Reliability Assessment of Competitive Failure Systems Based on Three Parameter Weibull Distribution and Wiener Process

As themany examples in this book illustrate,Markov chainMonte Carlo (MCMC)methods have revolutionized Bayesian statistical analyses. Rather than using off-the-shelf models and methods, we can use MCMC to fit application-specific models that are designed to account for the particular complexities of a problem. These complex multilevel models are becoming more prevalent throughout the natural, social, and engineering sciences largely because of the ease of using standard MCMC methods such as the Gibbs and MetropolisHastings (MH) samplers. Indeed, the ability to easily fit statistical models that directly represent the complexity of a data-generationmechanismhas arguably lead to the increased popularity of Bayesian methods in many scientific disciplines. Although simple standard methods work surprisingly well in many problems, neitherthe Gibbs nor theMH sampler can directly handle problemswith very high posterior correlations among theparameters. Themarginal distribution of a givenparameter ismuchmore variable than the corresponding full conditional distribution in this case, causing the Gibbs sampler to take small steps. With MH a proposal distribution that does not account for the posterior correlation either has far toomuchmass in regions of low posterior probability or has such smallmarginal variances that only small steps are proposed, causing high rejection rates and/or high autocorrelations in the resultingMarkov chains. Unfortunately, accounting for the posterior correlation requires more information about the posterior distribution than is typically available when the proposal distribution is constructed. Much work has been devoted to developing computational methods that extend theusefulness of these standard tools in the presence of high correlations. For Gibbs sampling, for example, it is nowwell known that blocking or grouping steps (Liu et al., 1994), nesting steps (van Dyk, 2000), collapsing or marginalizing parameters (Liu et al., 1994; Meng and vanDyk, 1999), incorporatingauxiliaryvariables (BesagandGreen, 1993), certainparameter transformations (Gelfand et al., 1995; Yu and Meng, 2011), and parameter expansion (Liu andWu,1999) canall beused to improve the convergenceof certain samplers.Byembedding an MH sampler within the Gibbs sampler and updating one parameter at a time (i.e. the well-known Metropolis-within-Gibbs sampler in the terminology of Gilks et al., 1995), the same strategies can be used to improve MH samplers.

Bayesian statistics has attracted people's interest again recently because of the application of Markov Chain Monte Carlo (MCMC) theory. In particle filtering, the diversification of particles disappears in the process of importance sampling. However, this problem can be solved using Metropolis-Hastings (MH) sampling usually used in MCMC theory. As a modification to added MCMC (AMCMC)—an improved MCMC particle filter that can track variable number of targets at the same time, a new approach to optimize those rejected samples in MH sampling process by Mean Shift algorithm is proposed in this paper. Because the operation rate of particles in AMCMC is increased, the circles of sampling needed for the convergence of Markov chain is reduced. It is shown by experiment that, the optimized algorithm has better tracking performance under the condition of fewer particles.

Three-parameter vs. two-parameter Weibull distribution for pultruded composite material properties

Using Bayesian inference to calibrate constitutive model parameters has recently seen a rise in interest. The Markov chain Monte Carlo (MCMC) algorithm is one of the most commonly used methods to sample from the posterior. However, the choice of which MCMC algorithm to apply is typically pragmatic and based on considerations such as software availability and experience. We compare three commonly used MCMC algorithms: Metropolis-Hastings (MH), Affine Invariant Stretch Move (AISM) and No-U-Turn sampler (NUTS). For the comparison, we use the Kullback-Leibler (KL) divergence as a convergence criterion, which measures the statistical distance between the sampled and the ‘true’ posterior. We apply the Bayesian framework to a Newtonian squeeze flow problem, for which there exists an analytical model. Furthermore, we have collected experimental data using a tailored setup. The ground truth for the posterior is obtained by evaluating it on a uniform reference grid. We conclude that, for the same number of samples, the NUTS results in the lowest KL divergence, followed by the AISM sampler and last the MH sampler.

What makes new variants of SARS-CoV-2 concerning is not where they come from, but the mutations they contain What do the new SARS-CoV-2 mutations mean, and how should we track...

Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters $ \mu $ and $ \sigma^2 $ were provided and the inverse estimation of parameters $ \eta $ and $ \beta $ were derived. The exact confidence intervals for parameters $ \mu $, $ \sigma^2 $, and $ \beta $ were obtained. Furthermore, the generalized confidence interval of parameter $ \eta $ was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the $ p $th percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.

Structural engineering makes a giant stride forward with this symposium. This book represents an international sym­ posium held in Stockholm honoring Prof. W. Weibull. It drew participants from 21 countries with a total of 55 invited papers. There were 7 scientific sessions. Most of our studies in school revolved about deterministic happenings. Random loading becomes more and more important when we equate events in our world with physical and mathematical reasoning. Probabilistic failure model and stochastic fatigue crack growth play important roles in the safety of nuclear structures, random loading due to wind turbulence, earthquake spectra, offshore structural design, and aircraft design. The initial paper summarizes Prof. Weibull's life and ac­ complishments including a mention of Weibull distribution used in fatigue studies. The initial session deals with extreme value theory (EVT). The first paper reviewed applications of extreme value theory to material behavior and indicates the importance of the accompanying statistical theory. The next paper generalizes Weibull distribution in matrix form. The following papers proclaim (a) extreme value theory as com­ patible to random field theory, (b) incomplete samples can be transformed to a mixed-exponential survival for a more com­ prehensive Weibull distribution, (c) EVT in endurance testing of ball and roller bearings, (d) use of the Slepian model identi­ fying the behavior of Gaussian noise near or between its zero crossing, (e) plastic movement of SDOF elastic plastic oscillator subjected to stationary Gaussian process excitation, and (/) generalized Hermitian polynomial used in the pertur­ bation method for nonlinear random vibrations. The final paper reports on Weibull distribution in large earthquake modelling. Session 2 reports on fatigue crack growth. Beginning with an analysis of stochastic equation models of crack growth, it continues with stochastic models of fatigue crack growth and propagation, prediction of crack growth under spectrum loading employing a cycle-by-cycle technique and fatigue life distribution in a random loading sense with interacting failures. The final paper considers a new approach to fatigue crack propagation under random loading and the use of Monte Carlo simulation in correlating micro fracture process and fatigue crack propagation. This session indicates that more experimental work is required in crack propagation to reach the same level as crack initiation. Session 3 comprises a total of 13 papers and dwells upon probabilistic failure models. The initial paper proposes a generalized probabilistic model for the fatigue life and reliability prediction of a structure containing noninteracting cracks and employs Poisson random sets. The next group of papers focuses upon (a) structural reliability treated as a single stochastic model, (Jb) fatigue life and reliability estimates of mineral pipe lines using probability theory, (c) Weibull-based equation employed in predicting the failure of a stressed brittle material, (d) a preliminary suggestion that specimens failed in proof tests may be used to estimate the parameters of the reliability function for the survivors, and (e) brittle material design using three parametric Weibull distributions. The next set of papers considers ring-in-ring tests on strength of cladded glass using the three-parameter Weibull distribution, prob­ abilistic analysis of plastic plates utilizing yield-line theory plus lower bound reliability analysis of plastic plates evaluated by inclusion-exc lusion theory. The final set of papers ques­ tions the possibility of flaws in the present probabilistic models for unidirectional fibrous composites, strength rela­ tions for cracks in unidirectiona l long fiber composites employing Weibull distribution and a stochastic approach in studying the fracture and fatigue of concrete. Session 4 delves into probabilistic fracture mechanics. The initial paper estimates the probable failure of PWR pressure vessels employing Weibull distribution and updates the wellknown Marshall A(a) and B(a) functions. The final set of papers covers probabilistic assessment of structures with weld defects, probability of fracture in main coolant pipes of PWR, and statistical modelling of shaft predictions (as measured by Charpy impact curve) in the reference temperature of pressure vessel welds. The authors suggested that U.S. Nuclear Regulatory Guide 1.99 be updated to remove excessive conservatism.

Markov Chain Monte Carlo (MCMC) is widely applied to fit complex, practical models in various disciplines, such as engineering, physics, agriculture, etc. Indeed, when the target density $f$ is not available in closed form, a Markov chain is constructed with stationary density $f$ for forming Monte Carlo estimators of means with respect to $f$. Although a Harris ergodic Markov chain will converge to the target distribution as the Monte Carlo sample size tends to infinity, and the population means can be consistently estimated by the corresponding MCMC estimators, in practice, it is important to estimate the standard errors associated with these Monte Carlo estimates due to the availability of only a finite sample. Therefore, the geometric ergodicity of a Markov Chain is highly desired as it is the most standard method for guaranteeing a central limit theorem (CLT) for MCMC estimators and finding its standard errors. Among the different MCMC algorithms, Metropolis-Hastings (MH) algorithms are predominant. This thesis establishes conditions under which MH algorithms with a position-dependent proposal covariance matrix will or will not have the geometric convergence rate. Indeed, we provide these results for MH algorithms where the normal proposal density has a general mean function $c(x)$ and a covariance matrix $G(x)$ depending on the current position $x$ of the Markov chain. For modern variants of the popular Metropolis adjusted Langevin algorithms (MALA), like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for the geometric ergodicity of different variations of the Langevin algorithms, e.g., MMALA, position-dependent MALA (PMALA), and the pre-conditioned MALA (PCMALA). We also provide conditions guaranteeing geometric ergodicity of the pre-conditioned unadjusted Langevin algorithms (PCULA). Generalized linear mixed models (GLMMs), where the response variable is allowed to follow a non-Gaussian distribution, are popular statistical models, although their likelihood functions are not available in closed form. GLMMs can handle the over-dispersion often present in real-world data, and MCMC methods are generally used to fit frequentist and Bayesian GLMMs. Since the likelihood function for GLMMs is not available in closed form, Monte Carlo EM or Monte Carlo maximum likelihood methods are often used for making inferences from GLMMs. Simulation from the conditional (on the data) distribution of the random effects is, in turn, needed for implementing the Monte Carlo EM or Monte Carlo maximum likelihood methods. In this thesis, our general conditions for geometric ergodicity of variants of Langevin algorithms are also verified in the context of conditional simulation from the two most popular GLMMs, namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. These results have important practical implications as users can now confidently fit these models by attaching asymptotically valid standard errors with their Markov chain-based Monte Carlo estimates. Spatial generalized linear mixed models (SGLMMs) are GLMMs where the random effects form a spatial process and are useful for modeling spatially correlated data. We compare the performance of different variants of MALA mentioned above as well as other popular MH algorithms like Random Walk Metropolis in the context of conditional simulation from SGLMMs. Next, we consider MCMC algorithms for full Bayesian analysis of GLMMs. Blocking and reparameterization are techniques used to improve the efficiency and convergence properties of MCMC algorithms. We combine these strategies with pre-conditioning to develop efficient Langevin algorithms for Bayesian GLMMs and SGLMMs. We illustrate our algorithms using extensive real and simulated data analyses.

Patz and Junker (1999) describe a general Markov chain Monte Carlo (MCMC) strategy, based on Metropolis-Hastings sampling, for Bayesian inference in complex item response theory (IRT) settings. They demonstrate the basic methodology using the two-parameter logistic (2PL) model. In this paper we extend their basic MCMC methodology to address issues such as non-response, designed missingness, multiple raters, guessing behavior and partial credit (polytomous) test items. We apply the basic MCMC methodology to two examples from the National Assessment of Educational Progress 1992 Trial State Assessment in Reading: (a) a multiple item format (2PL, 3PL, and generalized partial credit) subtest with missing response data; and (b) a sequence of rated, dichotomous short-response items, using a new IRT model called the generalized linear logistic test model (GLLTM).

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

Monte Carlo Markov chain methods and model selection in genetic epidemiology

The operation of a large industrial or other complex mechanical system incorporates a variety of mechanical and electrical subsystems to perform a given task, some of which require interaction by an operator or worker to oversee and control the process. As with anything mechanical, the system also requires periodic maintenance and replacement of worn parts. During the operation of industrial systems, problems can occur and result in catastrophic failures and/or injury. When applying forensic analysis to such failures, a methodical approach is necessary to allow for a deeper understanding of the overall operation of the system. However, once the forensic analysis has been performed, conveying the findings of such a complex system can be challenging. To assist in describing the system and failure, the use of visualization is a powerful tool to clearly convey the findings as well as normal operation. The following paper outlines the process of building a methodology to investigate and lay the proper foundation for visually presenting the findings. To demonstrate the methodology outlined in this paper, a case study involving a boiler system will be used.

Groundwater pollution source identification using Metropolis-Hasting algorithm combined with Kalman filter algorithm

This chapter provides a brief introduction to the use of Bayesian graphical models in R. In these models, parameters are treated as random quantities on an equal footing with the random variables. This allows complex stochastic systems to modeled, often using Markov chain Monte Carlo (MCMC) sampling methods. We first consider a series of examples, ranging from simple repeated sampling, linear regression models, random coefficient regression models, to the chest clinic example of Chap. 3. We formulate these as Bayesian graphical models, and represent them graphically in a compact form using plates. We then describe a special case, in which the unknown parameters are all discrete, and explain how probability propagation methods described in Chap. 3 may be used to compute the posterior distributions. We then turn to the general case, when MCMC sampling is required, and explain the computations involved in Metropolis-Hastings sampling and Gibbs sampling. Finally we illustrate a Bayesian linear regression analysis using the R2WinBUGS package.

The vibrational frequency method was used to measure the elastic modulus of laminated veneer lumber (LVL), and the feasibility of using Weibull distribution to analyze the elastic modulus data of LVL was considered. Samples were randomly selected as test pieces at the factory. The sponge support structure was used to realize the free beam state, and the modal test results verified the accuracy of realizing the free beam. Under transient excitation, the elastic modulus of the specimen was obtained by testing the first-order bending frequency. The Weibull distribution fitting test, Weibull distribution K-S test, and normal distribution K-S test were used for the test data. The probability of LVL elastic modulus was calculated under a given value. The results showed that the LVL elastic modulus did not obey the two-parameter Weibull distribution (Eu=0). The LVL elastic modulus fit to the three-parameter Weibull distribution (Eu) was greater than half of the minimum test value and the normal distribution. When 9 GPa and 8 GPa were used as the setting values of Eu, the calculated probability value was relatively stable. At this time, Eu was 81% and 92% of the minimum elastic modulus 9.815 GPa. Therefore, it was recommended to use 80% to 90% of the minimum value of the measured data as the setting value of the position parameter Eu. The three-parameter Weibull distribution and the normal distribution calculated LVL elastic modulus have the same probability under the given value.