# 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  
The problem we study is how to predict the vibration response of some locations which cannot be measured by using the known vibration response under unknown uncorrelated multi-source loads in the frequency domain. For a linear time invariant (LTI) dynamic structure in the frequency domain, the vibration response is split into two parts: measured vibration responses and unmeasured vibration responses. The measured vibration response means the vibration response value could be measured, and the vibration response value is known; the unmeasured vibration response means that the vibration response value is unknown, and is to be predicted in the model. \(n\) vibration responses can be classified into \(n_{1}\) measured vibration response points \(\overrightarrow{y}_{ms}(\omega)\triangleq[y_{1}(\omega),y_{2}(\omega),\cdots, y_{j}(\omega)\cdots,y_{n_{1}}(\omega)]\), \(j=1,2,\ldots,n_{1}\), and \(n_{2}\) unmeasured vibration response points \(y_{ms}(\omega)\triangleq[y_{n_{1}+1}(\omega),y_{n_{1}+2}(\omega),\cdots,y_{n _{1}+h}(\omega)\cdots,y_{n_{1}+n_{2}}(\omega)]\), \(h=1,2,\cdots,n_{2}\), where \(n_{1}+n_{2}=n\). For an LTI dynamics structure in the frequency domain, under the condition that \(m\) multi-source loads \(\overrightarrow{f}(\omega)\triangleq[f_{1}(\omega),f_{2}(\omega),\cdots,f_{ l}(\omega)\cdots,f_{m}(\omega)]\) are an uncorrelated random process with zero mean and second-order multiple-joint stationary characteristics, the relationship between the self-power spectrum \(\overrightarrow{s}_{ff}(\omega)\triangleq[s_{f_{1}f_{1}}(\omega),s_{f_{2}f_{ 2}}(\omega),\cdots,s_{f_{l}f_{l}}(\omega)\cdots,s_{f_{m}f_{m}}(\omega)]^{T}\) of the loads \(\overrightarrow{f}(\omega)\) and the self-power spectrum \(\overrightarrow{s}_{yy\_ms}(\omega)\triangleq[s_{y_{1}y_{1}}(\omega),s_{y_{2} y_{2}}(\omega),\cdots,s_{f_{l}f_{l}}(\omega)\cdots,s_{y_{n_{1}}y_{1}}(\omega)]^{T}\) of the vibration response \(\overrightarrow{y}_{ms}(\omega)\) can be expressed as [38]  
\[\overrightarrow{s}_{yy\_ms}(\omega)=\mathbf{H}_{ms}(\omega)\overrightarrow{s }_{ff}(\omega) \tag{1}\]  
where  
\[\mathbf{H}_{ms}(\omega)\triangleq\left[ \begin{array}{cccc}\left|H_{1,1}( \omega)\right|^{2}&\cdots&\left|H_{1,j}(\omega)\right|^{2}&\cdots&\left|H_{1,m }(\omega)\right|^{2}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \left|H_{j,1}(\omega)\right|^{2}&\cdots&\left|H_{j,i}(\omega)\right|^{2}&\cdots &\left|H_{j,m}(\omega)\right|^{2}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ \left|H_{n_{1},1}(\omega)\right|^{2}&\cdots&\left|H_{n_{1},i}(\omega)\right|^{2 }&\cdots&\left|H_{n_{1},m}(\omega)\right|^{2}\\ \end{array} \right],\]  
\(H_{j,i}(\omega)\), \(i=1,2,\cdots,m\), \(j=1,2,\cdots,n_{1}\) denotes the frequency characteristics of the transfer function from input \(f_{i}(\omega)\) to response \(y_{j}(\omega)\), and \(\left|H_{j,i}(\omega)\right|^{2}\) is the modulo square of \(H_{j,i}(\omega)\). \(\omega=\Delta f,2\Delta f,\cdots,l\Delta f,\cdots,\Omega\Delta f\). \(\Delta f\) is the frequency resolution. \(\Omega\) is the total number of frequencies. Using the least squares generalized inverse for Equation (1), an estimate \(\overset{\rightarrow}{s}_{ff}(\omega)\) of the self-power spectrum \(\overset{\rightarrow}{s}_{ff}(\omega)\) can be obtained from the self-power spectrum \(\overset{\rightarrow}{s}_{yy\_ms}(\omega)\) and \(\mathbf{H}_{ms}(\omega)\):  
\[ \begin{array}{c}\overset{\rightarrow}{s}_{ff}(\omega)=\left(\mathbf{H}_{ms} ^{T}(\omega)\mathbf{H}_{ms}(\omega)\right)^{-1}\mathbf{H}_{ms}^{T}(\omega) \overset{\rightarrow}{s}_{yy\_ms}(\omega)\\ \hskip 14.226378 pt\triangleq\mathbf{H}_{ms}(\omega)^{+}\overset{\rightarrow}{ s}_{yy\_ms}(\omega)\end{array}