Non-stationary functional series modeling and analysis of hardware reliability series: a comparative study using rail vehicle interfailure times

Abstract

A novel reliability modeling and analysis framework based upon the distinct class of non-stationary Functional Series (FS) models is introduced. This framework allows for non-stationary reliability modeling, evolution assessment, analysis (including non-stationarity assessment, dependency assessment, as well as cycle detection), and prediction. The Functional Series framework is used for the modeling and analysis of two rail vehicle reliability series (Times Between Failures, TBFs), while comparisons with alternative (ARIMA, adaptive RARMA–RML, and Bayesian) modeling approaches are also made. The results indicate the advantages and usefulness of the Functional Series framework, as the TBF modeling accuracy is improved, its non-stationarity and serial dependency are established, the presence of cyclic patterns is revealed, and reliability evolution is assessed. It is conjectured that the cycles revealed in the TBF series may be related to maintenance policies. Finally, reliability prediction is shown to be feasible, although the “larger” excursions in the TBF series are difficult to accurately predict.

Introduction

Time series models constitute mathematical representations, which are useful for describing the evolution of reliability measures in repairable systems. They have been pioneered by authors such as Singpurwalla [1], [2], Walls and Bendell [3], and Soyer [4], and have been demonstrated to be capable of describing the stochastic dependencies present in reliability data series obtained from repairable systems. Time series models thus constitute useful tools for reliability modeling, evolution assessment, underlying data structure analysis, exogenous effect analysis, as well as prediction.

The reliability of a repairable system is a expectedly deteriorating (for hardware systems) or improving (for software systems) stochastic function of time, although these trends may be reversed by human intervention, including maintenance related actions. Coupled with dependencies, which are inevitably present in reliability data series, such trends necessitate the use of non-stationary time series models for the proper representation of reliability data. Non-stationary time series models may be broadly used for the following practical purposes:

• (i) Gaining insight into the reliability evolution. Assessment and characterization of random effects and dependencies over time. Detection of the presence of cycles that may be associated with exogenous conditions. Detection of reliability trends.

• (ii) Prediction. Prediction of reliability evolution for a given system based upon retrospective data. In the study of the times between failures (TBFs), the time to the next failure is to be predicted. Prediction is important for planning purposes.

• (iii) Detection and assessment of exogenous effects. Detection and assessment of “environmental” effects (depending upon the type of hardware this may be weather effects, working conditions, and so on). Detection and assessment of scheduled maintenance effects. Appraisal of the effectiveness of maintenance policies.

The present study focuses on the following specific issues which are of fundamental importance within the context of non-stationary time series reliability models:

• (a) Modeling. What model classes are appropriate? What is the required model structure? What is the achievable model quality?

• (b) Reliability evolution assessment. How to assess reliability evolution? How to detect the presence of trends?

• (c) Analysis. How to assess the degree and type of non-stationarity? How to formally assess the dependency structure? How to detect cyclic patterns that may be present?

• (d) Prediction. Is prediction possible? What is the achievable prediction accuracy?

Parametric non-stationary time series models that may be potentially suitable in addressing the above issues include the class of Integrated AutoRegressive Moving Average (ARIMA) models of Box and Jenkins [5], the adaptive model class (of either the recursive model or the stochastic parameter evolution subclasses) [4], [6], and the class of Functional Series models [7], [8].

The ARIMA model class has been the first one used [1], [2], [3], [9], but is known to be limited to a rather restrictive type of non-stationarity (referred to as “homogeneous” [5]), and has been thus criticized for failing to capture a sufficiently large portion of the series variability [3]. The adaptive model class is significantly “wider”, as adaptive models allow for the evolution (over time) of the model parameters. In a notable subclass (referred to as the recursive model subclass) parameter evolution is essentially “unstructured” (“unrestricted”), with the parameter estimates being updated via Kalman Filter type recursions every time a new data sample becomes available [6]. In an alternative subclass (referred to as stochastic parameter evolution model subclass), parameter evolution is more “structured” with the parameters being bound to obey certain stochastic smoothness constraints. With a single recent exception [10], the former subclass remains essentially unexplored in reliability data analysis, whereas the latter has been developed in a series of studies [4], [11], [12] focusing on software reliability and using a Bayesian framework which places increased emphasis on prior assumptions.

The main goal of this paper is the introduction and assessment of a reliability modeling and analysis framework based upon the distinct class of non-stationary Functional Series (FS) models [7], [8]. Functional Series models constitute conceptual extensions of conventional stationary models, such as the ARMA models, in which their parameters are allowed to evolve with time while remaining subject to certain functional constraints. The latter are typically enforced by requiring the model parameters to belong to a functional subspace spanned by a suitably selected set of functions (basis functions). Compared to adaptive models, Functional Series models are characterized by significantly more “structured” parameter evolution, but are flexible enough to capture a broad spectrum of non-stationarities while also leading to a degree of statistical parsimony significantly higher than that of adaptive models. An additional characteristic of Functional Series models is that, in contrast to their adaptive counterparts, they may be estimated via batch (non-recursive) schemes. The potential usefulness of a restricted (first-order with specific parameter form) Functional Series model in reliability analysis has been conjectured by Singh [13], but no further studies or actual applications have been reported.

The Functional Series models used in this study are referred to as Time-dependent ARMA (TARMA) models, and are of the conventional ARMA form with parameters belonging to a proper functional subspace, while their estimation is based upon the recently introduced Polynomial-Algebraic (P-A) method [8]. The proposed Functional Series reliability modeling and analysis framework includes tools for: (a) reliability modeling; (b) evolution assessment; (c) model-based analysis; and (d) prediction.

The focus of the study is on hardware reliability, and the fundamental issues (a)–(d) posed earlier, by using the paradigm of rail vehicle reliability. The reliability data employed are chronologically ordered retrospective series of interfailure times (Times Between Failures, TBFs, with time expressed in km traveled) for vehicles of the Athens Electric Railways. Beyond the assessment of the Functional Series framework, the study has a significant comparative component in which the effectiveness of alternative modeling approaches is also assessed. These include the ARIMA, adaptive Recursive ARMA–Recursive Maximum Likelihood (RARMA–RML) [6], and the Bayesian stochastic parameter evolution approach of Singpurwalla and Soyer [11] (S&S). Model assessment and comparison are primarily based upon the ability of the various models to predict the time to next failure, although issues such as model parsimony are also considered through the Akaike Information Criterion (AIC) [6].

The rest of this paper is organized as follows. A description of the TBF series used in the study is, along with preliminary analysis, presented in Section 2. The Functional Series reliability modeling and analysis framework is presented in Section 3, while the alternative non-stationary time series reliability modeling approaches used are briefly reviewed in Section 4. Two cases of rail vehicle reliability modeling and analysis, using the Functional Series framework and the aforementioned alternative approaches, are presented in Section 5, and the conclusions of the study are summarized in Section 6.