Interval approach for encoding words into interval type-2 fuzzy sets based on normal distribution and free parameter

Constructing fuzzy set models from data is a key step in computing with words (CWW). The enhanced interval approach (EIA) is one of the most important approach for constructing interval type-2 fuzzy set (IT2 FS) from data intervals. This paper proposes an interval approach (NIA), which introduces a normal distribution associated with a free parameter (FP) and Gaussian footprint of uncertainty (FOU) as the supplement of uniform distribution and trapezoidal FOU. The NIA includes a data part, fuzzy set (FS) part and FOU part. The data part maps data intervals to probability distributions. The FS part encodes the distributions in the data part to fuzzy membership functions. Gaussian membership function (MF) is discussed and the parameter transformation is obtained. In the FOU part, the FOU of IT2 FS is built by collecting the obtained T1 FSs. The way to construct a Gaussian FOU is, for the first time, proposed in this paper. Owing to FP, the FOUs could be consist with data statistics. The effectiveness of NIA is validated by modeling a group of words, showing that NIA has greater flexibility and accuracy when approximating the distributions of data intervals.