‘Expert information’ versus ‘expert opinions’. Another approach to the problem of eliciting/ combining/using expert knowledge in PRA

In the traditional approach to eliciting expert knowledge for use in risk assessment and decision analysis, the expert is asked for his opinion about, say, the numerical value of some unknown parameter λ. This opinion is then expressed as a point estimate, λ_i, or a probabilistic estimate, P_i(λ). Much attention and debate is then given, in the traditional approach, to methods of weighing and combining the opinions from the individual experts.The present paper advocates another approach in which we ask each expert, instead, for his body of evidence, E_i, relevant to the value of λ. In this way, the approach first arrives at a consensual body of evidence, E − {E_i}, and second, at a consensual curve p(λ|E) that expresses our knowledge about λ based on that body of evidence.The essential difference between this ‘expert information’ approach and the traditional ‘expert opinion’ approaches may be captured in the slogan: lsWeigh evidence, not experts!’     

A Bayes approach to reliability prediction utilizing data from accelerated life tests and field failure observations

A Bayes approach is proposed to improve product reliability prediction by integrating failure information from both the field performance data and the accelerated life testing data. It is found that a product's field failure characteristic may not be directly extrapolated from the accelerated life testing results because of the variation of field use condition that cannot be replicated in the lab‐test environment. A calibration factor is introduced to model the effect of uncertainty of field stress on product lifetime. It is useful when the field performance of a new product needs to be inferred from its accelerated life test results and this product will be used in the same environment where the field failure data of older products are available. The proposed Bayes approach provides a proper mechanism of fusing information from various sources. The statistical inference procedure is carried out through the Markov chain Monte Carlo method. An example of an electronic device is provided to illustrate the use of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.     

Accelerated Reliability Demonstration Testing Design Based on Reliability Allocation of Environmental Stresses

The traditional reliability demonstration testing based on statistical method requires a large number of samples and long testing time, failing to satisfy the demand for short cycle and low cost. This paper proposes a new accelerated approach for determining reliability target of each environment stress, accelerated test profile, and comprehensive acceleration factor for multi‐failure mode product to conduct assembly level accelerated demonstration testing under multiple stresses and levels. By decomposing the product from four levels, namely function, structure, mechanism, and stress, the products' weaknesses can be identified. The main failure modes and sensitive environmental stresses are determined based on environmental profile and FMMEA. In this design, the reliability is apportioned to each actual environmental stress by AHP. And the SSI model is used to establish accelerated stress profile. The overall acceleration factor can be derived from the model of assembly level accelerated testing. Combined with a statistical plan, the accelerated reliability demonstration testing plan and test profile is built. A case example is presented to illustrate the effectiveness of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd.     

An Approach for Reliability Demonstration Test Based on Power‐Law Growth Model

Reliability demonstration test (RDT) is a critical and necessary step before the acceptance of an industrial system. Generally, a RDT focuses on designing a test plan through which one can judge whether the system reliability indices meet specific requirements. There are many established RDT plans, but few have incorporated the reliability growth aspects of the corresponding products. In this paper, we examine a comprehensive test plan that involves information concerning the reliability growth stage. An approach for RDT under the assumption of the power‐law model is proposed. It combines data related to the growth stage with those pertaining to the test stage of the product to reduce the cost of the test. Through simulation studies and numerical examples, we illustrate the characteristics of the test plan and significant reduction in test costs through our approach. Copyright © 2017 John Wiley & Sons, Ltd.     

A general Bayes weibull inference model for accelerated life testing

This article presents the development of a general Bayes inference model for accelerated life testing. The failure times at a constant stress level are assumed to belong to a Weibull distribution, but the specification of strict adherence to a parametric time-transformation function is not required. Rather, prior information is used to indirectly define a multivariate prior distribution for the scale parameters at the various stress levels and the common shape parameter. Using the approach, Bayes point estimates as well as probability statements for use-stress (and accelerated) life parameters may be inferred from a host of testing scenarios. The inference procedure accommodates both the interval data sampling strategy and type I censored sampling strategy for the collection of ALT test data. The inference procedure uses the well-known MCMC (Markov Chain Monte Carlo) methods to derive posterior approximations. The approach is illustrated with an example.     

Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.     

Assessing high reliability via Bayesian approach and accelerated tests

Sometimes the assessment of very high reliability levels is difficult for the following main reasons:

the high reliability level of each item makes it impossible to obtain, in a reasonably short time, a sufficient number of failures;

the high cost of the high reliability items to submit to life tests makes it unfeasible to collect enough data for ‘classical’ statistical analyses.

In the above context, this paper presents a Bayesian solution to the problem of estimation of the parameters of the Weibull–inverse power law model, on the basis of a limited number (say six) of life tests, carried out at different stress levels, all higher than the normal one.The over-stressed (i.e. accelerated) tests allow the use of experimental data obtained in a reasonably short time. The Bayesian approach enables one to reduce the required number of failures adding to the failure information the available a priori engineers' knowledge. This engineers' involvement conforms to the most advanced management policy that aims at involving everyone's commitment in order to obtain total quality.A Monte Carlo study of the non-asymptotic properties of the proposed estimators and a comparison with the properties of maximum likelihood estimators closes the work.     

Goodness-of-fit tests in parametric regression based on the estimation of the error distribution

Consider a heteroscedastic regression model Y=m(X)+σ(X)ε, where m(X)=E(Y|X) and σ ²(X)=Var (Y|X) are unknown, and the error ε is independent of the covariate X. We propose a new type of test statistic for testing whether the regression curve m(⋅) belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.     

Uncertainty in counts and operating time in estimating Poisson occurrence rates

When quantifying a plant-specific Poisson event occurrence rate λ in PRA studies, it is sometimes the case that either the reported plant-specific number of events x or the operating time t (or both) are uncertain. We present a Bayesian Markov chain Monte Carlo method that can be used to obtain the required average posterior distribution of λ which reflects the corresponding uncertainty in x and/or t. The method improves upon existing methods and is also easy to implement using hierarchical Bayesian software that is freely available from the Web.     

Parametric estimation of time-dependent failure rates for probabilistic risk assessment

This article describes a method for modeling a time-dependent failure rate, based on field data from similar components that are repeatedly restored to service after they fail. The method assumes that the failures follow a Poisson process, but allows many parametric models for the time-dependent failure rate λ(t). Ways to check many of the assumptions form an important part of the method. When the assumptions are satisfied, the maximum likelihood estimate of λ(t) is approximately lognormal, and this lognormal distribution is appropriate to use as a Bayesian distribution for λ(t) in a probabilistic risk assessment.     

How much information can be obtained from tracking the position of the leading edge in a scratch assay?

Moving cell fronts are an essential feature of wound healing, development and disease. The rate at which a cell front moves is driven, in part, by the cell motility, quantified in terms of the cell diffusivity D, and the cell proliferation rate λ. Scratch assays are a commonly reported procedure used to investigate the motion of cell fronts where an initial cell monolayer is scratched, and the motion of the front is monitored over a short period of time, often less than 24 h. The simplest way of quantifying a scratch assay is to monitor the progression of the leading edge. Use of leading edge data is very convenient because, unlike other methods, it is non-destructive and does not require labelling, tracking or counting individual cells among the population. In this work, we study short-time leading edge data in a scratch assay using a discrete mathematical model and automated image analysis with the aim of investigating whether such data allow us to reliably identify D and λ. Using a naive calibration approach where we simply scan the relevant region of the (D, λ) parameter space, we show that there are many choices of D and λ for which our model produces indistinguishable short-time leading edge data. Therefore, without due care, it is impossible to estimate D and λ from this kind of data. To address this, we present a modified approach accounting for the fact that cell motility occurs over a much shorter time scale than proliferation. Using this information, we divide the duration of the experiment into two periods, and we estimate D using data from the first period, whereas we estimate λ using data from the second period. We confirm the accuracy of our approach using in silico data and a new set of in vitro data, which shows that our method recovers estimates of D and λ that are consistent with previously reported values except that that our approach is fast, inexpensive, non-destructive and avoids the need for cell labelling and cell counting.     

The effect of the loading rate on the fracture toughness of Poly(methyl methacrylate), Polyacetal,Polyetheretherketone and modified PVC

The mode-l-plane-strain fracture toughness at initiation of various engineering plastics was determined for test speeds between 10^–4ms^–1 and 1.0 ms^–1, using a high-speed, servohydraulic testing apparatus. At high rates of testing, the transient acceleration of the specimen was reduced by the use of a damping technique aimed at overcoming dynamic effects. The results obtained are correlated with fractographic analysis performed by scanning electron microscopy (SEM) and by digital image analysis of macroscopic fracture surfaces. The results show that for some materials the high values of K _lc measured at low testing rates are associated with deformation detectable by either method. The extent of such deformation tends to be reduced as the testing velocity is increased and the fracture toughness drops.     

Possible Experiment for Determination of the Role of Microscopic Vortex Rings in the λ-transition in He-II

It is suggested that microscopic vortex rings (MVR) play an important role in the λ-transition in helium-II and substantially determine the value of T_λ. For very thin films of He-II, with thickness d less than the size of the smallest MVR, the rings do not fit in and, therefore, do not exist in such films. Consequently, for superfluid films of He-II, a peculiarity in the form of a smoothed-out jump should be observed in the curve T_m(d) at the values of thickness approximately equal to the size of the smallest M V R, d ≈ 6 Å ±3, Å (T_m is the temperature of the maximum of the broad peak in the curve of the dependence of the specific heat on temperature). The absence of a similar peculiarity will be an evidence that MVR do not influence the values of T_λ and T_m, and do not play any key role in the λ-transition. The currently available experimental data are insufficient for revealing the predicted peculiarity.     

Lifetime prediction and reliability estimation methodology for Stirling-type pulse tube refrigerators by gaseous contamination accelerated degradation testing

Lifetime and reliability are the two performance parameters of premium importance for modern space Stirling-type pulse tube refrigerators (SPTRs), which are required to operate in excess of 10 years. Demonstration of these parameters provides a significant challenge. This paper proposes a lifetime prediction and reliability estimation method that utilizes accelerated degradation testing (ADT) for SPTRs related to gaseous contamination failure. The method was experimentally validated via three groups of gaseous contamination ADT. First, the performance degradation model based on mechanism of contamination failure and material outgassing characteristics of SPTRs was established. Next, a preliminary test was performed to determine whether the mechanism of contamination failure of the SPTRs during ADT is consistent with normal life testing. Subsequently, the experimental program of ADT was designed for SPTRs. Then, three groups of gaseous contamination ADT were performed at elevated ambient temperatures of 40 °C, 50 °C, and 60 °C, respectively and the estimated lifetimes of the SPTRs under normal condition were obtained through acceleration model (Arrhenius model). The results show good fitting of the degradation model with the experimental data. Finally, we obtained the reliability estimation of SPTRs through using the Weibull distribution. The proposed novel methodology enables us to take less than one year time to estimate the reliability of the SPTRs designed for more than 10 years.     

Confidence bounds for the reliability of binary capacitated two-terminal networks

Binary capacitated two-terminal reliability at demand level d (2TR_d) is defined as the probability that network capacity, generated by binary capacitated components, between specified source and sink nodes is greater than or equal to a demand of d units. For the components that comprise these networks, reliability estimates are usually obtained from some source of testing. For these estimates and depending on the type of testing, there is an associated uncertainty that can significantly affect the overall estimation of 2TR_d. That is, an accurate estimate of 2TR_d is highly dependent on the uncertainty associated to the reliability of the network components. Current methods for the estimation of network reliability and associated uncertainty are restricted to the case where the network follows a series-parallel architecture and the components are binary and non-capacitated. For different capacitated network designs, an estimate on 2TR_d can only be approximated for specific scenarios. This paper presents a bounding approach for 2TR_d by explaining how component reliability and associated uncertainty impact estimates at the network level. The proposed method is based on a structured approach that generates a α-level confidence interval (CI) for binary capacitated two-terminal network reliability. Simulation results on different test networks show that the proposed methods can be used to develop very accurate bounds of two-terminal network reliability.     

Acceleration Factor Constant Principle and the Application under ADT

Accelerated degradation test (ADT) has become an efficient approach to evaluate the reliability for highly reliable products. However, when modeling accelerated degradation data with degradation models, it is difficult to exactly figure out the changing rules of parameters with stress variables varying. At present, the changing rules are mainly assumed according to engineering experience or subjective judgements, which probably results in inaccurately extrapolating the reliability. To figure out the changing rules of parameters with stress variables varying, the acceleration factor constant principle and its application under ADT are studied in the paper. It is well known that the acceleration factor between any two different stress levels should be a constant under an effective ADT. For each degradation model, its parameters should obey special changing rules to satisfy that the acceleration factor maintains a constant throughout an ADT. Taking three extensively used stochastic process models as examples, including Wiener process model, gamma process model, and inverse Gaussian process model, the method of deducing the changing rules of the parameters according to the acceleration factor constant principle was demonstrated. A simulation test was conducted to validate the deduced changing rules of the parameters for the three stochastic process models. An illustrative example involving self‐regulating heating cables was used to illustrate the application of the acceleration factor constant principle under ADT. Results indicate that the acceleration factor constant principle offers an appealing and credible approach to help model accelerated degradation data. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermal conductivity estimation of the organic and inorganic refrigerants in the saturated liquid state

The use of a general correlation, proposed for organic liquid thermal conductivity λ prediction and checked in the case of organic refrigerant fluids, is extended to inorganic refrigerant fluids whose λ decreases when the reduced temperature, T_r, increases. Suggestions are given for the refrigerants fluids whose λ increases with the reduced temperature.     

Refractive index measurements on bismuth tellurium oxide (Bi2TeO5) single crystal

The refractive indices of the biaxial Bi₂TeO₅ single crystal have been measured for the first time at seven wavelengths in the range of 457.9–632.8 nm at room temperature with an error of about 0.0003 (at λ=632.8 nm, n₁=2.3203, n₂=2.3678 and n₃=2.4022). The optical plane was found to be the (010) plane and the crystal is shown to be optically negative. The crossing angle of the optical axes was found to be between 79° and 86°. The parameters of Sellmeier's one- and two-term dispersion equations have been determined by the least squares method. Resonance wavelengths λ₁=219 nm for the first and λ₁=182 nm and λ₂=318 nm for the second one were found. The conditions of phase-matched SHG are discussed.     

A semi-Markovian model allowing for inhomogenities with respect to process time

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q_ij(t), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ_ij( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N² integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.     

A complementary note on Gegenbauer polynomial approximation for random response problem of stochastic structure

Recently, Gegenbauer polynomial approximation was proposed for solving the evolutionary random response problem of a random structure with bounded random parameters under evolutionary random excitations. The bounded random parameters used there are supposed to be proportional to a random variable with λ-PDF (probability density function). For this kind of random parameter, the Gegenbauer polynomial approximation is the unique correct choice for transforming a random structure into its deterministic equivalent system, which plays a central role in solving the response problem. Actually, the Gegenbauer polynomial approximation bridges the gap between the random structural response problem and the conventional methods. Just through its deterministic equivalent system, the random response problem of a random structure can be solved by any available, analytical and numerical method developed for deterministic systems. But the simple assumption on proportionality to λ-PDF may bring some unnecessary limitation on symmetry. Since λ-PDF is symmetrical about its center axis, so are these random parameters. However, not all random parameters have this kind of symmetry. Then, what can we do to lessen the limitation, if random parameters are non-symmetric per se? Besides, the graph of λ-PDF (see Fig. 1 in the text) shows its features’ dependence on λ, that is, the smaller the λ, the more dispersive the PDF. Then, what is the influence of different values of λ on the evolutionary random responses of the stochastic system? Moreover, is there any qualitative information about the dispersion in response with respect to every individual random parameter? This note is devoted to answering the above two questions and giving a supplementary note on Gegenbauer polynomial approximation. Our study shows that the symmetric limitation can be partly lessened by putting the random parameter into a polynomial function, even a quadratic one, of a random variable with λ-PDF. On the other hand, the second-order moment of the random responses of the stochastic structure relative to that of a nominal one can be used to describe qualitatively their relative dispersion with respect to each individual random parameter.