20.1 Regression Models and Fuzzy Data
As mentioned at the beginning of this book, all measurement results of continuous quantities are not precise numbers. Therefore regression models also have to take care of fuzzy data. A sample therefore is given by n vectors of fuzzy numbers, i.e.
Another kind of fuzzy data are given if the vector of independent variables is a fuzzy vector xi*. Then the sample takes the following form:
It is important to note that the fuzzy vector xi* is essentially different from the vector of fuzzy numbers (xi1*,…,xik*). The former is defined by a vector-characterizing function, whereas the latter is defined by a vector of characterizing functions.In applications both kinds of fuzzy data appear.Another kind of fuzziness in this context is the fuzziness of parameters in regression models.There are several situations possible if fuzziness is taken into account in regression models:
a. The parameters and the independent variables xi are assumed to be classical real valued, but the dependent variable is fuzzy, i.e. yi* are fuzzy numbers.
b. The independent variables xi as well as the values of the dependent variables yi are classical real numbers but the parameters are fuzzy numbers θj*, i.e.
Here ⊕ denotes the generalized addition operation for ...