Aparna V. Huzurbazar

Bayesian networks for multilevel system reliability

Bayesian networks have recently found many applications in systems reliability; however, the focus has been on binary outcomes. In this paper we extend their use to multilevel discrete data and discuss how to make joint inference about all of the nodes in the network. These methods are applicable when system structures are too complex to be represented by fault trees. The methods are illustrated through four examples that are structured to clarify the scope of the problem.

Stochastic Network Models for Survival Analysis

Abstract We present methodology giving highly accurate approximations for Bayesian predictive densities and distribution functions of first passage times between states of a semi-Markov process with a finite number of states. When the states describe a degenerative disorder with an absorbing end state, such predictive distributions are the survival distributions of a patient. We illustrate these methods with a variety of examples, including data from the San Francisco AIDS study. We achieve our approximations using a three-step sequence. First, we introduce advanced concepts of flowgraph theory, which allow us to compute the moment generating function of the first passage time given the model parameters. Next, we use saddlepoint approximations to convert this into a density or distribution function conditional on the model parameter. Finally, we use Monte Carlo methods to remove dependence on the model parameter. These methods apply quite generally to all finite-state semi-Markov models in discrete or con...

Modeling and Analysis of Engineering Systems Data Using Flowgraph Models

An innovative approach to data analysis for complicated stochastic systems involves modeling based on flowgraph methods. This article introduces flowgraph and associated saddlepoint methods for problems in systems engineering and reliability. The methods are especially useful for analyzing time-to-event data and finding predictive distributions of such data. Applications to a cellulartelephone network are given. Advantages of flowgraph models over direct simulation are presented. Methods of likelihood construction for incomplete data are given.

Flowgraph Models for Generalized Phase Type Distributions Having Non‐Exponential Waiting Times

. Aalen (1995) introduced phase type distributions based on Markov processes for modelling disease progression in survival analysis. For tractability and to maintain the Markov property, these use exponential waiting times for transitions between states. This article extends the work of Aalen (1995) by generalizing these models to semi-Markov processes with non-exponential waiting times. The generalization allows more realistic modelling of the stages of a disease where the Markov property and exponential waiting times may not hold. Flowgraph models are introduced to provide a closed form for the distributions in situations involving non-exponential waiting times. Flowgraph models work where traditional methods of stochastic processes are intractable. Saddlepoint approximations are used in the analysis. Together, generalized phase type distributions, flowgraphs, and saddlepoint approximations create exciting and innovative prospects for the analysis of survival data.

Multistate Models, Flowgraph Models, and Semi-Markov Processes

Abstract Semi-Markov models play an important role in the analysis of time to event data. However, in practice, data analysis for semi-Markov processes can be quite difficult and many simplifying assumptions are made. Markovian multistate models are popular for event history analysis and repeated events analysis for survival data. Semi-Markov processes provide a rich class of models applicable to this area. Flowgraph models provide a link between traditional multistate models and semi-Markov processes. Flowgraphs model semi-Markov processes and provide a data analytic method for estimating densities, survivor and reliability functions, and hazard functions for waiting times of interest in the presence of censored and incomplete data. While multistate models have been used primarily in medical research, flowgraph models are useful for time to event data arising from a variety of problems in many areas including engineering, medicine, operations research, and demography.

System Health Assessment

ABSTRACT Complex systems are increasingly confronted by two conflicting sets of requirements: on the one hand, demands for continuous operational readiness with high reliability and availability; on the other, the need to minimize life cycle cost, implying reduced inspections, maintenance, and logistics support. An emerging paradigm to address this challenge is prognostics and health management (PHM), where measures of system health are used to determine needs for preventive and corrective maintenance, to optimize maintenance scheduling and parts stocking, and to forecast when a system will reach the end of its useful life. Two key components of PHM are a definition of system health and a strategy for how it is to be measured as part of system health assessment (SHA). In this article we discuss system health as a general concept, illustrate its application with examples, and describe how the use of system health metrics as part of an SHA program can facilitate PHM.

A Censored Data Histogram

ABSTRACT A histogram is an important tool for exploratory data analysis. No matter what the ultimate analysis of a given set of data, it is always important to plot the data. However, when data are censored, this becomes problematic. This article presents a method for constructing a histogram for censored data using the Kaplan–Meier survivor function. The method is intended to be a practical, easy-to-use data analytic tool that provides a censored data histogram for the user. Program code for constructing such a censored data histogram is provided on the authors website. 1 www.stat.unm.edu/ aparna/cdh.html

A case study for quantifying system reliability and uncertainty

The ability to estimate system reliability with an appropriate measure of associated uncertainty is important for understanding its expected performance over time. Frequently, obtaining full-system data is prohibitively expensive, impractical, or not permissible. Hence, methodology which allows for the combination of different types of data at the component or subsystem levels can allow for improved estimation at the system level. We apply methodologies for aggregating uncertainty from component-level data to estimate system reliability and quantify its overall uncertainty. This paper provides a proof-of-concept that uncertainty quantification methods using Bayesian methodology can be constructed and applied to system reliability problems for a system with both series and parallel structures.

Incorporating Covariates in Flowgraph Models: Applications to Recurrent Event Data

Modeling recurrent event data is of current interest in statistics and engineering. This article proposes a framework for incorporating covariates in flowgraph models, with application to recurrent event data in systems reliability settings. A flowgraph is a generalized transition graph (GTG) originally developed to model total system waiting times for semi-Markov processes. The focus of flowgraph models is expanded by linking covariates into branch transition models, enriching the toolkit of available data analysis methods for complex stochastic systems. This article takes a Bayesian approach to the analysis of flowgraph models. Potential applications are not limited to engineering systems, but also extend to survival analysis.

System reliability and safety assessment using non-parametric flowgraph models:

Multi-state Markov models are widely used for prediction in reliability, safety, and risk analysis. Systems typically pass through different states between correct, reliable operation and failure, as a result of external events or internal ageing, and Markov models provide an effective compromise between realism and mathematical tractability. Statistical flowgraphs analyse these models using transforms of the transition time distributions between states, which are combined and inverted to obtain quantities of interest for the entire model. This paper presents an approach to flowgraphs using empirical transforms based on historical or testing data, with no assumption of parametric probability models for transition times. The non-parametric method is illustrated with an application to predicting cumulative earthquake damage to structures.