## 1 Introduction  
Brake systems subjected to friction-induced vibration are widely used in many industrial applications such as the automobile, aircraft, and railway fields [1]. One of the important vibration characteristics caused by friction is instability mechanisms, which can be classified into four categories: Variable friction coefficient, stick-slip, spray-slip, and mode-coupling [2]. Among these various types of instabilities, mode-coupling is considered one of the most crucial mechanisms [3]; one phenomenon that is observed when this mechanism occurs is that the real parts of the eigenvalues bifurcate, while the associated imaginary parts are merged. Since the stability of the system can be determined from the eigenvalues, many studies have been conducted to investigate the phenomenon through complex eigenvalue analyses (CEA) [4; 5; 6]. In carrying out CEA, the computational model is usually considered as deterministic, and the fixed friction coefficient is applied. However, the friction coefficient is known to have strong dispersion characteristics depending on the operation conditions [7], and should be modeled as a random parameter. In addition, most physical systems are subject to uncertainties concerning the input parameters, such as the material properties and geometry, and the contact conditions. Therefore, it is necessary to consider uncertainties to ensure that a robust analysis of brake systems can be made. The stochastic eigenvalue problem is an extension of the deterministic problem to study response variability under parametric input uncertainties. Several methods have been proposed, such as MonteCarlo simulation (MCS), a sensitivity-based approach, polynomial chaos expansion (PCE), and kriging (Gaussian process). MCS [8] is a sampling-based technique which is considered the most robust method for uncertainty quantifications. However, since a large number of samples must be drawn to obtain sufficient accuracy, the computational cost becomes prohibitive. The sensitivity-based method [9] is a low-order Taylor expansion method that approximates the solution near the input parameters. Although this method is computationally efficient, the obtained solution will be inaccurate under large parameter variabilities. PCE and kriging are metamodel (surrogate model) -based techniques that replace the original model with an easy-to-evaluate function. PCE [10], also known as a spectral approach, is a regression-based methods that approximates the model by multivariate polynomials. Kriging [11] is an interpolation-based approach in which the model is assumed to be a realization of a Gaussian process. Since metamodel-based methods can be applied to various levels of uncertainty, numerous studies have utilized them to solve stochastic eigenvalue problems in brake systems [12; 13; 14; 15]. However, despite previous studies, it remains challenging to apply PCE and kriging to brake systems with the mode-coupling phenomenon. Since eigenvalues exhibit nonsmooth behavior around the bifurcation point, PCE is not adequate for capturing local solution accuracy. Several studies have tackled this issue by applying multi-element PCE [16], a wavelet basis [17]. Although kriging interpolates each sample and reflects the local characteristics, to the authors' knowledge, previous studies have all been based on the stationarity assumption, which means that the smoothness of response is entirely correlated throughout the random input space. Therefore, for mode-coupling problems where the stationarity assumption is violated, the results are highly sensitive to the samples, and accuracy and robustness will be compromised. In this paper, to address this issue, we propose a new combined nonstationary kriging and support vector machine method for stochastic analyses of the brake systems. The main difference from conventional stationary kriging is that the proposed algorithm utilizes a nonstationary kernel to reflect the bifurcation characteristics of the eigenvalues.