Multiple performance characteristics (PCs) are common in modern products with complex structures and diverse functions. These PCs are usually dependent, with significant unit-specific variability among the multivariate degradation processes. Therefore, the associated degradation modeling for dependent multivariate degradation processes is important. This article proposes a novel multivariate$t$degradation model for this purpose. Specifically, the dependence between multivariate degradation processes is captured by random drift parameters that follow a multivariate normal distribution, and the variation in diffusion parameters and variance–covariance is characterized by a gamma distribution. An expectation-maximization (EM) algorithm is employed for likelihood inference, and confidence intervals of the model parameters are constructed by normal approximation and bootstrap method. A theoretical exploration investigating the effects of model misspecification in multivariate degradation modeling is addressed. Monte Carlo simulation studies are performed to validate the effectiveness of the EM algorithm and the theoretical properties of the multivariate$t$model. Finally, two illustrative examples are used to demonstrate the applicability and advantages of the proposed methods.

Local Bayesian optimization is a promising practical approach to solve the high dimensional black-box function optimization problem. Among them is the approximated gradient class of methods, which implements a strategy similar to gradient descent. These methods have achieved good experimental results and theoretical guarantees. However, given the distributional properties of the Gaussian processes applied on these methods, there may be potential to further exploit the information of the Gaussian processes to facilitate the BO search. In this work, we develop the relationship between the steps of the gradient descent method and one that minimizes the Upper Confidence Bound (UCB), and show that the latter can be a better strategy than direct gradient descent when a Gaussian process is applied as a surrogate. Through this insight, we propose a new local Bayesian optimization algorithm, MinUCB, which replaces the gradient descent step with minimizing UCB in GIBO. We further show that MinUCB maintains a similar convergence rate with GIBO. We then improve the acquisition function of MinUCB further through a look ahead strategy, and obtain a more efficient algorithm LA-MinUCB. We apply our algorithms on different synthetic and real-world functions, and the results show the effectiveness of our method. Our algorithms also illustrate improvements on local search strategies from an upper bound perspective in Bayesian optimization, and provides a new direction for future algorithm design.

Degradation has become the dominant failure mode for highly reliable engineering systems. Cracking, a fatigue phenomenon composed of sequential phases of crack initiation and propagation, is a major concern for critical aircraft structures. Traditional fracture mechanics analysis cannot fully meet the requirements for assessing reliability indicators from a reliability analysis perspective. Alternatively, the empirical Lifetime Delayed Degradation Process (LDDP) provides an explanatory framework for sequential hard&soft failure mode. This study further generalizes the LDDP framework by introducing the Bayesian method as a Bayes-LDDP model, which incorporates a weakly informative prior derived from historical data of similar systems for both non-destructive and destructive inspections. Additionally, we compare our proposed method to the LDDP approach using specific inspection datasets. Two practical applications are conducted to demonstrate the effectiveness of the Bayes-LDDP model for reliability monitoring and remaining useful life (RUL) prediction in critical aircraft structures using field data. The crack inspection datasets of a transport aircraft and an aircraft core automated maintenance system (CAMS) are utilized for non-destructive and destructive inspections, respectively. The Markov Chain Monte Carlo (MCMC) sampling algorithm is adopted for the Bayes-LDDP, improving the computational efficiency of model parameters estimation compared to the stochastic expectation maximum (SEM) algorithm. Furthermore, the Bayes-LDDP model enables precise inference including the mean time to failure (MTTF) of cracks for destructive inspections and the RUL for non-destructive inspections under the selected optimal model. This extended novel framework provides a clear depiction of the lifetime delayed degradation process from a Bayesian perspective.

Global market competition has led to manufacturers encountering a decision problem as the reliability of their products exceeds a given standard. Acceptance sampling tests (ASTs) can be expensive and time-consuming, especially for products with high reliability or quality. Thus, careful planning of the acceptance rule and determination of a suitable sample size for the test are important. In this article, an exact method for the optimal planning of ASTs is proposed for cases in which the product lifetimes follow a distribution in the general log-location-scale family of distributions. A novel procedure using the pivotal quantity is established based on the method-of-moments estimators of the model parameters, and the distribution of the reliability estimator is derived. An algorithm for obtaining the optimal sample size and the corresponding acceptability constant is presented in which the producer's and consumer's risks are constrained according to certain acceptance criteria. We apply the proposed method to a censored AST by generating censored observations with a quantile-filling method under Type-II censoring. The performance of the proposed AST with a complete or censored sample is studied and compared with an AST based on the maximum likelihood method. The robustness of the proposed AST is also examined under model misspecification. The results show that the proposed AST performs well under different scenarios.

By combining robust regression and prior information, we develop an effective robust regression method that can resist adaptive adversarial attacks. Due to the widespread existence of noise and data corruption, it is necessary to recover the true regression parameters when a certain proportion of the response variables have been corrupted. Methods to overcome this problem often involve robust least-squares regression. However, few methods achieve good performance when dealing with severe adaptive adversarial attacks. Based on the combination of prior information and robust regression via hard thresholding, this paper proposes an algorithm that improves the breakdown point when facing adaptive adversarial attacks. Furthermore, to improve the robustness and reduce the estimation error caused by the inclusion of a prior, the idea of Bayesian reweighting is used to construct a more robust algorithm. We prove the theoretical convergence of proposed algorithms under mild conditions. Extensive experiments show that, under different dataset attacks, our algorithms achieve state-of-the-art results compared with other benchmark algorithms, demonstrating the robustness of the proposed approach.

As reliability requirements become increasingly demanding for many engineering systems, conventional system reliability demonstration testing (SRDT) based on the number of failures depends on a large sample of system units. However, for many safety critical systems, such as missiles, it is prohibitive to perform such testing with large samples. To reduce the sample size, existing SRDT methods utilize test data from either system level or component level. In this paper, an aggregation-based SRDT methodology is proposed for hierarchical systems by utilizing multilevel reliability information of components, subsystems, and the overall system. Analytical conditions are identified for the proposed method to achieve lower consumer risk. The performances of different SRDT design strategies are evaluated and compared according to their consumer risks. A numerical case study is presented to illustrate the proposed methodology and demonstrate its validity and effectiveness.

Summary Software reliability growth modeling plays an important role in software reliability evaluation. To incorporate more information and provide more accurate analysis, modeling software fault detection and correction processes has attracted widespread research attention recently. In modeling software correction processes, the assumption of fault correction time is relaxed from constant delay to random delay. However, stochastic distribution of fault correction time brings more difficulties in modeling and corresponding parameter estimation. In this paper, a framework of software reliability models containing both information from software fault detection process and correction process is studied. Different from previous extensions on software reliability growth modeling, the proposed approach is based on Markov model other than a nonhomogeneous Poisson process model. Also, parameter estimation is carried out with weighted least‐square estimation method, which emphasizes the influence of later data on the prediction. Two data sets from practical software development projects are applied with the proposed framework, which shows satisfactory performance with the results. Copyright © 2016 John Wiley & Sons, Ltd.

Software reliability is one of the most significant attributes in various safety-critical systems. One popular approach to estimate the software reliability is based on Bayesian statistics. However, there are few attempts on Bayesian approaches considering fault correction process. In this paper, we present parameter estimation method based on Bayesian statistics for combined fault detection and correction processes. A practical example is devoted to investigating the fitting performance of the proposed Bayesian approach.

Software defects play a key role in software reliability, and the number of remaining defects is one of most important software reliability indexes. Observing the trend of the number of remaining defects during the testing process can provide very useful information on the software reliability. However, the number of remaining defects is not known and has to be estimated. Therefore, it is important to study the trend of the remaining software defect estimation (RSDE). In this paper, the concept of RSDE curves is proposed. An RSDE curve describes the dynamic behavior of RSDE as software testing proceeds. Generally, RSDE changes over time and displays two typical patterns: 1) single mode and 2) multiple modes. This behavior is due to the different characteristics of the testing process, i.e., testing under a single testing profile or multiple testing profiles with various change points. By studying the trend of the estimated number of remaining software defects, RSDE curves can provide further insights into the software testing process. In particular, in this paper, the Goel-Okumoto model is used to estimate this number on actual software failure data, and some properties of RSDE are derived. In addition, we discuss some theoretical and application issues of the RSDE curves. The concept of the proposed RSDE curves is independent of the selected model. The methods and development discussed in this paper can be applied to any valid estimation model to develop and study its corresponding RSDE curve. Finally, we discuss several possible areas for future research.

Software reliability modeling & estimation plays a critical role in software development, particularly during the software testing stage. Although there are many research papers on this subject, few of them address the realistic time delays between fault detection and fault correction processes. This paper investigates an approach to incorporate the time dependencies between the fault detection, and fault correction processes, focusing on the parameter estimations of the combined model. Maximum likelihood estimates of combined models are derived from an explicit likelihood formula under various time delay assumptions. Various characteristics of the combined model, like the predictive capability, are also analyzed, and compared with the traditional least squares estimation method. Furthermore, we study a direct, useful application of the proposed model & estimation method to the classical optimal release time problem faced by software decision makers. The results illustrate the effect of time delay on the optimal release policy, and the overall software development cost.

Generally, software reliability models can provide accurate reliability measurement in the later phase of testing. However, predictions in the early phase of software testing are useful as cost-effective and timely feedback. Early prediction is also feasible in practice with information from previous releases or similar projects. Such information has been utilized well for early reliability prediction with NHPP models by assuming the same failure rate between two similar projects. Alternatively, in this paper, we propose to "reuse" failure data from past projects/releases with ANN models to improve early reliability for current project/release. To illustrate the proposed approach, two numerical examples are developed. Better prediction performance is observed in early phase of testing compared with original ANN model without failure data reuse. Furthermore, the optimal switching point from proposed approach to original ANN model in the whole testing phase is studied, with specific analysis on the two examples

It is well-known that accurate reliability estimates can be obtained by using software reliability models only in the later phase of software testing. However, prediction in the early phase is important for cost-effective and timely management. Also this requirement can be achieved with information from previous releases or similar projects. This basic idea has been implemented with nonhomogeneous Poisson process (NHPP) models by assuming the same testing/debugging environment for similar projects or successive releases. In this paper we study an approach to using past fault-related data with artificial neural network (ANN) models to improve reliability predictions in the early testing phase. Numerical examples are shown with both actual and simulated datasets. Better performance of early prediction is observed compared with original ANN model with no such historical fault-related data incorporated. Also, the problem of optimal switching point from the proposed approach to original ANN model is studied, with three numerical examples

Most of the models for software reliability analysis are based on reliability growth models which deal with the fault detection process only. In this paper, some useful approaches to the modeling of both software fault detection and fault correction processes are discussed. Since the estimation of model parameters in software testing is essential to give accurate prediction and help make the right decision about software release, the problem of estimating the parameters is addressed. Taking into account the dependency between the fault correction process and the fault detection process, a new explicit formula for the likelihood function is derived and the maximum likelihood estimates are obtained under various time delay assumptions. An actual set of data from a software development project is used as an illustrative example. A Monte Carlo simulation is carried out to compare the predictive capability between the LSE method and the MLE method

Software inspection has been broadly accepted as a cost effective approach for defect removal during the whole software development lifecycle. To keep inspection under control, it is essential to measure its effectiveness. As human-oriented activity, inspection effectiveness is due to many uncertain factors that make such study a challenging task. Bayesian networks modeling is a powerful approach for the reasoning under uncertainty and it can describe inspection procedure well. With this framework, some extensions have been explored in this paper. The number of remaining defects in the software is proposed to be incorporated into the framework, with expectation to provide more information on the dynamic changing status of the software. In addition, a different approach is adopted to elicit the prior belief of related probability distributions for the network. Sensitivity analysis is developed with the model to locate the important factors to inspection effectiveness.