# MIMO LS-SVR-Based Multi-Point Vibration Response Prediction in the Frequency Domain
## 2 MIMO LS-SVR-Based Vibration Response Prediction under Unknown Uncorrelated Loads
### Theoretical Inference of the Relationship between Multi-Point Vibration Responses
As shown in Figure 1, the history measured vibration response \(\overset{\rightarrow}{s}_{yy\_ms}^{p}(\omega)\) is the input and the history unmeasured vibration response points \(\overset{\rightarrow}{s}_{yy\_ums}^{p}(\omega)\) are the output, \(p=1,2,\ldots,P\), where \(P\) is the number of training samples. Similarly, from the historical data for each frequency (\(\omega=1,2,\cdots,l,\cdots,\Omega\), \(\Omega\) is the total number of frequencies), the multi-point vibration response prediction model can be built for every frequency using multiple regression analysis, such as multiple linear regression model, MIMO neural network, or LS-SVR. After the model is trained, the multi-point vibration response prediction model can be used to predict the unmeasured multi-point vibration response in a real-work condition. The measured multi-point vibration responses \(\overset{\rightarrow}{s}_{yy\_ms}^{work}(\omega)\) are the input of the MIMO model, and the output of the model \(\overset{\rightarrow}{s}_{yy\_ums}^{work}(\omega)\) is the predicted unmeasured multi-point vibration response data. Clearly, the greater the amount of the history dataset that is used to train the data-driven multi-point vibration response model, the better the results the model acquires for all multiple regression analysis methods. Procedure for Building and Applying the MIMO LS-SVR-Based Multi-Point Vibration Response Prediction Model  
It can be seen from Section 2.2 that when \(\mathbf{D}(\omega)\) cannot be obtained, the problem can be transformed into a data-driven problem. The data-driven problem for an LTI structure can be solved using a MIMO LS-SVR with a linear or nonlinear kernel function. Thus, the next step is to use MIMO LS-SVR to acquire the complex relation \(\mathbf{D}(\omega)\) for each frequency. LS-SVR [40, 41] is a regression prediction model, which is developed and improved from Support vector machine (SVM). It is widely used in classification, numerical estimation, and density estimation. Its basic principle is to map the inputs into the high-dimensional space through the kernel function, and then find a hyperplane in the high-dimensional space to build a regression model in the hyperplane. Figure 1: Data-driven multi-point vibration response prediction under unknown uncorrelated multi-source loads. In the linear model, the predicted response \(\hat{\mathbf{S}}_{yy\_ums}(\omega)\) can be expressed as Equation (6), and the prediction model can be obtained by solving \(\mathbf{W}(\omega)\). \[\hat{\mathbf{S}}_{yy\_ums}(\omega)=\mathbf{W}(\omega)\mathbf{S}_{yy\_ms}( \omega)+\overset{\rightarrow}{b}_{0}(\omega) \tag{6}\]  
where \(\hat{\mathbf{S}}_{yy\_ums}(\omega)\triangleq\left[ \begin{subarray}{c}\to 1\\ \overset{s}{sy\_ums}(\omega),\overset{\rightarrow}{s}_{yy\_ums}(\omega), \ldots,\overset{\rightarrow}{s}_{yy\_ums}^{p}(\omega),\ldots,\overset{ \rightarrow}{s}_{yy\_ums}^{p}(\omega)\end{subarray} \right]\), \(\mathbf{W}(\omega)\) is \([n_{2}\times n_{1}]\) matrix of regression coefficients, \(\mathbf{S}_{yy\_ms}(\omega)\triangleq\left[\overset{\rightarrow}{s}_{yy\_ms} ^{1}(\omega),\overset{\rightarrow}{s}_{yy\_ms}^{2}(\omega),\ldots,\overset{ \rightarrow}{s}_{yy\_ms}^{p}(\omega),\ldots,\overset{\rightarrow}{s}_{yy\_ ms}^{p}(\omega)\right]\), \(\overset{\rightarrow}{b}_{0}(\omega)\) is \([n_{2}\times 1]\) vector of model offset. SVR uses a kernel function to map \(\mathbf{S}_{yy\_ms}(\omega)\) to a high dimensional feature space to obtain the kernel matrix \(\mathbf{K}(\omega)\):  
\[\mathbf{K}(\omega)=\left( \begin{array}{ccc}k_{1,1}(\omega)&\cdots&k_{1,P}( \omega)\\ \vdots&\ddots&\vdots\\ k_{P,P}(\omega)&\cdots&k_{P,P}(\omega)\end{array} \right) \tag{7}\]  
\(k(\omega)\) is the kernel function. In this study, the radial basis function (RBF) kernel function is used. \[k_{i,j}(\omega)=\exp(\frac{-\|\overset{\rightarrow}{s}_{yy\_ums}^{i}(\omega )-\overset{\rightarrow}{s}_{yy\_ums}^{j}(\omega)\|}{2\sigma(\omega)^{2}}) \tag{8}\]  
where \(\sigma(\omega)\) means the kernel width parameter.