Robust parameter design optimization for type‐I right censored data

Robust parameter design (RPD) aims to build product quality in the early design phase of product development by optimizing operating conditions of process parameters. A vast majority of the current RPD studies are based on an uncensored random sample from a process distribution. In reality, censoring schemes are widely implemented in lifetime testing, survival analysis, and reliability studies in which the value of a measurement is only partially known. However, there has been little work on the development of RPD when censored data are under study. To fill in the research gaps given practical needs, this paper proposes response surface–based RPD models that focus on survival times and hazard rate. Primary tools used in this paper include the Kaplan‐Meier estimator, Greenwood's formula, the Cox proportional hazards regression method, and a nonlinear programming method. The experimental modeling and optimization procedures are demonstrated through a numerical example. Various response surface–based RPD optimization models are proposed, and their RPD solutions are compared.     

Hazard rate estimation from incomplete and unclean warranty data

Warranty data are a rich source of information for feedback on product reliability. However, two-dimensional automobile warranties that include both time and mileage limits, pose two interesting and challenging problems in reliability studies. First, warranty data are restricted only to the reported failures within warranty coverage and such incompleteness can lead to inaccurate estimates of field failure rate or hazard rate. Second, factors such as inexact time/mileage data and vague reported failures in a warranty claim make warranty data unclean that can suppress inherent failure pattern. In this paper we discuss two parameter estimation methods that address the incompleteness issue. We use a simulation-based experiment to study these estimation methods when departure from normality and varying amount of truncation exists. Using a life cycle model of the vehicle, we also highlight and explore issues that lead to warranty data not being very clean. We then propose a five-step methodology to arrive at meaningful component level empirical hazard plots from incomplete and unclean warranty data.     

Merging expert and empirical data for rare event frequency estimation: Pool homogenisation for empirical Bayes models

Empirical Bayes provides one approach to estimating the frequency of rare events as a weighted average of the frequencies of an event and a pool of events. The pool will draw upon, for example, events with similar precursors. The higher the degree of homogeneity of the pool, then the Empirical Bayes estimator will be more accurate. We propose and evaluate a new method using homogenisation factors under the assumption that events are generated from a Homogeneous Poisson Process. The homogenisation factors are scaling constants, which can be elicited through structured expert judgement and used to align the frequencies of different events, hence homogenising the pool. The estimation error relative to the homogeneity of the pool is examined theoretically indicating that reduced error is associated with larger pool homogeneity. The effects of misspecified expert assessments of the homogenisation factors are examined theoretically and through simulation experiments. Our results show that the proposed Empirical Bayes method using homogenisation factors is robust under different degrees of misspecification.     

The method of separation for evolutionary spectral density estimation of multi-variate and multi-dimensional non-stationary stochastic processes

The method of separation can be used as a non-parametric estimation technique, especially suitable for evolutionary spectral density functions of uniformly modulated and strongly narrow-band stochastic processes. The paper at hand provides a consistent derivation of method of separation based spectrum estimation for the general multi-variate and multi-dimensional case. The validity of the method is demonstrated by benchmark tests with uniformly modulated spectra, for which convergence to the analytical solution is demonstrated. The key advantage of the method of separation is the minimization of spectral dispersion due to optimum time- or space–frequency localization. This is illustrated by the calibration of multi-dimensional and multi-variate geometric imperfection models from strongly narrow-band measurements in I-beams and cylindrical shells. Finally, the application of the method of separation based estimates for the stochastic buckling analysis of the example structures is briefly discussed.     

Reliability models for existing structures based on dynamic state estimation and data based asymptotic extreme value analysis

The problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations is considered. The study assumes that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes’ theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. In all these studies, the governing stochastic differential equations are discretized using the strong forms of Ito–Taylor’s discretization schemes. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplified by considering the reliability analysis of a few low-dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on a limited amount of pertinent Monte Carlo simulations.