Use modified ABC-MCMC algorithm to obtain posterior samples of \(\theta = (\mu_a, \sigma_a, \mu_b, \sigma_b, \mu_c, \sigma_c, \mu_n, \sigma_n, \mu_{\sigma_0} , \sigma_{\sigma_0})\), given ramp and constant load failure time data.
abcMCMC(n, numBurning, numThining, inputD, dataNames, verbose = FALSE)number of posterior samples
number of burn-in iterations
number of thining iterations
bandwidth \(\delta\) for ABC approximation
a vector of strings of the names of the datasets, which must be in the format "ID_(\(\tau_c\))_(t_c)Y" (see Details and Example)
displays information messages to console if TRUE
Returns a matrix of posterior samples where each row is one \(\theta\), and if verbose is TRUE, prints the acceptance rate.
The generated posterior samples are the parameters associated with (a, b, c, n, \(\eta\)), which are the random effects in the Canadian Model for load duration, $$\frac{d}{dt} \alpha(t) = [(a\tau_s)(\tau(t)/\tau_s - \sigma_0)_+]^b + [(c\tau_s)(\tau(t)/\tau_s - \sigma_0)_+]^n\alpha(t),$$ where
- \(a|\mu_a, \sigma_a \sim Log-Normal(\mu_a, \sigma_a)\);
- \(b|\mu_b, \sigma_b \sim Log-Normal(\mu_b, \sigma_b)\);
- \(c|\mu_c, \sigma_c \sim Log-Normal(\mu_c, \sigma_c)\);
- \(n|\mu_n, \sigma_n \sim Log-Normal(\mu_n, \sigma_n)\);
- \(\eta|\mu_{\sigma_0}, \sigma_{\sigma_0} \sim Log-Normal(\mu_{\sigma_0}, \sigma_{\sigma_0})\) and set \(\sigma_0 = \frac{\eta}{1+\eta}\).
* \((x)_+ = max(x, 0)\).
* \(\sigma_0\) serves as the stress ratio threshold in that damage starts to accumulate only when \(\tau(t)/\tau_s > \sigma_0\).
* When sample pieces are subject to the load profile
\(\tau(t) = kt\) if \(t \le T_0\)
\(\tau(t) = \tau_c\) if \(t > T_0\)
where \(\tau_c\) is the selected constant-load level under the ramp-loading rate k, and \(T_0\) is the time required for the load to reach \(\tau_c\) under the ramp-loading rate k.
* The constant load level is assumed to be reached at the ramp-loading rate (k). The ramp-loading rate is 388,440 psi/hour.
* The constant load test ends at time \(t_c\) (in years).
* To achieve a ramp-load test, set \(\tau_c\) to Inf.
Foschi, R. O., Folz, B., and Yao, F. (1989), Reliability-Based Design of Wood Structures (Vol. 34), Vancouver, BC: Department of Civil Engineering, University of British Columbia.
Wong, S. W., & Zidek, J. V. (2018). Dimensional and statistical foundations for accumulated damage models. Wood science and technology, 52(1), 45-65.
Yang, C. H., Zidek, J. V., & Wong, S. W. (2019). Bayesian analysis of accumulated damage models in lumber reliability. Technometrics, 61(2), 233-245.
# NOT RUN {
# run the abc-mcmc algorithm to obtain 10 posterior samples
# example only, more iterations needed for convergence
resTheta = abcMCMC(10, 100, 10, 0.3, c("constLoad_4500_1Y"), TRUE)
# }
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