MCMC simulation

The adaptive MCMC simulation method is applied to updating the robust reliability for a two-story structural frame, depicted in Figure 2.21. The bay width and story height are 5.0 m and 2.5 m, respectively. The Young s modulus and mass density are taken to be 200 GPa and 7800 kg/m, respectively. The beams have a cross-sectional area of 0.01 vn5 and a moment of inertia of 6.0 X 10 m but they are 0.02 m and 1.5 x 10 m" for the columns. As a result, the structure has modal frequencies of 5.20 and 15.4 Hz. The structure is assumed to have 1.0% of critical damping for all modes. A simple model with two degrees of freedom is used in the system identification in order to estimate the inter-story stiffnesses and to assess the reliability of the structure. Specifically, the stiffness matrix is given by ... 

Keywords Reliabihty growth Model Bayesian analysis Dirichlet distribution MCMC simulation Gibbs... 

Stochastic Network Model for the Pore Phase 689 24.3.5.1 MCMC Simulation for Edge Rearrangement... 

Figure 24.16 Edge length distributions (a) from the modified graph and a fitted (shifted) P-distribution (black solid line) (b) from the graph generated by the Hakimi-Havel algorithm (c) from the graph after MCMC-simulation. Copyright (2011) ]. Mater. Sd.

The resulting distribution of edge lengths after the MCMC simulation as described in the previous section can be seen in Figure 24.16c, where the fitted gamma distribution is also plotted. 

A realization of the random geometric graph model, where the edge lengths have been fitted to real data by the above-described MCMC simulation, can be seen in Figure 24.17. Note that Figure 24.17 just shows a small cutout of a realization of the network model describing the pore phase of non-woven GDL. 

MARKOV CHAIN MONTE CARLO (MCMC) SIMULATION... 

Generating Samples in Robust Reliability Based on Adaptive MCMC Simulation Technique... 

Therefore, the data do appear to be consistent with an elementary reaction. Below, we test this hypothesis more rigorously using MCMC simulation with the exact posterior. 

While analytical manipulation of this formula is difficult, MCMC simulation is a powerful tool to obtain posterior expectations of the form... 

When the dimension of 9, a) space is small, it may be possible to compute (8.152) by quadrature, but MCMC simulation is generally more efficient. 

To compute an expectation [g y] using MCMC simulation, we rewrite (8.149) to integrate over all values of [Pg.404]

We use MCMC simulation to generate a sequence (0[ "1, o-t " ) that is distributed according to 7Ts(9, a y), such that for a large number Ns of samples, the expectation is approximately... 

MCOPTS is a structure that controls the operation of the MCMC simulation and contains the following fields (and default values) ... 

We now compute the probability that 02 > 5 that is, that the protein expression of the mutant strain is five units above that of the wild-type strain. We nse MCMC simulation to compute the expectation of the indicator... 

Figure 8.6 Margiaal 2-D posterior density for the protein expression data, computed fiom MCMC simulation.

We next use the MCMC approach to study again the hypothesis that the reaction A + B C is elementary, given the data in Table 8.1. We compute the 95% HPD for the 2-D maiginal posterior density p(92,di y). If this HPD region contains the point ( 2 = 1, 3 = 1), we cannot support the conclusion that the hypothesis is false (i.e., the reaction is not elementary). We compute this marginal density using MCMC simulation 1 ... 

MCMC simulation with the multiresponse marginal posterior density... 

Figure 8.9 Marginal 1-D density for ki computed from multiresponse data of a chemical reaction using MCMC simulation.

D marginal posterior densities and HPD regions are computed from MCMC simulation for multiresponse data using the routines... 

From MCMC simulation, we can compute posterior expectations and 1-D and 2-D... 

Here we provide an approximation of the Bayes factor for single-response data that does not require MCMC simulation. Let the sum of squared errors for a model be... 

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