## 3 Online Processing Parameters Prediction and Reverse Design Framework
### Online Accelerated Support Vector Regression  
For a training set \(T=(x_{i},y_{i}),i=1,\ldots,l\), constructing a linear regression equation on the feature space \(F\), as shown in Equation (1). \[f(X)=\left\langle w\cdot\Phi(X)\right\rangle+b \tag{1}\]  
where \(w\) is a vector in the F space, \(\Phi(x)\) maps the input \(x\) to a vector in \(F\) and \(\left\langle\cdot,\cdot\right\rangle\) denotes the inner product in reproducing kernel Hilbert space (RKHS). Here, optimization techniques can be employed to obtain \(w\) and \(b\) for solving problems. \[ \begin{split}\min_{w,b}P&=\frac{1}{2}\|w\|^{2}+C \Bigg{(}\epsilon+\frac{1}{l}\sum_{i=1}^{l}(\xi_{i}+\xi_{i}^{*})\Bigg{)}\\ s.t.(\left\langle w,\Phi(X_{i})\right\rangle+b)-Y_{i}\leq\epsilon+ \xi_{i}\\ Y_{i}-(\left\langle w,\Phi(X_{i})\right\rangle+b)\leq\epsilon+\xi_ {i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0,\epsilon\geq 0,i=1,\ldots,l\end{split}  \tag{2}\]  
where \(\|w\|\) is a regularization term that characterizes the complexity of the regression model. The optimization criterion is used to penalize those values of \(Y\) that differ from \(f(x)\) by more than \(\epsilon\). Then, the original problem Equation (2) is converted into a pair of functions:  
and  
\[ \begin{split}&\min_{w_{2},\epsilon_{2},b_{2},\xi_{2}}\Bigg{\{} \frac{1}{2}\|w_{2}\|^{2}+C_{2}\Bigg{(}-\epsilon_{2}+\frac{2}{N}\sum_{i=1}^{N} \xi_{2i}\Bigg{)}\Bigg{\}}\\ s.t.(w_{2}\cdot\Phi(X_{i}))+b_{2}-Y_{i}\leq\epsilon_{2}-\xi_{2i} \text{ and }\xi_{2i}\geq 0,\epsilon_{2}\geq 0,i=1,\ldots,N\end{split}  \tag{4}\]  
Each of these two equations determines the upper and lower bounds. Meanwhile, to make the constraints independent of the training size, the objective function of Equation (3) is multiplied by the size of the training sample set. Therefore, the problem of seeking the  
Figure 1: Online processing parameters design framework. Apply concept drift detection on the streaming data collected from industrial equipment, eliminate data affected by drift and incorporate these refined data into the OASVR model for online learning. Utilize Bayesian optimization based on the predictions of model for processing parameter design. The industrial devices will automatically adjust according to the recommended process parameters to achieve enhanced manufacturing outcomes. upper bound function \(f_{1}(x)=\langle w_{1}\cdot\Phi(X_{i})\rangle+b_{1}\) is transformed into solving the following primal problems (QPPs):  
\[ \begin{split}&\min_{w_{1},\epsilon_{1},b_{1},\xi_{1i}}\left\{ \frac{N}{2}\|w_{1}\|^{2}+C_{1}\Bigg{(}B_{1}N+\sum_{i=1}^{N}\xi_{1i}\Bigg{)} \right\}\\ & s.t.\langle w_{1}\cdot\Phi(X_{i})\rangle+B_{1}\geq Y_{i}-\xi_{ 1i}\text{ and }\xi_{1i}\geq 0,i=1,\ldots,N\end{split}  \tag{5}\]  
where \(B_{1}\) is denoted as \(b_{1}-\epsilon_{1}\). Its corresponding pairwise problem is shown in Equation (6). \[ \begin{split}\max\Bigg{\{}-\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{ N}\alpha_{1i}\alpha_{1j}Q_{ij}+\sum_{i=1}^{N}\alpha_{1i}Y_{i}\Bigg{\}}\\ & s.t.\sum_{i=1}^{N}\alpha_{1i}=C_{1}N\\ &\alpha_{1i}\in[0,C_{1}],i=1,\ldots,N\end{split}  \tag{6}\]  
where \(Q\) is a positive definite matrix \(Q_{ij}=(1/N)\cdot K(X_{i},X_{i})\). The lower bound function \(f_{2}(X)=\langle w_{2}\cdot\Phi(X_{i})\rangle+b_{2}\) can be obtained by the same modification. Based on Equations (3) and (5), this algorithm solves a pair of quadratic programming problems (QPPs) instead of one QPP. This means that a data point only needs to satisfy the conditions of one of these functions. This indicates that there is only one set of constraints on all data points in OASVR. This strategy of solving two smaller-sized QPPs instead of one large QPP means that OASVR is simpler than classical SVR. With these modifications, the classical AONSVR is transformed into OASVR, where two smaller-sized QPPs need to be solved. Furthermore, the estimated upper and lower bounds of a regression model by OASVR effectively encapsulate the data distribution's properties. This can enhance the robustness of the model in many situations.