## 3 A Partial Multiplicative Dimensional Reduction-Based Reliability Estimation Method
### Method Presentation  
Given a structural system's performance function \(G\) (\(\mathbf{X}\), \(\mathbf{Y}\)) containing both random variables and interval variables as the inputs, where \(\mathbf{X}=[X_{1},\ldots,X_{N}]^{T}\) and \(\mathbf{Y}=[Y_{1},\ldots,Y_{M}]^{T}\) are the random input variables and interval input variables, respectively, and \(G=0\), \(G>0\), or \(G<0\) indicates that the system is in the limit safety state, safe state, or failure state, respectively, the existence of cross terms between random variables and interval variables contributes considerable difficulty to the reliability estimation. Hence, a method that can reasonably eliminate these cross terms with a subtle influence on the final reliability would be quite helpful for the corresponding reliability estimation since it decorrelates the random variables \(\mathbf{X}\) from the interval variables \(\mathbf{Y}\), which is the core idea for the proposed method. In order to obtain such a decorrelation method, let us first consider the bivariate function \(G\) (\(X_{1}\), \(Y_{1}\)). By using Equation (6), its M-DRM form becomes  
\[G(X_{1},Y_{1})\approx G(X_{1},Y_{1}^{c})\cdot G(X_{1}^{c},Y_{1})\cdot G(X_{1}^ {c},Y_{1}^{c})^{-1} \tag{9}\]  
where \(X_{i}^{c}\) is the mean value of \(X_{i}\) and \(Y_{i}^{c}\) is the center of \(Y_{i}\). Now, consider \(G=G\) (\(X_{1}\), \(X_{2}\), \(Y_{1}\)). Similarly, by applying Equation (6) to \(G\) (\(X_{1}\), \(X_{2}\), \(Y_{1}\)) while treating \(X_{2}\) as a parameter, the following is obtained:  
\[G(X_{1},X_{2},Y_{1})\approx G(X_{1},X_{2},Y_{1}^{c})\cdot G(X_{1}^{c},X_{2},Y _{1})\cdot G(X_{1}^{c},X_{2},Y_{1}^{c})^{-1} \tag{10}\]  
At this point, it is noted that the above approximation holds since Equation (6) is applicable to any reference point. Further, by substituting Equation (6) into \(G\) (\(X_{1}^{c}\), \(X_{2}\), \(Y_{1}\)), the following is obtained:  
\[G(X_{1}^{c},X_{2},Y_{1})\approx G(X_{1}^{c},X_{2},Y_{1}^{c})\cdot G(X_{1}^{c}, X_{2}^{c},Y_{1})\cdot G(X_{1}^{c},X_{2}^{c},Y_{1}^{c})^{-1} \tag{11}\]Substituting Equation (11) into Equation (10) yields  
\[G(X_{1},X_{2},Y_{1})\approx G(X_{1},X_{2},Y_{1}^{c})\cdot G(X_{1}^{c},X_{2}^{c},Y _{1})\cdot G(X_{1}^{c},X_{2}^{c},Y_{1}^{c})^{-1} \tag{12}\]  
Through the similar process, \(G\) (\(X_{1}\), \(X_{2}\), \(X_{3}\), \(Y_{1}\)) is obtained as  
\[G(X_{1},X_{2},X_{3},Y_{1})\approx G(X_{1},X_{2},X_{3},Y_{1}^{c})\cdot G(X_{1}^{ c},X_{2}^{c},X_{3}^{c},Y_{1})\cdot G(X_{1}^{c},X_{2}^{c},X_{3}^{c},Y_{1}^{c})^{-1} \tag{13}\]  
By using such a recursive method, \(G\) (\(X_{1}\),..., \(X_{N}\), \(Y_{1}\)) is obtained as  
\[G(X_{1},\ldots,X_{N},Y_{1})\approx G(X_{1},\ldots,X_{N},Y_{1}^{c})\cdot G(X_{ 1}^{c},\ldots,X_{N}^{c},Y_{1})\cdot G(X_{1}^{c},\ldots,X_{N}^{c},Y_{1}^{c})^{-1} \tag{14}\]  
Now, consider \(G=G\) (\(X_{1}\),..., \(X_{\rm N}\), \(Y_{1}\), \(Y_{2}\)). The similar operation yields  
\[ \begin{gathered} G(X_{1},\ldots,X_{N},Y_{1},Y_{2})\\ \approx G(X_{1},\ldots,X_{N},Y_{1}^{c},Y_{2})\cdot G(X_{1}^{c}, \ldots,X_{N}^{c},Y_{1},Y_{2})\cdot G(X_{1}^{c},\ldots,X_{N}^{c},Y_{1}^{c},Y_{2 })^{-1}\end{gathered}  \tag{15}\]  
and \(G\) (\(X_{1}\),..., \(X_{\rm N}\), \(Y_{1}^{c}\), \(Y_{2}\)) is transformed into  
\[ \begin{gathered} G(X_{1},\ldots,X_{N},Y_{1}^{c},Y_{2})\\ \approx G(X_{1},\ldots,X_{N},Y_{1}^{c},Y_{2}^{c})\cdot G(X_{1}^{ c},\ldots,X_{N}^{c},Y_{1}^{c},Y_{2})\cdot G(X_{1}^{c},\ldots,X_{N}^{c},Y_{1}^{c},Y_{2 }^{c})^{-1}\end{gathered}  \tag{16}\]  
By substituting Equation (16) into Equation (15), one can obtain  
\[ \begin{gathered} G(X_{1},\ldots,X_{N},Y_{1},Y_{2})\\ \approx G(X_{1},\ldots,X_{N},Y_{1}^{c},Y_{2}^{c})\cdot G(X_{1}^{ c},\ldots,X_{N}^{c},Y_{1},Y_{2})\cdot G(X_{1}^{c},\ldots,X_{N}^{c},Y_{1}^{c},Y_{2 }^{c})^{-1}\end{gathered}  \tag{17}\]  
Accordingly, \(G\) (\(X_{1}\),..., \(X_{\rm N}\), \(Y_{1}\),..., \(Y_{M}\)) is finally expressed as  
\[