## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
In this example, a total of 40 stiffness parameters corresponding to each floor are updated using measured modal data, i.e., \(K_{1}\sim K_{40}\), and the mass properties, i.e., \(M_{1}\sim M_{40}\), are well-known and not updated in order to avoid an unidentifiable situation resulting from the coupling effect [66]. We assume that twenty independent measurements are performed. For each measurement, only the first eight natural frequencies and mode shapes are available due to the limited number of sensors and the inaccuracy and difficulty in identifying higher modes. To simulate realistic measurement, Gaussian white noise with zero mean and 1% COV of frequencies and mode shapes is added to exact frequencies and mode shapes [67; 68]. The sample variance \(\sigma_{f}^{2}\) of each natural frequency and sample covariance matrix \(\Sigma_{\mathbf{\phi}}\) of each mode shape in Equation (9) can be calculated using twenty sets of modal data. In addition, mode shapes are usually incomplete with missing components; that is, only a few DOFs at test points are measured, as practical measurement is usually taken at restricted locations. For the purpose of vibration-based Bayesian model updating and damage detection, it is beneficial to expand reduced mode shapes onto complete mode shapes with full DOFs. In this example, multiple measurement setups are designed to identify full modal shapes by assembling local ones. Figure 2 schematically shows the setup plan. The 40 DOFs are covered by three setups, each sharing four reference sensors and including twelve rover sensors. Partial/local modal shapes corresponding to different measurement setups contain 12 components related to 12 DOFs. The acquired local mode shapes are then assembled to obtain global mode shapes with 40 DOFs using the least squares method [69]. Figure 1: Forty-story shear building and 40-DOF model. In this example, a single damage scenario with multiple damaged locations is intentionally introduced, as shown in Table 1. The negative sign indicates the stiffness reduction. Herein, we define the 40 stiffness parameters \(\theta_{1}\sim\theta_{40}\) from the bottom to the top floor, and each parameter is a ratio of true stiffness to nominal stiffness (356 MN/m). Hence, the ground truths of other parameters at undamaged locations are in unity. The first eight natural frequencies and modal shapes corresponding to the damage scenario are used for damage detection with DRAM, TMCMC, and DREAM. #### 4.1.2 Results

The initialization of DRAM, TMCMC, and DREAM is specified as follows. For DRAM, the initial covariance is defined as a 40-dimensional unit diagonal matrix scaled by a factor of 0.05. The parameter dimension in this example is 40, as recommended by [PERSON] et al. [18], and when the number of dimensions exceeds 20, the non-adaptation period for AM can be taken as 1000. Otherwise, it is taken as 200. The two-stage DR process is implemented with a scaling parameter of 0.1. The initial values for all unknown parameters are set to unity (1), and a sample size of 40,000 is generated. For TMCMC, this example sets the lower and upper bounds of uncertainty parameters as 0.7 and 1.2, respectively. The length of the Markov chain is set as 40,000. The scaling factor \(\beta\) in Equation (22) is defined as a default value of 0.2. For DREAM, ten Markov chains are used in parallel to sample parameters. Each chain has a length of 40,000. The initial sample population is generated by a Latin hypercube \(\theta\in[0.7,1.2]^{40}\). Other parameters in DREAM are used as default settings as suggested by [PERSON] et al. [50], e.g., \(n_{cr}=3\) in Equation (25). Figures 3-5 illustrate the resulting trace plots by DRAM, TMCMC, and DREAM for all uncertainty parameters. It can be observed that all stiffness parameters are estimated to be approximately one by DRAM, indicating that DRAM fails to accurately localize and quantify damage scenarios.