## 1 Introduction  
The gear system is a complex elastic mechanical system that is widely used for power transmission in industries. Various gear failures and environmental factors can cause the increase in vibration and noise as described, for example, in [1]. As they affect the performance and durability of the entire system, it is important to precisely measure the noise and vibration produced by gears; see [2], for example, which measures vibrations and noise of a chipper with an internal combustion drive to identify the scale and location of origin of vibration. The transmission error (TE) is a well-known representative criterion for such measurement. Thus, accurate prediction of the TE is indispensable for precise and reliable transmission system design. However, it is not easy because a typical gear system has numerous uncertainties, such as lubrication, material nonlinearity, contact, geometrical tolerance, and nonlinear input loading condition [3, 4]. While many studies, e.g., [5, 6, 7, 8], on geometric measurement have succeeded in enhancing the accuracy of geometric specification, gear manufacturing, or wear evaluation, it is still inevitable to admit the existence of geometric errors in gears induced by diverse uncertain factors. Uncertainty quantification (UQ) measures the possible uncertainties and estimates their relations with one another or the effects on the entire system. UQ involves specifying appropriate uncertain factors, quantifying the uncertainties, estimating their effects on the quantity of interest (QoI), and evaluating the sensitivity of the QoI to the uncertain factors. Ultimately, it facilitates adjustment of sensitive uncertain parameters and sophistication of the model design. Many studies have applied UQ to gear systems in order to examine the influence of uncertain parameters on the TE. The finite element method (FEM) serves as a powerful tool for calculating the tooth deflection and mesh stiffness of the gear system, which has various uncertainties. For example, [PERSON] [9] quantitatively investigated the effects of machining errors, pitch errors, and tooth modifications on the load sharing ratio and TE by using specially developed FEM software. [PERSON] and [PERSON] [10] comprehensively generated a gear model with machining and pitch errors and analyzed the TE using FEM software. Specifically, the helix error and the error in the cutter profile were determined using the sine function. With all the errors considered, the three-dimensional geometry of the tooth surface was generated via GEMTE software. Then, by varying the degree of the uncertainties, the static TE was investigated to analyze the interactions between the gear uncertainties and their effects on the TE. For reliable analysis of the tooth surface contact, a highly refined FEM mesh is required due to the nonlinearity of contact problems. Numerous studies have developed model reduction algorithms to improve computational efficiency while retaining accuracy (e.g., [11]). Still, for uncertainty quantification, non-intrusive methods such as Monte Carlo simulations require a considerable number of simulations. Thus, despite its advantages, FEM could be inappropriate for the gear design under various uncertainties. By contrast, an analytical method can rapidly calculate the TE via numerical kinematic simulation [12; 13]. This method is computationally efficient and provides an analytical function of the gear mesh stiffness consisting of the design parameters of the gear [14; 15]. For example, [PERSON] [16] generated a gear model with tooth modification and analyzed the TE using the analytical method. The modified tooth profile was obtained from the modified cutter profile and used for modeling the tooth surface. However, the uncertainties in thermal deformation have not been studied sufficiently in the case of both FEM and analytical approaches. Owing to various scientific and industrial advancements, gears can currently operate under heavy loads and high speeds.