## 3 Methodology of Adaptive Gaussian Mixture Model Using Virtual Sample Generation
### Gaussian Component Nonlinearity Detecting  
It is noted that only the linear propagation can maintain the Gaussian property, and any deviation in the nonlinear dynamics from the linearized solution may indicate nonlinear effects. This deviation can be handled by specifying a threshold and monitoring the nonlinearity [48]. That is to say, when the nonlinearity causes a more significant non-Gaussian property, a component of the GMM can be split into finer Gaussian components. A split GMM component is considered \"finer\" because Gaussians with smaller covariances would remain more Gaussian compared with ones with larger covariances, which will make the uncertainty propagation more representative. Thus, an adaptive GMM with splitting technologies is necessary, especially for long-term propagation. In this section, a novel unscented transformation and Cubature rule-based (UT-CR) method for monitoring (or detecting) the nonlinearity is introduced, and the Gaussian component splitting will be detailed in the next section. Figure 1: Flowchart of AGMM-VGS. Usually, the term entropy is used for detecting nonlinearity [42]. For a Gaussian distribution with the determinant of a scaled form of the covariance matrix, the corresponding differential entropy (DE), in terms of the logarithm, is given as:  
\[H(\mathbf{x})=\frac{1}{2}\log|2\pi e\mathbf{P}| \tag{7}\]  
where the temporal derivative of the DE is given by:  
\[\dot{H}(x)=\frac{1}{2}\textit{trace}\Big{\{}\mathbf{P}^{-1}\dot{\mathbb{P}}\Big{\}} \tag{8}\]  
where \(\dot{\mathbf{P}}\) is the temporal derivative of the covariance matrix \(\mathbf{P}\). In case of a linearized representation, the time rate of the DE is written as:  
\[\dot{H}(\mathbf{x})=\frac{1}{2}\textit{trace}\left\{\mathbf{f}_{x}\right\} \tag{9}\]  
where \(\mathbf{f}_{x}=\partial\mathbf{f}/\partial\mathbf{x}\) represents the linearized dynamics Jacobian. It is worth noting that for linearized dynamics, \(\dot{H}(\mathbf{x})=0\), meaning that the differential entropy is constant for the linearized system, making the DE unable to be used for splitting the Gaussian components for the linearized system. In this paper, a UT-CR-based splitting strategy is utilized to detect nonlinearity. The novel dynamic nonlinearity index measures the departure from linearity as [49]:  
\[v(t,t_{0})=\frac{\|\mathbf{\Phi}_{\text{CRS}}(t,t_{0})-\mathbf{\Phi}_{\text{UT}}(t,t_{ 0})\|}{\|\mathbf{\Phi}_{\text{UT}}(t,t_{0})\|} \tag{10}\]  
where \(v(t,t_{0})\) is the dynamic nonlinearity index and \(\|\cdot\|\) represents the norm. The \(\mathbf{\Phi}_{\text{CRS}}(t,t_{0})\) and \(\mathbf{\Phi}_{\text{UT}}(t,t_{0})\) are the STM obtained by fifth-order CR (CRS) and UT, respectively. It is noted that the UT could capture the solutions accurately to the third order while the CRS could capture accurately to the fifth order; thus, the nonlinearity with fourth and fifth orders are introduced with the index \(v(t,t_{0})\). The STM \(\mathbf{\Phi}_{\text{CRS}}(t,t_{0})\) and \(\mathbf{\Phi}_{\text{UT}}(t,t_{0})\) are obtained as follows. First, the posterior covariances (transformed distribution) \(\mathbf{P}_{\text{CRS}}(t,t_{0})\) and \(\mathbf{P}_{\text{UT}}(t,t_{0})\) are calculated using fifth-order CR and UT, given the same prior probability \(\mathbf{P}(t_{0})=\mathbf{P}_{0}\). Consider the following first-order approximation of the covariance of the transformed distribution [50]:  
\[ \begin{array}{l}\mathbf{P}_{\text{CRS}}(t)=\mathbf{\Phi}_{\text{CRS}}(t,t_{0})\mathbf{P} _{0}\mathbf{\Phi}_{\text{CRS}}^{T}(t,t_{0})\\ \mathbf{P}_{\text{UT}}(t)=\mathbf{\Phi}_{\text{UT}}(t,t_{0})\mathbf{P}_{0}\mathbf{\Phi}_{\text {UT}}^{T}(t,t_{0})\end{array}