## 3 Online 3D Fuzzy Modeling
### Spatial MSVR Learned 3D Fuzzy Model
Each fuzzy basis function satisfies [PERSON]'s theorem, so their weighted sum also satisfies symmetry and positive definiteness, thus ensuring the effectiveness of the SFBF as a kernel function [31]. Specifically, the SFBF can be rewritten as shown below:  
\[\psi^{l}(t)=\sum_{j=1}^{P}a_{j}^{l}\phi^{l}(t) \tag{6}\]  
\[\phi^{l}(t)=\prod_{i=1}^{J}\exp\left(-\left(\left(y\left(z_{j},t-i\right)-c_{ ij}^{l}\right)/\sigma_{ij}^{l}\right)^{2}\right)\times\prod_{i=1}^{m}\prod_{k=1}^{ K}\exp\left(-\left(\left(u_{i}(t-k)-d_{i}^{kl}\right)/\delta_{i}^{kl}\right)^{2}\right) \tag{7}\]  
where \(\phi^{l}(t)\) is the traditional fuzzy basis function, and each \(\phi^{l}(t)\) itself satisfies [PERSON]'s theorem, so they can be used as kernel functions. Since the linear combination of kernel functions is still a kernel function, the 3D spatial fuzzy basis function, as a linear combination of traditional fuzzy basis functions, also has the properties of kernel functions. Therefore, the 3D SFBF can be used as the SKF in MSVR for regression modeling in high-dimensional feature space. Spatial fuzzy basis functions as kernel functions in MSVR can effectively model and predict a nonlinear DPS. According to the basic principle of SVR, the input data are mapped to a high-dimensional feature space, and a linear model is constructed in the feature space so that the model can find the best hyperplane to fit the data in this space. The formula for the dual optimization problem is as follows:  
\[\min_{w,b}\frac{1}{2}\sum_{i=1}^{q}||\omega_{i}||^{2}+c\sum_{i=1}^{q}\sum_{k=1 }^{N}\ell_{\varepsilon}\left(f_{i}\left(x^{k}\right)-\hat{g}_{i}^{k}\right) \tag{8}\]  
where \(N\) is the number of samples. \(q\) is the dimension of the output variable. \(\ell_{\varepsilon}\) is the insensitive loss function. \[\ell_{\varepsilon}\left(f_{i}\left(x^{k}\right)-\hat{y}_{i}^{k}\right)=\left| f_{i}\left(x^{k}\right)-\left(w\left(x^{k}\right)^{T}\Phi\left(x^{k}\right)+b \right)\right|-\varepsilon_{i} \tag{9}\]  
Then, the multivariable support vector regression problem is transformed into the following optimization problem by introducing the relaxed variable \(\xi_{i}^{k},\xi_{i}^{k*}\):  
\[\min_{w,b,\xi_{i},\xi_{i}^{k*}}\frac{1}{2}\sum_{i=1}^{q}||\omega_{i}||^{2}+C \sum_{i=1}^{q}\sum_{k=1}^{N}\left(\xi_{i}^{k}+\xi_{i}^{k*}\right) \tag{10}\]  
\[s.t.\quad \begin{cases}\hat{y}_{i}^{k}-\left\langle\omega_{i},x^{k}\right\rangle -b_{i}\leq\varepsilon_{i}+\xi_{i}^{k}\\ \left\langle\omega_{i},x^{k}\right\rangle+b_{i}-\hat{g}_{i}^{k}\leq\varepsilon _{i}+\xi_{i}^{k*}\\ \xi_{i}^{k}\geq 0,\xi_{i}^{k*}\geq 0,i=1,\ldots q\end{cases}  \tag{11}\]The Lagrange function is introduced as:  
\[ \begin{array}{rl}L=\frac{1}{2}\sum\limits_{i=1}^{q}||\omega_{i}||^{2}+&C\sum \limits_{i=1}^{q}\sum\limits_{k=1}^{N}\left(\xi_{i}^{k}+\hat{\sigma}_{i}^{k*} \right)+\sum\limits_{i=1}^{q}\sum\limits_{k=1}^{N}\alpha_{i}^{k}\left(\hat{ \sigma}_{i}^{k}-\left<\omega_{i},x^{k}\right>-b_{i}-\varepsilon_{i}-\bar{\xi}_ {i}^{k}\right)\\ &+\sum\limits_{i=1}^{q}\sum\limits_{k=1}^{N}\hat{\alpha}_{i}^{k}\left(\left< \omega_{i},x^{k}\right>+b_{i}-\hat{\sigma}_{i}^{k}-\varepsilon_{i}-\hat{\sigma }_{i}^{k*}\right)-\sum\limits_{i=1}^{q}\sum\limits_{k=1}^{N}\mu_{i}^{k}\xi_{i} ^{k}\\ &-\sum\limits_{i=1}^{q}\sum\limits_{k=1}^{N}\hat{\mu}_{i}^{k}\xi_{i}^{k*} \end{array}  \tag{12}\]  
The conditions for optimality are given by:  
\[\left\{ \begin{array}{rl}\frac{\partial L}{\partial\omega_{i}}=\omega_{i}- \sum\limits_{k=1}^{N}\alpha_{i}^{k}x^{k}+\sum\limits_{k=1}^{N}\hat{\alpha}_{i} ^{k}x^{k}=0\rightarrow\omega_{i}=\sum\limits_{k=1}^{N}\left(\alpha_{i}^{k}- \hat{\alpha}_{i}^{k}\right)x^{k}\\ \frac{\partial L}{\partial b_{i}}=-\sum\limits_{k=1}^{N}\alpha_{i}^{k}+\sum \limits_{k=1}^{N}\hat{\alpha}_{i}^{k}=0\rightarrow\sum\limits_{k=1}^{N}\left( \alpha_{i}^{k}-\hat{\alpha}_{i}^{k}\right)=0\\ \frac{\partial L}{\partial\bar{\xi}_{i}^{k}}=C-\alpha_{i}^{k}-\mu_{i}^{k}=0\\ \frac{\partial L}{\partial\bar{\alpha}_{i}^{k}}=C-\hat{\alpha}_{i}^{k}-\hat{ \mu}_{i}^{k}=0\\ i=1,2,\ldots,q,\ k=1,2,\ldots,N\end{array} \right.