## 4 Dynamic BLINs Based on Response Surface Method
### Response Surface Method Using GPR  
To calculate the dynamic BLINs of each bearing with different combinations of vertical displacements, the response surface method based on GPR [28; 29] was used to establish the mapping relationship between the elastic displacement and the dynamic BLINs. The training sample set \(\mathbf{L}\) for the response surfaces is:  
\[\mathbf{L}=\left\{(\mathbf{x}_{i},\mathbf{y}_{i})|i=1,2,3,\ldots,n\right\}=(\mathbf{X},\mathbf{y}) \tag{7}\]  
where \(\mathbf{y}\) is the output vector of training, which refers to the BLINs; \(\mathbf{x}_{i}\in\mathbf{X}\) are the input variables, \(\mathbf{X}=\left[\Delta y_{1},\Delta y_{2},\Delta y_{3}\right]^{T}\). \(\Delta y_{1}\), \(\Delta y_{2}\), and \(\Delta y_{3}\) are the equivalent vertical displacements of the three bearings. The mean function and covariance function of the response surfaces are denoted as \(u(\mathbf{x})\) and \(r(\mathbf{x},\mathbf{x}^{\prime})\), respectively, and the Gaussian process can be expressed as:  
\[\mathbf{y}\sim GP(u(\mathbf{x}),r(\mathbf{x},\mathbf{x}^{\prime})) \tag{8}\]  
During the training process, \(u(\mathbf{x})\) was considered as zero, and the squared exponential covariance function was used for training, which can be expressed as:  
\[r_{SE}(\mathbf{x}_{i},\mathbf{x}_{j})=\sigma_{f}^{2}\exp(\frac{1}{2|^{2}}\left\|\mathbf{x }_{i}-\mathbf{x}_{j}\right\|^{2})+\sigma_{n}^{2}\delta_{ij} \tag{9}\]where \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) are the input vectors of the training sample set; and \(l\) is the local correlation parameter, which takes values within [0, 1] and indicates the data correlation between two sample points. During the process of GPR, better local correlation parameters can be adaptively obtained through error analysis and iterative optimization. \(\sigma_{f}=0.5\) is the local correlation control parameter; \(\epsilon\sim N(0,\sigma_{n}^{2})\) represents the random data error during training process and \(\sigma_{n}\) represents its standard deviation; \(\delta_{ij}\) is the Kronecker delta. Use the numerical model of the shifting to generate sample points. For the \(i\)-th sample point, \(\mathbf{x}_{i}\) and \(\mathbf{y}_{j}\) can be written as:  
\[\mathbf{x}_{i}=\left[\Delta y_{1}^{i},\,\Delta y_{2}^{i},\,\Delta y_{3}^{i}\right] ^{T} \tag{10}\]  
\[\mathbf{y}_{i}= \begin{bmatrix}\Delta F_{11}^{i}&\Delta F_{12}^{i}&\Delta F_{13}^{ i}\\ \Delta F_{21}^{i}&\Delta F_{22}^{i}&\Delta F_{23}^{i}\\ \Delta F_{31}^{i}&\Delta F_{32}^{i}&\Delta F_{33}^{i}\end{bmatrix}  \tag{11}\]  
where \(\Delta F_{jk}^{i}\) (\(j\), \(k=1\), \(2\), \(3\)) are the elements of the BLINs matrix of the \(i\)-th sample point. With the zero-mean constraint of the output vector \(\mathbf{y}\) imposed, according to Equation (8), the covariance matrix \(\mathbf{R}(\mathbf{X},\mathbf{X})\) can be written as:  
\[\mathbf{y}\sim N(0,\mathbf{R}(\mathbf{X},\mathbf{X})) \tag{12}\]  
\[\mathbf{R}(\mathbf{X},\mathbf{X})= \begin{bmatrix}r(\mathbf{x}_{1},\mathbf{x}_{1})&r(\mathbf{x}_{1},\bm {x}_{2})&\cdots&r(\mathbf{x}_{1},\mathbf{x}_{n})\\ r(\mathbf{x}_{2},\mathbf{x}_{1})&r(\mathbf{x}_{2},\mathbf{x}_{2})&\cdots&r(\mathbf{x}_{2},\mathbf{x}_{ n})\\ \vdots&\vdots&\ddots&\vdots\\ r(\mathbf{x}_{n},\mathbf{x}_{1})&r(\mathbf{x}_{n},\mathbf{x}_{2})&\cdots&r(\mathbf{x}_{n},\mathbf{x}_{ n})\end{bmatrix}  \tag{13}\]  
The following relations are defined:  
\[\mathbf{R}[\mathbf{X}_{s},\mathbf{X}]=\left[r(\mathbf{x}_{s},\mathbf{x}_{1})\quad r(\mathbf{x}_{s}, \mathbf{x}_{2})\quad\cdots\quad r(\mathbf{x}_{s},\mathbf{x}_{n})\right] \tag{14}\]  
\[\mathbf{R}[\mathbf{X}_{s},\mathbf{X}_{s}]=r(\mathbf{x}_{s},\mathbf{x}_{s}) \tag{15}\]  
The output vector \(\mathbf{y}\) and the regression-predicted vector \(\mathbf{y}_{s}\) then follow the joint distribution:  
\[ \begin{bmatrix}\mathbf{y}\\ \mathbf{y}_{s}\end{bmatrix} \sim N\Bigg{(}0, \begin{bmatrix}\mathbf{R}(\mathbf{X},\mathbf{X})+ \sigma_{n}^{2}\mathbf{E}&\mathbf{R}(\mathbf{X}_{s},\mathbf{X})^{T}\\ \mathbf{R}(\mathbf{X}_{s},\mathbf{X})&\mathbf{R}(\mathbf{X}_{s},\mathbf{X}_{s})\end{bmatrix}