Fuzzy Hypothesis Testing Research Articles

Testing fuzzy hypotheses with crisp data based on p-value

In this paper, we propose the p-value of fuzzy hypothesis by the concept of projection of the fuzzy relation, the small p-value gives evidence that research hypothesis is true. For the crisp hypothesis, we give the fuzzy set description of the test and prove that the rejection region of the test is the cut set of the fuzzy set. Finally, the p-values of fuzzy hypothesis are given by specific examples.

Employing fuzzy hypothesis testing to improve modified p charts for monitoring the process fraction nonconforming

Although 3σ control limits of p control charts are quite common in practical applications, they are not appropriate when the process mean is wandering with some degree of consistency. For this reason, we firstly present modified control limits for the binomial p chart based on interval-valued hypothesis testing to allow the process fraction nonconforming to vary over a specified interval without appreciably affecting the overall performance of the process. This modified p chart is able to support extreme quality levels, but it neglects to consider economy in production of units between both these extremes, i.e., gradual deterioration or gradual improvement of the product quality with respect to the average quality. With the aim of achieving a reasonable balance between both these extremes, we additionally introduce a modified p chart based on fuzzy-valued hypothesis testing. The fuzzy modified p chart is not only able to adequately meet the trade-off between the common and the modified p chart, but also enables to incorporate economic aspects of producing better/worse acceptable quality. To demonstrate the benefits of the proposed fuzzy modified p chart, we conduct a comprehensive case study on monitoring of medical insurance claims.

Asynchronous cooperative spectrum sensing via sequential detection in fuzzy hypothesis testing under noise uncertainty in cognitive radio networks

Spectrum sensing is a critical part in Cognitive radio networks to detect the available frequency resources, and its timeliness and accuracy are extremely important. The conventional detectors usually fail to be robust in the presence of noise uncertainty, hence an asynchronous cooperative spectrum sensing approach based on sequential probability ratio detection in fuzzy hypothesis testing is proposed. In the approach, sequential detection in fuzzy hypothesis testing is done by each cognitive radio to form local decisions, then fusion centre sequentially accumulates the local decisions of cooperative cognitive radios to make a global decision. Simulation results show that the proposed approach can reduce the average number of samples and overcome noise uncertainty when compared with conventional or selective asynchronous cooperative sensing and non-cooperative sensing.

Fuzzy supplier selection model based on lifetime performance index

In view of the impact of global warming, the concept of a circular economy (CE) is receiving much attention from corporations and governments. Many studies have suggested that when products and components are of high quality and long lifetime, they are more easily repaired after breakdown, making it more likely that they will be re-used. Longer product lifetimes also reduce carbon emissions and the cost of maintenance. Therefore, improving the quality and lifetime of products is key to the implementation of CE. However, in supplier selection, product lifespan is often overlooked in favor of product quality and the ability to deliver on time. This study therefore used a lifetime performance index as a tool for supplier selection to ensure the reliability of the final product. Due to the fact that the index contains unknown parameters, we derived the confidence interval of the index and examined the influence of sample size on confidence interval length and statistical inference accuracy. Based on cost and effectiveness considerations, we constructed a fuzzy membership function using the confidence interval of the index to increase testing accuracy and overcome uncertainty in measurement data. We further propose a fuzzy hypothesis testing method to aid in the selection of suppliers with good product lifetime performance. This method is grounded on the confidence interval of the index and can thus lower the chance of erroneous judgment caused by sampling error. At the same time, it increases testing accuracy. We also present a numerical example to demonstrate the efficacy of the proposed method.

Generalized two-tailed hypothesis testing for quantiles applied to the psychosocial status during the COVID-19 pandemic.

Nonparametric tests do not rely on data belonging to any particular parametric family of probability distributions, which makes them preferable in case of doubt about the underlying population. Although the two‐tailed sign test is likely the most common nonparametric test for location problems, practitioners face serious drawbacks, such as its lack of statistical power and its inapplicability when information regarding data and hypotheses is uncertain or imprecise. In this paper, we generalize the two‐tailed sign test by embedding fuzzy hypotheses caused by uncertainty/imprecision regarding linguistic statements on fractions of underlying quantiles. By achieving this objective, (1) crucial limitations of the common two‐tailed sign test are mitigated/overcome, (2) various further strengths are incorporated into the sign test (e.g., meeting the trade‐off between point‐ and interval‐valued hypotheses, facilitated formulation of fuzzy hypotheses, standardization of membership functions), and (3) shortcomings that often come along with fuzzy hypothesis testing are avoided (e.g., higher complexity, fuzzy test decision, possibilistic interpretation of test results). In addition, we conduct a comprehensive case study using a real data set on the psychosocial status during the COVID‐19 pandemic. The results of the case study clearly indicate that the generalized two‐tailed sign test is preferable to the two‐tailed sign test with point‐ or interval‐valued hypotheses.

Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing

In this paper, we develop fuzzy, possibilistic hypothesis tests for testing crisp hypotheses for a distribution parameter from crisp data. In these tests, fuzzy statistics are used, which are produced by the possibility distribution of the estimated parameter, constructed by the known from crisp statistics confidence intervals. The results of these tests are in much better agreement with crisp statistics than the ones produced by the respective tests of a popular book on fuzzy statistics, which uses fuzzy critical values. We also present an error that we found in the implementation of the unbiased fuzzy estimator of the variance in this book, due to a poor interpretation of its mathematical content, which leads to disagreement of some fuzzy hypotheses tests with their respective crisp ones. Implementing correctly this estimator, we produce test statistics that achieve results in hypotheses tests that are in much better agreement with the results of the respective crisp ones.

Nonparametric fuzzy hypothesis testing for quantiles applied to clinical characteristics of COVID-19.

The sign test is one of the most popular nonparametric tests for location problems and allows testing for any quantile of a population. However, the common sign test has serious drawbacks such as loss of information by considering solely signs of observations but not their magnitudes, various problems related to handling of ties in the data, and the lack of embedding uncertainty regarding the fraction of underlying quantile. To address these issues, we present an extended sign test based on fuzzy categories and fuzzy formulated hypotheses that improves the generality, versatility, and practicability of the common sign test. This generalized test procedure is neat in theory and practice and avoids disadvantages that are often associated with fuzzy tests (e.g., a considerably higher complexity of the underlying model, a fuzzy test decision, and a possibilistic instead of a probabilistic interpretation of test results). In addition, we perform a comprehensive case study on COVID‐19 in HIV‐infected individuals with a focus on human body temperature and related measurement problems. The results of the study clearly indicate that fuzzy categories and fuzzy hypotheses improve the performance of the sign test.

Fuzzy Hypotheses Testing in the Framework of Type-2 Fuzzy Numbers

The estimation of various components in science always faces with errors and inaccuracies that are presented in the form of an uncertainty in identifying the considered components. This uncertainty can be represented in different shapes, one of which is the representation of estimated components in the form of fuzzy numbers. In a situation where a shortage of information or ambiguity in the studied system can be seen, fuzzy numbers can be used to express the ambiguity available in the components. In this study, we will introduce new approach for the formation of type-2 fuzzy numbers in which the mean and standard deviations of data are used to determine the uncertainty zone. Finally, with proposed type-2 fuzzy numbers, in proportion to any situation we have, we face with the claims related to the mean and assess with the subject of the statistical hypothesis test with fuzzy approach.

Monitoring fuzzy linear quality profiles: A comparative study

Quality profiles representing the quality of a process or product as the functional relationship between one or more dependent variables and one or more explanatory variables are nowadays widely recognized in statistical process control (SPC) applications by both researchers and practitioners. On the other hand, in many real-world cases, evaluation of process or product characteristics is carried out with ambiguity or conducted using linguistic values. The theory of fuzzy sets provides an appropriate approach to deal with uncertainty due to ambiguity in human subjective evaluations or vagueness in linguistic variables. The purpose of this study is to introduce two novel methods based on fuzzy regression modeling for monitoring fuzzy linear profiles in phase II of SPC. To accomplish this, fuzzified Hoteling’s T2 statistic and fuzzy hypothesis testing are used. Moreover, a simulation study is used to compare the performance of the proposed methods compared with previous methods, based on the average run length (ARL) criterion in order to assess the detectability of charts with regard to the step shifts in profile parameters. Finally, the results of a real-world example in the tile and ceramic industry are presented.

Data driven supplier selection as a circular economy enabler: A Taguchi capability index for manufactured products with asymmetric tolerances

Prior studies have failed to account for the effect of product quality on the sustainability of supply chain networks. To address this gap, this study develops a Taguchi capability index that considers the economic, environmental, and social losses incurred due to poor product quality that leads to process losses. These product quality measures give a foundation for the operationalization of the circular economy concept, which is now receiving a growing level of attention from both firms and governments throughout the world. Sustainable supply chain management models that integrate upstream and downstream operations are increasing in popularity due to globalized competition, where the quality of components from each supplier affects the quality of the final product. One practical challenge is that many quality characteristics possess asymmetric tolerances. This study develops a comprehensive supplier selection model based on process quality, using a Taguchi capability index with asymmetric tolerances as an instrument for data driven supplier selection. Mathematical programming is used to obtain the confidence intervals of the indices to construct fuzzy membership functions. A fuzzy hypothesis testing method is then used to select suppliers taking into account uncertainty associated with measurement data. A case study illustrates the effectiveness of the methodology to assist firms in selecting the suppliers, and thus improve the sustainability of supply chain networks as a circular economy enabler.

Two-Tailed Fuzzy Hypothesis Testing for Unilateral Specification Process Quality Index

The quality characteristics with unilateral specifications include the smaller-the-better (STB) and larger-the-better (LTB) quality characteristics. Roundness, verticality, and concentricity are categorized into the STB quality characteristics, while the wire pull and the ball shear of gold wire bonding are categorized into the LTB quality characteristics. In terms of the tolerance, zero and infinity (∞) can be viewed as the target values in line with the STB and LTB quality characteristics, respectively. However, cost and timeliness considerations, or the restrictions of practical technical capabilities in the industry, mean that the process mean is generally far more than 1.5 standard deviations away from the target value. Researchers have accordingly proposed a process quality index conforming to the STB quality characteristics. In this study, we come up with a process quality index conforming to the LTB quality characteristics. We refer to these two types of indices as the unilateral specification process quality indices. These indices and the process yield have a one-to-one mathematical relationship. Besides, the process quality levels can be completely reflected as well. These indices possess unknown parameters. Therefore, sample data are required for calculation. Nevertheless, interval estimates can lower the misjudgment risk resulting from sampling errors more than point estimates can. In addition, considering cost and timeliness in the industry, samples are generally small, which lowers estimation accuracy. In an attempt to increase the accuracy of estimation as well as overcome the uncertainty of measured data, we first derive the confidence interval for unilateral specification process quality indices, and then propose a fuzzy membership function on the basis of the confidence interval to establish the two-tailed fuzzy testing rules for a single indicator. Lastly, we determine whether the process quality has improved.

Development and Application of a Performance Evaluation Matrix: A Case Study on Exploring the Items Considered Critical to Quality

Abstract A performance evaluation matrix is based on questionnaires collected from users or customers to pinpoint the performance of a system operation and furthermore to locate the items that need to be improved to upgrade the performance of system operations. To achieve this, this article proposes a discrimination index and applies statistics inference to deduce confidence intervals of the discrimination index. Meanwhile, we refer to Buckley’s fuzzy testing method to construct a fuzzy affiliation function and address a fuzzy evaluation criterion for exploring the items considered critical to quality to overcome the complicated problems of the questionnaires’ sampling error and interviewees’ fuzzy linguistics. The advantages of the method in this article are that it keeps the simple filling pattern of Likert’s scale and provides efficacy of data collection. Subsequently, we reduce the fuzzy linguistics and questionnaires’ sampling error by statistics inference and fuzzy hypothesis testing. Finally, we use a case study with the computer-assisted language learning system to demonstrate an application of the proposed method.

A fuzzy approach to determine process quality for one-sided specification with imprecise data

Process quality is a crucial determinant of client satisfaction and affects a product’s value; therefore, maintaining quality control is a vital aspect of corporate sustainability and development. Many researchers have developed evaluation models for process quality. The majority of such research assumes that collected measurement data are precise, but fuzziness and stochastic uncertainty are unavoidable features of any collected data. When the measurements of a quality characteristic are insufficiently precise, a crisp-based approach is not suitable for the assessment of process quality. This study endeavored to use one-sided Six Sigma quality indices as measurement tools to accurately reflect process yield and quality levels. Taking Buckley’s approach into consideration, we extend the crisp estimators from the indices into fuzzy estimators. We then develop a fuzzy hypothesis testing method for one-sided Six Sigma quality indices, with the intent of increasing reliability of evaluation for process quality levels. Finally, we present a real-world case to illustrate implementation of the proposed approach, demonstrating its effectiveness and practical applicability.

Fuzzy possibility regression integrated with fuzzy adaptive neural network for predicting and optimizing electrical discharge machining parameters

An electrical discharge machining (EDM) is one of the special production methods that are widely used in moldings, repairs and production of specific industrial components. Due to extensive production costs, optimal machining specifications are significant. Machining specifications are effective on output quality and thus attract more customers leading to higher profits. In this study, the impact of EDM parameters on surface roughness, material removal rate and electrode corrosion percentage have been investigated. In order to consider uncertainty of real production environments, the fuzzy theory is employed. Also, using the design of experiment (DOE) parameters calibration is performed and mathematical programming approach is applied for optimization purpose. The relationship between the machining parameters and the output process specification is examined by a fuzzy possibility regression model. Then, the mathematical relation of exact inputs and fuzzy outputs of the EDM process are extracted. The effectiveness of the three outputs is evaluated by interfacing models and fuzzy hypothesis testing. To determine the optimal levels of each output, a fuzzy adaptive neural network is used and appropriate models are prepared to be adapted with a fitted model of fuzzy possibility regression for comparison purposes. Validation tests imply the effectiveness of the proposed method. The integrated model is implemented in real case study. The results show that, fitted models can predict the material removal rate, surface fineness, and corrosion percentage of the electrode. The prediction accuracy of the proposed method is shown in comparison with the optimal fuzzy adaptive neural network outputs considering error value. Also, the proposed method is successful in identifying the optimal process parameters for EDM with reliable accuracy. The proposed integrated prediction and optimization model can be used as a calibration decision support in production systems to handle dynamic data structures and provide real time machining specifications to increase the output quality.

Developing a Fuzzy Verification Method of Performance Improvement Using Satisfaction Index

Performance evaluation matrix (PEM) has captured many scholars’ attention and has been widely used to a variety of industry fields to evaluate their efficiency performance. The data mainly come from the collection of questionnaires in which customers’ varied opinions are enquired; however, Likert scale will be modified to fuzzy linguistic scale due to some uncertainty aspect in the linguistic data. In contrast, the major flaw of some researches was that they failed to present a verification approach with statistical inference to confirm the effectiveness of the improvement. Regarding to the related issues, Chen et al. (2018a) use discrimination index in reliance section to establish fuzzy membership function; subsequently, they propose fuzzy hypothesis testing to locate critical-to-quality (CTQ) so as to make further improvement. Following the same logic, this paper uses the identical process to develop the fuzzy verification method of the improvement efficiency for CTQ. Finally, this paper puts forward a numerical example to demonstrate application of the proposed verification model. The result means that there is sufficient evidence showing that performance improvement is poor and non-effective. Thus, manager should adopt otherwise measures to improve. Obviously, the method proposed particularly highlights the reduction of sampling error, shies away from the complexity inherent in fuzzy semantic collection, overcomes the uncertainty in the collected data, and increases the reliability of evaluation method.

Construction and fuzzy hypothesis testing of Taguchi Six Sigma quality index

By taking process targeting as well as process variability into consideration, the Taguchi capability index gives a reasonable indication of process loss. This makes it an ideal tool for practical applications that depend on the evaluation of process quality. The ability of Six Sigma quality management to reduce process defect rates has led a number of researchers to investigate the relationship between and Six Sigma quality levels. Unfortunately, previous efforts indicate the quality level using only a range, rather than a specific value. Our objective in this study was to develop a Taguchi Six Sigma quality index that retains the advantages of in the assessment of process performance, while providing a specific value for the quality level associated with the process in question. To ensure rigorous quality assessments, we employed the upper confidence limit of in the design of a testing model for use by manufacturers. Fuzziness and stochastic uncertainty are unavoidable aspects of data collection. We, therefore, adopted a right half triangular-shaped fuzzy number for to deal with imprecise data. We also developed a method of the fuzzy hypothesis testing for to make reliable decisions for process quality assessment.

Quality Control Process Based on Fuzzy Random Variables

Quality control processes are very important for monitoring applications where the aim is to identify special causes of variation and improve quality and productivity. However, in many real-world cases, elements of quality control processes may be observed or defined imprecisely. Many studies have been focused on quality control processes dealing with normal random variables. Notably, none of such research has come to a unified approach to investigate all aspects of quality control processes in fuzzy environment, such as control charts, process capability, confidence interval, and hypothesis testing. In the present study, almost all processes required by the classical statistical quality control are developed in a fully fuzzy environment with fuzzy observations, fuzzy mean, fuzzy variance, and fuzzy control limits. The results are presented in three parts: 1) fuzzy control charts, 2) fuzzy process capability, and 3) fuzzy hypothesis testing for fuzzy mean and fuzzy variance. For this purpose, first, the concept of sample variance and associated large sample were extended in a fuzzy environment. Then, a new methodology was formulated for constructing fuzzy confidence intervals for the fuzzy mean and fuzzy variance. Common control charts were also developed based on particular fuzzy control limits. Moreover, a popular process capability indicator and its estimator were developed based on fuzzy random variables. Finally, a method was proposed for statistical hypotheses testing regarding fuzzy process capability. To this end, we began with developing the classical critical value at a certain significance level. Then, applying the credibility index, a fuzzy test was introduced to accept/reject the fuzzy hypotheses to a particular degree of credibility. In order to clarify the subject matter, illustrative numerical examples were provided, and feasibility and effectiveness of the proposed methods were compared to those of existing counterparts. The results indicated that the proposed method is a powerful tool for quality control processes when fuzzy observations are taken from a normal distribution with fuzzy parameters.

Fuzzy hypothesis testing for a population proportion based on set-valued information

In this paper we propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one-tailed hypergeometric test with fuzzy (non-)complementary hypotheses, and incorporate the conventional hypergeometric test with crisp hypotheses into its framework. In particular, we formulate hypotheses as ontic or epistemic (fuzzy) sets, since we assume that presumptions regarding the true population proportion can be expressed in the conjunctive or disjunctive reading. Further, we suggest modeling of membership functions in terms of cost or frequency due to considerations of user's priorities or incomplete knowledge in relation to hypotheses formulation. Considering a hypothesis as union of its crisp and fuzzy areas, we illustrate via real-life examples that in contrast to classical test theory, fuzzy hypothesis testing provides an additional partial and gradual consideration of the indifference zone for both complementary and non-complementary hypotheses. The test concept is introduced for both test methods, a test of significance and an alternative test, and leads to a crisp test decision considering further cost aspects like the sample size. The generalized error criteria are derived and interpreted in relation to both hypotheses. Additionally, a corresponding sensitivity analysis is presented for one of the most promising test types.

Randomized versus non-randomized hypergeometric hypothesis testing with crisp and fuzzy hypotheses

This paper is concerned with fuzzy hypothesis testing in the framework of the randomized and non-randomized hypergeometric test for a proportion. Moreover, we differentiate between a test of significance and an alternative test to control the type I error or both error types simultaneously. In contrast to classical (non-)randomized hypothesis testing, fuzzy hypothesis testing provides an additional gradual consideration of the indifference zone in compliance with expert opinion or user priorities. In particular, various types of hypotheses with user-specified membership functions can be formulated. Additionally, the proposed test methods are compared via a comprehensive case study, which demonstrates the high flexibility of fuzzy hypothesis testing in practical applications.

Inspection tables for single acceptance sampling with crisp and fuzzy formulation of quality limits

PurposeThe purpose of this paper is to construct innovative exact and approximative sampling plans for acceptance sampling in statistical quality control. These sampling plans are determined for crisp and fuzzy formulation of quality limits, various lot sizes and common α- and β-levels.Design/methodology/approachThe authors use generalized fuzzy hypothesis testing to determine sampling plans with fuzzified quality limits. This test method allows a consideration of the indifference zone related to expert opinion or user priorities. In addition to the exact sampling plans calculated with the hypergeometric operating characteristic function, the authors consider approximative sampling plans using a little known, but excellent operating characteristic function. Further, a comprehensive sensitivity analysis of calculated sampling plans is performed, in order to examine how the inspection effort depends on crisp and fuzzy formulation of quality limits, the lot size and specifications of the producer’s and consumer’s risks.FindingsThe results related the parametric sensitivity analysis of the calculated sampling plans and the conclusions regarding the approximation quality provide the user a comprehensive basis for a direct implementation of the sampling plans in practice.Originality/valueThe constructed sampling plans ensure the simultaneous control of producer’s and consumer’s risks with the smallest possible inspection effort on the one hand and a consideration of expert opinion or user priorities on the other hand.