In this article, an overview of Nozer Singpurwalla's work in reliability and risk analysis is provided. Rather than presenting a chronological review of his work, the emphasis is given to those areas of his research which better reflect Nozer's scientific personality.
In the context of Industry 4.0, a production line must be flexible and adaptable to stochastic or real-world environments. As a result, the assembly line balancing (ALB) problem involves managing uncertainty or stochasticity. Several methods have been proposed, including stochastic mathematical programming models and simulations. However, programming models can only incorporate a few sources of uncertainty that result in impractical or unfeasible solutions to implement due to overlooked complexities, while simulation is only used to test solutions from deterministic approaches or adjust parameters without maintaining their optimum value. The proposed methodology uses a deterministic mathematical model to minimize the cycle time, followed by the simulation to measure the impact of selected sources of uncertainty on the cycle time. Finally, the optimum value of the stochastic parameters is computed using simulation-based optimization to maintain the average cycle time close to the deterministic one. A real-life assembly line balancing problem for a motorcycle manufacturing company is solved to test the proposed methodology. The sources of uncertainty are the tasks' stochastic processing times, inter-arrival time, the number of workers in each station, and the speed of the material handling system. Results show that the average cycle time is above 2.7% from the deterministic value computed by the programming model when the inter-arrival time is set to 270 60 s; the processing times are allowed to increase or decrease by 3 s; the material handling system's speed is 1.5 m/s; and the number of workers in cells is between 4 and 6, with a speed of 2 m/s. The reader can download the source code and the simulation model to apply the methodology to other instances.
We consider gamma processes of homogeneous type, which live in a random environment or media represented by a pure jump Markov process. The aim of this paper is to approximate such gamma processes by a diffusion. Since gamma processes are increasing, the diffusion approximation requires an average approximation first. This averaged process will serve as an equilibrium to the initial gamma process. We present two main results: averaging and normal deviation. An application for degradation systems in reliability modeling is discussed.
Consider multi-state series and multi-state parallel systems consisting of independent components each. It is assumed that (i) each component and both the systems take values in the set , (ii) each system and each component start out in state 2, the perfect state, and they make the transition to state 1, depending upon system configuration, and, eventually, each system enters state 0, the failed state. This multi-state nature of components and systems leads to scenarios under which each of the systems makes the transition from state 2 to state 1, and eventually to state 0. The joint probability function for times spent in state 2 and state 1 is obtained based on these scenarios for each of the systems. It is interesting to note that by merely changing set of a standard binary series (parallel) system to a set of a multi-state series (multi-state parallel) system, renders expression of the joint probability function of system spending times in state 2 and state 1 of a multi-state series (multi-state parallel) system is quite complex as compared to the univariate survival probability of the binary series (parallel) system being in the functioning state. As a proof of concept, graphical comparison between these analytical joint probability functions and joint e
The paper deals with the estimation procedures for the proportional hazard class of distributions under a two-sample balanced joint progressive censoring scheme. The baseline hazard function is assumed to be piecewise constant, instead of any specific form. This adds flexibility to the proposed model, and the shape of the underlying hazard function is completely data-driven. Since the complicated form of the likelihood function does not yield closed-form estimators, we propose a variant of the Expectation-Maximization algorithm, known as the Expectation Conditional Maximization (ECM) algorithm, for obtaining maximum likelihood estimates of the model parameters. This leads to explicit expressions for the iterative constrained maximization steps of the algorithm. An extension to the case when the cut points are unknown has also been considered for dealing with problems involving real data. Simulation results and illustrations using real data have also been presented.
In the development of devices for cutting Chinese yams into chunks for use as seeds, accurately measuring the yam's shape with a simple mechanism is crucial. In our prior study, we introduced a statistical approach for predicting the shape of a Chinese yam based on its key diameters. This method involves organizing sample data, estimating diameters at discrete points along the central axis, and constructing a predictive model based on these estimated diameters. However, the initial predictive model relied on separate regression models for each point, potentially leading to instability. In this article, we enhance our previous approach by incorporating a new step that refines the estimation of regression coefficients through Bayesian linear modeling methods. This modification allows for the simultaneous estimation of regression coefficients, ensuring greater stability in the reconstructed model. Additionally, we modify the method for locating key diameters. To validate the performance of the enhanced approach, we apply it to a set of samples and compare the output of the reconstructed model with that of our initial method. The results demonstrate improved stability and performance, highlighting the efficacy of the refined modeling technique.
Limited failure or cure rate models provide generalization of lifetime models which allow the possibility of subjects or units to be cured or be failure-free. While modeling and analysis of such models are extensively studied, we study the important question of identifiability of these models. We discuss the latent and hierarchical activation cure models and establish a series of results on stochastic ordering within these models. We also establish a series of results on identifiability of these models under various conditions. Further, we demonstrate multiple cases where there models are not identifiable and illustrate the potential difficulty with these models in a simulation study.
In the early days of machine learning (ML), the emphasis was on developing complex algorithms to achieve best possible predictive performance. To understand and explain the model results, one had to rely on post hoc explainability techniques, which are known to have limitations. Recently, with the recognition in regulated industries that interpretability is also important, researchers are studying algorithms that compromise on small increases in predictive performance in favor of being more interpretable. While doing so, the ML community has rediscovered the use of low-order functional ANOVA (fANOVA) models that have been known in the statistical literature for some time. This paper starts with a description of challenges with post hoc explainability. This is followed by a brief review of the fANOVA framework with a focus on models with just main effects and second-order interactions (called generalized additive models with interactions or GAMI = GAM + Interactions). It then provides an overview of two recently developed GAMI techniques: Explainable Boosting Machines or EBM and GAMI-Net. The paper proposes a new algorithm that also uses trees, as in EBM, but does linear fits instead of piecewise constants within the partitions. We refer to this as GAMI-linear-tree (GAMI-Lin-T). There are many other differences, including the development of a new interaction filtering algorithm. The paper uses simulated and real datasets to compare the three fANOVA ML algorithms. The results show that GAMI-Lin-T and GAMI-Net have comparable performances, and both are generally better than EBM.
Negative probabilities arise primarily in physics, statistical quantum mechanics, and quantum computing. Negative probabilities arise as mixing distributions of unobserved latent variables in Bayesian modeling. Our goal is to provide a link between these two viewpoints. Bartlett provides a definition of negative probabilities based on extraordinary random variables and properties of their characteristic function. A version of the Bayes rule is given with negative mixing weights. The classic half-coin distribution and Polya-Gamma mixing are discussed. Heisenberg's principle of uncertainty and the duality of scale mixtures of Normals is also discussed. A number of examples of dual densities with negative mixing measures are provided including the Linnik and Wigner distributions. Finally, we conclude with directions for future research.
We consider several survival models in heterogeneous settings. Heterogeneity in the failure rates of subpopulations results (as a specific case) in the famous failure rate paradox when the failure rate of a mixture of items with constant failure rates is decreasing. Random failure rate that is due to a point process that increases it at random times on fixed values also results in the “bending down” of the population failure rate. Similar effect is observed while analyzing the extreme shock models with shock processes that possess memory. Finally, another paradox when, due to heterogeneity in a vital parameter of a model, a terminating point process with decreasing rate after “mixing” becomes a non-terminating one with increasing rate is described. Those are the impacts of heterogeneity that are discussed from the unified perspective that employs the “principle”: the weaker subpopulations are dying out first.
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