Statistical analysis using the Fuzzy approach

Introduction

At this stage of the Information Society, one of the most pressing issues is how to regulate the cognitive process while considering its inherent qualities of uncertainty, such as imprecision and vagueness. It has theoretical and practical consequences in fields like technology, economics, and biomedicine. Real-life events are, in fact, the primary source of inspiration for this type of management to be considered. Information and uncertainty are two concepts that are strongly connected from a theoretical standpoint. A lack of information causes uncertainty.

On the other hand, information may be viewed to reduce uncertainty, but this is only one potential, albeit significant, perspective on the idea. As a result, each conceptual expansion in the realm of information is accompanied by the requirement to manage new forms of uncertainty. The uncertainty theory has significantly expanded its conceptual breadth and methodological tools due to the combined influence of two developments in mathematical thinking during the previous several decades. On the one hand, applying the classical theory of additive measures (such as probability measures)  to monotone non-additive measures (e.g. possibility measures, belief functions, interval-valued probabilities). On the other hand, the application of classical set theory to the study of fuzzy sets in both standard and nonstandard forms.

Statistical reasoning is a method of reasoning in the context of uncertainty and incomplete knowledge. The advancements above in the fields of information and uncertainty have naturally influenced Statistical Science. The widespread use of the basic theory of fuzzy sets in logic, mathematics, and engineering has supplied statistical methods with exciting ideas and new tools. A constant stream of contributions has expanded statistical reasoning to incorporate fuzzy data and fuzzy uncertainty since the late 1960s. However, these advancements have occurred somewhat haphazardly, with contributions from various scientific communities and stimuli ranging from minor to major issues (from control systems to medical diagnosis, from marketing to environmental studies). In light of these contributions, there is a strong desire to systematise statistical reasoning, and this Special Issue is intended to be a step forward in that direction. To define the overall meaning of this endeavour, we need a broad framework that can accommodate both classic statistical information and uncertainty concepts as well as new ones developed from the fuzzy approach in its broadest sense.

Contributions of Fuzzy thinking to Statistics

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i) Fuzzy sets and numbers

The most fundamental concept in Fuzzy Sets Theory is a fuzzy set of a given referential set (universe of discourse) T, which is defined by a mapping Ấ.

ii) Fuzzy random variables

Consider the case when the observed statistical data is imprecise, loosely defined, or refers to language labels for ambiguous notions (such as good, large, and so on). A suitable approach to dealing with such data is to “fuzzify” it by creating appropriate “fuzzy-valued variables” that may convey the imprecision/vagueness connected with each observation. For the sake of example, we will refer to the univariate situation in the following sections. The advances in this example, however, may be appropriately extended to the multivariate scenario. A measurability requirement should be given to formally define the random process leading to a fuzzy-valued random variable inside the probabilistic context. This condition has been built up so that the formalisation guarantees that the random variable and random set concepts are extended.

iii) Fuzzy statistical models

Another significant contribution of “fuzzy thinking” to statistical analysis is the way statistical models are created. The focus of imprecision in this scenario is on theoretical informational components, such as the parameters or other model characteristics. In this context, fuzzy clustering and fuzzy regression analysis are two well-known examples.

iv) Fuzzy “if-then” rules in statistical analysis

The so-called “Fuzzy Inference Systems” are another source of statistical analysis inspiration in the realm of “fuzzy thinking.” A Fuzzy Inference System (FIS) is a logical framework based on “if… then” rules, with fuzzified premises and effects. It offers the reasoning methods some flexibility, allowing us to derive meaningful assertions from ambiguous or imprecise premises.

Applications

The use of fuzzy statistical approaches in diverse substantive areas is the last area of focus. The application articles in this issue are focused on the technological and economic sectors. Although real-life challenges inspire each contribution, the methodological elements are highlighted so that the proposed models and analytical techniques may be productively applied to similar real-life circumstances.

References

[1] Coppi, Renato, Maria A. Gil, and Henk AL Kiers. “The fuzzy approach to statistical analysis.” Computational statistics & data analysis 51, no. 1 (2006): 1-14.

[2] Hanna, Awad S., Wafik B. Lotfallah, and Min-Jae Lee. “Statistical-fuzzy approach to quantify cumulative impact of change orders.” Journal of computing in civil engineering 16, no. 4 (2002): 252-258.