Dynamic Fuzzy Machine Learning

8Appendix

8.1Dynamic fuzzy sets

8.1.1Definition of dynamic fuzzy sets

Definition 8.1 A map defined in the domain of discourse $U.(\overleftarrow{A,}\overrightarrow{A}):(\overleftarrow{U,}\overrightarrow{U})\to [0,1]\times$$[\leftarrow ,\to ],(\overleftarrow{u},\overrightarrow{u})\mapsto (\overleftarrow{A},(\overrightarrow{u}),\overleftarrow{A},(\overrightarrow{u})),$, marked as $(\overleftarrow{A},\overrightarrow{A})=\overleftarrow{A}\text{or}\overrightarrow{A}.$ Then we call $(\overleftarrow{A},\overrightarrow{A})$ as the dynamic fuzzy sets in $(\overleftarrow{U},\overrightarrow{U}),$, shorten for DFS. We call $(\overleftarrow{A},(\overrightarrow{u}),\overleftarrow{A},(\overrightarrow{u}))$ as the membership function of membership degree $(\overleftarrow{A},\overrightarrow{A}).$

*Explanation: For any a ∈ [0, 1], we can make it dynamic fuzzilization as $a\stackrel{DF}{=}(\overleftarrow{a},\overrightarrow{a}),$$\begin{array}{l}a\stackrel{DF}{=}\overleftarrow{a}\text{or}\overrightarrow{a},\max (\overleftarrow{a},\overrightarrow{a})\stackrel{\Delta }{=}\overrightarrow{a},\min (\overleftarrow{a},\overrightarrow{a})\stackrel{\Delta }{=}\overleftarrow{a}.\\ \end{array}$ So that we can visually present variation trends of number