## 3 Methodology of Adaptive Gaussian Mixture Model Using Virtual Sample Generation  
### Overall Procedure of AGMM-VSG  
Before introducing the methodology of the proposed AGMM-VSG, some concepts in AGMM-VSG are explained as follows:  
Definition 1: (point, response, and sample): As is shown in Table 1, the point is a kind of pure coordinate. In AGMM-VSG, point refers to any vector of state \(\mathbf{x}(t_{k})\) at time \(t_{k}\). The response is taken as the output of the black-box model (2), i.e., the state vector \(\mathbf{x}(t_{k+1})\), which corresponds to the point \(\mathbf{x}(t_{k})\). A _sample_ is a combination of a _point_ and its response.  
Definition 2: (real sample _and_ virtual sample): The real sample _represents a combination of a point and its real_ response _generated by invoking expensive black-box simulations while the _virtual sample_ refers to the combination of a _point_ and its virtual_ response _generated without calling expensive simulations. The virtual responses are only generated using the active KRG model. Note that the number of real samples indicates the computational expense of the uncertainty propagation. Moreover, in this paper, the virtual response is generated by the active sampling-based KRG model, and the expense to generate _virtual samples_ is negligible.  
Definition 3: (cheap point _and_ superior cheap point): Cheap point is a point generated according to the given PDF. Note that the actual response of a _cheap point_ is not calculated, and therefore, the computational cost is negligible. The _cheap points_ are drawn as a candidate pool for constructing the AS-KRG model. The _superior cheap point_ has a promising performance for improving the global accuracy of the KRG model, which is classified on basis of the variance predicted using the KRG model. The process of using both the cheap point and _superior cheap point_ will be detailed in Section 3.3.  
The flowchart of AGMM-VSG is shown in Figure 1, and the procedure is explained in detail as follows:  
_Step 1_: The parameters of the problem are determined, including the black-box discrete dynamics, the initial PDF (or initial mean and covariance matrix), the propagating period \(t_{f}\), and discrete-time \(\mathrm{d}t\). Other necessary items, such as the maximal number of samples for constructing the KRG model and splitting library, will be discussed in the next section. The current time epoch is set to be zero, i.e., \(t=0\).  
_Step 2_: In the first iteration, the single Gaussian model is used, with its mean and covariance matrix equaling the initial PDF. The number of Gaussian mixture components is one.  
_Step 3_: The KRG model is first constructed using initial real samples and then is gradually refined with newly added real samples. The constructed KRG model is utilized for generating virtual samples.  
_Step 4_: Propagate each component using fifth-order CR. Note that only virtual samples are used when propagating; thus, the expense of this step is negligible. Renew the current time epoch, i.e., \(t=t+dt\).  
_Step 5_: For each component, determine whether the splitting process should be executed. If it should, then apply a univariate splitting library in a specified direction, i.e., the principal (eigenvector) direction of the covariance matrix.  
_Step 6_: If \(t<t_{f}\), jumps to Step 3. Otherwise, it outputs the results.  
The algorithms of Gaussian components splitting and virtual sample generation-based propagation are detailed in Sections 3.2 and 3.3, respectively.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Symbol** & **Description** & **Other Statement** \\ \hline _Point_ & \(\mathbf{x}(t_{k})\) & Input of the black-box model & Pure coordinate \\ _Response_ & \(\mathbf{x}(t_{k+1})\) & Output of the black-box model & - \\ _Sample_ & \((\mathbf{x}(t_{k}),\mathbf{x}(t_{k+1}))\) & Combination of a _point_ and its _response_ & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Description of point, response and sample.