## 3 Online 3D Fuzzy Modeling
### Spatial MSVR Learned 3D Fuzzy Model  
Spatial MSVR is a machine learning method that extends single-output support vector regression, aiming at predicting multiple correlated output variables at the same time. MSVR deals with the correlation and interdependence among individual output variables by constructing a regression model in spatial multi-output space. Compared with traditional SVR methods, MSVR demonstrates higher prediction accuracy and stability in multidimensional regression problems [29]. Assume that the entire spatial area of the complex distributed parameter system is \(\overline{Z}\), and collect data samples from the DPS as a dataset \(D=\left\{u(t),y(z_{i},t)\right\}_{i=1,t=1}^{S,L}\cdot z_{i}\) is the spatial position of the \(i\)-th sensor, \(S\) is the number of sensors, and \(L\) is the sampling time length. Here, \(u(t)\in R^{n}\) is the input variable, and \(y(z,t)=\left\{y(z_{1},t),y(z_{2},t),\ldots,y(z_{q},t)\right\}\in R^{q}\) is the output variable. \(n\) and \(q\) are the dimensions of the input and output variables, respectively. The 3D fuzzy rules in 3D-OMSVR-SGD are as follows:  
\[ \begin{split}\overline{R}_{l}:if\ y(z,t-1)\text{is}\ \overline{C}_{l1}\text{and}\ y(z,t-2)\text{is}\ \overline{C}_{2l}\text{and}\ \ldots\ \text{and}\ y(z,t-J)\text{is}\ \overline{C}_{Il}\\ \text{and}\ u_{1}(t-1)\text{is}\ U_{1}^{ll}\text{and}\ u_{1}(t-2 )\text{is}\ U_{2}^{2l}\text{and}\ \ldots\ \text{and}\ u_{1}(t-K)\text{is}\ U_{2}^{ Kl}\\ \text{and}\ u_{m}(t-1)\text{is}\ U_{m}^{ll}\text{and}\ u_{m}(t-2 )\text{is}\ U_{m}^{2l}\text{and}\ \ldots\ \text{and}\ u_{m}(t-K)\text{is}\ U_{m}^{ Kl}\\ \text{then}\ y(\overline{Z},t)\text{is}\ \varphi_{l}(\overline{Z}) \end{split}  \tag{4}\]  
where \(\overline{C}_{Il}\) and \(U_{m}^{kl}\) are denoted as the 3D fuzzy set and the traditional fuzzy set, respectively. \(\overline{R}_{l}\) represents the \(l\)-th 3D fuzzy rule (\(l=1,2,\ldots,NR\)). \(\varphi_{l}(\overline{Z})\) denotes a spatial basis function. The combination of MSVR and 3D fuzzy systems can further enhance the ability of DPS modeling, especially in dealing with spatial multi-output and nonlinear relationships. Combining these two approaches can fully exploit the interpretability of 3D fuzzy systems and the spatial multi-output regression capabilities of MSVR, providing an effective modeling framework for solving complex spatiotemporal distribution problems. The rules of 3D fuzzy systems have strong linguistic interpretability and can flexibly handle complex dynamic changes without relying on traditional dimensionality reduction techniques. Combined with the online learning capability of MSVR, the model parameters can be continuously adjusted to adapt to changes in the environment and process conditions in the distributed parameter system, thereby enhancing the real-time online modeling capability. The kernel function (KF) in support vector regression satisfies the [PERSON] theorem. If the spatial fuzzy basis functions (SFBFs) in the 3D fuzzy system satisfy the [PERSON] theorem, they can be used as the spatial kernel function (SKF) in MSVR [30]. Since (5) satisfies [PERSON]'s theorem, the SFBF can be used as the kernel function of the SVM [31; 32]. The SFBF of the 3D fuzzy model is as follows:\[\psi^{l}(t)=\sum_{j=1}^{P}a_{j}^{l}\prod_{i=1}^{J}\exp\left(-\left(\left(y\left(z_{ j},t-i\right)-c_{ij}^{l}\right)/\sigma_{ij}^{l}\right)^{2}\right)\times\prod_{i=1}^{m} \prod_{k=1}^{K}\exp\left(-\left(\left(u_{i}(t-k)-d_{i}^{kl}\right)/\delta_{i}^{ kl}\right)^{2}\right) \tag{5}\]  
where \(c_{ij}^{l}\) and \(\sigma_{ij}^{l}\) represent the center and width of the Gaussian 3D fuzzy set \(\overline{c}_{il}\), and \(d_{i}^{kl}\) and \(\delta_{i}^{kl}\) represent the center and width of the traditional Gaussian fuzzy set \(U_{i}^{kl}\). Specifically, the SFBF can be expressed as a linear combination of multiple traditional fuzzy basis functions.