abcMCMC {abcADM}R Documentation

ABC-MCMC for load duration experiment data

Description

Use modified ABC-MCMC algorithm to obtain posterior samples of θ = (μ_a, σ_a, μ_b, σ_b, μ_c, σ_c, μ_n, σ_n, μ_σ0 , σ_σ0), given ramp and constant load failure time data.

Usage

abcMCMC(n, numBurning, numThining, inputD, dataNames, verbose = FALSE)

Arguments

n

number of posterior samples

numBurning

number of burn-in iterations

numThining

number of thining iterations

inputD

bandwidth δ for ABC approximation

dataNames

a vector of strings of the names of the datasets, which must be in the format "ID_(τ_c)_(t_c)Y" (see Details and Example)

verbose

displays information messages to console if TRUE

Details

The generated posterior samples are the parameters associated with (a, b, c, n, η), which are the random effects in the Canadian Model for load duration,

d/dt α(t) = [(a * τ_s) * (τ(t) / τ_s - σ_0)_+]^b + [(c * τ_s) * (τ(t) / τ_s - σ_0)_+]^n * α(t),

where

- a|μ_a, σ_a ~ Log-Normal(μ_a, σ_a);

- b|μ_b, σ_b ~ Log-Normal(μ_b, σ_b);

- c|μ_c, σ_c ~ Log-Normal(μ_c, σ_c);

- n|μ_n, σ_n ~ Log-Normal(μ_n, σ_n);

- η|μ_σ0, σ_σ0 ~ Log-Normal(μ_σ0, σ_σ0) and set σ_0 = η / (1 + η).

* (x)_+ = max(x, 0).

* σ_0 serves as the stress ratio threshold in that damage starts to accumulate only when τ(t)/τ_s > σ_0.

* When sample pieces are subject to the load profile

τ(t) = kt if t <= T_0

τ(t) = τ_c if t > T_0

where τ_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 τ_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 τ_c to Inf.

Value

Returns a matrix of posterior samples where each row is one θ, and if verbose is TRUE, prints the acceptance rate.

References

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.

Examples

# 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)



[Package abcADM version 1.0 Index]