how to choose a suitable design leading to their realization. Undesired three-way approximations might be produced if the selected mechanism is unsuitable to data distribution. In this study, the principles for constructing three-way approximations of fuzzy sets are summarized. The following taxonomy of these principles is provided, namely (i) uncertainty balance-based principle, (ii) prototype-based principle, and (iii) model-based invoking the tradeoff between classification error and the number of data that have to be classified. A number of detailed optimization models are discussed in detail. To evaluate the performance of different construction principles, a general unsupervised learning framework based on three-way approximations of fuzzy sets is exhibited. Some synthetic data sets along with sixteen data sets from UCI repository are involved for experiments. Friedman testing followed by Holm-Bonferroni testing are exploited to test the performance significance of the proposed criteria, which can provide insights and deliver guidance when choosing a principle for constructing three-way approximations of fuzzy sets in the real-world scenarios. The research methods in this paper can also be extended to supervised and semi-supervised learning areas. [ABSTRACT FROM AUTHOR]

Under vague and imprecise environment, data analysis methods are usually integrated with the fuzzy set theory. Intuitionistic methods sets are the extension of ordinary fuzzy sets whose aim is to incorporate the hesitancy of experts into their model. In this paper, novel intuitionistic fuzzy parametric approaches to statistical hypotheses testing for the population mean based on intuitionistic fuzzy data and intuitionistic fuzzy hypotheses are presented. First, bootstrap method is applied and then intuitionistic fuzzy test statistics is obtained. Applications of one sided and two sided hypotheses are provided. [ABSTRACT FROM AUTHOR]

Hypothesis tests are a statistical decision-making tool for testing if a hypothesized parameter value is supported by the sample data or not. Vagueness and impreciseness in the sample data require fuzzy techniques to be employed in the analysis. These techniques can be based on intuitionistic fuzzy sets, hesitant fuzzy sets, type-2 fuzzy sets, neutrosophic sets, or spherical fuzzy sets. In this paper, Z-fuzzy numbers are used to capture the vagueness in the sample data and develop Z-fuzzy hypothesis testing. A Z-fuzzy number is represented by a restriction function that is usually a triangular or trapezoidal fuzzy number and a reliability function representing the confidence level to the restriction function. Illustrative examples for left and right sided hypothesis testing and sensitivity analyses are presented. [ABSTRACT FROM AUTHOR]

Set‐theoretic multimethod research (SMMR) using Qualitative Comparative Analysis (QCA) formalizes the choice of cases based on a truth table analysis. We make three recommendations for improving SMMR. First, current standards can lead to faulty case selection if causal inference on a conjunction is the goal. Case selection needs to take into account that the non‐members of a conjunction might be empirically diverse and that only selected types of non‐members are ideal for causal inference. Second, we formally show that cases with similar fuzzy‐set memberships in a term and the outcome are the superior choice for process tracing. They minimize the expected membership in the mechanism and make it most difficult to pass a hypothesis test. Third, we propose formulas that comply with all SMMR principles and identify the best pairs of cases for analysis. We illustrate our arguments with a study of the effectiveness of sanctions against authoritarian regimes. [ABSTRACT FROM AUTHOR]

In this paper, we examine intuitionistic fuzzy soft sets arithmetical structure and give the definition of intuitionistic fuzzy soft subhemirings. We characterize operations on intuitionistic fuzzy soft subhemiring of hemiring and demonstrate a few outcomes on them. Moreover, meanings of fuzzy soft functions and fuzzy soft homomorphism are characterized and the hypotheses on homomorphic images and homomorphic pre images are given. From that point onward, the meaning of intuitionistic normal fuzzy subhemiring of hemiring is given what's more, some of its essential properties are considered and we made an endeavor to consider the logarithmic idea of an intuitionistic fuzzy soft subhemiring of a hemiring. [ABSTRACT FROM AUTHOR]

In testing the capability of industrial processes, the researcher considers and tests the vague hypothesis "the capability index is low" against the vague hypothesis "the capability index is high". But, two fuzzy concepts low and high are usually formulated by two crisp hypotheses in traditional quality tests. In this paper, we formulate these two fuzzy concepts by considering two complement fuzzy sets. Afterwords, a new p-value-based approach is considered for testing the mentioned fuzzy hypotheses which is constructed on the basis of two capability indices Cp and Cpm. This new approach has several advantages over the common p-value methods for testing fuzzy hypotheses. The main one is depending the result of this new approach on both null and alternative fuzzy hypotheses, while the common p-value-based methods are according to only the null fuzzy hypothesis. To clarify the potential of the proposed approach in the process of capability analyses, two applied examples are given based on two real-world data set. [ABSTRACT FROM AUTHOR]

In this paper, a new approach for testing fuzzy hypotheses based on likelihood ratio test statistic is presented. In this approach, we first formulate the hypotheses of interest by fuzzy sets, and then, a likelihood ratio statistic is defined based on the $$\delta $$ -cut of null fuzzy hypothesis. Also, in order to establish critical region, we give new definitions for the probabilities of type I and type II errors. [ABSTRACT FROM AUTHOR]

Anomalies are subsequences that exhibit departures from normal state of operation. In this paper, to solve the problems of unknown data distribution, control limit determination, multiple parameters, training data and fuzziness of 'anomaly', a self-adaptive and unsupervised model is developed for finding anomalies in data streams. A salient feature is a synergistic combination of both statistical and fuzzy set-based techniques. Anomaly detection problem is viewed as a certain statistical hypothesis testing which is realized in an unsupervised mode. At the same time, 'anomaly' is a much more complex concept and as such can be described with fuzzy set theory. Fuzzy sets bring a facet of robustness to the overall scheme and play an important role in the successive step of hypothesis testing. Because of the fuzzification, parameters determination is self-adaptive and no parameter needs to be specified by the user, what's more, there is no need to consider the data distribution in statistical hypothesis testing in this paper. The approach is validated with a number of experiments, which help to quantify the performance of constructed algorithm. [ABSTRACT FROM AUTHOR]

Fuzzy-set theory has provided researchers with a new perspective on many social-scientific problems. In particular, the method of fuzzy-set Qualitative Comparative Analysis (fsQCA) has gained in popularity across various disciplines. However, while the methodological development of fsQCA has progressed on a number of fronts, sensitivity diagnostics have only recently been put on the agenda. This article analyses how coverage as an important descriptive statistic in fsQCA is influenced by the interaction between membership function form and crossover threshold choice. Depending on the relative location of the latter, changes in the former influence coverage in either a negative or a positive direction, and to different magnitudes. This influence is not uniform but varies in relation to cases’ distance from unique peak points. Although the orientation of this article is theoretical, its results have implications for empirical research. Most importantly, the influence of membership functions should become part of routine sensitivity checks. [ABSTRACT FROM AUTHOR]

Abstract: A hypothesis test for the inclusion of the expected value of a fuzzy-valued random element in a given fuzzy set is provided. The exclusion, or the empty intersection of the expected value of a random fuzzy set and a fuzzy set, is also tested, that allows us to define one-sided tests for this expected value without the need of considering any restrictive ordering. Both tests are developed by using a measure of the intersection between fuzzy sets. Asymptotic and bootstrap techniques are established. Some simulations are included to show the performance of the bootstrap approaches. Finally, the methodology proposed is applied on a real-life situation related to the field of sensorial analysis. [Copyright &y& Elsevier]

Buckley's approach (Buckley (2004), (2005), (2006)) uses sets of confidence intervals by taking into consideration both of the uncertainty and impreciseness of concepts that produce triangular shaped fuzzy numbers for the estimator. This approach produces fuzzy test statistics and fuzzy critical values in hypothesis testing. In addition, the sample size is fixed for this test. When data comes sequentially, however, it is not suitable to study with a fixed sample size test. In such cases, sequential and group sequential tests are recommended. Unlike a sequential test, a group of sequential test provides substantial savings in sample and enables us to make decisions as early as possible. This intends paper to combine the benefits of group sequential test and Buckley's approach using α-cuts. It attempts to show that using α-cuts can be used within the group sequential tests. To illustrate the test more explicitly a numerical example is also given. [ABSTRACT FROM AUTHOR]

For many scientific experiments computing a p-value is the standard method for reporting the outcome. It is a simple way of summarizing the information in the data. One theoretical justification for p-values is the Neyman-Pearson theory of hypotheses testing. However, the decision making focus of this theory does not correspond well with the desire, in most scientific experiments, for a simple and easily interpretable summary of the data. Fuzzy set theory with its notion of a membership function gives a non-probabilistic way to talk about uncertainty. Here, we argue that for some situations, where a p-value is computed, it may make more sense to formulate the question as one of estimating a membership function of the subset of special parameter points which are of particular interest for the experiment. Choosing the appropriate membership function can be more difficult than specifying the null and alternative hypotheses but the resulting payoff is greater. This is because a membership function can better represent the shades of desirability among the parameter points than the sharp division of the parameter space into the null and alternative hypotheses. This approach yields an estimate which is easy to interpret and more flexible and informative than the cruder p-value. [ABSTRACT FROM AUTHOR]

How to account for variation in economic performance across countries and periods? This paper proposes a new substantive and methodological approach. Substantively, we focus on variation in overall performance: the combination of growth, employment, and debt. We construct eight models using these variables and analyze countries' fit to them. Methodologically, we employ fuzzy-set qualitative comparative analysis (fsQCA). The analysis of 19 OECD countries between 1975 and 2005 fails to support traditional hypotheses about partisanship, central bank independence, trade openness, and corporatism. Instead, different combinations of conditions are conducive to good performance and these differ across periods and countries. [ABSTRACT FROM AUTHOR]

This paper extends the Wilcoxon signed-rank test to the case where the available observations are imprecise quantities, rather than crisp. To do this, the associated test statistic is extended, using the α-cuts approach. In addition, the concept of critical value is generalized to the case when the significance level is given by a fuzzy number. Finally, to accept or reject the null hypothesis of interest, a preference degree between two fuzzy sets is employed for comparing the observed fuzzy test statistic and fuzzy critical value. [ABSTRACT FROM AUTHOR]

Process capability indices (PCIs) have been popularly used in the manufacturing industry for measuring process potential and performance. When our data are imprecise, classical PCIs are improper. In this paper, a graphical approach has been presented for monitoring whether or not our process is capable by using the process capability plots for two families of the one-sided PCIs based on fuzzy data. The presented methodology takes into consideration the certain degree of imprecision on the sample data and leads to the three-decision rule that has been presented on the process capability plot. Also, the advantages of this method in comparison to the fuzzy hypothesis test have been discussed. Two real examples have been presented to demonstrate this approach. [ABSTRACT FROM AUTHOR]

This work proposes a fuzzy statistical test of fuzzy hypotheses usinglinguistic variables. The method is based on Zadeh's principle: the fuzzy population mean in the null hypothesis is converted to fuzzy numbers using conversion scales proposed by Chen and Hwang (1992). The method proposed by Chen et al [10] reflects thereal situation than conventional methods [ABSTRACT FROM AUTHOR]

A lifetime capability index L tp has been proposed to measure the business lifetime performance, wherein output lifetime measurements are assumed to be precise from the Pareto model with censored information. In the present study, we study a more realistic situation where the lifetime output data are imprecise. The approach developed by Buckley [Fuzzy system, Soft Comput. 9 (2005), pp. 757–760; Fuzzy statistics: Regression and prediction, Soft Comput. 9 (2005), pp. 769–775] incorporated with some extensions (a set of confidence intervals, one on top of the other), is used to construct the triangular-shaped fuzzy number for the fuzzy estimate of the L tp. With the sampling distribution of the unbiased estimator of the L tp, two useful fuzzy inference criteria, its critical value and fuzzy p-value are obtained to assess the lifetime performance. The presented methodology can handle the lifetime performance assessment on the condition that sample lifetime data are involved with imprecise information, classifying the lifetime performance with the three-decision rule. With different preset requirements and a certain degree of imprecise data, we also develop a four quadrants decision-making plot where managers can easily simultaneously visualize several important features of lifetime performance for making a decision. An example of business lifetime data is given to illustrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We extend the notion of confidence region to fuzzy data, by defining a pair of fuzzy inner and outer confidence regions. We show the connection with previous proposals, as well as with recent studies on hypothesis testing with low quality data. [Copyright &y& Elsevier]

Abstract: A new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed. Under the considerations of fuzzy parameters and fuzzy arithmetic operations (fuzzy addition and multiplication), we propose a fuzzy linear regression model which has the similar form as that of conventional one. We conduct the h-level (conventional) linear regression models of fuzzy linear regression model for the sake of invoking the statistical techniques in (conventional) linear regression analysis for real-valued data. In order to determine the sign (nonnegativity or nonpositivity) of fuzzy parameters, we perform the statistical testing hypotheses and evaluate the confidence intervals. Using the least squares estimators obtained from the h-level linear regression models, we can construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is well-known in fuzzy sets theory. In order to obtain the membership degree of any given estimate taken from the fuzzy least squares estimator, optimization problems have to be solved. We also provide two computational procedures to deal with those optimization problems. [Copyright &y& Elsevier]

Abstract: The diagnosis of Sleep Apnoea–Hypopnoea Syndrome requires the visual inspection of a recording containing a large number of physiological parameters of the patient – the polysomnogram. The purpose of this visual inspection is the identification and characterization of different types of pathological events that occur over these parameters. These events are defined by a set of morphological criteria. Based on these criteria, commercial tools have been developed to support clinicians in the task of visually inspecting polysomnograms. This article argues that the standard morphological criteria are just guiding recommendations that experienced physicians often adapt to each specific diagnostic context. Thus, tools that support the analysis of polysomnograms ideally should use flexible criteria that could be easily customizable by clinicians. In this paper, we propose algorithms capable of identifying pathological events relevant in the diagnosis of SAHS using custom criteria that are acquired directly from the clinician. These algorithms take advantage of the Fuzzy Set Theory to capture and handle the vagueness and uncertainty that are characteristics of medical knowledge. Knowledge acquisition using the traditional linguistic approach of the Fuzzy Sets Theory is supported by a desktop tool. However, the authors feel that some of the criteria that need to be acquired are more visual in nature than linguistic. An alternative mechanism for the visual acquisition of these criteria is proposed. Finally, when presenting the pathological events that have been identified, the tool uses several visual metaphors designed to simplify visual inspection of the polysomnogram. We have validated our proposal over 69h of polysomnographic recordings arising from 12 patients that were subjected to a sleep study. 95.7% of the events identified were correct detections. The rate of false negatives was 1.6%. [Copyright &y& Elsevier]

Abstract: This paper first defines the profitability to be the probability of achieving a target profit under the optimal ordering policy, and introduces a new index (achievable capacity index; I A ) which can briefly analyze the profitability for newsboy-type product with normally distributed demand. Note that since the level of profitability depends on the demand mean μ and the demand standard deviation σ if the related costs, selling price, and target profit are given, the index I A is a function of μ and σ. Then, we assess level performance which examines if the profitability meets designated requirement. The results can determine whether the product is still desirable to order/manufacture. However, μ and σ are always unknown, and the demand quantity is common to be imprecise, especially for new product. To tackle these problems, a constructive approach combining the vector of fuzzy numbers is introduced to establish the membership function of the fuzzy estimator of I A . Furthermore, a three-decision testing rule and step-by-step procedure are developed to assess level performance based on fuzzy critical values and fuzzy p-values. [ABSTRACT FROM AUTHOR]

An approach to test the crisp statistical hypotheses upon the fuzzy test statistic is developed. The proposed approach is investigated for two cases: the case without nuisance parameter, and the case with a nuisance parameter. For rejecting or accepting the null hypothesis of interest, an evidential point of view is proposed, based on a concept called degree of consistency of observations with the hypothesis. The approach is applied to test the parameters of the normal, Bernoulli, Poisson, and exponential distributions, as well as to test the difference of means and the ratio of variances of two normal distributions. [ABSTRACT FROM AUTHOR]

The aim of this work is to introduce ψ-operations on fuzzy topological spaces and to use them to study fuzzy generalized ψρ-closed sets and fuzzy generalized ψρ-open sets. Also, we introduce some characterizations and properties for these concepts. Finally we show that certain results of several publications on the concepts of weakness and strength of fuzzy generalized closed sets are considered as corollaries of the results of this research. [ABSTRACT FROM AUTHOR]

Abstract: The application of fuzzy sets theory to statistical confidence intervals for unknown fuzzy parameters is proposed in this paper by considering fuzzy random variables. In order to obtain the belief degrees under the sense of fuzzy sets theory, we transform the original problem into the optimization problems. We provide the computational procedure to solve the optimization problems. A numerical example is also provided to illustrate the possible application of fuzzy sets theory to statistical confidence intervals. [Copyright &y& Elsevier]

Human Response to a question with multiple alternatives can be imprecise or fuzzy for various reasons. Part of this fuzziness can be attributed to the lack of sincerity and clarity in the thought process of the respondent. A formal measure of this aspect of fuzziness is formulated in this work on the basis of consistency of the respondent's response to the same question repeated in multiple scales. Discarding haphazard and insincere respondents can improve the quality of data resulting in more efficient survey analysis. This may be achieved in the framework of statistical testing of hypothesis using the probability distribution of the proposed fuzziness measure. Similarly, an attribute can be fuzzy, and it may generate inconsistent response from many respondents. The application of the proposed methodology extends to identifying such fuzzy or unclear attributes. The paper also proposes an algorithm for screening inconsistent respondents and fuzzy attributes simultaneously. [ABSTRACT FROM AUTHOR]

Presents a response by Charles J. Geyer and Glen D. Meeden to several comments on their paper published in this same issue which introduces a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of uniformly most powerful (UMP) one-tailed and UMP unbiased two-tailed randomized tests. New fuzzy procedures proposed by the commentators; Claim that the authors' fuzzy procedures are not really new; Purpose of the authors in developing fuzzy confidence intervals.

Critiques a paper by Charles J. Geyer and Glen D. Meeden published in this same issue which introduces a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of uniformly most powerful (UMP) one-tailed and UMP unbiased two-tailed randomized tests. Ways to represent randomized confidence intervals; Issue of exact versus approximate coverage; Idea of abstract randomized confidence intervals.

The optimal hypothesis tests for the binomial distribution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two-tailed randomized tests. Conventional confidence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of UMP and UMPU tests. We also introduce a new P-value notion, called fuzzy P-values or abstract randomized P-values, that also inherits the same exactness and optimality. [ABSTRACT FROM AUTHOR]

The concept of set inclusion has remained insufficiently developed in the fuzzy set literature to be of much use to social scientists. However a fully fledged concept of fuzzy set inclusion, along with appropriate statistical methods for evaluating it, could be very useful in the social sciences. This article combines fuzzy set and statistical methods, in the form of a cumulative distribution-based approach to evaluating fuzzy set inclusion without making strong assumptions about measurement levels. It establishes criteria for distinguishing an "inclusion relation "from independence plus skew as well as from other kinds of relationships. A measure of inclusion is developed that is sensitive to the degree to which individual cases violate a strict inclusion rule. A technique for modeling localized inclusion relations in contingency tables and scatter plots is also presented. Finally, the connections between the fuzzy set approach to set inclusion and mainstream statistical techniques are briefly adumbrated. [ABSTRACT FROM AUTHOR]

Initially, geographical information systems (GIS) concentrated on two issues: automated map making, and facilitating the comparison of data on thematic maps. The first required high quality graphics, vector data models and powerful data bases, the second is based on grid cells that can be manipulated by suites of mathematical operators collectively termed “map algebra”. Both kinds of GIS are widely available and are taught in many universities and technical colleges. After more than 20 years of development, most standard GIS provide both kinds of functionality and good quality graphic display, but until recently they have not included the methods of statistics and geostatistics as tools for spatial analysis. Recently, standard statistical packages have been linked to GIS for both exploratory data analysis and statistical analysis and hypothesis testing. Standard statistical packages include methods for the analysis of random samples of cases or objects that are not necessarily co-located in space—if the results of statistical analysis display a spatial pattern then that is because the underlying data also share that pattern. Geostatistics addresses the need to make predictions of sampled attributes (i.e., maps) at unsampled locations from sparse, often expensive data. To make up for lack of hard data geostatistics has concentrated on the development of powerful methods based on stochastic theory. Though there have been recent moves to incorporate ancillary data in geostatistical analyses, insufficient attention has been paid to using modern methods of data display for the visualization of results. GIS can serve geostatistics by aiding geo-registration of data, facilitating spatial exploratory data analysis, providing a spatial context for interpolation and conditional simulation, as well as providing easy-to-use and effective tools for data display and visualization. The value of geostatistics for GIS lies in the provision of reliable interpolation methods with known errors, methods of upscaling and generalization, and for supplying multiple realizations of spatial patterns that can be used in environmental modeling. These stochastic methods are improving understanding of how errors in models of spatial processes accrue from errors in data or incompleteness in the structure of the models. New developments in GIS, based on ideas taken from map algebra, cellular automata and image analysis are providing high level programming languages for modeling dynamic processes such as erosion or the development of alluvial fans and deltas. Research has demonstrated that these models need stochastic inputs to yield realistic results. Non-stochastic tools such as fuzzy subsets have been shown to be useful for spatial analysis when probabilistic approaches are inappropriate or impossible. The conclusion is that in spite of differences in history and approach, the linkage of GIS, statistics and geostatistics provides a powerful, and complementary suite of tools for spatial analysis in the agricultural, earth and environmental sciences. [ABSTRACT FROM AUTHOR]

Examines how statistical significance levels are treated and interpreted by graduate students who use hypothesis-testing in their scientific investigation. Use of fuzzy set theory to identify uncertain points in judgments; Provision of a technique that is more natural for examining and understanding imprecise boundaries of human decisions.

Fuzzy rough set theory is a powerful tool to deal with uncertainty information, which has been successfully applied to the fields of attribute reduction, rule extraction, classification tree induction, etc. In order to comprehensively investigate attribute reduction methods in fuzzy rough set theory, this paper first briefly reviews the related concepts of fuzzy rough set theory. Then, all methods are summarized through six different aspects including data sources, preprocessing methods, fuzzy similarity metrics, fuzzy operations, reduction rules, and evaluation methods. Among them, reduction rules are reviewed in three categories, i.e., fuzzy dependency-based, fuzzy uncertainty measure-based, and fuzzy discernibility matrix-based. These three types of reduction rules are compared and analyzed through experiments. The experimental results clarify that these three reduction rules can retain fewer attributes and improve or maintain the classification accuracy of a classifier. Moreover, the statistical hypothesis test is conducted to evaluate the statistical difference of these methods. The results show that these algorithms are statistically significantly different. Finally, some new research directions are discussed. • Attribute reduction methods based on fuzzy rough set theory are comprehensively reviewed. • All methods are summarized through six different aspects. • The experimental results clarify that the three types of reduction methods are effective. • Statistical hypothesis test verified that the performance of these algorithms is different. [ABSTRACT FROM AUTHOR]

Taking advantage of the possibility of fuzzy test statistic falling in the rejection region, a statistical hypothesis testing approach for fuzzy data is proposed in this study. In contrast to classical statistical testing, which yields a binary decision to reject or to accept a null hypothesis, the proposed approach is to determine the possibility of accepting a null hypothesis (or alternative hypothesis). When data are crisp, the proposed approach reduces to the classical hypothesis testing approach. [ABSTRACT FROM AUTHOR]

Comments on an article by Charles J. Geyer and Glen D. Meeden published in this same issue which introduces a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of uniformly most powerful (UMP) one-tailed and UMP unbiased two-tailed randomized tests. Fuzzy procedures proposed by Geyer and Meeden; Description of the fuzzy confidence sets; Interpretations of the fuzzy P-values.

Critiques a paper by Charles J. Geyer and Glen D. Meeden published in this same issue which introduces a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of uniformly most powerful (UMP) one-tailed and UMP unbiased two-tailed randomized tests.

Critiques a paper by Charles J. Geyer and Glen D. Meeden published in this same issue which introduces a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of uniformly most powerful (UMP) one-tailed and UMP unbiased two-tailed randomized tests. Paper's potential to improve statistical practice for both simple and complex models; Problem of inference for binomial probabilities; Ways in which the classical p-value can be generalized when moving to p-values that depend on unknown parameters in composite hypotheses.