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Modern Engineering Asset Management (EAM) requires the accurate assessment of current and the prediction of future asset health condition. Suitable mathematical models that are capable of predicting Time-to-Failure (TTF) and the probability of failure in future time are essential. In traditional reliability models, the lifetime of assets isestimated using failure time data. However, in most real-life situations and industry applications, the lifetime of assets is influenced by different risk factors, which are called covariates. The fundamental notion in reliability theory is the failure time of a system and its covariates. These covariates change stochastically and may influence and/or indicate the failure time. Research shows that manystatistical models have been developed to estimate the hazard of assets or individuals with covariates. An extensive amount of literature on hazard models with covariates (also termed covariate models), including theory and practical applications, has emerged. This paper is a state-of-the-art review of the existing literature on these covariate models in both the reliability and biomedical fields.One of the major purposes of this expository paper is to synthesise these models from both industrial reliability and biomedical fields and then contextually group them into non-parametric and semiparametric models. Comments on their merits and limitations are also presented. Another main purpose of this paper is to comprehensively review and summarise the current research on the development ofthe covariate models so as to facilitate the application of more covariate modelling techniques into prognostics and asset health management.

Key Words: Covariate model, Hazard, Reliability analysis, Survival analysis, Asset health, Life prediction 1 INTRODUCTION

In recent years, the emphasis on prognostics and asset life prediction has increased in the area of Engineering Asset Management(EAM) due to longer-term planning and budgeting requirements. One essential scientific research problem in EAM is the development of mathematical models that are capable of predicting Time-To-Failure (TTF) and the probability of failures in future time. In most real-life situations and industry applications, the hazard (failure rate) of assets is influenced and/or indicated by different riskfactors, which are often termed as covariates. Probabilistic modelling of assets lifetime using covariates (i.e. diagnostic factors and operating environment factors) is one of the indispensible scientific research problems for prognostics and asset life prediction. Until now, a number of statistical models have been developed to estimate the hazard of an asset/individual with covariates in both thereliability and biomedical fields. Most of these models are developed based on the Proportional Hazard Model (PHM) theory which was proposed by Cox in 1972 [1]. The basic theory of these covariate models is to build the baseline hazard function using historical failure data and the covariate function using covariate data. There are few review papers on the covariate models which have been reported inthe literature. Kumar and Klefsjo [2] reviews the existing literature on the proportional hazard model. Kumar and Westberg [3] provides a review of some reliability models for analysing the effect of operating conditions on equipment lifetime. Ma [4] discusses new research directions for Condition Monitoring (CM) and reviews some prognostic models in EAM. Almost all existing covariate models havebeen applied in the biomedical field. However, some of them have been applied in the reliability area. This expository paper is a collective review of the existing literature on covariate models in both the reliability and biomedical fields. In this paper, each individual covariate model has been contextually grouped into non-

parametric and semi-parametric models. Moreover, comments on...
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