# Bayesian Survival Analysis Model for Girth Weld Failure Prediction
## 3 Survival Probability and Cases  
Section 2 analyzes several key factors closely related to a girth-weld accident based on the \"tree-type\" accident cause theory. The prediction method is discussed in Section 2.3. It is well accepted that the rupture of a girth weld is often caused by multiple factors simultaneously. If each factor is independent of time, it will occur only under random conditions or conditions that cause fragility. It is assumed here that an undamaged pipe girth weld does not fail in service, and only a defective weld can be predicted to fail, as a consequence of the 7 factors mentioned in Section 2.1. The influence of different factors on the failure of a girth weld was analyzed by predicting the survival function factors. It is assumed that the largest number of factors causing a certain type of pipeline accident is m, and the pipeline state is found to have k factors. According to the occupation rate, the probability of occurrence of the accident is k/m. However, we should distinguish the extent to which each factor affects the failure. The survival function uses the staged Bayesian result in Figure 3. The 14 girth-weld failures with detailed information were selected for setting up the survival probability modeling in the \"infant mortality\" stage. These 14 failure cases, from a total of 64 failure cases, s occurred after less than 10 years of service. Table 2 shows the factors related to each case, with the top 7 factors as described in Section 3; 1 means that the corresponding factor is satisfied, 0 means that it is not satisfied, and 2 means that two elements of the same factors are satisfied. Firstly, according to the 14 accident samples, the life curve of every single factor leading to the failure of the girth weld was calculated, and the influence of each factor on the failure probability of the new pipe girth weld was determined. Then, the life curve of the girth failure was analyzed considering \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Lifetime (Year)** & **0–10** & **11–40** & **41–63** \\ \hline \multirow{2}{*}{**Type**} & Weibull distribution & Uniform distribution & Exponential distribution \\  & \(\exp(-\lambda t^{a})\) & \(Kt+C\) & \(a\exp(-\lambda t)\) \\ \hline
**Parameter** & \(\alpha\) & \(\lambda\) & K & C & \(a\) & \(\lambda\) \\ \hline
**Value** & 0.7298 & 0.1118 & \(-\)0.005796 & 0.642026 & 13.08 & 0.09138 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Survival function parameter of different stages. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Failure Casing** & **Failure Lifetime** & **Case 1** & **Case 2** & **Case 3** & **Case 4** & **Case 5** & **Case 6** & **Case 7** \\ \hline
1 & 0.35 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
2 & 0.5 & 1 & 1 & 0 & 1 & 2 & 0 & 0 \\
3 & 0.5 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\
4 & 0.  
all factors and comparing the effects on girth-weld failure of various combinations of them, that is, the probability of \"survival\". According to these assumptions, to directly consider the influence of multiple factors on a girth-weld function, let \(\lambda\) of staged Bayesian be:  
\[\lambda=r_{0}+r_{1}z_{1i}+r_{2}z_{2i}+r_{3}z_{3i}+r_{4}z_{4i}+r_{5}z_{5i}+r_{6}z _{6i}+r_{7}z_{7i} \tag{6}\]  
where:  
\(r_{1}\sim r_{7}\) Indicates the influence factor of seven different cases on the failure of the girth weld, \(r_{0}\) is a correlation factor between different factors,  
\(z_{1i}\sim z_{7i}\) represents the values corresponding to seven different Cases (see Table 2). The girth-weld survival function based on the failure case sample is determined by calculating \(r_{0}\) and \(r_{1}\sim r_{7}\) in Equation (6). Figure 4 shows the survival function of the girth weld under the influence of a single factor. As can be seen from Figure 4, the lowest probability of girth weld survival is when the physical properties of the weld do not meet the specifications, which means that it is the most critical of the seven factors; and the impact of Case1 is relatively small.