In analyzing and classifying data from a statistical perspective, fuzzy sets and logic have become a valuable tool either to model and handle imprecise data or to establish flexible techniques to deal with precise data.

From the very beginning of his 52 years-old theory, Professor Zadeh highlighted that “Probability theory/statistics and fuzzy logic should be viewed as complementary rather than competitive,” and he anticipated and encouraged the materialization of such a complementarity. Nowadays, this assertion is a reality, as shown by the many related papers, specialized conferences, special sessions and tracks in general conferences, and so on.

This special issue started in 2015, with the 50th anniversary of the seminal paper on fuzzy sets by Zadeh (1965), aiming to collect a sample of research papers about the current trends on the combination of fuzzy sets/logic and data analysis/classification.

When this special issue was almost ready for publication, Zadeh unfortunately passed away at age 96 (February 4, 1921–September 6, 2017). We wish this special issue to be dedicated to Professor Zadeh, as a modest part of the many tributes that he will receive, and intending to show that fuzzy sets/logic and data analysis/classification can certainly work in synergy.

1 Introduction

This issue means a contribution of a well-known rather young journal from a scientific field that, sometimes and especially at the beginning of the introduction of fuzzy theory, put it into question. However, Zadeh has been permanently trying to build bridges between statistics/probability and fuzzy logic. We have thought that this was a cordial and gentle way to combate criticisms, in accordance with his well-known friendly attitude, by demonstrating the potential benefits of collaboration between both fields. But we have been ‘probably’ wrong. Zadeh’s fondness and affection towards Probability and Statistics is neither incidental nor temporary, but distant in time.

To illustrate the last assertion, in his 2015’s paper (Zadeh 2015) Professor Zadeh has taken a look back over his past in research and he says: “\(\ldots \) The early years in my academic career coincided with the birth of the age of computers and information. It was an exciting period, spurred by competition between the United States and the Soviet Union. At Columbia, my research was focused on system theory and information systems. Probability theory had a position of centrality in my work. My first paper, published in 1949 in the Journal of Applied Physics, was entitled, Probability criterion for the design of servomechanisms. My second paper, published also in the Journal of Applied Physics in 1950, was entitled, An extension of Wiener’s theory of prediction. I had a close relationship with the Department of Mathematical Statistics and its Chair, Herbert Robbins, a brilliant mathematician, who became my best friend...” Actually, many of Zadeh’s first publications’ titles involve unequivocally statistical terms such as stochastic operators, correlation functions, prediction, time-series, and so on. The same applies to several communications presented in meetings of the American Mathematical Society.

Anyway, as it often happens with new theories and approaches, there has been more than two decades in which probabilistic and statistical journals scarcely publish papers concerning fuzzy material, but maybe some eventual debates. In this respect, as outlined in Belohlavek et al. (2017) and Ross et al. (2002), one should at least mention the following:

  • First, the debate along the Conference on the Calculus of Uncertainty in Artificial Intelligence and Expert Systems held in 1984. In this conference, Lotfi Zadeh, as supporter of fuzzy logic, Dennis Lindley, as supporter of subjective probability, Glenn Shafer, as supporter of evidence theory, and David Spiegelhalter, as supporter of applications of expert systems, acted as invited speakers. This debate was mostly gathered in the Issue 1 of Volume 2 of the journal (Statistical Science 1987), although Zadeh could not send his written presentation in due time but he often referred to what was discussed there in different papers.

  • Secondly, the invited paper with discussions by Laviolette et al. (1995) in the journal Technometrics, where Laviolette et al. reviewed some basic ideas in fuzzy theory and offer what they consider to be simpler alternatives based on traditional probability and statistical theory; the corresponding Zadeh’s discussion (Zadeh 1995) argued that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real-world settings, but allowing them to coexist is much more effective and reasonable.

  • Finally, another interesting debate is the one around the invited paper with discussions by Singpurwalla and Booker (2004) in the Journal of the American Statistical Association, in which authors articulated that probability theory has a sufficiently rich structure for incorporating fuzzy sets within its framework, and that probability and fuzzy set theories can work in concert. Zadeh’s discussion about this last paper (Zadeh 2004) suggested to restructure probability theory by involving a shift in its foundations from bivalent to fuzzy logic.

As we have already commented, nowadays the controversy has been substantially diminishing and ideas of reconciliation and coexistence of fuzzy and probability/statistics theories and the development of hybrid models and methods have been thriving. Kruse et al. have pointed out (Kruse et al. 2013) that “... We must differentiate between fuzzy data analysis and fuzzy data analysis. The former deals with the analysis of classical data using methods based on fuzzy set theory. These methods, e.g., fuzzy clustering or fuzzy regression analysis, have been used successfully in lots of industrial applications. The second approach tries to analyze fuzzy data by using statistical methods...” In other words, Fuzzy Data Analysis and Classification studies are mainly focussed

  • either on developing concepts, results and methods to deal with classical (non-fuzzy) data, where fuzziness is involved in the construction of the analysis/classification procedures,

  • or on developing/extending concepts, results and methods concerning data analysis and classification of fuzzy-valued data,

  • or on both.

2 On the fuzzy analysis and the fuzzy classification of non-fuzzy/standard data

The development of fuzzy approaches to classify ‘crisp’ data started soon after the formalization of fuzzy sets (Zadeh 1965). In fact, Zadeh along with Bellman and Kalaba were the first in suggesting fuzzy sets as a theoretical basis to develop clustering algorithms (Bellman et al. 1966). Some of the most influential pioneer works on the subject are, among others, those by Ruspini (1969, 1970), Tamura et al. (1971), Dunn (1973, 1974), Bezdek (1973, 1974, 1980), and Bezdek et al. (1984), which have inspired both applications and many further methodologies. At present, this is one of the most successful topics involving Fuzzy Sets and Statistical theories, and the number of research papers on it is unquestionably growing [among the most recent ones see, for instance, the approaches in Liu et al. (2013), Gong et al. (2014), Yamashita and Mayekawa (2015), Ruan et al. (2016), and Nguyen-Trang and Vo-Van (2017)], and it appears often either combined with or supporting other data analysis problems.

In more detail, useful references to the extensive literature on the fuzzy clustering (from both theoretical and applicative points of view) can be found in the chapter on the fuzzy clustering by D’Urso (2016), the seminal monograph by Bezdek (1981), the books by Jain and Dubes (1988), De Oliveira and Pedrycz (2007), Miyamoto et al. (2008) and, e.g., the following journals: Fuzzy Sets and Systems, IEEE Transactions on Fuzzy Systems, Information Sciences, Pattern Recognition, Applied Soft Computing, Soft Computing, Advances in Data Analysis and Classification, Computational Statistics and Data Analysis, Chemometrics and Intelligent Laboratory Systems, Pattern Recognition Letters, etc.

As remarked by D’Urso (2017a), there are different uncertainty-based clustering methods that can be considered extensions, variants and alternatives of the fuzzy clustering for non-fuzzy/standard data, like

  • possibilistic clustering [see, for instance, Krishnapuram and Keller (1993)],

  • shadowed clustering [see, for instance, Pedrycz (1998)],

  • rough sets-based clustering [see, for instance, Lingras and West (2004)],

  • intuitionistic fuzzy clustering [see, for instance, Hung et al. (2004)],

  • evidential clustering, credal clustering or belief clustering [see, for instance, Denoeux and Masson (2004a)],

  • credibilistic clustering [see, for instance, Zhou et al. (2007)],

  • type-2 fuzzy clustering [see, for instance, Hwang and Rhee (2007)],

  • neutrosophic clustering [see, for instance, Shan et al. (2012)],

  • hesitant fuzzy clustering [see, for instance, Chen et al. (2013)],

  • interval-based fuzzy clustering [see, for instance, Silva et al. (2015)],

  • picture fuzzy clustering [see, for instance, Son (2015)].

Fuzzy approaches to analyze crisp/standard data, have not been carried out as exhaustively as fuzzy clustering ones for the same data. And they were developed several years after fuzzy sets were introduced. Among them, one can highlight

  • the fuzzy linear regression ideas between non-fuzzy input and output data, by considering the problem as a linear programming one [see, for instance, the first formulation by Tanaka et al. (1982) and Tanaka and Watada (1988)],

  • hypothesis fuzzy testing, testing of fuzzy hypotheses, and fuzzy estimation regarding non-fuzzy parameters on the basis of non-fuzzy data [see, for instance, Watanabe and Imaizumi (1993), Arnold (1998), Buckley (2004), Hryniewicz (2006), Parchami et al. (2009)],

  • fuzzy statistical quality control [see, for instance, Grzegorzewski and Hryniewicz (2000)],

  • statistical decision problems with fuzzy utilites/losses [see, for instance, Gil and Jain (1992), Gil and López-Díaz (1996)].

3 On the analysis and classification of fuzzy data

On the other hand, approaches to classify fuzzy-valued data are becoming a challenging topic. Among the first published approaches one should mention those by Esogbue (1986), Hathaway et al. (1996), and Pedrycz et al. (1998) and, among the recent ones, see those by Coppi et al. (2012), D’Urso and De Giovanni (2014), D’Urso et al. (2015a), Ansari et al. (2017) and Ferraro and Giordani (2017).

The analysis of fuzzy-valued data is also a topic receiving an increasing attention along the years. Some of the developed methodologies aiming to analyze fuzzy data consider a descriptive view and do not refer to models associated with the probabilistic framework. Nevertheless, most of the methodologies are based on the modeling of the random mechanisms generating fuzzy data within a probabilistic setting. In this respect, we can mention, among some of these methodologies the following:

  • The methodologies based on the notion of fuzzy information system, introduced by Okuda et al. (1978), who consider the available information from a classical random experiment associated with a real-valued random variable to be fuzzy [i.e., they consider an epistemic viewpoint in accordance with the distinction made by Couso and Dubois (2014)] and assume that this available information constitutes a fuzzy partition (in Ruspini’s sense Ruspini 1970) of the sample space of the variable, and probabilities are based on Zadeh’s probabilistic definition of fuzzy events (Zadeh 1968). Some data analysis developments using this model can be seen, for instance, in Gil et al. (1988), Gil (1992) and, more recently, Denoeux (2011).

  • The methodologies based on the notion of fuzzy random variable, introduced by Kwakernaak (1978, 1979) and later formalized by Kruse and Meyer (1987). As Kruse et al. pointed out in Kruse et al. (2013), this deep and wide data analysis with vague data was a fruit of the encouragement by Professors Lotfi Zadeh and Heinz Skala [editor of the Series Theory and Decision Library of the D. Reidel Publishing Co., see, e.g., Skala (1975)], which was mostly prompted by the development by Kruse and Meyer of useful fuzzy methods and a software tool for statistical applications for the Siemens AG. The model refers to the epistemic perspective and fuzzy random variable is viewed as the fuzzy perception of an original non-fuzzy random variable. Statistical developments with fuzzy data coming from the fuzzy perception of real-valued ones will be mainly based on propagating the associated imprecision to the distribution function, parameters, etc., through Zadeh’s extension principle (Zadeh 1975a, b, c). It should be remarked that, albeit based on fuzzy information, statistical conclusions with Kruse and Meyer’s fuzzy random variables always concern the original random variable and its parameters. Among the studies based on Kruse and Meyer’s fuzzy random variables one can refer, for instance, to Kruse (1984, 1987), Grzegorzewski (1998), Wang (2004), and Wu (2005).

  • The methodologies based on the notion of random fuzzy sets, introduced by Féron (1979), and to some extent anticipated by Fréchet (1948), and later formalized by Puri and Ralescu (1985, 1986). The model, which was initially coined as fuzzy random variables, refers to a kind of ontic perspective, since a random fuzzy set (or random fuzzy number if values are fuzzy numbers) is viewed as a mapping associating experimental outcomes with fuzzy values in a Borel-measurable way, so that the induced distribution associated with the random fuzzy set is immediate, the stochastic independence between random fuzzy sets is also trivially induced, and so on. It should be remarked that, in contrast to the Kruse and Meyer’s approach, statistical conclusions with Puri and Ralescu random fuzzy sets always concern the fuzzy-valued random element and the parameters associated with its induced distribution. An interesting distinctive feature of the statistical methodology based on this approach to generate fuzzy data is that most of the classical ideas in data analysis can be immediately preserved without needing to either define or adapt them expressly. Among the statistical developments involving this approach one can refer, for instance, to Bandemer and Näther (1992) and Näther (1997, 2006) and, more recently, Blanco-Fernández et al. (2014a, b).

Other developments and approaches can be found in the literature (for instance, those by Viertl (2006), Grzegorzewski and Szymanowski (2014), etc.). Among the scientific journals publishing papers on the topic, one can mention those indicated in Sect. 2 for the fuzzy clustering and analysis of standard data.

4 Additional related literature

It is pertinent to state at this point that some other procedures have been suggested in the literature to categorize non-standard data in a fuzzy manner, and some of them have been gathered in Table 1 [see also, D’Urso (2016)].

Table 1 Some relevant references on fuzzy clustering of non-standard data
Full size table

In a similar way, Table 2 collects some of the most relevant references on the methodological statistical studies on the analysis of fuzzy data (see also D’Urso 2017b).

Table 2 Some relevant references on the analysis of fuzzy data
Full size table

5 On this special issue

The papers in this special issue are to be considered as a sample of recent advances in data analysis and classification involving fuzziness, illustrating the need of taking advantage of other topics like Fuzzy Logic to enrich and to widen statistical methodologies. Although small samples are not usually informative enough from a statistical perspective, and this special issue is certainly a very small sample, we trust that readers can get a flavour of some of the current trends about.

The first four papers in the issue concern fuzzy data analysis or classification:

  • The paper “A fuzzy approach to robust regression clustering”, by Dotto, Farcomeni, García-Escudero and Mayo-Iscar, proposes a fuzzy regression clustering method based on a maximum likelihood approach but in such a way that the method resists well to data contamination.

  • The paper “A novel method for forecasting time series based on fuzzy logic and visibility graph”, by Zhang, Ashuri and Deng, relates to a new suggestion to forecast time series, which is based on fuzzy logic, visibility graph and link prediction.

  • The paper “Fuzzy rule based classification systems for big data with MapReduce: granularity analysis”, by Fernández, del Río, Bawakid and Herrera, aims to discuss the effect of the granularity level and the number of selected Maps on the performance of the Chi-Fuzzy Rule Based Classification Systems with a MapReduce approach for big data.

  • The paper “On ill-conceived initialization in archetypal analysis”, by Suleman, addresses the problem of initialization and the performance of fuzzy clustering by means of an archetypal analysis.

The last two papers in the issue concern fuzzy data analysis or classification:

  • The paper “Robust scale estimators for fuzzy data”, by de la Rosa de Sáa, Lubiano, Sinova and Filzmoser, regards the introduction of some robust location-based scale measures/estimates for random fuzzy numbers, along with the analysis of their robustness.

  • The paper “Parametric classification with soft labels using the evidential EM algorithm. Linear discriminant analysis vs logistic regression”, by Denoeux, Quost and Li, analyzes the problem of partially supervised classification when learning instances are labeled by means of Dempster–Shafer mass functions (which include fuzzy sets as a particular case).