# Uncertainty Analysis Based on Kriging Meta-Model for Acoustic-Structural Problems
## 4 Surrogate Modeling Techniques
### Kriging Meta-Model  
Generally, the Kriging method is very applied today because of its robustness and the quality of the approximation it can provide. In addition, it provides an estimate of the least squares error of the process that it estimates. The modern framework of Kriging (also called Gaussian process modeling) is based on the fact that the substitution function is assumed to be an achievement of a Gaussian process denoted \(y(\mathbf{x})\). Thus, the response \(y(\mathbf{x})\) can be expressed as follows:  
\[y(\mathbf{x})=G(\mathbf{x})+Z(\mathbf{x}) \tag{10}\]  
where \(G(\mathbf{x})\) is usually given as a linear combination of basic polynomial functions:  
\[G(\mathbf{x})=\sum_{i=1}^{p-1}\beta_{i}f_{i}(\mathbf{x}) \tag{11}\]  
where \(\mathbf{f}(\mathbf{x})=[f_{0}(\mathbf{x}),f_{1}(\mathbf{x}),\ldots,f_{p-1}(\mathbf{x})]^{\rm T}\) represents the vector of regression basis function and the column vector \(\beta=[\beta_{0},\beta_{1},\ldots,\beta_{p-1}]^{\rm T}\) is the vector of regression coefficients and  
Figure 3: Flowchart of surrogate models construction. \(Z(\mathbf{x})\) is a stochastic process (random function) with zero mean \((\mathbb{E}[Z(\mathbf{x})]=0)\), a variance \(\sigma^{2}\) and non-zero covariance given by:  
\[Cov(Z(\mathbf{x})-Z(\mathbf{x}^{\prime}))=\sigma^{2}R(\mathbf{x},\mathbf{x}^{\prime}) \tag{12}\]  
where \(R(\mathbf{x},\mathbf{x}^{\prime})\) denotes the correlation function between the estimated sampling points \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\) in the design space. It controls the smoothness of the resulting model. The correlation function in Equation (12) affects the smoothness of the model. A product of correlation functions is generally used to construct n dimensional ones:  
\[R(\mathbf{x},\mathbf{x}^{\prime})=\prod_{i=1}^{m}R(x_{i},x_{i}^{\prime}) \tag{13}\]  
where \(m\) is the dimension of design variables and \(x_{i}\) and \(x_{i}^{\prime}\), respectively, are the \(i\)th component of the sampling design point \(\mathbf{x}\) and \(\mathbf{x}^{\prime}\), respectively. According to [46] two commonly applied functions are the exponential (Equation (14)) and the Gaussian correlation functions (Equation (15)):  
\[R(\mathbf{x},\mathbf{x}^{\prime})=\exp\left[-\sum_{i=1}^{m}\theta_{i}|x_{i}-x_{i}^{ \prime}|\right] \tag{14}\]  
\[R(\mathbf{x},\mathbf{x}^{\prime})=\exp\left[-\sum_{i=1}^{m}\theta_{i}|x_{i}-x_{i}^{ \prime}|^{2}\right] \tag{15}\]  
with \(\mathbf{\theta}=[\theta_{1},\dots,\theta_{m}]^{\mathrm{T}}\) denotes the vector of unknown parameters of the correlation function. The Kriging estimator at a given point \(\mathbf{x}\) is by definition a Gaussian random variable obtained as a linear combination of the observations and which is called the best linear unbiased estimator (BLUE) [47]:  
\[\hat{y}(\mathbf{x})=f^{\mathrm{T}}(\mathbf{x})\mathbf{\beta}+\mathbf{r}^{\mathrm{T}}(\mathbf{x})R^ {-1}(\mathbf{y}_{S}-\mathbf{F}^{\mathrm{T}}\mathbf{\beta}) \tag{16}\]  
where \(\mathbf{f}^{\mathrm{T}}(\mathbf{x})\) is the regression basis function vector of the untired site \(\mathbf{x}\), \(\mathbf{\beta}\) denotes the vector of estimated regression coefficients, \(\mathbf{r}(\mathbf{x})=[R(\mathbf{x},\mathbf{x}^{(1)}),R(\mathbf{x},\mathbf{x}^{(2)}),\dots,R(\mathbf{x}, \mathbf{x}^{(n)})]^{\mathrm{T}}\) represents the vector of correlation functions between the untired sites \(\mathbf{x}\) and the \(m\) sample points, \(\mathbf{R}\) is the \(m\times m\) matrix of correlation functions for the fitting sample, \(\mathbf{y}_{S}\) represents the vector of the observed responses in the fitting sample and the term \((\mathbf{y}_{S}-\mathbf{F}^{\mathrm{T}}\mathbf{\beta})\) represents the vector of residuals for all fitting points when the stochastic term of the model is neglected. \(\mathbf{F}\) represents the model matrix as follows:  
\[\mathbf{F}=