The term "fuzzy set" is general and includes type-1 and type-2 fuzzy sets (and even higher-type fuzzy sets). All fuzzy sets are characterized by MFs. A type-1 fuzzy set is characterized by a two-dimensional MF, whereas a type-2 fuzzy set is characterized by a three-dimensional MF.
As an example, suppose that the variable of interest is eye contact, which we denote as x. Let's put eye contact on a scale of values 0–10. One of the terms that might characterize the amount of perceived eye contact (for example, during flirtation) is "some eye contact." Suppose that we surveyed 100 men and women and asked them to locate the ends of an interval for "some eye contact" on the scale of 0–10. Surely, we will not get the same results from all of them because words mean different things to different people.
One approach to using the 100 sets of two endpoints is to average the endpoint data and use the average values for the interval associated with "some eye contact." We could then construct a triangular (or other shape) MF whose base endpoints (on the x-axis) are at the two average values and whose apex is midway between the two endpoints. This type-1 triangle MF can be displayed in two dimensions and can be expressed mathematically as follows:
{(x, MF(x))| x an element of X}
Unfortunately, this MF has completely ignored the uncertainties associated with the two endpoints.
A second approach is to make use of the average values and the standard deviations for the two endpoints. By doing this, we are blurring the location of the two endpoints along the x-axis. Now locate triangles so that their base endpoints can be anywhere in the intervals along the x-axis associated with the blurred average endpoints. Doing this leads to a continuum of triangular MFs sitting on the x-axis—for example, picture a whole bunch of triangles all having the same apex point but different base points, as in Figure
1.
Figure 1 Triangular MFs when base endpoints (l and r) have uncertainty intervals associated with them.
For the purposes of this discussion, suppose that there are exactly N such triangles. Then at each value of x, there can be up to N MF values: MF1(x), MF2(x), …, MFN(x). Let's assign a weight to each of the possible MF values, say wx1, wx2, …, wxN (see Figure
1). We can think of these weights as the possibilities associated with each triangle at this value of x. The resulting type-2 MF can be expressed as follows:
{(x, {( MFi(x), wxi)| i = 1, …, N}| x an element of X}
Another way to write this is:
{(x, MF(x, w)| x an element of X and w an element of Jx}
MF(x, w) is a type-2 MF. It is three-dimensional because MF(x, w) depends on two variables, x and w.
Dear Adriana,
The fuzzy set of type 1 is a special case of the fuzzy set of type 2, the latter generalizes the fuzzy set of type 1. In addition, the fuzzy set of type 2 makes it possible to integrate the uncertainty about the functions membership and other options.
For more details please see links and attached files in topics.
Type-1 Fuzzy Logic | SpringerLink
https://link.springer.com/chapter/10.1007/978-3-540-32378-5_2
Comparative Study of type-1 and Type-2 Fuzzy Systems - International ...
ijergs.org.managewebsiteportal.com/files/.../COMPARATIVE23.p...
A type-1 fuzzy set is a special case of a type-2 fuzzy set; its secondary membership function is a subset with only one element—unity.
- Badges
- Science topic
- Similar topics
- Mathematics