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Non-convex fuzzy data and fuzzy statistics: a first descriptive approach to data analysis

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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.

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Notes

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

  2. 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.

  3. 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).

  4. The fuzzy sets were obtained from the histograms of the empirical responses by maximizing the entropy of the data (Avci and Avci 2009; Cheng and Chen 1997; Medasani et al. 1998; Nieradka and Butkiewicz 2007).

  5. We used the following abbreviations, \(\hbox {B} = \hbox {Belgium}, \hbox {C} = \hbox {Czech Republic}, \hbox {E} = \hbox {Estonia}, \hbox {D} = \hbox {Germany}, \hbox {G} = \hbox {Greece}, \hbox {H} = \hbox {Hungary}, \hbox {P} = \hbox {Portugal}, \hbox {K} = \hbox {Slovak Republic}, \hbox {S} = \hbox {Sweden}\).

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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
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  2. L. Lombardi
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  3. E. Pascali
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Correspondence to A. Calcagnì.

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Communicated by E. Viedma.

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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

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