α-plane representation for type-2 fuzzy sets: theory and applications

  • Authors:

  • Jerry M. Mendel;Feilong Liu;Daoyuan Zhai

  • Affiliations:

  • Signal and Image Processing Institute, Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA;Chevron Corporation, Richmond, CA and University of Southern California, Los Angeles, CA;Signal and Image Processing Institute, Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA

  • Venue:
  • IEEE Transactions on Fuzzy Systems
  • Year:
  • 2009

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Abstract

This paper 1) reviews the α-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the α-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple α-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using α-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each α-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 α-planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.