Dynamic Fuzzy Machine Learning

Assumptions: (1) Response variables are fully observed, with $\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)$$|\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\left(i=1,2,\mathrm{...},n\right)$ independent mutual conditions. The conditional density function is then $p\left(\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)|\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)\right)=a\left(\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right)|{\left(\overleftarrow{\sigma },\overrightarrow{\sigma }\right)}^2\right)$$\exp \left\{-\frac{1}{2{\left(\overleftarrow{\sigma },\overrightarrow{\sigma }\right)}^2}{d}_i\left({\overleftarrow{y}}_i,{\overleftarrow{y}}_i\right);\left({\overleftarrow{\mu }}_i,{\overrightarrow{\mu }}_t\right)\right\}\left(i=1,2,\mathrm{...},n\right);$

(2) The vector $\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right),$ which is composed of k explanatory variables, is a discrete random vector whose distribution density function is $p\left(\left({\overleftarrow{x}}_i,{\overleftarrow{x}}_i\right)|\left(\overleftarrow{\gamma },\overrightarrow{\gamma }\right)\right),\left(\overleftarrow{\gamma },\overrightarrow{\gamma }\right)$$=(\left({\overleftarrow{\gamma }}_1,{\overleftarrow{\gamma }}_1\right),\left({\overleftarrow{\gamma }}_2,{\overleftarrow{\gamma }}_2\right),\mathrm{...},({}_{}$