Dynamic Fuzzy Machine Learning

$\{[({\stackrel{\leftharpoonup }{v}}_1,{\stackrel{\rightharpoonup }{v}}_1),({\stackrel{\leftharpoonup }{v}}_2,{\stackrel{\rightharpoonup }{v}}_2)),[({\stackrel{\leftharpoonup }{v}}_2,{\stackrel{\rightharpoonup }{v}}_2),({\stackrel{\leftharpoonup }{v}}_3,{\stackrel{\rightharpoonup }{v}}_3)),\dots ,[({\stackrel{\leftharpoonup }{v}}_{k-1},{\stackrel{\rightharpoonup }{v}}_{k-1}),({\stackrel{\leftharpoonup }{v}}_k,{\stackrel{\rightharpoonup }{v}}_k))\}.$

Each interval is replaced with a dynamic fuzzy number $\left(\stackrel{\leftharpoonup }{\alpha },\stackrel{\rightharpoonup }{\alpha }\right),$ thus completing the discretization of $\left({\stackrel{\leftharpoonup }{A}}_i,{\stackrel{\rightharpoonup }{A}}_i\right).$ A division of the range of $\left({\stackrel{\leftharpoonup }{A}}_i,{\stackrel{\rightharpoonup }{A}}_i\right)$ is called a discretization scheme. If two discretization schemes have the same division, then the two discretization schemes are equivalent.

As $\left|\left(\stackrel{\leftharpoonup }{U},\stackrel{\rightharpoonup }{U}\right)\right|\triangleq n,$ we have $\max \left(\left|{\overleftarrow{V}}_{{\overleftarrow{A}}_1},{\overrightarrow{V}}_{{\overrightarrow{A}}_1}\right|\right)=n.$ This is because there can be multiple separation points for the same value interval, and the resulting discretization schemes are equivalent. ...