# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 3 Reliability Analysis Approach under Multiple Failure Modes  
### Adaptive Multiple Response Gaussian Process  
Unlike the Kriging model, which regards a single output variable as a Gaussian stochastic process, the MRGP model can treat multi-dimensional output variables \(\mathbf{y}=[y_{1},y_{2},\cdots,y_{m}]\) as m-dimensional Gaussian stochastic process with separable covariance functions. Then, the mathematical expression of the MRGP model is described as follows:  
\[y^{m}(\bullet,\bullet)\sim GP\left(\mathbf{\mathrm{h}}^{m}(\bullet,\bullet)\mathbf{ \mathrm{B}}^{m},\sum^{m}\mathbf{\mathrm{R}}^{m}((\bullet,\bullet),(\bullet, \bullet))\right) \tag{7}\]  
where \(GP(\bullet,\bullet)\) is the Gaussian stochastic process, which is used to construct the global approximate relationship between input variables and output variables  
Further, to better describe the correlation of multidimensional output response variables, an isotropic Gaussian model is adopted to quantify the correlation of spatial location (Figure 1), which is given as follows:  
\[R^{m}\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}=\exp\{-\sum_{k=1}^{n}\mathbf{\omega}_{k}^{ m}\big{(}x_{k}-x_{k}^{\prime}\big{)}^{2}\} \tag{8}\]  
where \(\mathbf{\omega}_{k}^{m}=(\omega_{1},\omega_{2},\cdots,\omega_{n})\) is the vector of roughness parameters, which represent the rate at which the correlation between multiple failure models decays to zero. The lower values of \(\mathbf{\omega}_{k}^{m}\), the smother MRGP model for the response \(y^{m}(\bullet,\bullet)\). For Equation (7), \(\mathbf{\mathsf{B}}^{m}\) and \(\sum^{m}\) are related to the spatial correlation function \(R^{m}(\bullet,\bullet)\), which is usually expressed by the following equation:  
\[\left\{ \begin{array}{l}\hat{\mathbf{\mathsf{B}}}^{m}=\left[(\mathbf{\mathsf{h}}^{m}) ^{\mathsf{T}}\mathbf{\mathsf{R}}^{-1}\mathbf{\mathsf{h}}^{m}\right]^{-1}(\mathbf{\mathsf{ h}}^{m})^{\mathsf{T}}\mathbf{\mathsf{R}}^{-1}\mathbf{\mathsf{Y}}\\ \hat{\mathbf{\mathsf{\Sigma}}}^{m}=\frac{1}{p}\Big{[}(\mathbf{\mathsf{Y}}-\mathbf{\mathsf{ h}}^{m}\mathbf{\mathsf{B}}^{m})^{\mathsf{T}}\mathbf{\mathsf{R}}^{-1}(\mathbf{\mathsf{Y}}-\mathbf{ \mathsf{h}}^{m}\mathbf{\mathsf{B}}^{m})\Big{]}\end{array} \right. \tag{9}\]  
where \(\mathbf{\mathsf{R}}\) is the correlation matrix of \(R^{m}(\mathbf{x},\mathbf{x}^{\prime})\), \(\mathbf{\mathsf{Y}}\) is the corresponding matrix of response values, and \(p\) is the number of sample points. Due to the solution of roughness parameters \(\mathbf{\omega}_{k}^{m}\) being the key to building the MRGP model, a logarithmic inverse function of the maximum likelihood function \(\ln\big{[}p\big{(}\text{vec}(\mathbf{\mathsf{Y}})\big{|}\mathbf{\mathsf{B}}^{m},\sum ^{m},\mathbf{\omega}_{k}^{m}\big{)}\big{]}\) is employed. \[ \begin{array}{rl}\ln\big{[}p\big{(}\text{vec}(\mathbf{\mathsf{Y}})\big{|}\mathbf{ \mathsf{B}}^{m},\sum^{m},\mathbf{w}_{k}^{m}\big{)}\big{]}=&-\frac{mp}{2}\ln(2\pi)- \frac{p}{2}\ln(|\Sigma^{m}|)-\frac{m}{2}\ln(|\mathbf{\mathsf{R}}|)\\ &-\frac{1}{2}\text{vec}(\mathbf{\mathsf{Y}}-\mathbf{\mathsf{h}}^{m}\mathbf{\mathsf{B}}^{m} )^{\mathsf{T}}\times(\sum^{m}\otimes\mathbf{\mathsf{R}})^{-1}\text{vec}(\mathbf{ \mathsf{Y}}-\mathbf{\mathsf{h}}^{m}\mathbf{\mathsf{B}}^{m})\end{array}  \tag{10}\]  
where \(\text{vec}(\bullet)\) represents sorting a matrix by a column vector, and \(\otimes\) is the Kronecker product. The static MRGP model can be built based on Equation (10), which can predict the multidimensional response value \(\hat{\mathbf{\mathsf{y}}}(x^{*})\) at any unknown point \(x^{*}\).