Fuzzy regression analysis: Systematic review and bibliography

Applied Soft Computing Journal 84 (2019) 105708

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

Review article

Fuzzy regression analysis: Systematic review and bibliography Nataliya Chukhrova, Arne Johannssen

∗

University of Hamburg, Faculty of Business Administration, 20146 Hamburg, Germany

article

info

Article history: Received 15 March 2019 Received in revised form 22 July 2019 Accepted 9 August 2019 Available online 14 August 2019 Keywords: Fuzzy least squares regression Fuzzy linear regression Fuzzy nonlinear regression Interval regression Machine learning techniques Possibilistic regression

a b s t r a c t Statistical regression analysis is a powerful and reliable method to determine the impact of one or several independent variable(s) on a dependent variable. It is the most widely used of all statistical methods and has broad applicability to numerous practical problems. However, various problems can arise, when for instance the sample size is too small, distributional assumptions are not fulfilled, the relationship between independent and dependent variables is vague or when there is an ambiguity of events. Moreover, the complexity of real-life problems often makes the underlying models inadequate, since information is frequently imprecise in many ways. To relax these rigidities, numerous researchers have modified and extended concepts of statistical regression analysis by means of concepts of fuzzy set theory. By now, there is a large number of papers on the topic of fuzzy regression analysis, especially concerning possibilistic, fuzzy least squares or machine learning approaches. Additionally, the variety of approaches includes probabilistic, logistic, type-2 and clusterwise fuzzy regression methods, among many others. Besides papers mainly devoted to advances in methodology, there are also several papers presenting case studies in various research fields. To structure this diversity of papers, proposals and applications we give in this paper a comprehensive systematic review and provide a bibliography on the topic of fuzzy regression analysis. Thus, the paper intends to consolidate the topic in order to aid new researchers in this area, focuses the field’s attention on key open questions, and highlights possible directions for future research. © 2019 Elsevier B.V. All rights reserved.

Contents 1. 2.

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Introduction......................................................................................................................................................................................................................... Methodology ....................................................................................................................................................................................................................... 2.1. Research questions ................................................................................................................................................................................................ 2.2. Search strategy and inclusion/exclusion criteria ............................................................................................................................................... Systematic structuring of the literature .......................................................................................................................................................................... 3.1. Major and minor fields of fuzzy regression analysis ........................................................................................................................................ 3.2. Some information about articles, journals and publishers ............................................................................................................................... Possibilistic regression analysis ........................................................................................................................................................................................ 4.1. Linear and non-linear programming methods ................................................................................................................................................... 4.1.1. Criticism of Tanaka’s approach............................................................................................................................................................. 4.1.2. Direct enhancements of Tanaka’s approach........................................................................................................................................ 4.1.3. Developments in possibilistic regression analysis based on linear programming ......................................................................... 4.1.4. Non-linear programming approaches .................................................................................................................................................. 4.1.5. Piecewise necessity regression analysis .............................................................................................................................................. 4.2. Goal programming approaches ............................................................................................................................................................................ 4.3. Interval regression analysis .................................................................................................................................................................................. Fuzzy least squares and fuzzy least absolutes methods ............................................................................................................................................... 5.1. Fuzzy least squares methods................................................................................................................................................................................ 5.1.1. Direct enhancements of Celmin’s and Diamond’s approaches ......................................................................................................... 5.1.2. Developments in fuzzy least squares methods .................................................................................................................................. 5.1.3. Separate sub-models for core and spreads .........................................................................................................................................

∗ Corresponding author. E-mail addresses: [email protected] (N. Chukhrova), [email protected] (A. Johannssen). https://doi.org/10.1016/j.asoc.2019.105708 1568-4946/© 2019 Elsevier B.V. All rights reserved.

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5.1.4. Defuzzification approaches and h-level least squares estimates...................................................................................................... 5.1.5. Interval and intuitionistic fuzzy regression models........................................................................................................................... 5.1.6. Asymptotic properties of least squares estimators ............................................................................................................................ 5.2. Fuzzy least absolutes methods............................................................................................................................................................................. Machine learning techniques in fuzzy regression analysis ........................................................................................................................................... 6.1. Evolutionary algorithms ........................................................................................................................................................................................ 6.2. Support vector machines ...................................................................................................................................................................................... 6.3. Neural networks .................................................................................................................................................................................................... 6.4. Other machine learning techniques .................................................................................................................................................................... Various minor fields of fuzzy regression analysis .......................................................................................................................................................... 7.1. Robust fuzzy regression analysis ......................................................................................................................................................................... 7.2. Fuzzy probabilistic approach ................................................................................................................................................................................ 7.3. Fuzzy logistic regression ....................................................................................................................................................................................... 7.4. Type-2 fuzzy regression analysis ......................................................................................................................................................................... 7.5. Fuzzy clusterwise regression ................................................................................................................................................................................ 7.6. Fuzzy regression analysis combined with time series analysis ....................................................................................................................... 7.7. Further approaches ................................................................................................................................................................................................ 7.7.1. Fuzzy entropy approaches .................................................................................................................................................................... 7.7.2. Non-parametric fuzzy regression analysis .......................................................................................................................................... 7.7.3. Monte Carlo methods ............................................................................................................................................................................ 7.7.4. Bootstrap techniques ............................................................................................................................................................................. 7.7.5. Regression analysis based on fuzzy prior information ...................................................................................................................... 7.7.6. Surveys .................................................................................................................................................................................................... Practical applications of fuzzy regression analysis......................................................................................................................................................... Critical discussion and future directions ......................................................................................................................................................................... Conclusions.......................................................................................................................................................................................................................... Declaration of competing interest.................................................................................................................................................................................... Acknowledgments .............................................................................................................................................................................................................. References ...........................................................................................................................................................................................................................

1. Introduction Regression analysis is a powerful method in evaluating the functional relationship between one dependent variable (output/response variable) and one or a set of independent variables (input/explanatory variables). It is widely used to describe, control, and forecast values of the response variable with the help of observations for the explanatory variables. The conventional approach in regression modeling is based on crisp data and a crisp relationship between dependent variable and independent variables. However, in the case of imprecise phenomena, or if the considered phenomenon has a vague variability instead of a stochastic variability, it seems to be a more natural way to assume a fuzzy relationship (see Coppi [1]). A fuzzified regression approach is also a promising alternative, when distributional assumptions of the underlying regression model are not satisfied or they cannot be tested (for instance caused by a small sample size). In addition, there are many situations in practical applications where the observations cannot be measured as crisp quantities, because information is often imprecise, incomplete, linguistic, noisy, qualitative or vague. Therefore, fuzzy modeling approaches provide appropriate techniques for dealing with those various types of uncertain information (see Zadeh [2,3]). In the framework of a fuzzy regression approach, the fuzzy uncertainty that measures the ambiguity or vagueness of a phenomenon (which cannot be expressed by randomness) is evaluated by a measure called possibility (see Dubois and Prade [4] and Klir [5]). Thus, fuzzy regression analysis as a non-statistical method is not based on probability theory but on possibility theory and fuzzy set theory (see Zadeh [6,7]). For this reason, fuzzy regression models do not have error terms, instead they are contained in the fuzzy coefficients. In the general fuzzy regression model both input and output data are fuzzy, the functional relationship between independent variables and the dependent variable is given by a fuzzy function, and the distribution of the data is possibilistic.

10 10 10 10 10 10 11 11 11 12 12 12 13 13 13 13 13 14 14 14 15 15 15 15 15 20 20 20 20

In the last few decades, many approaches have been proposed by numerous researchers to combine statistical regression analysis with concepts of fuzzy set theory. Besides papers focused on advances in methodology, there are also numerous applications of the proposed methods in various fields of research. In addition, many papers are exclusively devoted to case studies, where existent methodology is applied, likewise in many different research fields. Since fuzzy modeling approaches in regression analysis have experienced tremendous growth in recent years, it gets harder to track and integrate the varying results into a cohesive whole. We believe that the topic of fuzzy regression analysis has now grown to the point that a consolidation is needed, in order to aid new researchers in this area and to identify potentially fruitful future directions for the topic. Therefore, this paper intends to give a comprehensive systematic review, which is separated into a methodology and an application part, and a bibliography on the relevant articles. The systematic review follows the methodology suggested by Kitchenham [8] and focuses on key research questions regarding fuzzy regression analysis. We systematically have searched the literature to identify the papers that propose advances in the field of fuzzy regression analysis and its applications, and then we have reviewed each included paper through the lenses of our research questions. Based on this content, the paper is structured as follows (see also the table of contents above). In Section 2, we formally construct our research questions, and explain the procedure to identify relevant primary sources. In Section 3, we specify major and minor fields of fuzzy regression analysis and give information about articles included in the systematic review. Section 4 presents papers regarding the possibilistic approach, especially linear, non-linear, and goal programming methods as well as interval regression analysis. In Section 5 fuzzy least squares and fuzzy least absolutes methods are considered, and in Section 6 we discuss machine learning techniques like evolutionary algorithms, support vector machines and neural networks embedded in fuzzy regression analysis. Afterwards, Section 7 deals with

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various minor fields of fuzzy regression analysis, while Section 8 catalogs papers presenting practical applications and case studies. In Section 9, we provide a critical discussion of the presented methods and identify potential directions for future research. Finally, in Section 10 the paper concludes with an overview of study results. 2. Methodology The overall goal of the systematic review is to consolidate the research area of fuzzy regression analysis. To accomplish this goal, we formulate in this section specific research questions, the search strategy, and inclusion/exclusion criteria for primary sources. 2.1. Research questions Since there is a nearly unmanageable number of papers on the topic of fuzzy regression analysis, we have to organize the various fields in a reasonable way. Thus, first of all, we need to structure the papers and identify major fields, where regression analysis is considered in fuzzy environments. Moreover, we have to subdivide these fields into subareas to arrange the numerous approaches in a constructive manner. This directly leads us to the first research question: Q1: What are major fields and their subareas, where regression analysis is implemented in fuzzy environments? Besides the consideration of major fields, there are many further fields of fuzzy regression analysis that need attention. That is, we have to identify these additional fields, which cannot be assigned to the major fields. This then points us to a straightforward formulation of the second research question as: Q2: What are minor fields, where regression analysis is implemented in fuzzy environments? There is a wide variety of journals edited by various publishers that are concerned with publishing papers on fuzzy regression analysis and its applications. As a guideline, especially to aid new researches in this area, it is important to localize appropriate journals for publishing papers with regard to certain major or minor fields of fuzzy regression analysis. This gives us our third research question: Q3: What are the journals, where specific topics of fuzzy regression analysis are usually published? Given the major and minor fields, the contribution of each publication in the respective field has to be specified. Thus, the proposed approaches, advances and/or extensions of each publication should be briefly discussed and arranged in the overall context of the field. This leads to our fourth and superordinate research question: Q4: What is the contribution of the individual publications with regard to their corresponding field? Regression analysis as the most widely used statistical technique has a broad applicability to practical problems. Hence, it is not surprising that there are numerous practical applications of fuzzy regression analysis performed in the papers. Moreover, several papers present comprehensive case studies based on fuzzy regression methods. The purpose of bringing together the hitherto existing applications based on fuzzy regression analyses directly provides us the opportunity for the fifth research question: Q5: What are the areas, where fuzzy regression analysis has been applied up to now, and which specific applications are involved?

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After identifying major and minor fields as well as presenting methods, advances and extensions provided in the papers, there is the need for a subsequent critical discussion of the proposed techniques. In addition, potential directions for future research have to be figured out, resulting in our sixth and final research question: Q6: What are potential directions for future research? 2.2. Search strategy and inclusion/exclusion criteria The both key methodological steps in a systematic review comprise the determination of a search strategy to locate candidate primary sources and the formulation of inclusion/exclusion criteria to reliably identify truly relevant primary sources (see Yazdanbakhsh and Dick [9]). Fundamentally, our search strategy corresponds to the procedure suggested by Tranfield et al. [10] that seeks to create a reliable knowledge stock by synthesizing the relevant body of literature. That is, this research study does not collect primary data, but gathers secondary data from reliable database sources. There exists a multiplicity of citation databases, which are either curated (e.g. Scopus) or based on search algorithms (e.g. Google Scholar). Since we are interested in maximum coverage of the literature, and are going to examine each article for quality, it is reasonable to mainly rely on algorithm-based databases. We perform a Google Scholar search for the exact strings (i.e. in quotes) ‘‘fuzzy regression’’ OR ‘‘interval regression’’ OR ‘‘fuzzy linear regression’’ OR ‘‘possibilistic regression’’ OR ‘‘fuzzy least squares regression’’ OR ’’fuzzy nonlinear regression’’, which locate about 16 700 references (as of March 2019). This combined search has the benefit of avoiding numerous duplicates that result from separate searches for these strings (11 200, 5680, 4370, 563, 461, and 389 located references, respectively). The search results are ‘‘sorted by relevance’’, so the search for candidate primary sources can be focused on the first pages of located references (and ends on page 100). With regard to the inclusion/exclusion criteria, we first remove articles published in languages other than English. Then we exclude the gray literature (conference proceedings, magazine related articles, text books, theses, editorials, prefaces, poster sessions, panel discussions, viewpoints, commentaries, and unpublished working papers) apart from proceedings of IEEE and IFSA conferences/congresses, such that only academic peer-reviewed journal articles, publications in edited volumes, and peer-reviewed conference proceedings remain in the search results. Moreover, we only include articles published in journals or edited volumes that are ranked in the ‘‘Scimago Journal Rank’’ (SJR) and/or ranked in ‘‘Scopus’’ and/or from well-reputed publishers like Elsevier, Springer, Taylor & Francis, Wiley or IEEE. Additionally, we carefully examine the references from the selected papers to identify further relevant articles that were potentially missed in the database search. Our pool of candidate primary sources is thus the union of these searches without any duplicates. Note that there are no exclusion criteria regarding publication dates. The subsequent selection for inclusion is based upon the title and abstract of the paper. If these criteria state that a specific study is suitable for the systematic review (i.e. related to fuzzy regression analysis and its applications), we retain full-text copies of the respective papers. When the title and abstract provide insufficient information to evaluate the paper’s relevance, we retrieve and review a full-text copy of the study. For the final selection, we examine a full-text copy of the article to assess whether the inclusion criteria are satisfied. Once all the primary sources for inclusion in the systematic review are identified, we

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examine the contribution of each paper with respect to the above stated research questions. Plainly, this is a qualitative review, having more in common with a taxonomic classification than a quantitative meta-analysis (see Yazdanbakhsh and Dick [9]). 3. Systematic structuring of the literature

68.58% in the quarter Q1 of the journals with the highest values, 22.19% in Q2, 4.24% in Q3, and 2.74% in Q4, while 2.25% are not yet ranked. Additionally, we give in Table 1 more detailed information in relation to journals, their publishers, their ranking, and the number of papers published in these journals in relation to major fields (Sections 4, 5, and 6 in this paper) and minor fields (Section 7).

3.1. Major and minor fields of fuzzy regression analysis 4. Possibilistic regression analysis In this subsection, we first focus on research question Q1 by identifying major fields of fuzzy regression analysis and their subareas. Subsequently, we investigate additional minor fields in the context of research question Q2. Considering the findings with regard to major fields of fuzzy regression analysis, we get the following results: There are three major fields, namely

• Possibilistic regression analysis, • Fuzzy least squares and fuzzy least absolutes methods, and • Machine learning techniques in fuzzy regression analysis. Since each of these fields comprises a large number of articles, we propose the following subarea classification:

• Linear and non-linear programming methods, goal programming approaches, interval regression analysis

• Fuzzy least squares methods, fuzzy least absolutes methods • Evolutionary algorithms, support vector machines, neural networks, other machine learning techniques Moreover, we identify some minor fields, namely Robust fuzzy regression analysis, Fuzzy probabilistic approach, Fuzzy logistic regression, Type-2 fuzzy regression analysis, Fuzzy clusterwise regression, Fuzzy regression analysis combined with time series analysis, and • Further approaches.

• • • • • •

The classification scheme is shown in Fig. 1. Note that these major and minor fields need not be taken as mutually exclusive. Instead, the choice of the appropriate field was made considering the main approach used in the respective paper. 3.2. Some information about articles, journals and publishers In this subsection, we examine research question Q3 by exploring information regarding journals, where specific topics of fuzzy regression analysis are mainly published. The systematic review includes 401 articles published in peerreviewed academic journals, 36 articles appeared in edited volumes, and 18 IEEE/IFSA conference proceedings (that is, a total of 455 articles). These articles have been published in a period of 38 years (1982–2019, as of March 2019), and appeared in 131 peer-reviewed academic journals and 16 edited volumes. The majority of peer-reviewed journal articles is published in Fuzzy Sets and Systems (#84, 20.95%), followed by IEEE Transactions on Fuzzy Systems (#22, 5.49%), Information Sciences (#20, 4.99%), Soft Computing (#18, 4.49%), Applied Soft Computing (#17, 4.24%), Expert Systems with Applications (#14, 3.49%), European Journal of Operational Research and Computers & Mathematics with Applications (each #11, 2.74%). Most of the articles in peer-reviewed journals have been published by Elsevier (#215, 53.62%), followed by Springer (#52, 12.97%), IEEE (#33, 8.23%), and Taylor & Francis (#22, 5.49%). Having regard to the ranking of the publications according to the SJR 2017 of the respective journals, we find

In this section, we answer research question Q4 by specifying the contribution of the individual publications (in chronological order) within the first major field ‘‘possibilistic regression analysis’’. Note that we use in the following abbreviations for different types of input and output data: crisp input and crisp output (CICO), crisp input and fuzzy output (CIFO), fuzzy input and fuzzy output (FIFO). 4.1. Linear and non-linear programming methods The possibilistic approach in fuzzy regression analysis was introduced by Tanaka et al. [11]. This method minimizes the entire fuzziness of the model by minimizing the total spread of its fuzzy parameters, subject to the support of the estimations to cover the support of the observations for a particular h-level. They propose a fuzzy linear system as a regression model, and formulate a mathematical linear programming (LP) problem with symmetric triangular fuzzy parameters to estimate the fuzzy regression parameters. Since membership functions of fuzzy sets can be seen as possibility distributions, approaches in this category are referred to as ‘‘possibilistic regression analysis’’. The possibilistic approach has been investigated and improved by numerous authors, which we present below. Starting with Tanaka [12], who deals with possibilistic linear models for fuzzy data analysis using CIFO data, where fuzzy output data are defined by fuzzy numbers. He modifies the method of Tanaka et al. [11] by minimizing the sum of the fuzzy widths around the predicted values. Additionally, Tanaka and Watada [13] consider properties of possibilistic linear systems and formulate a fuzzy linear regression model for fuzzy data. Further, Tanaka et al. [14] present various formulations of the criterion to be optimized and the constraints to be fulfilled for possibility and necessity estimation models. They also discuss mutual relations of three formulations to investigate some properties of fuzzy data analysis. An early overview on possibilistic regression analysis based on LP methods can be found in Tanaka and Ishibuchi [15]. 4.1.1. Criticism of Tanaka’s approach The approaches presented so far have in common that the possibilistic distribution of the parameters is defined by minimum operators, i.e. the possibilistic parameters are non-interactive and they often become crisp. This problem occurs when the spread of a fuzzy regression parameter is zero, and is an inherent characteristic of LP problems. This is one of the points in a criticism formulated by Celmins [16,17], who develops one of the first fuzzy least squares (LS) approaches (see Section 5). Moreover, Jozsef [18] shows that the method of Tanaka et al. [11] is scale dependent (see also Hojati et al. [19]). Redden and Woodall [20] point out that the approaches of Tanaka et al. [11] and Tanaka [12] are not well-defined in the sense that the number of possible solutions can be infinite. Wang and Tsaur [21] criticize the model of Tanaka et al. [11] in the way that it provides too wide ranges in estimation, and therefore complicates practical applications. Further criticism on Tanaka’s approach comprises the data (see Redden and Woodall [22] and Sakawa and Yano

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Fig. 1. Classification scheme.

4.1.2. Direct enhancements of Tanaka’s approach To address the point of non-interactive parameters, Tanaka and Ishibuchi [30] present an extended approach that allows to identify interactive fuzzy parameters in a possibilistic linear model based on quadratic membership functions defined by Celmins [16,17], which can be reduced to LP. Peters [25] investigates a generalized form of the possibilistic regression model of Tanaka et al. [11] by presenting a fuzzy linear model with fuzzy intervals that leads to a fuzzy LP problem. Since this form comprises constants in the model constraints and there is no strategy provided to determine these constants, practical applications are considerably complicated. Yen et al. [31] propose a fuzzy linear regression (FLR) model with coefficients modeled by symmetric triangular fuzzy numbers to reduce inflexibility in the approach of Tanaka et al. [11]. Wang and Tsaur [32] are concerned with a variable selection method for an FLR equation with CIFO observational data. In addition, Hong et al. [33] discuss an approach for the evaluation of FLR models proposed by Tanaka et al. [11] using FIFO data and T ω-based fuzzy arithmetic operations. Chen [34] considers the fuzzy regression model of Tanaka [12] with additional constraints to reduce the impact of outliers. Because these constraints involve a constant, however, the same problem as in Peters [25] (and Özelkan and Duckstein [35], see Section 4.2) arises. For this reason, Hung and Yang [36] deal with another approach to resolve the outliers problem of Tanaka’s approach and present an omission method to detect outliers in FLR models.

Moskowitz and Kim [39] examine the relationship between the h-level, the spreads of fuzzy regression parameters, and the shape of the membership function in FLR models. Chang and Lee [24] investigate an unrestricted FLR model regarding two aspects, avoiding misinterpretation of data when there are conflicting trends (spread vs. modal value), and the sensitivity with respect to the h-level chosen by decision maker. Tanaka et al. [40,41] and Tanaka and Lee [42] consider interactive possibility distributions of the model parameters by using exponential possibility distributions. Kim et al. [27] and Kim and Chen [28] compare FLR to statistical linear and non-parametric linear regression and conclude that fuzzy regression is to prefer for small sample sizes, error terms with small variability, or when the functional relationship is not well specified. Ghoshray [43] presents an FLR model with symmetric or non-symmetric triangular coefficients, and propose parameter estimation via an LP problem. Modarres et al. [44] develop three FLR models (risk-seeking, risk-neutral and risk-averse models) to improve the predictability of LP methods and to simplify computational effort. Additionally, Modarres et al. [45] deal with a mathematical programming model for parameter estimation of FLR models using CIFO data. Guo and Tanaka [46] introduce dual possibilistic regression models (upper and lower regression models) with crisp input and interval output data, and propose as an extension possibilistic regression models with FIFO data. Ge and Wang [47] investigate the dependency between Gaussian noisy input and degree of fit in fuzzy regression models with non-symmetric fuzzy triangular coefficients based on an LP approach. Ge et al. [48] propose a solution for determining an appropriate threshold value in the framework of possibilistic linear models. Bisserier et al. [49,50] present a possibilistic FLR model with a total inclusion property, i.e. all data points are included in estimating model parameters. In addition, the model output is able to feature each kind of spread tendency. Parvathi et al. [51] deal with an intuitionistic FLR model, where the coefficients (modeled as symmetric triangular intuitionistic fuzzy numbers) are estimated with the help of an LP problem.

4.1.3. Developments in possibilistic regression analysis based on linear programming A modification of a possibilistic regression model is proposed in Savic and Pedrycz [26,37], who consider a two-phase construction of a linear regression model combining LS estimation and LP in different phases. Sakawa and Yano [38] are concerned with a class of FLR models for FIFO data by using indices for equalities between fuzzy numbers and LP methods for problem solving.

4.1.4. Non-linear programming approaches When some of the constraints and/or the objective function of an optimization problem are non-linear, we face a non-linear programming problem. In this direction, Bardossy [52] considers general regression models for fuzzy numbers, including the nonlinear case, and shows that this lead to a more general mathematical programming problem. Hayashi and Tanaka [53] propose the Group Method of Data Handling (GMDH, see Ivakhnenko [54]) for

[23]), the poor resistance to outliers (see Chang and Lee [24], Peters [25], and Redden and Woodall [20]), missing forecasting issues (see Savic and Pedrycz [26]), and the problem of multicollinearity (see Kim et al. [27] and Kim and Chen [28]). On the other hand, one of the main advantages of the model proposed by Tanaka et al. [11] is its simplicity in programming and computation (see, for instance, Wang and Tsaur [29]). The above shortcomings have been considered by many authors to enhance possibilistic regression approaches.

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Table 1 Journals, publishers, the highest quartiles regarding SJR 2017, and number of publications listed by sections. Journal

Acta Metallurgica Sinica (English Letters) Advances in Data Analysis and Classification Advances in Modeling & Simulation Analytical Chemistry Applied Energy Applied Mathematical Modeling Applied Mathematics & Information Sciences Applied Mathematics and Computation Applied Soft Computing Asia-Pacific Journal of Operational Research Austrian Journal of Statistics Automatic Control and Computer Sciences Automation and Remote Control Automation in Construction Behaviormetrika Biocybernetics and Biomedical Engineering Canadian Journal of Forest Research Chaos, Solitons & Fractals Chemical Engineering Communications Communications in Statistics — Simulation and Computation Communications in Statistics — Theory and Methods Communications of the Korean Mathematical Society Communications of the Korean Statistical Society Complex & Intelligent Systems Computational & Applied Mathematics Computational Economics Computational Statistics & Data Analysis Computers & Industrial Engineering Computers & Mathematics with Applications Computers & Operations Research Electric Power Systems Research Electronics and Communications in Japan (Part III: Fund. Electr. Sci.) Energy Energy Policy Engineering Applications of Artificial Intelligence Environmetrics European Journal of Operational Research Expert Systems with Applications Fibers and Polymers Flow, Turbulence and Combustion Fundamenta Informaticae Fuzzy Economic Review Fuzzy Optimization and Decision Making Fuzzy Sets and Systems Granular Computing Hydrological Sciences Journal IEEE Transactions on Fuzzy Systems IEEE Transactions on Industrial Informatics IEEE Transactions on Power Delivery IEEE Transactions on Power Systems IEEE Transactions on Systems, Man, and Cybernetics IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics IEEE Transactions on Systems, Man, and Cybernetics, Part C: Appl. and Rev. IEEE Transactions on Systems, Man, and Cybernetics: Systems IEEJ Transactions on Electrical and Electronic Engineering Indoor and Built Environment Information Sciences Insurance: Mathematics and Economics Intelligent Decision Technologies Int. Journal of Approximate Reasoning Int. Journal of Computational Intelligence Systems Int. Journal of Consumer Studies Int. Journal of Electrical, Computer, Energetic, Electronic and Comm. Eng. Int. Journal of Fuzzy Logic and Intelligent Systems Int. Journal of Fuzzy Systems Int. Journal of General Systems Int. Journal of Hydrology Science and Technology Int. Journal of Industrial and Systems Engineering Int. Journal of Industrial Ergonomics Int. Journal of Information Technology & Decision Making Int. Journal of Management Science and Engineering Management Int. Journal of Mathematics in Operational Research Int. Journal of Operational Research

Publisher

Kexue Chubaneshe/Science Press Springer AMSE Press American Chemical Society Elsevier Elsevier Natural Sciences Publishing Corporation Elsevier Elsevier World Scientific Publishing Co Austrian Society for Statistics Springer Maik Nauka/Interperiodica Publishing Elsevier Springer Polish Scientific Publishers PWN NRC Research Press Elsevier Taylor & Francis Taylor & Francis Taylor & Francis Korean Mathematical Society The Korean Statistical Society Springer Birkhauser Springer Elsevier Elsevier Elsevier Elsevier Elsevier Wiley Elsevier Elsevier Elsevier Wiley Elsevier Elsevier Korean Fiber Society Springer IOS Press SIGEF Springer Elsevier Springer Taylor & Francis IEEE IEEE IEEE IEEE IEEE IEEE IEEE IEEE Wiley SAGE Publications Elsevier Elsevier IOS Press Elsevier Atlantis Press Wiley World Academy of Sci., Eng. and Techn. Korean Institute of Intelligent Systems Springer Taylor & Francis Inderscience Publishers Inderscience Publishers Elsevier World Scientific Publishing Co Taylor & Francis Inderscience Publishers Inderscience Publishers

SJR 2017

#publications

Quartile

∑

Q1 Q1 Q3 Q1 Q1 Q1 Q3 Q1 Q1 Q2 Q4 Q3 Q2 Q1 No Q3 Q1 Q2 Q2 Q3 Q3 Q4 No No Q3 Q2 Q1 Q1 Q1 Q1 Q1 Q4 Q1 Q1 Q1 Q2 Q1 Q1 Q2 Q1 Q3 Q4 Q1 Q1 No Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q3 Q2 Q1 Q1 Q4 Q1 Q1 Q2 No No Q2 Q1 Q2 Q2 Q1 Q1 Q2 Q3 Q3

1 1 1 1 1 1 1 4 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 11 2 1 2 1 1 6 1 11 14 1 1 1 1 3 84 1 2 22 1 1 3 1 2 2 1 1 1 20 4 1 3 2 1 1 1 4 1 1 1 1 1 1 1 1

data

data data

data

data data

4

5

6

7

1 1 1 1 1 1 1 3 5 1

1

1 1

10 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

4

4 2 1 2

4

1

1 2 1 2

1 1 2 7 4

2 3 1

1 4

4 1 1 3

1 1 1 3 39

15

3

1 5

13

6 1

17 1 1 8 1 1

2 1 1 2

1 1 1

3 2

6 2

1 4

7 1 3 1

1 1

1 1 1

3 1

1 1 1 1 1 1 1 (continued on next page)

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Table 1 (continued). Journal

Int. Journal of Production Research Int. Journal of Services and Operations Management Int. Journal of Systems Science Int. Journal of Technology Management Int. Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Int. Mathematical Journal Int. Scholarly Research Notices Iranian Journal of Fuzzy Systems Iranian Polymer Journal Journal of Advanced Computational Intelligence and Intelligent Informatics Journal of Applied Mathematics and Computing Journal of Computational and Applied Mathematics Journal of Computing Science and Engineering Journal of Engineering Design Journal of Hydrology Journal of Intelligent & Fuzzy Systems Journal of Manufacturing Systems Journal of Materials Processing Technology Journal of Multiple-Valued Logic and Soft Computing Journal of Multivariate Analysis Journal of Optimization Theory and Applications Journal of Statistical Planning and Inference Journal of Textiles and Polymers Journal of the Chinese Institute of Engineers Journal of the Korean Mathematical Society Journal of the Operations Research Society of Japan Journal of Uncertain Systems Knowledge-Based Systems Kybernetika Materials & Design Mathematical and Computational Applications Mathematical and Computer Modeling Mathematical Problems in Engineering Mathematics and Computers in Simulation Measurement Science Review Metrika Metron Neural Computing and Applications Neurocomputing New Mathematics and Natural Computation Nutrition Palaeogeography, Palaeoclimatology, Palaeoecology Quality & Quantity Random Operators and Stochastic Equations Safety Science SIAM Journal on Scientific and Statistical Computing Soft Computing Software Quality Journal Solar Energy Statistical Papers Statistics Statistics & Decisions Technological Forecasting and Social Change The International Journal of Advanced Manufacturing Technology The Journal of Risk and Insurance The Scientific World Journal Transportation Planning and Technology Water Resources Research

a non-linear possibilistic regression, and formulate a fuzzy GMDH. Lee and Chen [55] discuss a generalized FLR model with FIFO data and suggest a non-linear programming method to estimate the fuzzy parameters (in this context, see also the note of Hong and Yi [56]). Nasrabadi and Nasrabadi [57] introduce a mathematical programming approach to FLR analysis which avoids the problem of increasing spreads by introducing additional arithmetic operations for symmetric fuzzy numbers. Chen and Hsueh [58] develop a mathematical programming model, which yields minimum total estimation error in terms of distance using L1 -norm. Moreover, the proposed approach avoids an overlapping area between observed and estimated responses. Further, Kocadagli [59] proposes a constrained non-linear programming approach for fuzzy multiple regression models with fuzzy output.

Publisher

Taylor & Francis Inderscience Publishers Taylor & Francis Inderscience Publishers World Scientific Publishing Co Hikari Hindawi University of Sistan & Baluchistan Springer Fuji Technology Press Ltd. Springer Elsevier Korean Institute of Inf. Sci. and Eng. Taylor & Francis Elsevier IOS Press Elsevier Elsevier Old City Publishing Inc. Elsevier Springer Elsevier Iranian Textile Ass. of Sci. and Techn. Taylor & Francis Korean Mathematical Society Operations Research Society of Japan World Academic Press Elsevier Academy of Sciences of the Czech Rep. Elsevier MDPI Elsevier Hindawi Elsevier Institute of Measurement Science Springer Springer Springer Elsevier World Scientific Publishing Co Elsevier Elsevier Springer Walter de Gruyter GmbH Elsevier Society for Industrial and Appl. Math. Springer Springer Elsevier Springer Taylor & Francis Walter de Gruyter GmbH Elsevier Springer Wiley Hindawi Taylor & Francis Wiley

SJR 2017

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Further, there can be found quadratic programming approaches, which combine the central tendency of LS and the possibilistic properties of fuzzy regression models. Tanaka and Lee [60] and Lee and Tanaka [61] are concerned with FLR models based on quadratic programming in order to minimize both the distances between the estimated output centers and the respective observations as well as the spreads of the estimated outputs. Chen [62], using three different ranking methods to measure the distance between two fuzzy numbers, deals with quadratic regression and non-linear programming to solve the resulting optimization problem. Lee and Tanaka [63] consider an FLR model with CIFO data and non-symmetric fuzzy coefficients based on quadratic programming, and construct two approximation models, a lower and an upper approximation model. Finally, Tanaka

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and Guo [64] give an overview on linear as well as quadratic programming methods in possibilistic regression analysis. 4.1.5. Piecewise necessity regression analysis Yu et al. [65] consider a general piecewise necessity regression model using LP with the aim of obtaining the necessity area (see for the necessity problem, for example, Tanaka [12]). This method implies that a practitioner has to determine the number and positions of change-points, which become complex when sample size increases. Therefore, Yu et al. [66] are concerned with an automatic change-point detection using the piecewise concept of Yu et al. [65] to obtain both the fuzzy regression model and the change-points based on a mixed-integer programming problem. Tseng et al. [67] apply the fuzzy piecewise regression approach with automatic change-point detection proposed by Yu et al. [66] to predict non-linear time series. In addition, Yu and Lee [68] suggest a piecewise fuzzy regression model to handle the large variation issues regarding FIFO data with an automatic change-point detection via quadratic programming. 4.2. Goal programming approaches There are also goal programming approaches (GPA) in fuzzy regression analysis that allow to consider multiple objective functions, and linearity is not a requirement. LP problems are therefore a special case of goal programming problems, when there is a single linear objective function and solely linear constraints. Sakawa and Yano [23,69,70] formulate three types of multi-objective programming problems to obtain FLR models based on FIFO data by using three indices for equalities between two fuzzy numbers (see Dubois and Prade [71] and Dubois [72]). However, Redden and Woodall [22] point out, inter alia, that these models are very sensitive to outliers. In contrast, Özelkan and Duckstein [35] consider a multi-objective fuzzy regression framework with some constants in the model constraints of the LP model, which leads to similar problems as in Peters [25]. Another multi-objective fuzzy regression approach, which unites central tendency and possibilistic properties, is presented by Tran and Duckstein [73] to overcome some of the drawbacks of LP methods. Hojati et al. [19] introduce a GPA to estimate the parameters of an FLR model by estimating the predicted band with the help of the endpoints of the observed intervals, which lead to the drawback that their approach is only applicable when symmetric triangular numbers or intervals are assumed for the fuzzy regression parameters. Nasrabadi et al. [74] propose a multi-objective FLR model to address some problems of FLR analysis (such as sensitivity to outliers and missing influence of some data points for parameter estimation) by introducing soft boundaries in the framework of the risk-neutral model of Modarres et al. [44,45]. Nasrabadi et al. [75] revisit the multi-objective FLR model of Nasrabadi et al. [74] and incorporate FIFO data in the modeling. Additionally, the GPA of Hassanpour et al. [76] minimizes total absolute deviations between central points of observed and estimated responses as well as absolute deviations between the spreads using CIFO data. Their GPA is based on a criterion of goodness of fit proposed by Kim and Bishu [77], and in contrast to the GPA of Hojati et al. [19] the model can also deal with non-symmetric data. In a further paper, Hassanpour et al. [78] estimate crisp regression parameters using FIFO (non-) symmetric triangular fuzzy numbers by quantifying the distance between the fuzzy numbers with the help of L1 -norm. On this basis, Hassanpour et al. [79] (also using FIFO data) introduce approximations for the product of two triangular fuzzy numbers, and select the best one for minimizing an objective function via a GPA to estimate the regression coefficients. Rafiei and Ghoreyshi [80] deal with a bi-objective GPA in FLR analysis with fuzzy parameters considering two cases, the CIFO and the FIFO case. This bi-objective formulation compensates the drawbacks of the proposed method by Özelkan and Duckstein [35].

4.3. Interval regression analysis Interval regression analysis is based on interval valued coefficients, and is regarded as the simplest variant of possibilistic regression models (see Tanaka and Lee [81]). Various methods for interval regression analysis have been considered, starting with LP. Ishibuchi and Tanaka [82], using the fuzzy regression model of Tanaka et al. [11], discuss approaches to identify the fuzzy parameters with symmetrical triangular form and with asymmetrical trapezoidal form based on interval regression expressions, respectively. Moreover, Ishibuchi and Tanaka [83] establish an LP approach to combine possibility and necessity models, and Inuiguchi et al. [84] measure interval errors by Minkowski difference. Lee and Tanaka [85] discuss two interval approximation models (lower and upper approximation models, see also Lee and Tanaka [63]) based on regression quantile techniques to select the majority of data. Additionally, Tanaka and Lee [86] develop interval regression models using polynomials with crisp input data and interval valued output based on possibility and necessity measures. However, the LP approach is known to have some shortcomings like non-centrality and crisp characteristics (see Section 4.1.1). Therefore, some promising alternatives have been proposed. Tanaka et al. [87] and Tanaka and Lee [81] deal with an interval regression model on the basis of quadratic programming to overcome the problem of crisp parameters. This model provides a higher central tendency compared to fuzzy regression models based on LP problems of their former works (see Tanaka and Ishibuchi [15]). Tanaka et al. [88] derive upper and lower approximation models for interval regression from the given data using possibility and necessity concepts, respectively. Wang and Tsaur [21] propose an approach to analytically perform interval regression analysis, data type analysis and variable selection. Inuiguchi and Tanino [89] deal with three interval regression models with crisp input and different assumptions on interval output using Minkowski difference, and the resulting estimation problems are formulated via quadratic or linear programming. Recently, there are a few works based on support vector machines to improve interval regression analysis, such as Jeng et al. [90], Hong and Hwang [91], Hwang et al. [92], Chuang [93], Hao [94,95], and Huang [96], as well as some works based on genetic algorithms, see Hu [97,98], or neural networks, see Huang et al. [99] (see Section 6). Moreover, Yu and Tzeng [100] extend the model of Yu et al. [66] with the purpose of determining the adequate number of change-points in interval piecewise regression models by using fuzzy multiple objective programming. Cerny and Rada [101] and Cerny et al. [102] present a possibilistic generalization of the ordinary LS estimator, a so called OLS-set for interval regression models. Hladik and Cerny [103], in contrast, discuss an approach that is motivated by tolerance analysis. This method overcomes some of the drawbacks of LP models, and moreover, it is easy and efficient to compute. Wang et al. [104] are concerned with the outlier problem in Tanaka’s method and propose two outlier detection approaches based on normalized upper and lower interval regression models. Likewise on the basis of the tolerance approach using prior information, Cerny and Hladik [105] examine two cases of interval regression, the case of crisp input and interval output as well as the case, where both input and output are interval valued. 5. Fuzzy least squares and fuzzy least absolutes methods This section is devoted to research question Q4 with regard to the second major field ‘‘fuzzy least squares and fuzzy least absolutes methods’’.

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5.1. Fuzzy least squares methods Fuzzy regression analysis has also been considered from the viewpoint of the least squares method, where the diversity between the predicted fuzzy values and the given fuzzy data is minimized with regard to various distance measures between two fuzzy numbers. So the fuzzy least squares (FLS) approach comprises the idea of goodness of fit and the residuals can be used to investigate the model accuracy (see Diamond and Kloeden [106,107] and Diamond and Tanaka [108]). The FLS approach was initially introduced by Celmins [16,17] and Diamond [109,110], who propose an estimation of the fuzzy model parameters by minimizing the total squared error of the output. Since then, the FLS approach has been extended and applied by a large number of researchers. In the following we present several of the main works in this field of research. 5.1.1. Direct enhancements of Celmin’s and Diamond’s approaches Celmins [111,112] extends the results of Celmins [16,17] to non-linear FLS regression models and the usage of FIFO data. Ming et al. [113] generalize the FLS model proposed by Diamond [110] in the way that all fuzzy numbers represented by single maxima piecewise continuous functions with compact support can be included. Xu [114,115] considers the LS approach with fuzzy data modeled via fuzzy numbers for a linear and an sshaped regression model, respectively. He uses a distance on a fuzzy number space by the integral of distance of every level set instead of the distance proposed by Diamond [110], where three vertices of the triangular fuzzy number are treated equally. Diamond and Körner [116] revisit the model of Diamond [110] to overcome the problem of negative spreads in the approach from Chang and Lee [24] by using Hukuhara difference, L2 -metric distance and LR fuzzy numbers. Based on the introduced distance by Xu [114,115], Xu and Li [117] are concerned with a multiple FLS linear regression model and an appropriate index to measure the goodness of fit. Hong and Hwang [118] extend three FLR models proposed by Diamond [110] to the multiple case using the technique of regularization. Arabpour and Tata [119] examine some metrics on trapezoidal and triangular fuzzy numbers (by generalizing the metric defined by Diamond [110]), and consider an estimation approach for the fuzzy parameters based on normal equations corresponding to LS models. 5.1.2. Developments in fuzzy least squares methods Kim and Bishu [77] point out that Tanaka et al. [11] and Savic and Pedrycz [26] use the spreads of the fuzzy regression parameters to evaluate the vagueness of the estimated output, i.e. the wider the spreads, the more vague is the model. To address this aspect, Kim and Bishu [77] introduce a so called ‘‘fuzzy membership LS regression model’’, which solves three LS problems for estimation of the fuzzy parameters based on a criterion of goodness of fit. Wang and Tsaur [29] consider the CIFO regression problem described by Tanaka et al. [11] and discuss a modified LS approach for its solution, which leads to better predictability compared to Tanaka et al. [11] and simplified computation compared to common FLS. Hong et al. [120] are focussed on an FLS approach with FIFO data, which uses shape preserving (T ωbased) operations on LR fuzzy numbers. Chang [121] deals with a hybrid FLS regression based on weighted fuzzy arithmetic, that allows to fit models to various types of data. Salas et al. [122] introduce a general squared distance between fuzzy numbers and extend the conventional LS method to the case of fuzzy data. Yang and Lin [123] propose two efficient estimation approaches based on FLS for FLR models with fuzzy parameters and by using FIFO data (as an alternative to the approach in Sakawa and Yano [23]). Yang and Liu [124] discuss a robust FLS algorithm with a

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noise cluster for interactive FLR models considered by Tanaka and Ishibuchi [30]. Domrachev and Poleshuk [125] concentrate on a linear regression model combining a fuzzy and a classical model, where coefficients are modeled as unimodal LR numbers and estimated via LS. Coppi et al. [126] investigate a linear regression model to study the dependence of an LR fuzzy dependent variable and crisp independent variables, along with an iterative LS estimation method (proposal related to D’Urso [127]). Nasibov [128] examines an FLS regression model using a weighted distance between fuzzy numbers. Bargiela et al. [129], using the standard LS criterion as a performance index, propose an iterative gradientdescent optimization algorithm to calculate the coefficients of a multiple regression model with fuzzy data. Chen and Hsueh [130] apply the LS approach to estimate the model parameters based on the concept of distance to increase the explanatory power of a fuzzy regression model. Hassanpour et al. [131] modify the approach of Kim and Bishu [77] to prevent negative spreads for triangular fuzzy output data. Yeh [132] investigates LS multiple regression with crisp coefficients and fuzzy data, and converts the primary regression problem to a 0–1 programming problem. Poleshchuk and Komarov [133] deal with a hybrid FLS regression model, where input and output data have qualitative characteristics. Shen et al. [134] discuss a fuzzy varying coefficient regression model with fuzzy coefficients that vary with a covariate, and propose a restricted weighted LS estimation approach to guarantee the non-negativity of the spreads of the coefficients (in contrast to, for example, Xu and Li [117], D’Urso [127], Coppi et al. [126]). Yoon and Choi [135] consider an FLS approach for fuzzy regression models using FIFO data with an error structure and express the estimators in one compact formula. D’Urso and Massari [136] study the dependence of an LR2 fuzzy response variable on a set of crisp or LR2 fuzzy explanatory variables using iterative weighted LS estimation (generalization of the model presented by Coppi et al. [126]). Chan et al. [137] propose a fuzzy stepwise regression with a polynomial structure and estimate the fuzzy coefficients via FLS. 5.1.3. Separate sub-models for core and spreads Chang and Lee [138] propose an LS approach, where the modal value and the spreads are estimated separately, and therefore interactions between them can be analyzed. Additionally, D’Urso and Gastaldi [139] consider a fuzzy regression model which is based on two linear sub-models, the so called ‘‘doubly linear adaptive fuzzy regression model’’. These sub-models explain the centers of the fuzzy observations (core model) and the spreads (spread model), respectively. Using three sub-models (for the centers and the left/right spreads separately), D’Urso and Gastaldi [140] discuss a numerical estimation method, when there are asymmetric spreads in the fuzzy data. Considering a polynomial regression model with three sub-models, D’Urso and Gastaldi [141] present a sequential procedure to find a suitable regression model for fuzzy data. Moreover, D’Urso [127] is concerned with extensions of the approach presented by D’Urso and Gastaldi [140] for various crisp/fuzzy data combinations and develop unconstrained as well as constrained (using inequality restrictions) FLS estimation methods. D’Urso and Giordani [142] consider FLS regression analysis following D’Urso and Gastaldi [140] with a multiple symmetric fuzzy dependent variable and symmetric crisp or fuzzy independent variables. Also based on the concept of three sub-models, D’Urso and Santoro [143] deal with an exploratory approach in a multiple FLR model, and introduce a coefficient of determination for symmetrical fuzzy variables as well as some variable selection procedures. Yoon and Choi [144] are focussed on a componentwise FLR model, which splits up response functions in modes and spreads of dependent variables. Choi and Yoon [145] construct a generalized fuzzy regression model using LS estimation, where the regression equations on the mode and the spreads of the predicted LR fuzzy numbers are also separated from each other.

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5.1.4. Defuzzification approaches and h-level least squares estimates In this subcategory of LS estimation the authors use defuzzification approaches combined with OLS estimation or h-level sets to express the estimators. Kao and Chyu [146] investigate a twostage approach to estimate the fuzzy model parameters. In the first stage, they defuzzify the fuzzy observations and estimate the model parameters by OLS, while in the second stage the error term of the FLR model is determined. In addition, Kao and Chyu [147] discuss an FLS regression method for CIFO/FIFO data based on the extension principle and a ranking of fuzzy numbers, and minimize the fuzzy sum of squared errors using a nonlinear programming method. Wu [148,149] obtains the h-level LS estimates with the help of conventional linear regression using the h-level real-valued data of the corresponding FIFO data. Here, the fuzzy coefficients are estimated according to the resolution identity proposed by Zadeh [150]. In contrast, Wu [151] develops FLS estimators based on the extension principle, where the membership functions of FLS estimators are constructed using conventional LS estimators. Chen and Dang [152] improve the model proposed by Kao and Chyu [147] and present a three-stage method to achieve a better performance in reducing the total estimation error based on the criterion of Kim and Bishu [77]. Wu [153] deals with an FLR model formulated via fuzzy scalar product, and constructs the membership functions of the FLS estimators by means of the resolution identity. Chachi et al. [154] propose an interval-based fuzzy regression approach for FIFO data, where h-level sets of fuzzy data are used to estimate the crisp parameters. Chen et al. [155] suggest a two-stage approach (following Kao and Chyu [146] and Chen and Hsueh [58]) for the construction of fuzzy regression models based on distance concept, which contain crisp coefficients estimated via OLS and a fuzzy adjustment term. 5.1.5. Interval and intuitionistic fuzzy regression models Rabiei et al. [156] develop an LS approach, where coefficients and input/output data are assumed to be triangular intervalvalued fuzzy numbers. Torkian et al. [157] investigate a multiple LS regression model with input and output data as intervalvalued fuzzy numbers by using an extended Yao–Wu signed distance. Arefi and Taheri [158] consider an intuitionistic FLS regression model, where input and output data as well as the model parameters are assumed to follow Atanassov’s intuitionistic fuzzy numbers. Chachi and Taheri [159] propose a multiple FLS regression model for FIFO data, and employ a distance on the space of interval-valued quantities for parameter estimation. 5.1.6. Asymptotic properties of least squares estimators Stahl [160] deals with an extended LS estimator within an FLS regression model with fuzzy input variables and fuzzy parameters, and prove its strong consistency. Krätschmer [161,162] √ shows strong consistency and n-consistency as well as limit distributions of generalized LS estimators. Kim et al. [163] investigate asymptotic properties of LS estimators in FLR analysis with FIFO triangular data, such as asymptotic normality and asymptotic relative efficiency with respect to OLS. Yoon et al. [164] prove asymptotic unbiasedness and asymptotic consistency of FLS estimators using a suitable metric. Additionally, Yoon et al. [165] consider a multiple FLR model with FIFO data modeled by triangular fuzzy numbers, and prove consistency and asymptotic normality of the LS estimators. 5.2. Fuzzy least absolutes methods Both, the approaches LS as well as LP, are sensitive to outliers, and more robust methods are needed for handling outliers. The method of least absolute deviations (LAD) based on medians is

a more robust approach, since it is superior to OLS when there are outliers in the data (see, for example, Dielman [166]). The main works in the field of fuzzy LAD methods are as follows. Chang and Lee [167] propose, besides an LS approach, a fuzzy LAD regression approach using a fuzzy difference ranking method, which leads to an LP problem. Kim et al. [168] deal with a twostage procedure, where in the first stage the fuzzy observations are defuzzified and the LAD estimators are applied, and in the second stage the fuzzy error term is specified to minimize the total deviation between observations and estimations. Torabi and Behboodian [169] suggest an optimization problem to get the LAD estimates of the regression parameters using the resolution principle and FIFO data. Choi and Buckley [170] develop a fuzzy regression model similar to the approach presented by Kao and Chyu [146], and estimate the fuzzy parameters using fuzzy least absolutes method. They also use a defuzzification of fuzzy data and estimation of crisp parameters in the first stage. Kelkinnama and Taheri [171] and Taheri and Kelkinnama [172] are concerned with two methods based on a novel metric on the space of LR fuzzy numbers and on the weakest triangular norm T ω to estimate the coefficients of a fuzzy regression model using LAD for CIFO and FIFO data, respectively. Chachi and Taheri [173] develop an LAD approach for CIFO data by estimating the model parameters with the generalized Hausdorff-metric on the space of LR fuzzy numbers (see Zimmermann [174]). Chachi et al. [175], also using the generalized Hausdorff-metric, present two LAD approaches for multiple regression analysis with CIFO data. Li et al. [176] and Zeng et al. [177] deal with LAD estimation using a new distance measure for triangular fuzzy numbers, and discuss different cases with various types of input and output data as well as regression coefficients. Hesamian et al. [178] are focussed on a semi-parametric partially linear regression model for FIFO data, fuzzy smooth function and crisp coefficients, and propose a two-phase procedure based on curve fitting methods and LAD for estimation of the smooth function and the fuzzy parameters. Hesamian and Akbari [179], also considering the semi-parametric partially linear model, but with crisp inputs and interval-valued outputs, extend the two-phase procedure for estimation of an interval-valued fuzzy smooth function. 6. Machine learning techniques in fuzzy regression analysis In this section, the focus lies on research question Q4 by identifying the contribution of each relevant publication in the context of the third major field ‘‘machine learning techniques in fuzzy regression analysis’’. The generalization capability of fuzzy regression analysis has been enhanced by incorporating machine learning techniques, such as evolutionary algorithms, neural networks or support vector machines. For general references in machine learning we refer to Vapnik [180] and Hastie et al. [181]. 6.1. Evolutionary algorithms Evolutionary algorithms are a part of evolutionary computation, a generic population-based metaheuristic optimization algorithm, which uses ideas and terminology of biological evolution, such as mutation, recombination, reproduction, and selection. Candidate solutions of the optimization problem are synonymous with individuals in a population, and a fitness function is used to determine the quality of these solutions. The most popular type of evolutionary algorithms are genetic algorithms, which are mostly used in optimization problems. Another type is genetic programming, where the fitness is determined by the ability to solve a computational problem. There are some papers in fuzzy regression analysis, that use evolutionary algorithms and genetic algorithms/programming. Buckley and Hayashi [182]

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propose fuzzy genetic algorithms to (approximately) solve fuzzy optimization problems, such as fuzzy regression. Yabuuchi and Watada [183] consider genetic algorithms in the framework of a robust fuzzy regression model (see Section 7.1). Buckley et al. [184] are concerned with fuzzy non-linear regression models, and use an evolutionary algorithm approach to fit multivariate fuzzy polynomials to CIFO data. Buckley and Feuring [185] extend this evolutionary algorithm for searching a library of fuzzy functions, such as linear, polynomial, exponential, and logarithmic, which lead to the best fit of the data. Aliev et al. [186] show that fuzzy regression analysis can be simplified in application and improved in performance by including genetic algorithms. Mogilenko et al. [187] compare fuzzy regression analysis based on genetic algorithms and LP considering CICO and CIFO data. Hu [97] presents a robust non-linear interval regression model and investigates a method based on genetic algorithms with the aim of determining two functional-link nets, one to identify the upper bound and the other to identify the lower bound of the data interval. Chan et al. [188] discuss a genetic programming-based fuzzy regression approach with an incorporated detection of outliers. Additionally, Chan et al. [189] consider an intelligent fuzzy regression approach for generating models via an evolutionary algorithm, which represents non-linear fuzzy relationships, and propose parameter estimation via LS. Hu [98], using a genetic algorithm as proposed in Hu [97], constructs a robust non-linear interval regression model based on multilayer perceptron. Moreover, Chan et al. [190] propose a fuzzy regression method that is based on genetic programming to develop non-linear model structures, where the coefficients are estimated by optimizing the fuzzy criteria. 6.2. Support vector machines Support vector machines are supervised learning models with associated learning algorithms that are used for pattern recognition and function estimation problems. Hong and Hwang [191] investigate the convex optimization problem of fuzzy multiple linear and non-linear regression models using support vector machines, and develop support vector fuzzy regression machines (SVFRM). Additionally, Hong and Hwang [192] propose an estimation of fuzzy multiple non-linear regression models for FIFO data using an LS support vector machine. Yao and Yu [193] deal with asymmetric support vector machines for evaluating functional relationship in linear and non-linear fuzzy regression models. Hao and Chiang [194,195] assume the parameters in support vector regression machines as fuzzy numbers and present an algorithm where various learning machines with different types of nonlinear regression functions can be constructed by using kernel functions. Wu and Law [196] present an SVFRM with the ability to penalize Gaussian noise on triangular fuzzy number space. For an SVFRM method with a loss function based on the Trutschnig distance as well as for a more detailed review on SVFRM we refer to Wieszczy and Grzegorzewski [197]. There are also some works combining interval regression analysis with support vector machines. Jeng et al. [90] apply support vector machines to interval regression analysis by considering a two-step approach with a construction of two radial basis function networks for identifying the lower and upper side of the data interval, respectively. Hong and Hwang [91] also deal with a support vector machine based method for interval (linear and non-linear) regression analysis, and combine possibility and necessity estimation with quadratic loss support vector machines. Hwang et al. [92] investigate a robust approach in interval regression analysis for CICO data by using support vector interval regression machine. Also considering interval regression analysis, Chuang [93] discusses an extended support vector interval

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regression machine for interval valued input and output data. Hao [94] uses support vector interval regression machines for the evaluation of (non-)linear regression models with CICO data. In addition, Hao [95] evaluates interval (non-)linear regression models by combining possibility and necessity estimation with the concept of support vector machines. Moreover, Huang [96] introduces a reduced support vector machine for interval regression analysis, which reduces the number of support vectors via random selection of sample subsets. 6.3. Neural networks Neural networks are biologically-inspired information processing paradigms that enable a computer to learn from observational data. There are a few approaches where back-propagation, radial basis function or random weight neural networks are applied to improve fuzzy regression models. Ishibuchi and Tanaka [198] are the pioneers in combining fuzzy regression analysis and back-propagation neural networks to handle more complex systems than the LP based methods. Ishibuchi et al. [199] investigate a back-propagation neural network with interval weights and interval biases and apply it to fuzzy regression analysis. Huang et al. [99] present two robust learning algorithms based on neural networks to determine a robust non-linear interval regression model. Cheng and Lee [200] combine a fuzzy inference system for fuzzy regression analysis with the learning ability of neural networks. Dunyak and Wunsch [201] generalize (linear and non-linear) fuzzy regression analysis by using neural network models with general fuzzy number inputs, outputs, biases, and weights. Cheng et al. [202] consider FLR with fuzzy intervals in the framework of a neural network classification model. Ishibuchi and Nii [203] utilize fuzzified neural networks, where the connection weights are asymmetric fuzzy numbers, as non-linear fuzzy models in fuzzy regression analysis. Cheng and Lee [204] propose a fuzzy regression approach based on radial basis function networks, where the connection weights between the hidden and the output layers are fuzzified. Jeng et al. [90] also consider radial basis function networks in the framework of fuzzy regression analysis (see Section 6.2). Alex [205] discusses a fuzzy regression neural network system for a fuzzy normal regression model. Zhang et al. [206] apply a fuzzy radial basis function network to fuzzy non-linear regression analysis for multiple LR-type fuzzy data. Mosleh et al. [207,208,209] and Otadi [210] present a hybrid approach based on fuzzy neural networks for approximate fuzzy coefficients of fuzzy (non-)linear and polynomial regression models with CIFO or FIFO data. Roh et al. [211] are concerned with an FLR model based on the design approach of polynomial neural networks. He et al. [212,213] use a random weight network to develop a fuzzy non-linear regression model, where inputs/outputs are triangular and trapezoidal fuzzy numbers, respectively. Pehlivan and Apaydin [214] utilize a fuzzified radial basis function network for obtaining estimations of fuzzy regression models with FIFO data. Recently, Liu et al. [215] deal with a special single-hidden layer feed forward neural network, the so called extreme learning machine, for fuzzy regression considering FIFO data. Further papers, where fuzzy regression analysis is combined with neural networks, are for instance Nasrabadi and Hashemi [216] (see Section 7.1), Khashei et al. [217] and Chaudhuri and De [218] (see Section 7.6). 6.4. Other machine learning techniques Beyond evolutionary algorithms, neural networks and support vector machines, there are further machine learning approaches that have been utilized to improve fuzzy regression analysis:

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Ramli et al. [219,220,221] propose an efficient real-time (switching) fuzzy regression approach using convex hull designed via a beneath-beyond algorithm. Zuo et al. [222] present a fuzzy regression transfer learning method using fuzzy rules by developing a Takagi–Sugeno fuzzy regression model to transfer knowledge from a source domain to a target domain. In extension of this work, Zuo et al. [223] discuss granular fuzzy regression domain adaptation methods that are also based on Takagi–Sugeno fuzzy models. Considering a semi-parametric regression approach (see also Hesamian et al. [178] and Hesamian and Akbari [179, 224]), Akbari and Hesamian [225] combine kernel smoothing and elastic net penalized methods for the construction of a variable selection approach within a multiple fuzzy regression model. Moreover, some authors are concerned with the topic of fuzzy ridge regression, for instance Hong and Hwang [226], Hong et al. [227], Donoso et al. [228], Suk and Hwang [229], Farnoosh et al. [230,231], and Zhang et al. [232].

contrast, Ferraro and Giordani [241] propose a robust regression model for fuzzy data exploiting Ferraro and Giordani [242] (see Section 7.2) in order to handle situations characterized by the presence of outliers. Based on the model introduced by Shen et al. [134], Yang et al. [243] present a robustified fuzzy varying coefficient model for FIFO data. Shakouri and Nadimi [244] deal with the outlier problem in fuzzy regression analysis with CICO data both in the sense of elimination and impact reduction considering linguistic variables and possibility concept. Choi et al. [245] consider Theil’s method (see Theil [246]) in fuzzy regression analysis, which is robust to outliers. Chachi and Roozbeh [247] describe a robust estimation method for CIFO data based on least trimmed squares, that helps to identify and ignore outliers as well as other irregular data. Finally, Chachi [248] discusses a robustified LS fuzzy regression model for CIFO data with a weighted objective function that prevents the drawbacks of OLS estimation in the presence of outliers.

7. Various minor fields of fuzzy regression analysis

7.2. Fuzzy probabilistic approach

In this section, we concentrate on research question Q4 by investigating papers and their contributions to various minor fields of fuzzy regression analysis.

When imprecise information (represented by fuzzy sets) is also affected by randomness, the concept of fuzzy random variables proposed by Puri and Ralescu [249] can be employed. Ferraro and Giordani [242] calls a fuzzy regression approach, where randomness is also taken into account for estimating the regression coefficients, a fuzzy probabilistic approach. A linear regression model containing both fuzziness and randomness is firstly considered by Diamond [250]. Näther and Albrecht [251], Näther [252,253], Körner and Näther [254] and Näther and Körner [255] develop a linear estimation theory for the parameters in linear regression models in the case of fuzzy random observations. Näther and Körner [256] investigate the linear regression problem with random fuzzy numbers and propose the LS approximation principle for fuzzy data. Wünsche and Näther [257] prove that the conditional expectation of a fuzzy random response variable (given measurable functions of the fuzzy input variables) is the best approximation for the fuzzy response variable by measurable functions of the fuzzy input variables. Krätschmer [258,259] deals with finding of a stochastic regression model with integrated physical vagueness of the items without changes in the initial relationships between the variables. Näther [260] considers different variants of regression models with fuzzy random data including a model where the response variable is modeled by means of a fuzzy random variable, and a regression model between two fuzzy random variables (following Wünsche and Näther [257]). Kwong et al. [261] develop a hybrid FLS regression to model both randomness and fuzziness, and introduce for this purpose a new form of weighted fuzzy arithmetic. Watada and Wang [262] discuss a regression model based on fuzzy random variables and expected value operators. Gladysz and Kuchta [263] show how the possibilistic distributions of the response variable and the model parameters can be determined, when the random component of the model is an LR fuzzy variable with given generative probabilistic distribution. Gonzalez-Rodriguez et al. [264] investigate a linear regression model between two general fuzzy random variables using a fuzzy-arithmetic approach. This model agrees with the respective models in Diamond [250], Diamond and Körner [116], Körner and Näther [254], Näther [253], Wünsche and Näther [257], and Krätschmer [258] in the way of considering the same conditional expectation under particular conditions. Watada et al. [265] deal with hybrid uncertain (fuzzy random) data and present the so called confidence-interval-based fuzzy random regression model. Ferraro et al. [266] introduce a linearity test for a simple regression model with an imprecise response formalized by LR fuzzy numbers. Ferraro et al. [267] are concerned with a generalization of the model proposed by Coppi

7.1. Robust fuzzy regression analysis Besides the LAD approach, the problem of outliers has been considered with regard to robust estimation methods and detection criteria for outliers. Some of the above discussed fuzzy regression approaches are also dedicated to the influence of outliers, such as Peters [25], Huang et al. [99], Özelkan and Duckstein [35], Chen [34], Yang and Liu [124], Modarres et al. [44, 45], Nasrabadi et al. [74], Hung and Yang [36], Nasrabadi et al. [75], and Wang et al. [104]. In addition, Yabuuchi and Watada [183] present a robust fuzzy regression model by employing the concept of distance to identify and remove the impact of outliers, and they use genetic algorithms to solve the resulting integer programming problem. Nazarko and Zalewski [233] propose multi-criteria programming to estimate the regression coefficients unaffected by outliers. Sohn [234] discusses a robust FLR approach with triangular fuzzy coefficients and CICO data using M-estimators. Hwang et al. [92] consider a robustified method in the framework of a support vector interval regression machine (see Section 6.2). Gladysz and Kuchta [235], based on the approach of Chen [34], insert dummy variables into the fuzzy regression model for labeling outliers, which leads to a better fit of the model. Varga [236] is concerned with two robust estimation procedures for fuzzy regression parameters, and compares these models with fuzzy and classical regression approaches. Nasrabadi and Hashemi [216] extend the risk-neutral model of Modarres et al. [44] to a non-linear robust fuzzy regression model with the help of fuzzy neural networks, where weights and biases as well as input and output variables are assumed to be LR fuzzy numbers. Kula and Apaydin [237] suggest a robust fuzzy regression based on a ranking of fuzzy sets regarding the residuals, and they define the weight matrix by the membership function of the residuals. On this basis, Kula and Apaydin [238] estimate the standard deviations of the parameters and present a hypothesis testing approach for the parameters in robust fuzzy regression models. Hu [97,98] considers robust non-linear interval regression using genetic algorithms (see Section 6.1). D’Urso et al. [239] and D’Urso and Massari [136] discuss a least median squares weighted LS approach based on the reference model of Coppi et al. [126], which leads to a robustified estimation of model coefficients. Yabuuchi and Watada [240] develop a robust fuzzy regression model with the help of possibility maximization. In

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708

et al. [126] and present an approach based on a random LR fuzzy-valued response variable and a scalar explanatory random variable, which overcomes the non-negativity condition imposed by Coppi et al. [126]. Ferraro et al. [268], based on the model of Ferraro et al. [267], discuss a determination coefficient to measure the goodness of fit, and prove its strong consistency. Ferraro and Giordani [242], also using LR fuzzy random variables in the linear regression framework, consider parameter estimation and the problem of hypothesis testing following a bootstrap approach. Additionally, Coppi et al. [269] are focussed on a class of linear regression models using LR fuzzy random variables and FIFO data. Ferraro [270] analyzes the generalization performance of a linear regression model for imprecise random variables, where the prediction error is estimated via a bootstrap technique. Jiang et al. [271] consider the randomness caused by particular explanatory variables by adopting probability density functions, where the parameter settings are specified with help of a chaos optimization algorithm. 7.3. Fuzzy logistic regression Logistic regression models are non-linear and the response variable is of categorical nature (see Agresti [272]). There are many situations, where the response categories are affected by fuzziness, and therefore a probability distribution for the response variable cannot be considered. Further problems of common logistic regressions arise, when the sample size is too small or the observations are non-precise (this problem often occurs in medical studies, see Pourahmad et al. [273]). For these reasons, fuzzy logistic regression has been considered from both main viewpoints in fuzzy regression analysis, the possibilistic and the LS approach. Yang and Chen [274] derive an algorithm for parameter estimation within a fuzzy class logistic regression model. Pourahmad et al. [273] introduce so called possibilistic odds by proposing a fuzzy logistic regression model with crisp input and binary response. Pourahmad et al. [275] deal with FLS estimation for the parameters of a fuzzy logistic regression, and use a criterion named ‘‘capability index’’ for model evaluation. Namdari et al. [276] present a possibilistic logistic regression approach for CIFO data, while Namdari et al. [277] suggest LAD and LS estimation for the parameters of a fuzzy logistic regression. Hesamian and Akbari [224] investigate a fuzzy logistic regression model with crisp inputs and intuitionistic fuzzy linguistic outputs, and improve fuzzy logistic regression models by using a semiparametric approach (see for similar approaches Hesamian et al. [178] and Hesamian and Akbari [179]). Salmani et al. [278], using LS estimation proposed by Diamond [109], consider a fuzzy logistic regression model to analyze the relationship between a fuzzy binary response variable and fuzzy input variables. 7.4. Type-2 fuzzy regression analysis In situations with high levels of uncertainty, type-2 fuzzy sets are more appropriate than type-1 fuzzy sets because of additional degrees of freedom (see Castillo and Melin [279]). There are some recent works regarding fuzzy regression analysis based on type-2 fuzzy sets: Wei and Watada [280] propose a fuzzy qualitative regression approach using type-2 fuzzy sets as well as type-2 fuzzy data. Poleshchuk and Komarov [281] present a fuzzy non-linear regression model for interval type-2 fuzzy sets using LS for parameter estimation. Further, Hosseinzadeh et al. [282] discuss a weighted GPA in FLR analysis (based on a metric introduced by Hassanpour et al. [78]) with CIFO data using type2 fuzzy sets to model output data. Darwish et al. [283] extend the approach presented by Poleshchuk and Komarov [281] by constructing an affinity measure for two interval type-2 fuzzy

13

sets. Wei and Watada [284] generalize their former work (see Wei and Watada [280]) and consider a type-2 fuzzy regression model based on credibility theory. In contrast, Bajestani et al. [285] deal with a linear and a piecewise type-2 fuzzy regression model in a possibilistic framework, and Bajestani et al. [286] propose a type-2 fuzzy regression model based on type-2 fuzzy time series concepts. 7.5. Fuzzy clusterwise regression The so-called clusterwise regression analysis combines cluster and regression analysis, and helps to overcome the problem of heterogeneous observations in traditional regression models. Clusterwise regression models have been considered by a few researchers in a fuzzy framework. The idea of fuzzy clusterwise linear regression (FCLR) goes back to Jajuga [287], who proposes a two-stage procedure for determining fuzzy clusters (first stage) and evaluating a linear fuzzy regression for each fuzzy cluster (second stage). This two-stage model is reconsidered and extended by Yang and Ko [288], where the cluster memberships (determined by a clustering algorithm) serve as weights in weighted FLR. This leads to the fact that the results strongly depend on the clustering algorithm. D’Urso and Santoro [289] and D’Urso et al. [290] introduce an FCLR model with symmetrical CIFO data to perform a fuzzy cluster analysis in the framework of an FLR model. Further, they present a fitting index to measure the goodness of fit of the FCLR model. Suk and Hwang [229] discuss a fuzzy clusterwise regression model which combines fuzzy clustering and ridge regression in a unified framework with the purpose of dealing with potential multicollinearity. Another hybrid approach based on cluster memberships is given by Tutmez [291] (see Section 7.7). 7.6. Fuzzy regression analysis combined with time series analysis There are also some approaches which combine time series analysis with fuzzy regression analysis. Watada [292,293] proposes fuzzy time series models by using the concept of intersection of fuzzy numbers. Chang [294] discusses a fuzzy forecasting method for seasonality in time series data based on fuzzy regression models. Tseng et al. [67] use a fuzzy piecewise regression approach for the prediction of non-linear time series (see Section 4.1.5), and Tseng and Tzeng [295] combine seasonal ARIMA models with fuzzy regression models to develop fuzzy seasonal ARIMA models. Roychowdhury and Pedrycz [296] combine granular regression models and fuzzy rules for modeling temporal systems using CICO data. Tsaur et al. [297] investigate fuzzy regression models to forecast seasonal time series. Tsaur [298] performs a forecasting analysis based on the so-called fuzzy gray regression model to solve limited time series data. Khashei et al. [217] and Chaudhuri and De [218] deal with hybrid fuzzy regression models that combine artificial neural networks and fuzzy regression for time series forecasting. Azadeh et al. [299,300] also consider a hybrid approach by using an integrated fuzzy regression and time series framework for estimation and forecasting with non-stationary data. Finally, Bajestani et al. [286] combine type-2 fuzzy regression analysis with type-2 fuzzy time series concepts (see Section 7.4). 7.7. Further approaches Yager [301], as one of the pioneers in applicating fuzzy set theory in regression analysis, uses a linguistic explanatory variable for prediction in the simple regression case. For early discussions in transferring fuzzy observations into fuzzy parameters see Bandemer [302]. Chen [303] establishes a relation of multiple

14

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708

fuzzy regression and gives some analytical methods by means of fuzzy set theory. Wang and Li [304] discuss two different FLR models with fuzzy valued variables based on possibility theory. Xizhao and Minghu [305] consider a generalized linear regression model, where the coefficients and the explanatory variables are fuzzy, and propose Minimax estimation. Inuiguchi et al. [306] are focussed on FLR analysis using mean absolute deviations, where the fuzzy regression function is estimated by minimizing the sum of absolute deviations of a level set. Papadopoulos and Sirpi [307,308] and Profillidis et al. [309] deal with the set of solutions of a fuzzy regression model as a metric space using similarity ratios in order to compare the spaces of solutions. Arnold and Gerke [310] propose an approach for testing fuzzy hypotheses in the linear regression model with crisp data that goes back to Arnold [311,312] (see also Chukhrova and Johannssen [313, 314]). Toyoura et al. [315] and Watada and Pedrycz [316] consider linguistic regression models against the background that expert opinion is often expressed by natural words rather than in a numerical form. Additionally, Alex [317] proposes a method to combine fuzzy regression and fuzzy inference modeling. Choi and Kim [318] suggest a censored fuzzy regression model based on censored data in order to solve the problem of crossing estimated fuzzy outputs. Guo et al. [319] focus on the estimation of a scalar fuzzy regression model using axiomatic credibility measure theory of Liu [320]. Lu and Wang [321] present an FLR model estimated by maximizing the average similarity between estimated and observed response variables, where the spreads of the estimated response variable fit the spreads of the observed input variables, and therefore enhance models proposed by D’Urso [127], Coppi et al. [126], and Chen and Dang [152] by reducing the spreads increasing problem. Shakouri and Nadimi [322] consider an FLR model for CIFO data, in which they formulate an objective function based on a non-equality possibility index. In this way they get a minimum degree of acceptable uncertainty. Mashinchi et al. [323] investigate an unconstrained global continuous optimization procedure using tabu and harmony search for the determination of an FLR model. A hybrid approach using FLR and fuzzy cognitive map is proposed by Azadeh et al. [324]. Tutmez [291] and Tutmez and Kaymak [325] also deal with hybrid fuzzy regression approaches by considering a weighted fuzzy regression on the basis of spatial dependence measure of the memberships and by using statistical confidence bounds as spreads of the fuzzy numbers, respectively. Su et al. [326] introduce the kernel based non-linear fuzzy regression model, which is able to deal with non-linearity between crisp inputs and fuzzy output. Su et al. [327] are concerned with linear and non-linear parametric regression analyses for imprecise and uncertain data represented by a fuzzy belief function (see Petit-Renaud and Denoeux [328]), where the coefficients are estimated with the help of the fuzzy evidential expectation maximization algorithm. The non-linear regression model is constructed via a kernel function as in Su et al. [326]. Jiang et al. [329] are focussed on a chaos-based fuzzy regression approach, where chaos variables are used for representing polynomial structures of fuzzy polynomial models. Liu and Chen [330], Liu et al. [331] and Chen et al. [332] propose systematic approaches to optimize the h-value for FLR models with symmetric and asymmetric triangular fuzzy coefficients, respectively. Jung et al. [333] deal with an h-level fuzzy regression model for CIFO/FIFO data, which is constructed with the help of the resolution identity, and apply the rank transform method to the model. Chan and Ling [334] are concerned with a forward selection based fuzzy regression to determine significant regressors. Roldan Lopez de Hierro et al. [335] utilize Bernstein polynomials for developing a fuzzy regression procedure which guarantees non-negative spreads. In addition, Roldan Lopez de Hierro et al. [336] introduce a fuzzy partial order and a family

Table 2 Automotive industry. Author(s)

Application

Area(s)

D’Urso [127]

Characteristics of different kinds of cars Modeling car ownership

5.1

Ögüt [353], Azadeh et al. [354] Suk and Hwang [229] Wu and Law [196] Roh et al. [211]

Retail price of general motors cars Car sale series forecast Automobile miles per gallon

4.1, 6.3 7.5 6.2 6.3

of fuzzy distance measures with the aim of solving the conflict of interpreting fuzzy data as possibility distributions and minimizing real error functions. Alfonso et al. [337] introduce a fuzzy regression method which is based on so called finite fuzzy numbers. Chan and Engelke [338] focus on a fuzzy regression method that uses a varying spread based on a third-order polynomial for the simulation of non-linear and non-symmetrical fuzziness. Shakouri et al. [339] apply a new possibility for equality of two fuzzy numbers to the objective function of an FLR model in order to estimate the regression coefficients. Boukezzoula et al. [340] discuss parametric interval-based regression methods in the light of ontic and epistemic visions of intervals (see Couso and Dubois [341]). Hose and Hanss [342] propose an intuitive and consistent data-driven approach to infer membership functions of fuzzy parameters in parameter affine models. Recently, Hesamian and Akbari [343] extend the quantile regression model for usage in fuzzy environments via a semi-parametric kernel-based method with crisp predictors, fuzzy responses and fuzzy smooth function. 7.7.1. Fuzzy entropy approaches Kao and Lin [344] introduce the idea of entropy for fuzzy regression analysis by decomposing fuzzy numbers into two parts, position and fuzziness. Kumar et al. [345] estimate the parameters of a restricted fuzzy weighted linear regression model using the concept of fuzzy entropy. Kumar and Bajaj [346] deal with an intuitionistic fuzzy weighted linear regression model based on the concept of fuzzy entropy, which is a generalization of the approach presented in Kumar et al. [345]. Moreover, an estimation approach for FLR models based on generalized maximum entropy is proposed by Ciavolino and Calcagni [347]. 7.7.2. Non-parametric fuzzy regression analysis The most widely used non-parametric fuzzy regression methods (i.e. regression without a predefined functional form) involve fuzzified neural networks and SVFRM, see Sections 6.2 and 6.3. Some other works include the following approaches: Firstly, Cheng and Lee [348] fuzzify the techniques k-nearest neighbor and kernel smoothing. Petit-Renaud and Denoeux [328] consider a non-parametric regression approach called evidential regression, which is based on fuzzy belief assignment and makes use of discounting and Dempster’s rule of combination (see also PetitRenaud and Denoeux [349] for fuzzy evidence theory applied to regression problems). Wang et al. [350] present a non-parametric fuzzy regression model for CIFO data based on weighted LS and a local linear smoothing technique with a cross-validation procedure. Chachi et al. [351,352] present a fuzzy regression model for CIFO data based on the (non-parametric) Multivariate Adaptive Regression Splines (MARS) approach. 7.7.3. Monte Carlo methods Abdalla and Buckley [373,374] apply the Monte Carlo method to the FLR model with the purpose of obtaining the optimal solution within a predetermined error bound. Additionally, the Monte Carlo method applied to non-linear fuzzy regression can be

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708

found in Abdalla and Buckley [375]. For a more general overview of the application of Monte Carlo methods in fuzzy optimization including various fuzzy regression models see Buckley and Jowers [376]. Recently, Icen and Demirhan [377] and Icen and Cattaneo [378] consider different distance measures for Monte Carlo methods to estimate the fuzzy parameters in FLR models. Moreover, Icen and Günay [379] improve the application of fuzzy expert systems for parameter estimation by Monte Carlo methods in FLR models. 7.7.4. Bootstrap techniques Akbari et al. [380] discuss an LS estimation of crisp regression coefficients using fuzzy data, and introduce confidence intervals and hypothesis testing for the coefficients with the help of bootstrap techniques (similar to the approach of Akbari and Rezaei [381]). Lin et al. [382] also consider hypothesis testing for regression coefficients, and use bootstrapping for the computation of standard errors and p-values. Ferraro et al. [383] deal with bootstrapping confidence intervals for the coefficients of an FLR model with fuzzy LR random variables. Lee et al. [384] propose statistical inference for the coefficients of fuzzy regression models with CIFO data for each h-level based on FLS estimation and bootstrapping. Moreover, Ferraro and Giordani [242] and Ferraro [270] also apply bootstrap methods, see Section 7.2. 7.7.5. Regression analysis based on fuzzy prior information Arnold and Stahlecker [385,386,387,388] consider estimation and prediction within linear regression models using fuzzy prior information. In addition, Arnold and Stahlecker [389,390] deal with an uniformly best estimator in linear regression analysis, when fuzzy prior information is given. 7.7.6. Surveys For other references on surveys to fuzzy regression analysis we refer to Kacprzyk and Fedrizzi [391], Wen and Lee [392], Chang and Ayyub [393], Taheri [394], Kahraman et al. [395], and Azadeh et al. [396]. For a general review paper on exploratory multivariate analysis methods using fuzzy information, including a section on fuzzy regression analysis, see D’Urso [397]. 8. Practical applications of fuzzy regression analysis In this section, we focus on second to last research question Q5, i.e. we are concerned with practical applications and case studies using techniques of fuzzy regression analysis. While most papers reveal numerical and/or experimental applications, there are also several papers regarding case studies

15

Table 4 Chemistry. Author(s)

Application

Area(s)

Gharpuray et al. [398]

Cellulose/enzymatic hydrolysis Calibration lines in analytical chemistry Coloration process in loom industrial Fault diagnosis of chemical plants

4.1

Pop and Sarbu [399] Tavanai et al. [400], Torkian et al. [157] Kimura et al. [401]

6.3

9. Critical discussion and future directions In this section, we provide a critical discussion of the presented methods, and moreover, we are concerned with our final research question Q6 and pursue the goal of investigating some potential directions for future research. Comparing ordinary and fuzzy regression, the deviations between observations and estimated values have different interpretations, i.e. we have random errors (due to observation inconsistency) on the one hand and fuzzy errors (due to system fuzziness) on the other hand. Since randomness and fuzziness measure differing types of uncertainty, regression analyses should be able to model both of these types adequately. We are of the opinion that fuzzy probabilistic approaches improve the capability of managing uncertainties due to randomness and fuzziness, which is why they deserve careful future attention. For instance, the

Author(s)

Application

Area(s)

Lee and Chen [55] Chen et al. [355], Chen and Chen [356], Fung et al. [357], Karsak [358], Kwong et al. [359], Sekkeli et al. [360], Sener and Karsak [361,362,363], Karsak et al. [364], Jiang et al. [271,329], Chan et al. [137,365], Liu et al. [331,366], Chan and Ling [334] He et al. [367]

Manpower forecasting Quality function deployment

4.1 4.1, 4.2, 5.1, 7.2, 7.7

Balancing productivity and consumer satisfaction Product life cycle prediction R&D project evaluation Evaluation of employees’ performance

4.1

Huang and Tzeng [368] Imoto et al. [369] Chen and Hsueh [130], Kelkinnama and Taheri [171], Taheri and Kelkinnama [172], Hesamian et al. [178], Akbari and Hesamian [225], Hesamian and Akbari [343] Chan et al. [189,190,370], Chan and Engelke [338,371] Ramli et al. [219]

Höglund [372]

4.1, 5.1

on various specific topics. Tables 2–17 show numerous of these practical applications structured according to areas of application and in chronological order with regard to the publications. In addition, the subsection(s) for the respective area(s) of the proposed applications are also given. Papers which solely present case studies without providing advances in methodology are merely listed in Tables 2–17, and thus are not discussed in the previous part of this paper. In particular, we have identified some main areas of application: automotive industry (Table 2), business administration (Table 3), chemistry (Table 4), economics (Table 5), electrical engineering (Table 6), environmental research (Table 7), finance (Table 8), housing (Table 9), hydrology (Table 10), information technology (Table 11), insurance (Table 12), manufacturing (Table 13), materials science (Table 14), medical science (Table 15), nutrition (Table 16), and further areas of application (Table 17). Note that these areas need not be taken as mutually exclusive.

Table 3 Business administration.

D’Urso and Massari [136]

7.7

Affective quality assessment Occupational health and safety management Advertising types: traditional vs. ‘‘creative’’ Earnings management

4.1 4.1 5.1, 5.2, 6.4, 7.7

5.1, 6.1, 7.7 6.4 5.1, 7.1 4.1

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N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708 Table 5 Economics. Author(s)

Application

Area(s)

Jajuga [287]

Employees outside the agricultural sector Student grades and family income

7.5

Exchange rates to the dollar

4.1, 4.3, 6.3

Research effects on regional technologies Economic indicators Gross domestic product Number of officials in mid-size cities Relationship between GNP and money Modeling deregulated electricity markets Fuzzy profit in an oligopolistic environment Business cycle analysis Modeling private consumption Expenditures on public education Forecasting gold price Oil consumption estimation Retail trade sales

4.1

Diamond [110], Ming et al. [113], Wang and Tsaur [29], Hong et al. [120], Hong and Hwang [118], Kelkinnama and Taheri [171], Choi et al. [245], Wieszczy and Grzegorzewski [197] Tanaka and Watada [13], Moskowitz and Kim [39], Tanaka et al. [87], Khashei et al. [217], Chaudhuri and De [218] Ramezani and Duckstein [402] Peters [25] Lee and Tanaka [63], Shen et al. [134], Yang et al. [243] Lee and Tanaka [85] Papadopoulos and Sirpi [307] Niimura and Nakashima [403] Aliev et al. [186] Wu and Tseng [404], Lin and Pai [405] Papadopoulos and Sirpi [308] Chen and Hsueh [58] Khashei et al. [217] Azadeh et al. [406,407] Ferraro et al. [267], Chachi and Taheri [173], Roldan Lopez de Hierro et al. [335,336] Chou et al. [408] Mosleh et al. [208] Lin et al. [382] Azadeh et al. [409] Cerny and Hladik [105]

Air cargo volume forecast Supply and demand in the labor market Early warning of macroeconomic index data Gasoline consumption of Iran Inflation–consumption model

5.1, 5.2, 6.2, 7.1

4.1 4.1, 5.1, 7.1 4.3 7.7 4.1 6.1 5.1, 6.2 7.7 4.1 6.3 4.1 5.2, 7.2, 7.7 4.1 6.3 7.7 6.3 4.1

Table 6 Electrical engineering. Author(s)

Application

Area(s)

Nazarko and Zalewski [233], Gladysz and Kuchta [235,263] Hong and Chao [410], Hong et al. [411] Soliman et al. [412] Yin et al. [413] Song et al. [414], Wi et al. [415], Hong and Wang [416] Chen et al. [417], Megri et al. [418] Azadeh et al. [299,300,396,419,420], Shakouri and Nadimi [322], Shakouri et al. [421], Rabbani et al. [422] Su et al. [326]

Electricity load Energy-loss modeling Evaluations in power networks Customer interruption cost evaluation Short-term load forecasting Modeling of thermal comfort Energy consumption

7.1, 7.2 7.5 4.1 4.1 4.1 4.1, 5.1, 6.2 4.1, 5.1, 7.1, 7.7

Monitoring parameter in power plant

7.7

robust FLR model discussed by D’Urso and Massari [136] could be improved by using fuzzy random variables. In the following, we would like to point out some assumptions that are frequently used but should be considered with more caution in our opinion.

• The most frequently used assumption of a triangular form with symmetric or asymmetric spreads for the fuzzy regression parameters leads to a series of limitations and there is no reasonable justification for this assumption (apart from the fact that triangular forms are easy to handle in theory and practice). • The fuzzy regression parameters are often assumed to be independent, also lacking a reasonable explanation. • Many approaches require expert knowledge for the derivation of membership functions. However, reliable expert opinion is not always available. • Fuzzy variables (with regard to fuzzy-valued input and/or output data) are rather seldom given in distributional form. Instead, sampling is needed, and the underlying sampling distribution has to be inferred. These problems are closely related and there is the need for methods to infer membership functions of fuzzy parameters, not

only in linear affine models. In this way, interdependency of fuzzy parameters could also be examined. Furthermore, one of the central aspects to be considered in fuzzy regression analysis is the sensitivity to outliers. Summarizing the literature concerning possibilistic models, there are mainly three approaches to detect outliers:

• No restriction in sign to increase the model fit via an unrestricted spread

• Adding auxiliary variables into the regression model with the purpose of relaxing the inclusion relationship between given and estimated data • Applying omission methods, where the number of LP problems is equal to the number of observations In our opinion, however, these approaches have shortcomings that should also be mentioned: A negative spread is in conflict with the extension principle (besides the fact that there is no compelling reason for a negative spread), adding auxiliary variables lacks theoretical foundation and increases computational effort, and omission methods clearly lead to higher computational complexity for larger samples. Instead of these approaches, we would like to recommend for future research robust methods based on least median squares or least trimmed squares (see for

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708

17

Table 7 Environmental research. Author(s)

Application

Area(s)

Bardossy et al. [423] Boreux et al. [424] Boreux et al. [425] Wen and Lee [392] Tseng et al. [67] Roychowdhury and Pedrycz [296] Mohammadi and Taheri [426], Chachi and Taheri [173], Rabiei et al. [156], Torkian et al. [157], Arefi and Taheri [158] Coppi et al. [126], Chuang [93], Wang et al. [350], Ferraro et al. [266], Shen et al. [134], D’Urso et al. [239], Ciavolino and Calcagni [347] Gladysz and Kuchta [235] Chang et al. [427] Gonzalez-Rodriguez et al. [264], Ferraro et al. [267,268], Ferraro [270] Shen et al. [134] Ramli et al. [220] Tutmez [291] D’Urso and Massari [136] Tutmez and Kaymak [325] An et al. [428] Ramedani et al. [429] Zuo et al. [223]

Earthquake prediction Radiocarbon dating of sediment layers Modeling of radial tree-growth Wastewater treatment systems Near-wall turbulent flows Wolfers sunspots data Pedomodels fitting in soil science

5.1 4.1 4.1 4.1 4.1 7.7 5.1, 5.2

Atmospheric concentration of carbon monoxide

5.1, 6.2, 7.1, 7.7

Air temperature variability Estimation of heat tolerance in plants Progress of reforestation

7.1 7.5 7.2

Average temperature and sunlight time Traffic-related air pollution Iris flower data set & lakes in Norway Daily variation of pollutant concentration Oasis Valley data Wind speed prediction Global solar radiation prediction Air quality dataset

5.1 6.4 7.7 5.1, 7.1 7.7 7.7 4.1 6.4

Table 8 Finance. Author(s)

Application

Area(s)

Roychowdhury and Pedrycz [296], Chuang [93], Huang [96], Bajestani et al. [430], Zuo et al. [223] Kocadagli [431]

Stock market indices

6.2, 6.4, 7.4, 7.7

Estimating CAPM beta of an asset Implied volatility smile function Technology credit scoring model Arbitrage pricing theory Chinese convertible bonds Istanbul stock exchange dataset

4.1

Muzzioli et al. [432] Sohn et al. [433] Wei and Watada [284] Alfonso et al. [337] Zuo et al. [223]

4.1, 6.3 7.3 7.4 7.7 6.4

the first attempts in these directions D’Urso et al. [239], D’Urso and Massari [136], and Chachi and Roozbeh [247]). These approaches are well-suited for neutralizing/smoothing disruptive effects of potential (fuzzy or crisp) outliers in the process of parameter estimation. Another fruitful direction to reduce the influence of outliers is the fuzzy regression approach based on genetic programming proposed by Chan et al. [188] to detect and exclude outliers. Moreover, the primary research focus regarding fuzzy regression analysis is on parametric regression models, particularly on linear models. However, the functional relationship between the output variable and input variables is often difficult to determine in practical applications. In cases, where the predefined parametric model structure hardly corresponds to the data generation process, a considerable estimation bias will be the consequence. To counter this problem, several fuzzy regression methods without a predefined parametric model structure have been developed (see Sections 6.2, 6.3, 7.7.2). Since non-parametric regression approaches make fewer assumptions on the underlying functional relationship, their flexibility in exploring hidden structures is considerably higher compared to parametric regression models. On the other hand, they are not able to incorporate available prior information and can lead to an invalid analysis when there is a large number of input variables (‘‘curse of dimensionality’’). Thus, there is the need for approaches that combine

the ‘‘best of both worlds’’, i.e. the simplicity of parametric (linear) models and the flexibility of non-parametric models. In this direction, the recent consideration of fuzzy semi-parametric partially linear models (see Hesamian et al. [178] and Hesamian and Akbari [179,224]) is up-and-coming in our opinion. However, up to now, these models have been proposed for CIFO or FIFO data and by focussing solely on triangular (interval-valued or intuitionistic) fuzzy numbers. For the above reasons regarding triangular fuzzy numbers, we recommend for future research the usage of other types of LR (interval-valued or intuitionistic) fuzzy numbers and also other types of imprecise data such as interval type-2 fuzzy sets within the framework of fuzzy semi-parametric partially linear models. As for the most popular and (even nowadays) widely-used approach in fuzzy regression analysis, i.e. Tanaka et al. [11] as well as several of its variants (e.g. Tanaka et al. [14], Tanaka and Ishibuchi [30], Ge and Wang [47], Guo and Tanaka [46], Hung and Yang [36]), they mainly lead to the same types of problems. In addition to the points often stated in the literature (see Section 4.1.1), there are other drawbacks that should be kept in mind: a change in the number of input variables leads to a reformulation of the entire set of constraints, and the nonnegativity constraints increase the number of unknown variables because their sign is restricted to be non-negative. Moreover, we would like to focus on three further types of problems by using this approach: 1. It is not ensured that the specified model has the least global fuzziness. 2. It is not guaranteed that all the observations are included in the predicted model output. 3. It is not possible to represent any tendency of the output spread. The first problem results from criteria that are solely based on observational data instead of an entire input domain of definition, the second problem is originated in the frequently assumed (symmetrical or asymmetrical) triangular form for the distribution of the fuzzy parameters, and the third problem arises from the fact that the output fuzziness varies in the same way as the absolute input values. Thus, we advocate more sophisticated fuzzy regression methods, especially recent approaches based on machine learning techniques.

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N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708 Table 9 Housing. Author(s)

Application

Area(s)

Tanaka et al. [11,40,41,87], Bardossy [52], Redden and Woodall [20], Lee and Tanaka [61,85], Tanaka and Lee [42,81], Alex [205,317], Kao and Lin [344], Yao and Yu [193], Donoso et al. [228], Hao and Chiang [194,195], Chen and Dang [152], Gladysz and Kuchta [263], Choi and Yoon [145], Mashinchi et al. [323] Hao [94], Roh et al. [211], Chen et al. [155], Zuo et al. [222,223] Azadeh et al. [324] Chung [434]

Price mechanism of prefabricated houses

4.1, 4.3, 5.1, 6.2, 6.3, 6.4, 7.2, 7.7

Boston housing dataset Housing market fluctuations Energy efficiency of commercial buildings House pricing problem Real estate re-auction data

5.1, 6.2, 6.3, 6.4 7.7 4.1

Wang et al. [104], Zhou et al. [435] Kim et al. [436]

4.3, 5.1 5.1

Table 10 Hydrology. Author(s)

Application

Area(s)

Bardossy et al. [437]

Hydraulic conductivity, electrical resistivity Water temperature Dose–response relationship Conceptual rainfall-runoff models Reservoir operations optimization problem Basin areas and drift distances Suspended load measurement

4.1

Hayashi and Tanaka [53] Bardossy et al. [438] Özelkan and Duckstein [439] Mousavi et al. [440] Yabuuchi and Watada [240] Chachi et al. [351,352], Rabiei et al. [156], Chachi and Roozbeh [247], Chachi [248], Hesamian and Akbari [179] Khan and Valeo [441] Amiri et al. [442]

4.1 4.1 4.2 4.1 7.1 5.1, 5.2, 7.1, 7.7

Dissolved oxygen prediction Estimating of reference evapotranspiration

5.1 4.1

Table 11 Information technology. Author(s)

Application

Area(s)

Heshmaty and Kandel [443], Celmins [17], Kandel and Heshmaty [444], Watada [292], Chang [294], Chen and Wang [445], Tseng and Tzeng [295] Hayashi and Tanaka [53] Chang et al. [446], D’Urso and Gastaldi [139], D’Urso [127] D’Urso and Gastaldi [140] D’Urso and Gastaldi [141], Wang et al. [350] Kahraman [447] Tsaur [448] Tsaur [298] Roh et al. [211] Yu and Tseng [449]

Computer sales forecasting

4.1, 5.1, 7.7

Amount of production of computers Video display terminal legibility Popular e-mail client Software reliability problem Sale levels of computer equipment Demand of internet users LCD TV demand CPU performance Growth trends of innovative TV products

4.1 4.1, 5.1 5.1 5.1, 7.7 4.1 4.1 7.7 6.3 7.3

In particular, we would like to accentuate the usage of random weight networks in the framework of fuzzy non-linear regression analysis for further directions of future research. In contrast to fuzzy non-linear regression models based on back-propagation or radial basis function networks, fuzzy non-linear regression using random weight networks does not need iterative adjustments of the input layer weights and hidden layer biases, and moreover, no learning parameter has to be determined. Such a learning algorithm provides a considerably higher training speed, a high generalization performance, and helps to avoid some difficulties that affect traditional gradient-based fuzzy non-linear regression models. Recently, the first promising works on this route have been presented by He et al. [212,213]. In addition, fuzzy regression transfer learning methods are also an auspicious direction for future research. In contrast to common data-driven machine learning techniques, transfer learning approaches exploit the knowledge accumulated from data in an auxiliary domain to improve predictive modeling in a related domain that is characterized by insufficient training data (see Zuo et al. [222]). There are only a few studies to date combining fuzzy regression and transfer learning methods to handle regression problems in domains where training data is inadequate, see Zuo

Table 12 Insurance. Author(s)

Application

Area(s)

De Andres-Sanchez and Gomez [450], De Andres-Sanchez [451,452,453,454], Apaydin and Baser [455] De Andres-Sanchez and Gomez [456,457] Koissi and Shapiro [458] Berry-Stölzle et al. [459]

Stochastic claims reserving

4.1, 4.2

Term structure of interest rates Mortality forecasting Insurer solvency surveillance Claims payments

4.1

Kula et al. [460]

5.1 4.1 7.1

et al. [222,223]. Fuzzy regression transfer learning methods also lead to the benefit that the distributions of training data and test data are not assumed to be the same, as usually in fuzzy regression analysis. Finally, we would like to point out some future directions for applications of fuzzy regression models. On the one hand, there are many fields, where fuzzy regression has been applied

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708

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Table 13 Manufacturing. Author(s)

Application

Area(s)

Ishibuchi and Tanaka [198], Dunyak and Wunsch [201], Aliev et al. [186] Lai and Chang [461] Diamond and Tanaka [108], Lee and Tanaka [63], Inuiguchi and Tanino [89], Wang et al. [350] Ip et al. [462,463], Kwong and Bai [464], Kahraman et al. [395], Kwong et al. [465], Chan et al. [188,466], Chan and Kwong [467] Tseng and Tzeng [295] Xue et al. [468], Sung et al. [469] Kwong et al. [261], Chan et al. [470], Chan et al. [370] Wang and Lin [471] Wang et al. [472] Ramli et al. [220] Shakouri and Nadimi [244] Zolfaghari et al. [473] Atalay et al. [474] Shakouri et al. [339] Gholizadeh et al. [475]

Quality evaluation of injection moldings

6.1, 6.3

Improve die casting quality Feed speed and surface roughness

4.1 4.1, 4.3, 5.1, 7.7

Modeling of microchip encapsulation

4.1, 6.1, 7.7

Production value of machinery industry Bead geometry in robotic welding process Modeling solder paste dispensing process Monitoring mechanism of production loss Optimizing sheet metal forming design Performance evaluation problem Determining cycle time in manufacturing Sensory evaluation of fried donut Time estimation for manufacturing systems SAFA Rolling and Pipe Mills Company Surface roughness for a turning process

7.7 4.1 7.2, 6.1 4.3 4.1 6.4 7.1 4.1 4.1 7.7 6.1

Table 14 Materials science. Author(s)

Application

Area(s)

Celmins [16] Celmins [17] Chen [303] Celmins [111] Xu and Li [117], D’Urso and Santoro [143] Petit-Renaud and Denoeux [328], Hao [94], Su et al. [326] Toyoura et al. [315] Pan et al. [476,477] Fattahi et al. [478], Chachi et al. [175] Karakasidis et al. [479] Tutmez and Kaymak [325] Gonzalez-Gonzalez et al. [480] Zuo et al. [222,223]

Plate perforation by projectiles Terminal ballistics problem Refractive index of benzene Vulnerability of armored vehicles Heat evolved in calories per gram of cement Impact of a motorcycle against an obstacle Experts damage assessment of a structure Bridge conditions and pavement performance Determining yarn properties Tensile strength of materials and hardness scales Lignite quality Reliability in a degradation process Concrete compressive strength dataset

5.1 5.1 7.7 5.1 5.1 6.2, 7.7 7.7 5.1 5.1, 5.2 4.1 7.7 7.7 6.4

Table 15 Medical science. Author(s)

Application

Area(s)

Savic and Pedrycz [26] McCauley-Bell and Wang [481], McCauley-Bell et al. [482] Cheng et al. [202] Hwang et al. [92] Choi and Yoon [145] Pourahmad et al. [273,275], Namdari et al. [276,277], Hesamian and Akbari [224], Salmani et al. [278] Roh et al. [211] Bajestani et al. [286,483]

Systolic blood pressure Cumulative trauma disorder risk Patients’ breast mass cytology information Ultrasonic calibration data Sustaining power of arthritis medicine Clinical studies

4.1 4.1 6.3 6.2 5.1 7.3

Medical imaging system Nephropathy and retinopathy in diabetic patients

6.3 7.4

Table 16 Nutrition. Author(s)

Application

Area(s)

D’Urso and Gastaldi [140] D’Urso [127], D’Urso et al. [290], Hassanpour et al. [78], Chachi and Taheri [159], Akbari and Hesamian [225] Tsutsumi and Yong-Sun [484] Nureize and Watada [485] Roldan et al. [486] Namdari et al. [277,487] Taheri et al. [488]

Wine quality trends Good-quality restaurant performance

5.1 4.2, 5.1, 6.4, 7.5

Studying eating-out behavior Oil palm fruit grading Bioproduct from olive tree pruning waste Nutrition for preschool children Dietary pattern and risk factors

4.1 4.1 7.7 7.3 7.3

to date, ranging from automotive industry to nutrition (see Tables 2–16). However, on the other hand, there are a lot of further fields, where applications are very rare or entirely missing, such as astronomy, behavioral science, neuroscience, physics, political science, psychology, and sociology. Beyond these additional fields for potential applications there are also numerous promising possibilities for future applications and case studies within the fields

presented in Section 8, for instance in business administration and economics, like:

• Development economics: analysis of socio-economic drivers for malnutrition of newborns in developing countries

• Hedonic pricing methods: consideration of temporal quality change of economic products in the determination of inflation

20

N. Chukhrova and A. Johannssen / Applied Soft Computing Journal 84 (2019) 105708 Table 17 Further areas of application. Author(s)

Application

Area(s)

Bardossy [52] Shimizu and Jindo [489] Kim and Bishu [77], Abdalla and Buckley [373], Mashinchi et al. [323], Kelkinnama and Taheri [171], Taheri and Kelkinnama [172], Chachi and Taheri [173], Jung et al. [333], Icen and Demirhan [377], Roldan Lopez de Hierro et al. [336], Icen and Cattaneo [378], Icen and Günay [379] Profillidis et al. [309] D’Urso and Santoro [289] D’Urso et al. [290] Ferraro and Giordani [242], Roldan Lopez de Hierro et al. [336], Ferraro [270]

Performance of female runners Modeling and evaluating human sensitivity Cognitive response times to an abnormal event

4.1 4.1 5.1, 5.2, 7.7

Airport of Rhodes Tone perception data Suitability of a given holiday period Students satisfaction of a course

7.7 7.5 7.5 7.2, 7.7

• Innovation research: investigation of the probability of ob-

• •

• • •

jection in the granting of patents by the European Patent Office Market research: quantifying the relationship between product sales and promotions Premium calculation: analysis of claims frequency and claims amount in motor vehicle insurance or analysis of the probability of death in life insurance for the calculation of insurance premiums Rent index: investigation of the dependency of the rent on type, location and condition of the flat Risk management: analysis of the creditworthiness of bank clients Solvency: comparison of the characteristics of financially strong and financially weak companies

10. Conclusions In this paper, we have provided a comprehensive systematic review and a bibliography on the topic of regression analysis in fuzzy environments. It can be stated that fuzzy modeling approaches in regression analysis have experienced tremendous growth and have become the most important field of research within the area of fuzzy statistics. Fuzzy regression analysis based on possibilistic approaches is most common (151 papers, 33.19%), followed by fuzzy least squares and fuzzy least absolutes methods (85 papers, 18.68%) as well as machine learning-based fuzzy regression analysis (64 papers, 14.07%). Thus, these three major fields comprise nearly two-thirds (65.94%) of the papers on fuzzy regression analysis. Additionally, there are numerous papers in the framework of robust (#17), probabilistic (#23), logistic (#11), type-2 (#9), clusterwise (#8), and time series analysisbased (#11) fuzzy regression methods. Further approaches include, for instance, the concept of fuzzy entropy, Monte Carlo methods or Bootstrapping techniques, among many others (#76). Our main intentions were to identify where tools of fuzzy statistics have been used for improving statistical regression analysis, to consolidate the topic in order to aid new researchers in this area, and to highlight possible directions for future research. Out of these objectives, we have structured the diversity in methodology of fuzzy regression analysis as well as its advancements, extensions, and modifications. Moreover, we have cataloged the variety of proposed practical applications and case studies based on fuzzy regression modeling. Ideally, this sound basis would assist researchers currently focused on regression analysis in fuzzy environments and lead to an expansion of areas requiring further research.

Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105708. Acknowledgments The authors would like to thank Bas van Vlijmen, Wojciech Froelich, and three anonymous reviewers for their valuable feedback and suggestions, which were very important and helpful to significantly improve the paper. References [1] R. Coppi, Management of uncertainty in statistical reasoning: The case of regression analysis, Internat. J. Approx. Reason. 47 (3) (2008) 284–305. [2] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Inform. Sci. 172 (1–2) (2005) 1–40. [3] L.A. Zadeh, Is there a need for fuzzy logic? Inform. Sci. 178 (13) (2008) 2751–2779. [4] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1988. [5] G.J. Klir, Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like? Fuzzy Sets and Systems 24 (2) (1987) 141–160. [6] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (3) (1965) 338–353. [7] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (3) (1978) 3–28. [8] B. Kitchenham, Procedures for Performing Systematic Reviews, Keele University, Keele, 2004. [9] O. Yazdanbakhsh, S. Dick, A systematic review of complex fuzzy sets and logic, Fuzzy Sets and Systems 338 (2018) 1–22. [10] D. Tranfield, D. Denyer, P. Smart, Towards a methodology for developing evidence informed management knowledge by means of systematic review, Br. J. Manag. 14 (3) (2003) 207–222. [11] H. Tanaka, S. Uejima, K. Asai, Linear regression analysis with Fuzzy model, IEEE Trans. Syst. Man Cybern. 12 (6) (1982) 903–907. [12] H. Tanaka, Fuzzy data analysis by possibilistic linear models, Fuzzy Sets and Systems 24 (3) (1987) 363–375. [13] H. Tanaka, J. Watada, Possibilistic linear systems and their application to the linear regression model, Fuzzy Sets and Systems 27 (3) (1988) 275–289. [14] H. Tanaka, I. Hayashi, J. Watada, Possibilistic linear regression analysis for Fuzzy data, European J. Oper. Res. 40 (3) (1989) 389–396. [15] H. Tanaka, H. Ishibuchi, Possibilistic regression analysis based on linear programming, in: J. Kacprzyk, M. Fedrizzi (Eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw, and Physica-Verlag, Heidelberg, 1992, pp. 47–60. [16] A. Celmins, Least squares model fitting to Fuzzy vector data, Fuzzy Sets and Systems 22 (3) (1987) 245–269. [17] A. Celmins, Multidimensional least-squares fitting of Fuzzy models, Math. Model. 9 (9) (1987) 669–690. [18] S. Jozsef, On the effect of linear data transformations in possibilistic Fuzzy linear regression, Fuzzy Sets and Systems 45 (2) (1992) 185–188. [19] M. Hojati, C.R. Bector, K. Smimou, A simple method for computation of Fuzzy linear regression, European J. Oper. Res. 166 (1) (2005) 172–184. [20] D.T. Redden, W.H. Woodall, Properties of certain Fuzzy linear regression methods, Fuzzy Sets and Systems 64 (3) (1994) 361–375.

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