## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
In addition, the samples of most parameters are not updated until the end of iterations (also referred to as the'standing still' phenomenon) and converge around unity. This may be explained by the fact that (1) DRAM cannot effectively adjust proposal variances for high-dimensional problems, although it combines DR and AM strategies; and (2) as a first example, the initial values greatly affect the performance of DRAM based on the observation that most parameters are estimated to be close to the initial value of unity. It therefore can be concluded that, in this example, DRAM is unable to accurately estimate parameters and detect damage given available modal data. Figure 1:3 Trace plots by DRAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations. Figure 1:4 Three-measurement setup for a concrete beam. On the other hand, it can be seen from Figures 12 and 13 that the Markov chains obtained through TMCMC and DREAM continue to fluctuate during the process of sampling and do not exhibit the phenomenon of'sampling stagnation' or'standing still', indicating the generation of valid candidate samples. It can also be seen that the TMCMC has a faster convergence to a stable state compared to DREAM. However, the samples of some parameters at damaged locations significantly diverge from the true values, see the green and cyan samples in Figure 12b, while the Markov chains by DREAM start to converge at around 110,000 iterations. Ultimately, samples of all 30 stiffness parameters fluctuate in close proximity to true values. The slighter fluctuation in Figure 13 compared to those in Figure 12 also illustrates smaller uncertainty and more reliability in parameter estimation using DREAM. Figure 14 displays the sampling process of five Markov chains using DREAM for stiffness parameters at damaged locations. It can be seen that each chain consistently converges to a similar value with another, which also proves the reliability of damage detection by DREAM. Figure 1:5 Trace plots by TMCMC: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value). Figure 1:6 Trace plots by DREAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value). The marginal posterior distributions of model parameters are shown in Figure 1:7 Note that only the stiffness parameters at damaged elements, e.g., \(\theta_{2}\), \(\theta_{6}\), \(\theta_{13}\), \(\theta_{19}\), \(\theta_{24}\), and \(\theta_{28}\), and one stiffness parameter at intact elements, e.g., \(\theta_{23}\), are present for the sake of brevity. The results of DRAM are ignored in Figure 15 due to its inability to achieve high-dimensional parameter estimation. As observed, all diagonal distributions obtained by DREAM cover the ground truth, while TMCMC results in remarkable deviations of parameter estimation, such as \(\theta_{23}\) and \(\theta_{28}\). In addition, the scatter plots in the upper triangle of TMCMC are more widespread compared to those in DREAM, implying that the resulting samples by TMCMC encompass larger uncertainties, which is also reflected by the trace plots exhibiting greater fluctuations in Figure 12 compared to those in Figure 1:8 A comparison of the estimation errors of the three algorithms is reported in Figure 1:9 DRAM performs worst for damage detection, giving a maximum error of over 40% and completely incorrect detection. The estimation error using TMCMC is up to around 10%. DREAM shows the best performance in damage detection, and its maximum estimation error is less than 5%. The results of damage detection demonstrate that the samples obtained by DREAM with fusing information from multi-chains have higher convergence accuracy. The interactions between Markov chains allow the enhancement of the quality of posterior samples. For computation, the three algorithms were implemented on the same personal laptop. DRAM is the least computationally costly, only requiring about 10 min to evaluate a model 40,000 times.