## 3 Kriging-Based Probability Density Estimation for FCTP
### Active Learning Function EDF  
After transforming input random variables into mutually independent standard normal variables [38], the first-order origin moments of the \(j\)th FCTP surrogate model can be expressed as follows:  
\[M_{l}^{(j)}=\int\cdots\int_{-\infty}^{+\infty}\left[\hat{\mathbf{\vartheta}}^{(j) }(\mathbf{Z})\right]^{l}\mathbf{\phi}(\mathbf{Z})d\mathbf{Z} \tag{22}\]  
where \(\mathbf{\phi}(\mathbf{Z})\) is the joint probability density function of mutually independent standard normal distributions. Since most of the integrated functions composed of LSF and PDF are not integrable [39], numerical methods can be used to approximate the integration process of Equation (22):  
\[M_{l}^{(j)}\approx\sum_{j=1}^{a}\lambda_{j}\Big{[}\hat{\mathbf{\vartheta}}^{(j)} \big{(}\theta_{j}\big{)}\Big{]}^{l} \tag{23}\]  
where \(\theta_{j}\) is the integration point of numerical integration, \(\lambda_{j}\) is the weight coefficient corresponding to the integration point, and \(a\) is the number of integration points. In this paper, the LPSS method is used to generate the integration points and weight coefficients. The key to efficient surrogate modeling lies in the sample selection strategy [40]. This paper summarizes two main factors influencing sample selection:  
(1) Model Error. The mean can measure the model error, variance, and joint PDF [41]. Since the first-order origin moments characterize the mean value, the samples selected should be generated as close as possible to the mean of FCTP. The approximate first-order origin moments of the \(j\)th FCTP surrogate model \(M_{1}^{(j)}\) can be obtained using Equation (23). Variance measures the degree of dispersion of the data and is an important basis for constructing active learning methods. The variance estimate of the Kriging surrogate model can be obtained by Equation (17), characterizing the prediction error of the model at the prediction point. The joint PDF \(f(x)\) of random variables can ensure that the selected points fall in a large probability region, effectively improving the estimation accuracy of the model. (2) Sample Distance. The distance between samples is an important principle in sample selection. On the one hand, the selected samples should be as sparse as possible to ensure global accuracy of the model; on the other hand, the selected samples should be preventedfrom bunching to avoid reducing the utilization rate of samples. In the \(s\) dimensional space \(R^{s}\), the Euclidean distance \(d_{a}\) from any point \(x\) to point \(x_{a}\) is  
\[d_{a}=\sqrt{\sum_{i=1}^{s}\left(x^{(i)}-x_{a}^{(i)}\right)^{2}} \tag{24}\]  
It is assumed that current surrogate model training set \(\xi\) has \(n_{\xi}\) sample points \(x_{c}\in\xi\), \(c=1,2,\ldots,n_{\xi}\), The distance from any point \(x\) to each point in set \(\xi\) is  
\[d_{\xi}(x)=\left\{d_{1}(x),d_{2}(x),\ldots,d_{n_{\xi}}(x)\right\} \tag{25}\]  
The average value \(\overline{d_{\xi}(x)}\) of distances from point \(x\) to each point in set \(\xi\) can be used as a measure of the sparsity of the samples:  
\[\overline{d_{\xi}(x)}=\frac{\sum_{c=1}^{n_{\xi}}d_{c}(x)}{n_{\xi}} \tag{26}\]  
Using the minimum distance \(d_{m}(x)\) from any point \(x\) to each point in set \(\xi\) can avoid bunching as follows:  
\[d_{m}(x)=min\left(d_{\xi}(x)\right)_{m} \tag{27}\]  
It is essential to choose a suitable active learning function for building highly accurate and efficient surrogate models. Too many adaptive selected samples can consume a lot of computational resources and reduce the benefits of the surrogate model method. Model accuracy cannot be guaranteed with too few samples.