## 4 Confidence Interval of Damage: Non-Stationary Case
### Confidence Interval of Damage for a Switching Loading with Two or More States  
Thanks to Equation (13), the results of Section 3.2 are now extended to a non-stationary stress-time history \(z(t)\) of duration \(T\) that switches between a finite number \(N_{S}\) of stationary states--an example with three states (1, 2, 3) is depicted in Figure 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{5}{c}{**\(i\)-th Random Variable \(X_{i}\)**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Expected value, \(\mu_{i}\) & 0 & 0 & 0 & 0 & 1 & 1 \\ Variance, \(\sigma_{i}^{2}\) & 1 & 5 & 10 & 100 & 1 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Combinations of expected value \(\mu_{i}\) and variance \(\sigma_{i}^{2}\) of the first six random variables \(X_{i}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Sample Size, \(n\)**} & \multicolumn{5}{c}{**Number of Random Variables, \(N\)**} \\ \cline{2-6}  & **2** & **3** & **4** & **5** & **6** \\ \hline
2 & 98.47\% & 98.24\% & 95.96\% & 95.82\% & 94.90\% \\
10 & 95.08\% & 95.05\% & 95.08\% & 95.07\% & 95.11\% \\
100 & 95.00\% & 95.01\% & 95.00\% & 95.02\% & 95.00\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Observed confidence level \(100(1-\hat{\alpha})\%\) as a function of the number of random variables \(N\) and sample size \(n\). Compared to the stationary case, the procedure now requires an additional step in which the distinct stationary states in \(z(t)\) are first identified. This identification can be performed by checking for abrupt changes, for example, in the root-mean-square (rms) value in \(z(t)\), as it will be demonstrated in Section 6. Since each stationary state represents a portion of the whole record \(z(t)\), the total time length of \(z(t)\) sums the length of individual states as \(T=\sum_{i=1}^{N_{S}}T_{S,i}\). In a similar way, the total damage of \(z(t)\) is the summation of the damage of individual loading states as \(D(T)\cong\sum_{i=1}^{N_{S}}D_{S,i}(T_{S,i})\)[31]. The approximation sign in this equality is due to the fatigue cycles lost when separating the entire stress-time history \(z(t)\) into stationary segments and omitting the transitions cycles between them (an example is the range between the global maximum and global minimum if they do not occur within the same segment--typical for the ground-air-ground cycle in the aviation industry). On the other hand, since the number of segments or stationary states is generally small, the amount of these lost cycles--and the damage they contribute--is negligible compared to that of the multitude of other cycles counted within individual states. Assume that in the non-stationary stress-time history \(z(t)\), a total of \(N_{S}\) distinct stationary states with lengths \(T_{S,i}\), \(i=1,2,\ldots N_{S}\) have been identified. As in the stationary case, each state is further subdivided into \(N_{B}\) blocks of length \(T_{B,i}=T_{S,i}/N_{B}\), see Figure 1. Since distinct states have different lengths but share a common \(N_{B}\), block lengths in every state are in general different \(T_{B,i}\), \(i=1,2,\ldots N_{S}\). It is noted that the choice of a common \(N_{B}\) for all states is not strictly necessary, though it simplifies the following theoretical solution. State identification and block subdivision give the damage values \(D_{B,ij}(T_{B,i})\), with \(i=1,2,\ldots N_{S}\) (state index) and \(j=1,2,\ldots N_{B}\) (block index). Damage \(D_{B,ij}(T_{B,j})\) refers to block \(j\) in state \(i\). Note that \(D_{B,ij}(T_{B,i})\), \(j=1,2,\ldots N_{B}\) represent a sample of \(N_{B}\) observed values for the random variable \(D_{B,i}(T_{B,i})\) (block damage of state \(i\)), with expected value \(E[D_{B,i}(T_{B,i})]\).