Dynamic Fuzzy Machine Learning

(2)When the input and output data contain measurement noise, the learning parameter cannot converge to the true value ${\left(\overleftarrow{\theta },\overrightarrow{\theta }\right)}^{*}$ using the LSE objective function.

When the input and output data contain noise, the input and output data pairs can be expressed as:

where ${\left({\overleftarrow{X}}_k,{\overleftarrow{X}}_k\right)}^{*},{\left({\overleftarrow{Y}}_k,{\overrightarrow{Y}}_k\right)}^{*}$ are the true values of the input and output data, respectively, $n_{\left({\overleftarrow{X}}_k,{\overrightarrow{X}}_k\right)},n_{\left({\overleftarrow{Y}}_k,{\overleftarrow{Y}}_k\right)}$ are the input and output data contained in the noise, respectively, and the variance is $\left\{{\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)}_{\left({\overleftarrow{x}}_k^1,{\overleftarrow{x}}_k^1\right)},i=1,2,\mathrm{...},n\right\},$