# An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function
## 3 The Proposed Approach: MSRBF-MCS
### The Proposed Learning Function  
In order to improve the efficiency and effectiveness of the adaptive model for acquiring LSF information, the location of its new training sample points should meet the following three requirements:  
1. Maintain sufficient distance from the existing training sample points (i.e., to alleviate the clustering problem);
2. Be in the sensitive region;
3. Be near enough to the LSF, i.e., \(\hat{g}(\mathbf{X})\) is close enough to zero. To achieve the above purpose, first, the sample points are transformed from the original space (\(\mathbf{X}\)-space) to the equivalent standard normal space (\(\mathbf{U}\)-space) with Nataf transform [31]. This is because the following facilities are available in \(\mathbf{U}\)-space: (1) the Euclidean distance from a sample point to the origin is directly related to the probability of that point, and (2) the effect of order-of-magnitude differences between the different dimensions of the sample on the distance is eliminated. Second, for the above requirement (1), a penalty function based on the distance between the sample points is proposed. Assuming the minimum distance between each sample point of the \(DoE\) in \(\mathbf{U}\)-space is \(d_{DoE}\) and the minimum distance from a sample point to the \(DoE\) in \(\mathbf{U}\)-space is \(d_{U}\), the penalty function can be expressed as follows:  
\[ \begin{array}{c}d_{penalty}(\mathbf{X})=\frac{d_{DoE}}{d_{U}}\\ \left\{\begin{array}{c}d_{\mathbf{U}}=norm(\mathbf{U}-\mathbf{U}_{nearest})\\ d_{DoE}=\min(norm(\mathbf{U}_{i}-\mathbf{U}_{j}))\\ i,j=1,2,\ldots,n_{DoE}\text{ and }i\
eq j\end{array}\right.\end{array}  \tag{14}\]  
where \(\mathbf{U}\) is the Nataf transform of \(\mathbf{X}\); \(\mathbf{U}_{nearest}\) is a sample point of \(DoE\) in \(\mathbf{U}\)-space, from which the Euclidean distance to \(\mathbf{U}\) is the minimum; \(\mathbf{U}_{i}\) and \(\mathbf{U}_{j}\) are the \(i\)-th and \(j\)-th sample points of \(DoE\) in \(\mathbf{U}\)-space; \(n_{DoE}\) is the number of sample points in \(DoE\); and \(norm(\cdot)\) is the two-norm function. The candidate sample point with a smaller predicted value is considered to be better. Hence, \(d_{penalty}>1\) (i.e., \(d_{U}>d_{DoE}\)) is regarded as a penalty and \(d_{penalty}<1\) (i.e., \(d_{U}<d_{DoE}\)) is considered as a reward. By employing \(d_{penalty}(\mathbf{X})\), the learning function will tend to select points far away from \(DoE\). However, an excessively large \(d_{U}\) may result in new training samples being in an insensitive region. To avoid losing the validity of LSF information due to the low probability of a new training sample point, \(p(\mathbf{X})\) is employed to satisfy the above requirement (2). Denote \(p(\mathbf{X})\) as follows:  
\[p(\mathbf{X})=norm(\mathbf{U}) \tag{15}\]  
Then, a distance control function \(dc(\mathbf{X})\) that can adjust the weight between the above requirement (1) and requirement (2) can be expressed as:  
\[dc(\mathbf{X})=p(\mathbf{X})d_{penalty}(\mathbf{X})^{2\alpha} \tag{16}\]  
where the adjustment coefficient \(\alpha\) is employed to adjust the influence of \(d_{penalty}(\mathbf{X})\). It is important to underline that the input variables must be normalized. Otherwise, when there are order-of-magnitude differences between different dimensions of \(\mathbf{X}\), the distance control function \(dc(\mathbf{X})\) may not work as intended. Finally, synthesizing the three requirements above, an active learning function \(L(\mathbf{X})\) with reference to \(U(\mathbf{X})\)[22] is proposed, as follows:  
\[L(\mathbf{X})=\frac{|mean^{w}|}{sigma^{w}}dc(\mathbf{X}) \tag{17}\]  
The optimization problem of finding a sample point with a minimum \(L(\mathbf{X})\) can be solved by direct MCS, which has the advantage of saving computational cost. This is because MCS would already need to be performed to predict failure probability. An alternative method is searching the variable space with meta-algorithms such as particle swarm optimization (PSO) [32] or Genetic algorithm [33].