Skip to main content
SpringerLink
Log in

A dimension reduction technique for two-mode non-convex fuzzy data

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Fuzzy modeling and fuzzy statistics provide useful tools for handling empirical situations affected by vagueness and imprecision in the data. Several fuzzy statistical models and methods (e.g., fuzzy regression, fuzzy principal component analysis, fuzzy clustering) have been developed over the years. Generally the standard LR-fuzzy data representation has been used in these methods. However, several empirical contexts, such as human ratings and decision making, may show more complex fuzzy structures which cannot be successfully modeled by the LR representation. In all these cases another type of fuzzy data representation, the so-called LHIR representation, should be preferred instead. In particular, this novel representation allows to handle with fuzzy data which are characterized by non-convex membership functions. In this paper, we address the problem of summarizing large datasets characterized by two-mode non-convex fuzzy data. We introduce a novel dimension reduction technique (NCFCA) based on the framework of Component Analysis and Least squares programming. Finally, to better highlight some important characteristics of the proposed model, we apply NCFCA to three empirical datasets concerning behavioral and socio-economic issues.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. Note, however, that the membership functions always contribute to the orientation of the axes in \(\mathbb {R}^p\) even if they are not directly illustrated in the graphical representation.

References

  • Abdi H, Williams LJ (2010) Principal component analysis. Wiley Interdiscip Rev Comput Stat 2(4):433–459

    Article  Google Scholar 

  • van den Berg RA, Hoefsloot HC, Westerhuis JA, Smilde AK, van der Werf MJ (2006) Centering, scaling, and transformations: improving the biological information content of metabolomics data. BMC Genomics 7(1):142

    Article  Google Scholar 

  • Bloch I (1999) On fuzzy distances and their use in image processing under imprecision. Pattern Recognit 32(11):1873–1895

    Article  Google Scholar 

  • Bro R, Smilde AK (2003) Centering and scaling in component analysis. J Chemom 17(1):16–33

    Article  Google Scholar 

  • Calcagnì A, Lombardi L (2014) Dynamic fuzzy rating tracker (DYFRAT): a novel methodology for modeling real-time dynamic cognitive processes in rating scales. Appl Soft Comput 24(1):948–961

  • Calcagnì A, Lombardi L, Pascali E (2014) Non-convex fuzzy data and fuzzy statistics: a first descriptive approach to data analysis. Soft Comput 18(8):1575–1588

  • Cazes P, Chouakria A, Diday E, Schektrman Y (1997) Entension de l’analyse en composantes principales à des données de type intervalle. Revue de Stat appliquée 45(3):5–24

    Google Scholar 

  • Ciavolino E, Salvatore S, Calcagnì A (2014) A fuzzy set theory based computational model to represent the quality of inter-rater agreement. Qual Quant 48(4):2225–2240

  • Coppi R, Giordani P, D’Urso P (2006) Component models for fuzzy data. Psychometrika 71(4):733–761

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Douzal-Chouakria A, Billard L, Diday E (2011) Principal component analysis for interval-valued observations. Stat Anal Data Min 4(2):229–246

    Article  MathSciNet  Google Scholar 

  • Dubois D, Prade HM, Farreny H, Martin-Clouaire R, Testemale C (1988) Possibility theory: an approach to computerized processing of uncertainty, vol 2. Plenum press, New York

  • Eriksson L (2006) Multi-and megavariate data analysis. MKS Umetrics AB

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

    Article  MATH  MathSciNet  Google Scholar 

  • Garibaldi JM, Musikasuwan S, Ozen T, John RI (2004) A case study to illustrate the use of non-convex membership functions for linguistic terms. In: Fuzzy systems, 2004. Proceedings 2004 IEEE International Conference on, IEEE, vol 3. pp 1403–1408

  • Giordani P (2010) Three-way analysis of imprecise data. J Multivar Anal 101(3):568–582

    Article  MATH  MathSciNet  Google Scholar 

  • Giordani P, Kiers HA (2004) Principal component analysis of symmetric fuzzy data. Comput Stat Data Anal 45(3):519–548

    Article  MATH  MathSciNet  Google Scholar 

  • Giordani P, Kiers HA (2007) Principal component analysis with boundary constraints. J Chemom 21(12):547–556

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Hanss M (2005) Applied fuzzy arithmetic. Springer

  • Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning theory. Springer

  • 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

    Article  MATH  Google Scholar 

  • Irpino A, Lauro C, Verde R (2003) Visualizing symbolic data by closed shapes. In: Between data science and applied data analysis, Springer, pp 244–251

  • Jahanshahloo GR, Lotfi FH, Izadikhah M (2006) Extension of the topsis method for decision-making problems with fuzzy data. Appl Math Comput 181(2):1544–1551

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Kaiser HF (1958) The varimax criterion for analytic rotation in factor analysis. Psychometrika 23(3):187–200

    Article  MATH  Google Scholar 

  • Kiers HA (1997) Techniques for rotating two or more loading matrices to optimal agreement and simple structure: a comparison and some technical details. Psychometrika 62(4):545–568

    Article  MATH  MathSciNet  Google Scholar 

  • Kiers HA (2002) Setting up alternating least squares and iterative majorization algorithms for solving various matrix optimization problems. Comput Stat Data Anal 41(1):157–170

  • Kiers HA (2004) Bootstrap confidence intervals for three-way methods. J Chemom 18(1):22–36

  • Kiers HA, ten Berge JM (1989) Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations. Psychometrika 54(3):467–473

  • 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 Quant 38(5):577–601

  • Lauro CN, Palumbo F (2000) Principal component analysis of interval data: a symbolic data analysis approach. Comput Stat 15(1):73–87

    Article  MATH  Google Scholar 

  • Le-Rademacher J, Billard L (2012) Symbolic covariance principal component analysis and visualization for interval-valued data. J Comput Graph Stat 21(2):413–432

    Article  MathSciNet  Google Scholar 

  • Magnuson JS (2005) Moving hand reveals dynamics of thought. Proc Natl Acad Sci USA 102(29):9995–9996

    Article  Google Scholar 

  • Meredith W, Millsap RE (1985) On component analyses. Psychometrika 50(4):495–507

    Article  MATH  MathSciNet  Google Scholar 

  • Millsap RE, Meredith W (1988) Component analysis in cross-sectional and longitudinal data. Psychometrika 53(1):123–134

    Article  MATH  Google Scholar 

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

  • Stöber J, Joormann J (2001) A short form of the worry domains questionnaire: construction and factorial validation. Personal Individ Differ 31(4):591–598

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Trefethen LN, Bau D (1997) Numerical linear algebra. Siam, p 50

  • Viertl R (2011) Statistical methods for fuzzy data. Wiley

  • Weber EU, Johnson EJ (2009) Mindful judgment and decision making. Ann Rev Psychol 60:53–85

    Article  Google Scholar 

  • Yang MS, Ko CH (1996) On a class of fuzzy c-numbers clustering procedures for fuzzy data. Fuzzy Sets Syst 84(1):49–60

    Article  MATH  MathSciNet  Google Scholar 

  • Zimmermann HJ (2001) Fuzzy set theory-and its applications. Springer

Download references

Author information

Authors and Affiliations

  1. Department of Psychology and Cognitive Science, University of Trento, 38068, Rovereto, TN, Italy

    A. Calcagnì & L. Lombardi

  2. Department of Mathematics and Physics ‘E. De Giorgi’, University of Salento, 73100, Lecce, Italy

    E. Pascali

Authors
  1. A. Calcagnì
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Lombardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. Pascali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Calcagnì.

Additional information

Communicated by V. Loia.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 110 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calcagnì, A., Lombardi, L. & Pascali, E. A dimension reduction technique for two-mode non-convex fuzzy data. Soft Comput 20, 749–762 (2016). https://doi.org/10.1007/s00500-014-1538-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-014-1538-8

Keywords

Search

Navigation