Olgierd Hryniewicz

Possibilistic decisions and fuzzy statistical tests

The paper deals with the problem of the interpretation of the results of statistical tests in terms of the theory of possibility. The well known concept in statistics of the observed test size p (also known as p-value and significance) has been given a new possibilistic interpretation and generalised for the case of imprecisely defined statistical hypotheses and vague statistical data. The proposed approach allows a practitioner to evaluate the test results using intuitive concepts of possibility, necessity or indifference.

Statistics with fuzzy data in statistical quality control

Statistical quality control (SQC) is an important field where both theory of probability and theory of fuzzy sets may be used. In the paper we give a short overview of basic problems of SQC that have been solved using both these theories simultaneously. Some new results on the applications of fuzzy sets in SQC are presented in details. We also present problems which are still open, and whose solution should definitely increase the applicability of fuzzy sets in quality control.

Fuzzy Sets in the Evaluation of Reliability

Theory of reliability is more than fifty years old. Its basic concepts were established in the 1950s as useful tools for the analysis of complex technical systems. The rapid development of the theory of reliability was closely related to the importance of its main field of applications - military and space. For this reason the origins of the research in the area of reliability are still not well known. Ralph A. Evans, one of the founders of the IEEE Transactions on Reliability, wrote in an Editorial in this journal that all important theoretical results published in the 1960s and 1970s had been already obtained even in the 1950s, and for many years remained classified. The authors of the most important publications on reliability from those years belonged to the group of the most important scientists working in theory of probability, mathematical statistics, electronics and computer sciences.

Lifetime Tests for Imprecise Data and Fuzzy Reliability Requirements

The paper deals with the problem of the estimation of the average lifetime when observed survival are described by fuzzy numbers. The fuzzy estimator of the average lifetime is given. A simple method for the verification of reliability requirement is proposed.

On the Robustness of the Shewhart Control Chart to Different Types of Dependencies in Data

Shewhart control charts were originally designed under the assumption of independence of consecutive observations. In the presence of dependence the authors usually assume dependencies in the form of autocorrelated and normally distributed data. However, there exist many other types of dependencies which are described by other mathematical models. The question arises then, how classical control charts are robust to different types of dependencies. This problem has been sufficiently well discussed for the case of autocorrelated and normal data. In the paper we use the concept of copulas to model dependencies of other types. We use Monte Carlo simulation experiments to investigate the impact of type and strength of dependence in data on the value of the ARL of Shewhart control charts.

User-preferred solutions of fuzzy optimization problems: an application in choosing user-preferred inspection intervals

The paper deals with the problem of fuzzy optimization in presence of flexible constraints that describe user preferences. Fuzzy optimization procedure is used to find a certain reference value. Then, the set of admissible solutions that are in a certain sense indistinguishable from the best attainable reference value is calculated using the known from the theory of possibility concept of the Necessity of Strict Dominance index. Admissible solutions are evaluated and ranked using methods developed for solving the flexible constraint satisfaction problems. The proposed algorithm is applied for solving a practical problem of choosing a user-preferred inspection interval. This problem is often encountered in statistical quality control and reliability testing.

Bayes statistical decisions with random fuzzy data—an application in reliability

Bayesian methods are widely accepted as the methodology for the analysis of reliability data. In practical applications such data are often expressed in an imprecise way. In such a case fuzzy sets are frequently used for modeling imprecision. In the paper we present some recent results on the application of the fuzzy Bayes methodology for the analysis of imprecise reliability data. We also present some useful approximations obtained using the concept of shadowed sets.

Possibilistic Approach to the Bayes Statistical Decisions

In the paper we consider the problem of Bayes verification of statistical hypotheses. We consider different cases where statistical data, assumed loss functions and considered hypotheses may be described vaguely. The formulae for the calculation of expected fuzzy risks are given. The tools of the possibility theory such as Possibility of Dominance (PD) and Necessity of Strict Dominance (NSD) indices are proposed for final decision making.

Sequential Signals on a Control Chart Based on Nonparametric Statistical Tests

The existence of dependencies between consecutive observations of a process makes the usage of SPC tools much more complicated. In order to avoid unnecessary costs we need to have simple tools for the discrimination between correlated and uncorrelated process data. In the paper we propose a new control chart based on Kendall’s tau statistic which can be used for this purpose. In case of normally distributed observations with dependence of an autoregressive type the proposed Kendall control chart is nearly as good as a well known autocorrelation chart, but outperforms this chart when these basic assumptions are not fulfilled.

Comparison of Fuzzy and Crisp Random Variables by Monte Carlo Simulations

Fuzzy random variables are used when randomness is merged with imprecision described by fuzzy sets. When we need to use computer simulations for the comparison of a classical probabilistic approach with that based on fuzzy random variables we need to establish the method for the generation of crisp random variables compatible with existing fuzzy data. In the paper we consider this problem, and propose some practical solutions.