## 3 Time-Dependent Reliability Analysis of an Aqueduct Based on Dual Power Transformation
### Proposed Fourth-Moment Method Based on Dual Power Transformation
#### 3.2.3 Estimation of the First Four Moments of the Transformation Performance Function Using HUT  
With the aid of Rosenblatt transformation [45], the first four moments, e.g., \(\mu_{\widetilde{g}}\), \(\sigma_{\widetilde{g}}\), \(\alpha_{3\widetilde{g}}\) and \(\alpha_{4\widetilde{g}}\) of the transformed performance function can be formulated as  
\[\mu_{\widetilde{g}}=\int_{D_{X}}\widetilde{g}(\mathbf{X})f_{\mathbf{x}}( \mathbf{x})d\mathbf{x}=\int_{D_{U}}\widetilde{g}_{u}(\mathbf{u})\phi( \mathbf{u})d\mathbf{u} \tag{16}\]  
\[\sigma_{\widetilde{g}}=\sqrt{\int_{D_{X}}\left[\widetilde{g}(\mathbf{x})-\mu_ {\widetilde{g}}\right]^{2}}f_{\mathbf{x}}(\mathbf{x})d\mathbf{x}=\sqrt{\int_{ D_{U}}\left[\widetilde{g}_{u}(\mathbf{u}))\right]-\mu_{\widetilde{g}}\right]^{2} }\phi(\mathbf{u})d\mathbf{u} \tag{17}\]  
\[\alpha_{3\widetilde{g}}=\frac{1}{\sigma_{\widetilde{g}}^{3}}\int_{D_{X}}\left[ \widetilde{g}(\mathbf{X})-\mu_{\widetilde{g}}\right]^{3}}f_{\mathbf{x}}( \mathbf{x})d\mathbf{x}=\frac{1}{\sigma_{\widetilde{g}}^{3}}\int_{D_{U}}\left[ G_{u}(\mathbf{u})-\mu_{\widetilde{g}}\right]^{3}\phi(\mathbf{u})d\mathbf{u} \tag{18}\]  
\[\alpha_{4\widetilde{g}}=\frac{1}{\sigma_{\widetilde{g}}^{4}}\int_{D_{X}}\left[ G(\mathbf{x})-\mu_{\widetilde{g}}\right]^{4}}f_{\mathbf{X}}(\mathbf{x})d \mathbf{x}=\frac{1}{\sigma_{\widetilde{g}}^{4}}\int_{D_{U}}\left[ \widetilde{g}_{u}(\mathbf{u})-\mu_{\widetilde{g}}\right]^{4}\phi(\mathbf{u})d \mathbf{u} \tag{19}\]  
where \(D_{X}=\) the domain of \(X\), \(D_{U}=\) the domain of \(U\), and \(U\) is a random variable with independent standard normal distribution; \(\widetilde{g}_{u}(\cdot)\) is the performance function in independent standard normal space using normal transformation technology; \(\phi(\cdot)\) is the joint PDF of independent standard normal random variables.  
To obtain the first four moments, the HUT is introduced to generate the sigma points and corresponding weights. Generally, there are \(p=2n^{2}+1\) sigma points in total. For simplicity, the derivation process for the sigma points and weights is not given, and the details of the derivation can be found by referring to [40]. The sigma points, weights, and the first four moments of the transformed performance function are provided in Appendix A.  
A flowchart for the first four moment estimates of the transformed performance function using dual power transformation and HUT is illustrated in Figure 5. The flowchart consists of two parts. The first part of the process involves applying the dual power transformation to the performance function. The dual power transformation applies a parameter \(q\), which is iteratively optimized to achieve the desired distribution properties, such as reduced skewness and improved symmetry. The second part focuses on calculating the first four central moments of the transformed performance function. Using the HUT approach, the first four moments are calculated. These moments include the mean, standard deviation, skewness, and kurtosis, which are crucial for assessing the reliability of the system.