# A Semi-Parallel Active Learning Method Based on Kriging for Structural Reliability Analysis
## 2 Review  
### Kriging  
In place of the performance function, which was initially designed for forecasts in mining engineering and geostatistics, Kriging is beneficial for creating intricate and highly nonlinear models [53]. The approximated performance function is a realization of a statistical Gaussian process \(G(x)\) as Equation (2):  
\[G(x)=f(x)^{T}\beta+Z(x) \tag{2}\]  
In the equation, \(f(x)=[f_{1}(x),\dots,f_{P}(x)]^{T}\) there is a vector of the basic functions. \(\beta\) is a vector of regression coefficients. \(f(x)^{T}\beta\) forms an approximation of the response of the performance function in mean. \(Z(x)\) is the stationary Gaussian process with zero mean and covariance between two points \(x_{i}\) and \(x_{j}\) calculated by Equation (3):  
\[Cov\big{[}Z(x_{i}),Z(x_{j})\big{]}=\sigma_{Z}^{2}\cdot R\big{(}x_{i},\ x_{j}, \theta\big{)} \tag{3}\]  
\(\sigma_{Z}^{2}\) is the variance of \(Z(x)\). \(R\big{(}x_{i},\ x_{j},\theta\big{)}\) is the stochastic process correlation function with respect to parameter \(\theta\). The correlation function can be defined using a variety of models, including exponential, linear, spline, spherical, Gaussian, and so on. The Gaussian model is the most popular among them. The model can be formulated by Equation (4):  
\[R\big{(}x_{i},\ x_{j},\theta\big{)}=\prod_{m=1}^{M}\exp\Big{[}-\theta_{m}\big{(} x_{i,m}-x_{j,m}\big{)}^{2}\Big{]} \tag{4}\]  
where \(x_{i,m}\), \(x_{j,m}\) and \(\theta_{m}\) are the \(m\)th dimension of \(x_{i}\), \(x_{j}\) and \(\theta\). \(\big{|}x_{i,m}-x_{j,m}\big{|}\) represents the distance in the dimension \(x_{m}\). \(\theta_{m}\), which is a scale parameter that indicates the correlation between the points within dimension \(m\)th. The larger \(\theta_{m}\) and \(\big{|}x_{i,m}-x_{j,m}\big{|}\) are representative of a shorter correlation length. \(M\) is the number of input variables. Based on a set of training points (i.e., \([x_{1},x_{2},\dots,x_{n}]\)) with \(x_{i}\in R^{n}\) the \(i\)th experiment, and \(Y\) with \(Y_{i}=G(x_{i})\in R\), \(\beta\) and \(\sigma^{2}\) are estimated according to Equation (5):  
\[\hat{\beta}=\Big{(}F^{\mathrm{T}}R^{-1}F\Big{)}^{-1}F^{\mathrm{T}}R^{-1}Y \tag{5}\]  
where \(\hat{\beta}\) is the generalized least square estimation of \(\beta\), \(F\) is a matrix with each row \(f(x)^{\mathrm{T}}\), \(R=R\big{(}x_{i},\ x_{j},\theta\big{)}\) is the matrix of correlation between each pair of points of the DoE. Based on Equation (2) to Equation (5), the prediction of the Kriging model of the performance function with untried points \(x\) can be described by Equation (6):  
\[\hat{G}(x)=f(x)^{\mathrm{T}}\hat{\beta}+r(x)^{\mathrm{T}}R^{-1}\big{(}Y-F\hat{ \beta}\big{)} \tag{6}\]\(r(x)\) is the correlation vector between the untried points \(x\) and each of the training points, which is estimated by Equation (7):  
\[r(x)=[R(x,\ x_{1},\theta),R(x,\ x_{2},\theta),\ldots,R(x,\ x_{n},\theta)]^{T} \tag{7}\]  
The Kriging predicted variance is evaluated by Equation (8):  
\[\hat{\sigma}_{G}^{2}(x)=\hat{\sigma}_{Z}^{2}[1+u^{T}\Big{(}F^{T}R^{-1}F\Big{)} ^{-1}u-r(x)^{T}R^{-1}r(x)] \tag{8}\]  
where \(u\) and \(\hat{\sigma}_{Z}^{2}\) can be estimated respectively by the following  
\[u(x)=F^{T}R^{-1}r(x)-f(x) \tag{9}\]  
\[\hat{\sigma}_{Z}^{2}(x)=\frac{1}{n}\big{(}Y-F\hat{\beta}\big{)}^{T}R^{-1} \big{(}Y-F\hat{\beta}\big{)} \tag{10}\]  
Based on the above analysis, the Kriging prediction at an untried point \(x\) is normally distributed [50] as Equation (11):  
\[G(x)\,\sim N\Big{(}\mu_{\hat{G}}(x),\ \hat{\sigma}_{\hat{G}}^{2}(x)\Big{)} \tag{11}\]  
The prediction \(\hat{G}(x_{i})\) in the training point \(x_{i}\) of the design of experiments is exact, i.e., \(\hat{G}(x_{i})=G(x_{i})\). The variance is zero in these points. Therefore, it is an exact interpolation method, while in unexplored areas, the Kriging variance is predicted based on Equation (8). It enables the quantification of the uncertainty of local predictions. This property is important in reliability studies, which relate to the index to improve the surrogate model.