## 1 Introduction  
The seakeeping performance of floating bodies such as ships in harsh sea conditions is one of most important hydrodynamic characteristics that need to be carefully investigated by engineers. Generally, ship motions can be acceptably predicted using inviscid linear potential theories except for roll response. The difficulty of predicting roll response originates from the nonlinearity of roll motion and viscous effects of water. As compared to other motions, the roll motion is inherently highly nonlinear when the roll amplitude is medium to large. The viscous effects always accompany with the roll motion, and they become increasingly significant when ships are fitted with bilge keels. To accurately predict the roll motion, both nonlinear and viscous effects should be considered in the analysis. As far back as a century ago, [PERSON] discovered that the roll motion of ships in regular beam waves is nonlinear in certain conditions and should be described using nonlinear equations [1]. Systematic studies on the nonlinearity of ship roll motion only started from 1950s, when nonlinear dynamics theories began to be applied to investigate the nonlinear roll characteristics. Typically, the roll response of ships in regular beam waves can be modeled using a single degree of freedom differential equation, which includes linear terms and nonlinear damping and stiffness terms. The nonlinear stiffness terms mainly characterize the shape of the static stability diagram, while the nonlinear damping terms reflect the viscous effects as well as the nonlinearity. The effect of these nonlinear terms is enhanced as the wave amplitude increases. However, it is hard to obtain the exact analytical solution for the nonlinear differential equation with respect to roll response. Therefore, various approximate analytical methods have been developed for solving the nonlinear roll equation to this day, such as the multi-scale method [2], perturbation-based method [3], Krylov-Bogoliubov asymptotic method [4], and so on. To model and solve the nonlinear differential equation for roll response, the nonlinear damping and stiffness terms must be determined beforehand. There are two approaches to achieve this: one is the parametric identification approach [5], and the other is the non-parametric identification approach [6]. In the parametric identification approach, one needs to presuppose the expressions of damping and stiffness, which are generally assumed to be odd-ordered polynomials, e.g., linear plus cubic damping (\(b_{1}\dot{\theta}+b_{3}\dot{\theta}^{3}\)), and linear plus cubic, quintic stiffness (\(c_{1}\theta+c_{3}\theta^{3}+c_{3}\theta^{3}\)) [7], where \(\theta\), \(\dot{\theta}\) are roll angle and roll angular velocity, respectively, and \(b_{i},\ c_{i}\) are damping and stiffness coefficients, respectively. However, as in some works, the even-ordered terms might also be adopted, such as linear plus quadric damping [1] or linear plus quadric, cubic, quintic stiffness [8]. The nonlinear stiffness could be evaluated using hydrostatic restoring curve fitting method, while nonlinear damping coefficients are difficult to obtain using the classical potential theories. Thereby, most studies that made parametric identification for nonlinear roll models concentrated on investigating the nonlinear damping coefficients. Generally, roll damping can be identified using either the extinction curve method [5] or the energy method [9] from the free roll decay curves. To enhance the numerical stability, some ingenious data processing tools such as support vector regression [10], kalman filter [11], and wavelets [12] are employed in the damping identification. In contrast to the parametric identification approach, which depends on the given nonlinear damping and stiffness models, the non-parametric identification approach does not make any priori assumption of the damping form. A pioneer work using this approach was made by [PERSON] et al.