Dynamic Fuzzy Machine Learning

Because the max function is non-continuous and differentiable, the partial derivative of the above equation is problematic. Here, we introduce an approximation function to overcome this problem.

We define a generalized p mean for the data ${\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=[{\left({\stackrel{\leftharpoonup }{a}}_1,{\stackrel{\rightharpoonup }{a}}_1\right)}^{\prime }{\left({\stackrel{\leftharpoonup }{a}}_2,{\stackrel{\rightharpoonup }{a}}_2\right)}^{\prime },$$\dots ,{\left({\stackrel{\leftharpoonup }{a}}_N,{\stackrel{\rightharpoonup }{a}}_N\right)}^{\prime }]:$

$M_p{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }$ is a good approximation function [53] for $\underset{i}{\max }{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime }\text{and}\underset{i}{\min }{\left({\stackrel{\leftharpoonup }{a}}_i,{\stackrel{\rightharpoonup }{a}}_i\right)}^{\prime },$ and has the following properties.

Lemma 1.2 [53]

(1)$M_0{\left(\stackrel{\leftharpoonup }{a},\stackrel{\rightharpoonup }{a}\right)}^{\prime }=\underset{p\to 0}{\lim }{M}_p{(\stackrel{\leftharpoonup }{a}}^{}$