## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
A reason for this is due to DRAM's incapability of dealing with high-dimensional problems, which also corresponds with the conclusion in [PERSON] et al. [50]. Furthermore, the performance of DRAM heavily depends on the choice of initial values [70]. For instance, the initial value of unity estimates all parameters as being close to unity in this example. Figure 2: Setup plan for the shear building. In contrast, it can be found in Figures 4 and 5, where dashed lines denote the true values, that TMCMC and DREAM exhibit better damage detection performance than DRAM. It can be seen that samples using the two methods quickly reach a stationary state, displaying satisfactory convergence. However, two parameters in TMCMC at damaged locations are not very close to the true values, but the two methods are not very close to the true values. Figure 4: Trace plots by TMCMC: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value). Figure 5: Trace plots by DREAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value). Figure 3: Trace plots by DRAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations. locations converge to around 0.9 in Figure 4b (black and green trace plots), implying deviation from true values in Table 1. The estimation of all parameters at damaged locations by DREAM in Figure 5b has a desirable agreement with the ground truth in Table 1. Figure 6 shows the trace plots of five Markov chains for four parameters at damaged locations. It can be observed that each Markov chain consistently exhibits fast convergence to the true values, demonstrating the collective power in sampling by DREAM due to the interaction of multiple chains. Figure 7 shows the marginal distributions of unknown model parameters using the last 10,000 samples obtained by TMCMC and DREAM. Note that the results by DRAM are not included in Figure 7, as all model parameters are estimated as being in unity by DRAM. In addition, for a clear visualization, only parameters at damaged locations, e.g., \(\theta_{1}\), \(\theta_{12}\), \(\theta_{25}\), and \(\theta_{38}\), and two parameters at undamaged locations, e.g., \(\theta_{4}\), \(\theta_{20}\), and \(\theta_{38}\), are manifested. It is revealed that the true values consistently cross posterior distributions at a diagonal by DREAM, illustrating a fairly good match between posterior means and true values. On the other hand, the posterior distributions by TMCMC noticeably deviate from true values for certain parameters, e.g., \(\theta_{4}\), \(\theta_{20}\), and \(\theta_{38}\), indicating that TMCMC may provide false alarms for damage detection. It can also be observed that the distributions of \(\theta_{4}\), \(\theta_{20}\), and \(\theta_{38}\) are spread across a relatively wide region, suggesting larger uncertainties surrounding these parameters, which is also reflected by significant fluctuations in Figures 4a and 5a. Figure 8 compares the error of parameter estimation by DRAM, TMCMC, and DREAM. It was found that DRAM performs worst in damage detection, giving the maximum error of about 25%. It is worth mentioning that in DRAM, small errors for parameters at undamaged locations mainly arise from the initial value of unity. TMCMC offers acceptable damage detection results for most parameters, but the maximum error is still around 15%, e.g., in \(\theta_{19}\) and \(\theta_{40}\), while DREAM works the best in damage detection with a maximum error of 4.8%, demonstrating that DREAM has superior identification capabilities for high-dimensional problems (40 model parameters), which is attributed to the use of multiple chains to sufficiently explore parameter space. For the computation of the three algorithms DRAM, TMCMC, and DREAM, we used a personal MacBook, 2017, with a 3.1 GHz Dual-Core Intel Core i5.