## 3 SPSA Learning-Based Three-Dimensional Fuzzy Modeling
### AP Clustering Learning for Preceding Components of Three-Dimensional Fuzzy Rule  
[PERSON] and [PERSON] introduced Affinity Propagation (AP) clustering [23], a novel clustering technique, in 2007. It can handle with large-scale data sets in a short period of time without initializations of the cluster number and cluster core. The process of AP clustering involves initially selecting specific samples known as exemplars, followed by linking each sample on the left to its closest examples. Two kinds of messages, namely, responsibility and availability, are passed in AP clustering. Let \(V=[r(i,\
u)]_{n\times n}\) be the responsibility matrix, \(A=[a(i,\
u)]_{n\times n}\) be the availability matrix, and \(S=[s(i,\
u)]_{n\times n}\) be the similarity matrix. The responsibility \(r(i,\
u)\) reflects how strongly the sample \(i\) wants to choose the candidate exemplar \(\
u\) as its exemplar. The presence of \(a(i,\
u)\) mirrors the collective data, demonstrating the appropriateness of sample \(i\) in selecting candidate \(\
u\) as the exemplar. The similarity \(s(i,\
u)\) represents the degree of similarity between the sample \(i\) and its nearest exemplar \(\
u\). Then, the message between \(V\) and \(A\) is propagated so that the sample similarity \(S\) in the cluster is the largest, and the sample similarity between the  
Figure 2: Scheme of SPSA learning-based three-dimensional fuzzy modeling.  
clusters is the smallest. The detailed algorithm description is presented in Appendix A, and the algorithm procedure is shown in Figure A3 of Appendix A.  
For a spatiotemporal data set \(\bar{S}\) from DPS, the AP clustering algorithm is employed to cluster the data set \(\bar{S}\) to \(Z^{\prime}\) clusters. Each cluster corresponds to the preceding components of a three-dimensional fuzzy rule. Hence, the preceding components of \(Z^{\prime}\) three-dimensional fuzzy rules are achieved. Assume that \(o^{1},\cdots,o^{Z^{\prime}}\) with \(o^{i}=\left(o^{i}_{X^{\prime}},o^{i}_{c}\right)\) are \(Z^{\prime}\) cluster cores. These cores will guide the construction of the preceding components of \(Z^{\prime}\) three-dimensional fuzzy rules. Utilizing a Gaussian-type membership function, the cluster's core \(o^{i}\) aligns with the center of the Gaussian-type MFs in the preceding components, while \(o^{i}_{X}\) and \(o^{i}_{c}\) represent the centers of three-dimensional MFs and conventional MFs, respectively.  
The breadth of the Gaussian-type membership function can be computed according to the interval range of the input variables from the input data set. Regarding the spatial output variable \(w(\bar{F},k)\), the breadth of the Gaussian-type three-dimensional MF at an identical spatial point is calculated as in (7):  
\[\begin{split}\sigma_{w}\big{(}f_{j}\big{)}\approx\max\left(\frac{ w^{\max}(f_{j})-w^{\min}(f_{j})}{\eta}\right),\\ \eta\in[0.01,2].\end{split} \tag{7}\]  
where \(w^{\max}\big{(}f_{j}\big{)}\) and \(w^{\min}\big{(}f_{j}\big{)}\) represent the highest and lowest \(w(\bar{F},k)\) values at the sensor's location \(f=f_{j}\); \(\eta\) is a variable set by the users.  
Regarding the variable \(Q(k)\), the breadth of the conventional Gaussian MF is outlined below:  
\[\sigma_{Q}=\big{[}\ \sigma_{\mu_{1}}\ \ \ \cdots\ \ \ \sigma_{\mu_{L}}\ \ \big{]}. \tag{8}\]  
where \(\sigma_{\mu_{i}}=\frac{q_{i}^{\max}-q_{i}^{\min}}{\eta_{i}}\), \(\eta_{i}\in[0.01,2]\); \(q_{i}^{\max}\) and \(q_{i}^{\min}\) represent the highest and lowest values of the \(i\)th input variable; and \(\eta_{i}\) is a variable set by the users; \(i=1,2,\cdots,L\).