1.2.3.1Geometric model of DFMLS [8, 11]

As shown in Fig. 1.3, we define the universe as a dynamic fuzzy sphere (large sphere), in which some small spheres represent the DFSs in the universe. Each dynamic fuzzy number is defined as a point in the DFS (small sphere). The membership degree of each dynamic fuzzy number is determined by the position and radius of the DFS (small sphere) in the domain (large sphere) and the position in the DFS (small sphere).

1.2.3.2Geometric model of DFML algorithm

In Fig. 1.4, the centre of the two balls is the expected value $\left({\overleftarrow{y}}_d,{\overleftarrow{y}}_d\right),$ the radius of the large sphere is $\left(\overleftarrow{\epsilon },\overrightarrow{\epsilon }\right),$ and the radius of the sphere is $\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)\left[\text{here},\left(\overrightarrow{\epsilon },\overleftarrow{\epsilon }\right)\right]$ and $\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)$ are the same as in Algorithm 1.2].

The geometry model can be described as follows:

(1)If the value $\left({\overleftarrow{y}}_k,{\overleftarrow{y}}_k\right)$ of the learning algorithm falls outside the ball, then this learning is invalid. Discard $\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)$ and feed the information back to the rules of the system library and learning part, then begin the process again;

(2)If the value $\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)$ of the learning algorithm falls between the big sphere and the small sphere, the information is fed back to the system rules and learning part of the library so that they can be amended. Proceed to the next step;

(3)If the value $\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)$ of the learning algorithm falls within the small sphere, it is considered that the precision requirement has been reached. Terminate the learning process and output $\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)$

1.3.1.1Problem statement

According to Algorithm 1.2, DFMLS adjusts and modifies the rules in the rule base according to the results of each learning process. The adjustment and modification of rules are mainly reflected in the adjustment of parameters in the rules. For this problem, this section derives a DFML algorithm that identifies the optimal system parameters.

Consider the DFMLS described in the previous section, which is formalized as $\left(\overleftarrow{y},\overrightarrow{y}\right)=f\left(\left(\overleftarrow{X},\overrightarrow{X}\right),\left(\overleftarrow{\theta },\overrightarrow{\theta }\right)\right),\text{Each}\text{element}\text{in}\left(\overleftarrow{\theta },\overrightarrow{\theta }\right)$ is

where ${\left(\overleftarrow{m},\overrightarrow{m}\right)}_{\left({\overleftarrow{A}}_1^l,{\overleftarrow{A}}_1^l\right)},{\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)}_{\left({\overleftarrow{A}}_1^l,{\overleftarrow{A}}_1^l\right)}\mathrm{and}{\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)}_{\left({\overleftarrow{b}}_1,{\overleftarrow{b}}_1\right)}$ are the mean and variance of the corresponding membership functions.

According to the given input and output data pairs $\left(\left({\overleftarrow{X}}_k,{\overrightarrow{X}}_k\right),\left({\overleftarrow{y}}_k,{\overrightarrow{y}}_k\right)\right)(k=$ 1, 2, . . ., N), the system parameters can be learnt using the least-squares error (LSE) objective function, which minimizes the output error of the system:

and modifies the parameters along the direction of steepest gradient descent:

where η is the training step length. Thus, the iterative optimization equation of parameter ${\left(\overrightarrow{m},\overleftarrow{m}\right)}_{\left({\overleftarrow{A}}_1^l,{\overleftarrow{A}}_1^l\right)},{\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)}_{\left({\overleftarrow{A}}_1^l,{\overleftarrow{A}}_1^l\right)}\mathrm{and}{\left(\overleftarrow{\delta },\overrightarrow{\delta }\right)}_{\left({\overleftarrow{b}}_i,{\overleftarrow{b}}_i\right)}$ can be obtained so as to minimize the sum of squares of the error in (1.11).

There are two issues worth considering:

(1)The choice of step size: If only a single fixed training step is used, it is difficult to take into account the convergence of different error variations, sometimes resulting in oscillations in the learning of the parameters, especially near the minimum point. Fixed training steps will reduce the convergence speed. In practical applications, there is often no universal, fixed training step for different parameter learning problems. Therefore, the following optimization steps are proposed to solve this problem [22]:

$\begin{array}{rl}& \mathrm{\eta <!--\; ?\; -->}(t)={\left[{\mathrm{\eta <!--\; ?\; -->}}_{{(\stackrel{\leftharpoonup <!--\; ?\; -->}{m},\stackrel{\rightharpoonup <!--\; ?\; -->}{m})}_{({\stackrel{\leftharpoonup <!--\; ?\; -->}{A}}_i^l,{\stackrel{\rightharpoonup <!--\; ?\; -->}{A}}_i^l)}}\left(t\right){,}^{\mathrm{\eta <!--\; ?\; -->}}{(\stackrel{\leftharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}},\stackrel{\rightharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}})}_{({\stackrel{\leftharpoonup <!--\; ?\; -->}{A}}_i^l,{\stackrel{\rightharpoonup <!--\; ?\; -->}{A}}_i^l)}(t){,}^{\mathrm{\eta <!--\; ?\; -->}}{(\stackrel{\leftharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}},\stackrel{\rightharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}})}_{({\stackrel{\leftharpoonup <!--\; ?\; -->}{b}}_i,{\stackrel{\rightharpoonup <!--\; ?\; -->}{b}}_i)}(t)\right]}^T\\ & {\mathrm{\eta <!--\; ?\; -->}}_{{(\stackrel{\leftharpoonup <!--\; ?\; -->}{m},\stackrel{\rightharpoonup <!--\; ?\; -->}{m})}_{({\stackrel{\leftharpoonup <!--\; ?\; -->}{A}}_i^l,{\stackrel{\rightharpoonup <!--\; ?\; -->}{A}}_i^l)}}\left(t\right)\approx \frac{1}{\sum _{i=1}^m\sum _{i=1}^n\left\{\frac{\mathrm{\partial }f\left({\stackrel{\leftharpoonup <!--\; ?\; -->}{\theta }}_i^l.{\stackrel{\rightharpoonup <!--\; ?\; -->}{\mathrm{\theta <!--\; ?\; -->}}}_i^l\right).\left({\stackrel{\leftharpoonup <!--\; ?\; -->}{X}}_l.{\stackrel{\rightharpoonup <!--\; ?\; -->}{X}}_l\right)}{\mathrm{\partial <!--\; ?\; -->}{\left(\stackrel{\leftharpoonup <!--\; ?\; -->}{m}.\stackrel{\rightharpoonup <!--\; ?\; -->}{m}\right)}_{{({\stackrel{\leftharpoonup <!--\; ?\; -->}{A}}_i^l,{\stackrel{\rightharpoonup <!--\; ?\; -->}{A}}_i^l)}^{\left(t\right)}}}\right\}}\\ & {\mathrm{\eta <!--\; ?\; -->}}_{{(\stackrel{\leftharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}}.\stackrel{\rightharpoonup <!--\; ?\; -->}{\mathrm{\delta <!--\; d\; -->}})}_{({\stackrel{\leftharpoonup <!--\; ?\; -->}{b}}_i,{\stackrel{\rightharpoonup <!--\; ?\; -->}{b}}_i)}}\left(t\right)\approx \frac{1}{\sum _{i=1}^m\sum _{i=1}^n\left\{\frac{\mathrm{\partial }f\left({\stackrel{\leftharpoonup }{\theta }}_i^l.{\stackrel{\rightharpoonup }{\theta }}_i^l\right).\left({\stackrel{\leftharpoonup }{X}}_l.{\stackrel{\rightharpoonup }{X}}_l\right)}{\mathrm{\partial }{\left(\stackrel{\leftharpoonup }{m}.\stackrel{\rightharpoonup }{m}\right)}_{{({\stackrel{\leftharpoonup }{b}}_i,{\stackrel{\rightharpoonup }{b}}_i)}^{\left(t\right)}}}\right\}}\\ & {\eta }_{{(\stackrel{\leftharpoonup }{\mathrm{\delta <!--\; d\; -->}}.\stackrel{\rightharpoonup }{\mathrm{\delta <!--\; d\; -->}})}_{({\stackrel{\leftharpoonup }{A}}_i^1,{\stackrel{\rightharpoonup }{A}}_l^i)}}\left(t\right)\approx \frac{1}{\sum _{i=1}^m\sum _{i=1}^n\left\{\frac{\mathrm{\partial }f\left({\stackrel{\leftharpoonup }{\theta }}_i^l.{\stackrel{\rightharpoonup }{\theta }}_i^l\right).\left({\stackrel{\leftharpoonup }{X}}_l.{\stackrel{\rightharpoonup }{X}}_l\right)}{\mathrm{\partial }{\left(\stackrel{\leftharpoonup }{\mathrm{\delta <!--\; d\; -->}}.\overrightarrow{\mathrm{\delta <!--\; d\; -->}}\right)}_{{({\stackrel{\leftharpoonup }{A}}_i^1,{\stackrel{\rightharpoonup }{A}}_l^i)}^{\left(t\right)}}}\right\}}\end{array}$