# 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
 \tag{2}\]  
Similarly, for the LTI structure in the frequency domain, the relationship between the self-power spectrum \(\overset{\rightarrow}{s}_{ff}(\omega)\) of the loads \(\overset{\rightarrow}{f}(\omega)\) and the self-power spectrum \(\overset{\rightarrow}{s}_{yy\_ums}(\omega)\triangleq\left[s_{y_{n_{1}+1}y_{n_ {1}+1}}(\omega),s_{y_{n_{1}+2}y_{n_{1}+2}}(\omega),\cdots,s_{f_{n_{1}+b/n_{1}+ b}}(\omega)\cdots,s_{y_{n_{1}+b/n_{1}+b}}(\omega)\right]^{T}\) of the vibration responses \(y_{ums}(\omega)\)[25] is  
\[\overset{\rightarrow}{s}_{yy\_ums}(\omega)=\mathbf{H}_{ums}(\omega)\overset{ \rightarrow}{s}_{ff}(\omega) \tag{3}\]  
where  
\[\mathbf{H}_{ums}(\omega)\triangleq\left[ \begin{array}{cccc}\left|H_{n_{1}+1, 1}(\omega)\right|^{2}&\cdots&\left|H_{n_{1}+1,i}(\omega)\right|^{2}&\cdots& \left|H_{n_{1}+1,m}(\omega)\right|^{2}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \left|H_{n_{1}+h,1}(\omega)\right|^{2}&\cdots&\left|H_{n_{1}+h,i}(\omega) \right|^{2}&\cdots&\left|H_{n_{1}+h,m}(\omega)\right|^{2}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \left|H_{n_{1}+n_{2},1}(\omega)\right|^{2}&\cdots&\left|H_{n_{1}+n_{2},i}( \omega)\right|^{2}&\cdots&\left|H_{n_{1}+n_{2},m}(\omega)\right|^{2}\end{array}  \right],\]  
Substituting Equation (2) into Equation (3):  
\[ \begin{array}{c}\overset{\rightarrow}{s}_{yy\_ums}(\omega)=\mathbf{H}_{ums }(\omega)\overset{\rightarrow}{s}_{ff}(\omega)\\ \approx\mathbf{H}_{ums}(\omega)\mathbf{H}_{ms}(\omega)^{+}\overset{ \rightarrow}{s}_{yy\_ms}(\omega)\end{array}  \tag{4}\]  
When \(\mathbf{H}_{ums}(\omega)\), \(\mathbf{H}_{ms}(\omega)\), and \(\overset{\rightarrow}{s}_{yy\_ms}(\omega)\) are known, the \(\overset{\rightarrow}{s}_{yy\_ums}(\omega)\) can be obtained using Equation (4). The frequency characteristics of the transfer function \(H_{j,i}(\omega)\)\(j=1,2,\cdots,n_{1}\)\(i=1,2,\cdots,m\) and \(H_{n_{1}+h,i}(\omega)\)\(h=1,2,\cdots,n_{2}\)\(i=1,2,\cdots,m\) can be obtained by experiment or the FEM method [36]. In this study, the experimental method is carried out through multi-point excitation to obtain transfer function. In order to get the transfer function, we need to collect the historical data of the load. The transfer function from a load point to a response point can be obtained by single point excitation, and then the transfer function from this point to all response points can be obtained by changing the position of the response measuring point. Then, we can move to another load point to give excitation and repeat the above actions to obtain all the transfer functions of the LTI structure. Data-Driven Multiple Regression Analysis Problem Description of Multi-Point Vibration Response Prediction under Unknown Uncorrelated Multi-Source Loads  
Let \(\mathbf{D}(\omega)\triangleq\mathbf{H}_{ums}(\omega)\mathbf{H}_{ms}(\omega )^{+}\in\mathbb{R}^{n_{2}\times n_{1}}\). Thus, Equation (4) can be rewritten as  
\[\overset{\rightarrow}{s}_{yy\_ums}(\omega)\approx\mathbf{D}(\omega) \overset{\rightarrow}{s}_{yy\_ms}(\omega) \tag{5}\]  
When \(\mathbf{D}(\omega)\in\mathbb{R}^{n_{2}\times n_{1}}\) cannot be obtained using transfer functions and least squares generalization, it can be obtained using the history training dataset and a data-driven multi-point vibration response prediction model. Using the measured multi-point vibration response in Equation (3) as the input directly and the unmeasured multi-point vibration response as the output, a multi-input and multi-output multi-point vibration response prediction model for an LTI dynamics structure under unknown uncorrelated multi-source loads can be established using the history training dataset and multiple linear regression model. As shown in Figure 1, after training, the model can be used to predict the unmeasured vibration response in reality.