How are type-2 fuzzy sets visualized?
We just indicated that type-2 fuzzy sets are three-dimensional, so they can be visualized as three-dimensional plots. Unfortunately, it is not as easy to sketch such plots as it is to sketch the two-dimensional plots of a type-1 MF. Another way to visualize type-2 fuzzy sets is to plot their so-called footprint of uncertainty (FOU). The type-2 MF, MF(x, w), sits atop a two-dimensional x-w plane. It sits only on the permissible (sometimes called “admissible”) values of x and w. This means that x is defined over a range of values (its domain) say, X. In addition, w is defined over its range of values (its domain) say, W. An example of an FOU is shown in Figure 2. It is an FOU for a Gaussian MF whose standard deviation is known with perfect certainty but whose mean, m, is uncertain and varies anywhere in the interval from m1 to m2, which can be expressed as m_[m1, m2]. The uniform shading over the entire FOU means that, for this example, we are assuming uniform weighting (possibilities).
