## Abstract

LR-fuzzy numbers are widely used in Fuzzy Set Theory applications based on the standard definition of convex fuzzy sets. However, in some empirical contexts such as, for example, human decision making and ratings, convex representations might not be capable to capture more complex structures in the data. Moreover, non-convexity seems to arise as a natural property in many applications based on fuzzy systems (e.g., fuzzy scales of measurement). In these contexts, the usage of standard fuzzy statistical techniques could be questionable. A possible way out consists in adopting ad-hoc data manipulation procedures to transform non-convex data into standard convex representations. However, these procedures can artificially mask relevant information carried out by the non-convexity property. To overcome this problem, in this article we introduce a novel computational definition of *non-convex fuzzy number* which extends the traditional definition of LR-fuzzy number. Moreover, we also present a new *fuzzy regression model for crisp input/non-convex fuzzy output data* based on the fuzzy least squares approach. In order to better highlight some important characteristics of the model, we applied the fuzzy regression model to some datasets characterized by convex as well as non-convex features. Finally, some critical points are outlined in the final section of the article together with suggestions about future extensions of this work.

### Similar content being viewed by others

## Notes

Note that: \(l=(m-lb)\) and \(r=(ub-m)\) where \(ub\) and \(lb\) mean the minimum and maximum of the support, respectively.

The name

*2-mode fuzzy number*is based on the intuition that fuzzy numbers can be represented by means of the convexity/non-convexity condition. Thus, LR-fuzzy numbers can be named*1-mode fuzzy number*because their \(\alpha \)-sets are compact and convex sets, whereas*k-modes fuzzy numbers*are fuzzy numbers which \(\alpha \)-sets are the result of the union of, at maximum, \(k\) disjoint components. It is clear that when \(k>1,\) the fuzzy numbers are non-convex fuzzy sets.In the following section we adopt the term

*estimation*to indicate the*interpolation procedure*without assuming any inferential meaning (that is to say this approach is based on a descriptive non-inferential rationale).We used the following abbreviations, \(\hbox {B} = \hbox {Belgium}, \hbox {C} = \hbox {Czech Republic}, \hbox {E} = \hbox {Estonia}, \hbox {D} = \hbox {Germany}, \hbox {G} = \hbox {Greece}, \hbox {H} = \hbox {Hungary}, \hbox {P} = \hbox {Portugal}, \hbox {K} = \hbox {Slovak Republic}, \hbox {S} = \hbox {Sweden}\).

## References

Avci E, Avci D (2009) An expert system based on fuzzy entropy for automatic threshold selection in image processing. Exp Syst Appl 36(2):3077–3085

Benítez JM, Martín JC, Román C (2007) Using fuzzy number for measuring quality of service in the hotel industry. Tour Manag 28(2):544–555

Bisserier A, Boukezzoula R, Galichet S (2010) A revisited approach to linear fuzzy regression using trapezoidal fuzzy intervals. Inf Sci 180(19):3653–3673

Biswas R (1995) An application of fuzzy sets in students’ evaluation. Fuzzy Sets Syst 74(2):187–194

Buckley JJ (2004) Fuzzy statistics, vol 149. Springer, Berlin

Celmiņš A (1987) Least squares model fitting to fuzzy vector data. Fuzzy Sets Syst 22(3):245–269

Chan L, Kao H, Wu M (1999) Rating the importance of customer needs in quality function deployment by fuzzy and entropy methods. Int J Prod Res 37(11):2499–2518

Chang YH, Yeh CH (2002) A survey analysis of service quality for domestic airlines. Eur J Oper Res 139(1):166–177

Cheng H, Chen J (1997) Automatically determine the membership function based on the maximum entropy principle. Inf Sci 96(3):163–182

Ciavolino E, Dahlgaard J (2009) Simultaneous equation model based on the generalized maximum entropy for studying the effect of management factors on enterprise performance. J Appl Stat 36(7):801–815

Colubi A, Santos Domınguez-Menchero J (2001) On the formalization of fuzzy random variables. Inf Sci 133(1):3–6

Coppi R, Durso P, Giordani P, Santoro A (2006) Least squares estimation of a linear regression model with LR fuzzy response. Comput Stat Data Anal 51(1):267–286

Coppi R, Gil MA, Kiers HA (2006) The fuzzy approach to statistical analysis. Comput Stat Data Anal 51(1):1–14

Diamond P (1988) Fuzzy least squares. Inf Sci 46(3):141–157

Dubois D, Prade H (2000) Fundamentals of fuzzy sets, vol 7. Springer, Berlin

Dubois D, Prade H, Harding E (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York

D’Urso P (2003) Linear regression analysis for fuzzy/crisp input and fuzzy/crisp output data. Comput Stat Data Anal 42(1–2):47–72

D’Urso P, Gastaldi T (2000) A least-squares approach to fuzzy linear regression analysis. Comput Stat Data Anal 34(4):427–440

Facchinetti G, Pacchiarotti N (2006) Evaluations of fuzzy quantities. Fuzzy Sets Syst 157(7):892–903

Freeman JB, Ambady N (2010) Mousetracker: software for studying real-time mental processing using a computer mouse-tracking method. Behav Res Methods 42(1):226–241

Garibaldi J, John R (2003) Choosing membership functions of linguistic terms. In: The 12th IEEE international conference on fuzzy systems, 2003. FUZZ’03, vol 1. IEEE, pp 578–583

Garibaldi J, Musikasuwan S, Ozen T, John R (2004) A case study to illustrate the use of non-convex membership functions for linguistic terms. In: 2004 IEEE international conference on fuzzy systems, 2004. Proceedings, vol 3. IEEE, pp 1403–1408

Gil MÁ, González-Rodríguez G (2012) Fuzzy vs. Likert scale in statistics. In: Combining experimentation and theory. Springer, Berlin, pp 407–420

Gil MÁ, López-Díaz M, Ralescu DA (2006) Overview on the development of fuzzy random variables. Fuzzy Sets Syst 157(19):2546–2557

Gill PE, Murray W, Wright MH (1981) Practical, optimization. Academic Press, New York

Golan A, Judge G (1996) Maximum entropy econometrics: robust estimation with limited data. Wiley, New York

González-Rodríguez G, Colubi A (2006) A fuzzy representation of random variables: an operational tool in exploratory analysis and hypothesis testing. Comput Stat Data Anal 51(1):163–176

Greene J, Haidt J (2002) How (and where) does moral judgment work? Trends Cognit Sci 6(12):517–523

Haidt J (2001) The emotional dog and its rational tail: a social intuitionist approach to moral judgment. Psychol Rev 108(4):814

Hanss M (2005) Applied fuzzy arithmetic. Springer, Berlin

Hesketh B et al (1989) Fuzzy logic: toward measuring Gottfredson’s concept of occupational social space. J Counsel Psychol 36(1):103–109

Hesketh T, Pryor R, Hesketh B (1988) An application of a computerized fuzzy graphic rating scale to the psychological measurement of individual differences. Int J Man Mach Stud 29(1):21–35

Johnson A, Mulder B, Sijbinga A, Hulsebos L (2012) Action as a window to perception: measuring attention with mouse movements. Appl Cognit Psychol 26(5):802–809

Kacprzyk J, Fedrizzi M (1992) Fuzzy regression, analysis, vol 1. Physica-Verlag, Heidelberg

Lalla M, Facchinetti G, Mastroleo G (2005) Ordinal scales and fuzzy set systems to measure agreement: An application to the evaluation of teaching activity. Qual Quan 38(5):577–601

Lee S, Kim S, Jang N (2008) Design of fuzzy entropy for non convex membership function. In: Advanced intelligent computing theories and applications with aspects of contemporary intelligent computing, techniques, pp 55–60

Lima Neto E, De Carvalho F (2010) Constrained linear regression models for symbolic interval-valued variables. Comput Stat Data Anal 54(2):333–347

McGuire J, Langdon R, Coltheart M, Mackenzie C (2009) A reanalysis of the personal/impersonal distinction in moral psychology research. J Exp Soc Psychol 45(3):577–580

Medasani S, Kim J, Krishnapuram R (1998) An overview of membership function generation techniques for pattern recognition. Int J Approx Reason 19(3):391–417

Nguyen HT, Wu B (2006) Fundamentals of statistics with fuzzy data. Springer, Berlin

Nichols S, Mallon R (2006) Moral dilemmas and moral rules. Cognition 100(3):530–542

Nieradka G, Butkiewicz B (2007) A method for automatic membership function estimation based on fuzzy measures. In: Foundations of fuzzy logic and, soft computing, pp 451–460

OECD (2011) Employment rate. http://www.oecd.org/statistics/

OECD (2013) Unemployment rates. http://www.oecd.org/statistics/

Pashler H, Wixted J (2002) Stevens’ handbook of experimental psychology, methodology. In: Experimental psychology, vol 4. Wiley, New York

Rai TS, Holyoak KJ (2010) Moral principles or consumer preferences? Alternative framings of the trolley problem. Cognit Sci 34(2):311–321

Reuter U (2008) Application of non-convex fuzzy variables to fuzzy structural analysis. In: Soft methods for handling variability and imprecision, pp 369–375

Ross T (2009) Fuzzy logic with engineering applications. Wiley, New York

Taheri SM (2003) Trends in fuzzy statistics. Aust J Stat 32(3):239–257

Trevino LK (1986) Ethical decision making in organizations: a person-situation interactionist model. Acad Manag Rev 11(3):601–617

Verkuilen J, Smithson M (2006) Fuzzy set theory: applications in the social sciences, vol 147. Sage Publications, Incorporated

Viertl R (1996) Real data and their mathematical description for parameter estimation. Math Model Syst 2(4):262–298

Viertl R (2006) Univariate statistical analysis with fuzzy data. Comput Stat Data Anal 51(1):133–147

Viertl R (2011) Statistical methods for fuzzy data. Wiley, New York

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by E. Viedma.

## Electronic supplementary material

Below is the link to the electronic supplementary material.

## Rights and permissions

## About this article

### Cite this article

Calcagnì, A., Lombardi, L. & Pascali, E. Non-convex fuzzy data and fuzzy statistics: a first descriptive approach to data analysis.
*Soft Comput* **18**, 1575–1588 (2014). https://doi.org/10.1007/s00500-013-1164-x

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00500-013-1164-x