## 3 Proposed Algorithm  
### Nonstationary Kriging  
As described in the previous section, complex eigenvalues show heterogeneous behavior near the bifurcation point. Since stationary kriging is based on the assumption that the response smoothness is fairly uniform, application to brake systems is still limited. To address this issue, the nonstationary kriging algorithm is presented. For simplicity, let \(\mathcal{M}(\zeta)\) be the model problem in Equation (8) and \(y\) the interested eigenvalue result. Using this notation, the random eigenvalue problem can be simply expressed as \(y=\mathcal{M}(\zeta)\). Given \(N\) samples \(\left\{\zeta^{1},\ldots\zeta^{N}\right\}\) in the \(M\) dimensional input space and their associated output \(\left\{y^{1},\ldots y^{N}\right\}\), the kriging assumes that the response of a computational model is a realization of the Gaussian random process as  
\[y=\mathcal{M}(\zeta)\simeq\mathbf{f}^{\mathrm{T}}(\zeta)\beta+Z(\zeta) \tag{9}\]  
where the first part \(\mathbf{f}(\zeta),\ \ \beta\) represents the trend function, and their product is the mean of the response. In constructing a global trend, ordinary kriging uses a constant value, while universal kriging uses polynomial functions. The second part \(Z(\zeta)\) is a Gaussian process with zero mean satisfying the following statistical property:  
\[\mathbb{E}\left[Z\left(\zeta^{i}\right)Z\left(\zeta^{i}\right)\right]=\sigma ^{2}R\left(\zeta^{i},\zeta^{j}\right) \tag{10}\]  
where \(\sigma^{2},\ \ R\left(\zeta^{i},\zeta^{j}\right)\), and \(\zeta^{i}\) are the process variance of \(Z(\zeta)\), the correlation function, and \(i\)-th input sample. In selecting the correlation function in the conventional stationary kriging, the Gaussian type \(R\left(\zeta^{i},\zeta^{j}\right)=\prod_{k=1}^{M}\exp\left(-\theta_{k}\left( \zeta^{i}-\zeta^{j}\right)^{2}\right)\) is widely used. However, rather than applying the stationary kernel, the following nonstationary Gibbs kernel [21] is adopted. \[R\left(\zeta^{i},\zeta^{j}\right)=\prod_{k=1}^{M}\left(\frac{2l_{k}\left( \zeta^{i}\right)l_{k}\left(\zeta^{j}\right)}{l_{k}^{2}\left(\zeta^{i}\right)+ l_{k}^{2}\left(\zeta^{j}\right)}\right)^{1/2}\exp\left(-\frac{\left(\zeta^{i}- \zeta^{j}\right)^{2}}{l_{k}^{2}\left(\zeta^{i}\right)+l_{k}^{2}\left(\zeta^{ i}\right)}\right) \tag{11}\]  
where \(l_{k}(\cdot)\) is the length-scale in \(k\)-th input dimension. The length-scale controls the response smoothness within the input domain, and can be any arbitrary positive function of random variables \(\zeta\). In this paper, to reflect the heterogeneous properties in each stable and unstable regions, the length-scale for the input sample is modeled by using step-wise hyperparameters as follows:  
\[l_{k}\left(\zeta^{i}\right)= \begin{cases}\theta_{k}^{\ 1}&g\left(\zeta^{i} \right)=0\\ \theta_{k}^{\ 2}&g\left(\zeta^{i}\right)>0\end{cases}  \tag{12}\]  
where \(g\left(\zeta^{i}\right),\ \ \theta\) are the criteria function and hyperparameter to be determined. Based on the fact that the real eigenvalue pair \(\lambda_{1},\ \ \lambda_{2}\) is bifurcate after the mode-coupling, the following stability criteria function is proposed. \[g\left(\zeta^{i}\right)=\left|\ \mathrm{Re}\left(\lambda_{1}\left(\zeta^{i} \right)\right)-\mathrm{Re}\left(\lambda_{2}\left(\zeta^{i}\right)\right)\right| \tag{13}\]  
The proposed function can be interpreted as a gap between two eigenvalues. If \(g\left(\zeta^{i}\right)=0\), the given eigenvalue pair is stable, and if \(g\left(\zeta^{i}\right)>0\), instability occurs. A simple comparison of nonstationary and stationary correlation functions is presented in Figure 5. For a stationary correlation function, the Gaussian type is considered. By comparing these two results, the proposed kernel and length scale function makes it possible to model the heterogeneity of the response. If the length-scale is relatively high, the response is smooth, whereas if the length-scale is low, the response changes rapidly. However, for a stationary correlation function, the entire input space is correlated.