A Study of Proportional Hazards Models: Its Applications in Prognostics

Abstract

Prognostics and health management technology is proposed to satisfy the requirements of equipment autonomous maintenance and diagnosis, which is a new technique relying on condition-based maintenance. It mainly includes condition monitoring, fault diagnostics, life prediction, maintenance decision-making, and spare parts management. As one of the most commonly used reliability statistical modeling methods, proportional hazards model (PHM) is widely used in the field of prognostics, because it can effectively combine equipment service age and condition monitoring information to obtain more accurate condition prediction results. In the past decades, PHM-based methods have been widely employed, especially since the twenty-first century. However, after the rapid development of PHM, there is no systematic review and summary particularly focused on it. Therefore, this chapter comprehensively summarizes the research progress of PHM in prognostics.

1. Introduction

With the rapid development of science and technology, the integration, complexity, and intelligence of industrial systems have increased sharply. The traditional fault diagnostics and maintenance support technology is gradually difficult to adapt the new operation and maintenance requirements. As early as the 1970s, prognostics and health management technology first appeared. It achieves condition monitoring, fault prediction, and health management of complex industrial systems by processing and analyzing various operating data generated in the industrial process. This technology can predict the failure of the system before it happens and make effective maintenance decisions or suggestions in combination with the current working conditions. The main implementation steps are shown in Figure 1. In the figure, equipment prognostics and health management are divided into the processes of equipment condition monitoring, data acquisition, data processing, state prediction, and health management. Equipment prognostics is to predict the current or future state of equipment by using data acquisition and data processing technology based on condition monitoring information, including equipment failure rate, service reliability, remaining useful life (RUL), and other reliability indicators. The following health management measures are scheduled based on the predicted results of the component or system degradation state, such as reducing the frequency of monitoring, minimizing the number of maintenance downtimes, optimizing spare parts ordering and inventory management. In recent years, prognostic and health management has attracted extensive attention from industry and academia. There have also been several types of health prediction methods proposed, including physical model-based methods, data-driven methods, and hybrid model-based methods. In practical application, the physical model-based methods need to obtain the physical mechanism of the mechanical equipment degradation process. However, the physical processes for complex modern equipment are usually difficult to obtain. The data-driven method does not need to consider the kinematic principle of the equipment, instead relying on the data generated in the industrial dynamic process and extracting and processing the data to achieve the purpose of prediction. PHM is one of the commonly used models in data-driven methods. The hybrid model-based methods can combine the advantages of physical model and data-driven model to enhance the prediction accuracy, but designing the fusion mechanism between disparate models is a complex issue.

Figure 1.

Flowchart of equipment prognostics and health management.

As early as 1972, David Cox [1] first proposed the PHM to characterize the effect of multiple factors on the mortality or failure rate at a given time. Initially, PHM was utilized in the biomedical domain to analyze the survival of cancer patients. The evolution of equipment failure rate has a certain similarity to human mortality, and PHM can better fit various risk processes over time. Therefore, PHM is also widely used in reliability modeling of industrial equipment. Compared with other reliability statistical models, PHM has the characteristics of universality, flexibility, and simplicity and can effectively incorporate information on equipment service age and condition monitoring data. This means that PHM can estimate the probability of equipment failure at any time in a given state and then evaluate the health state of the equipment. In addition, PHM is also suitable for dealing with censored data [2].

The remainder of this chapter is organized as follows. Firstly, Section 2 introduces the basic form of PHM, which is divided into four parts: baseline hazard function, link function, covariate process, and parameter estimation. Secondly, the research progress of PHM in prognostics is reviewed in Section 3, which mainly outlines the reliability evaluation and RUL prediction of PHM in various engineering fields. Finally, Section 4 summarizes the conclusions and discusses the advantages and current challenges of PHM.

2. Basic form of PHM

The PHM is used to describe the failure rate of equipment, which is related to time and covariates. It is usually represented by the product of two independent functions $h_0\left(t\right)$ and $\psi \left(\gamma Z_t\right)$. The failure rate function can be expressed as:

$$\mathrm{h}\left(\mathrm{t},{\mathrm{Z}}_{\mathrm{t}}\right)={\mathrm{h}}_0\left(\mathrm{t}\right)\mathrm{\psi }\left(\mathrm{\gamma }{\mathrm{Z}}_{\mathrm{t}}\right)$$E1

where $h_0\left(t\right)$ is the baseline hazard function, which is only related to the sequipment service time, and represents the failure rate of the equipment when it is not affected by covariates. $\psi \left(\gamma Z_t\right)$ is the link function, which is related to the value of the covariate $Z_t$ and the covariate coefficient $\gamma$, indicating the influence of the covariate on the failure rate. Given the covariate $Z_t$, the failure rate function for service time $t$ is defined as:

$$h\left(t,Z_t\right)=\underset{\mathrm{\Delta }t\to 0}{lim}Pr\left(t<T<t+∆t|T>t,Z_t\right)/∆t$$E2

where $T$ is the failure time of the system. According to the definition of equipment reliability, the equipment conditional reliability function can be derived as:

$$R\left(t|Z_u,0\le u\le t\right)=Pr\left(T>t|Z_u,0\le u\le t\right)=exp\left(-{\int }_0^{t}h\left(t,Z_u\right)\mathit{du}\right)$$E3

According to the definition of the remaining useful life $X_t$, $X_t=\left\{x_t:T-t|T>t,Z_t\right\}$. The probability density function of the RUL of the equipment at service time t can be expressed as:

$${\mathrm{f}}_{{\mathrm{x}}_{\mathrm{t}}}\left({\mathrm{x}}_{\mathrm{t}}|{\mathrm{Z}}_{\mathrm{t}}\right)=\frac{\mathrm{f}\left(\mathrm{t}+{\mathrm{x}}_{\mathrm{t}}|{\mathrm{Z}}_{\mathrm{t}}\right)}{\mathrm{R}\left(\mathrm{t}|{\mathrm{Z}}_{\mathrm{t}}\right)}=\mathrm{h}\left(\mathrm{t}+{\mathrm{x}}_{\mathrm{t}}|{\mathrm{Z}}_{\mathrm{t}}\right)\frac{\mathrm{R}\left(\mathrm{t}+{\mathrm{x}}_{\mathrm{t}}|{\mathrm{Z}}_{\mathrm{t}}\right)}{\mathrm{R}\left(\mathrm{t}|{\mathrm{Z}}_{\mathrm{t}}\right)}$$E4

Next, the baseline hazard function, link function, covariate process, and parameter estimation of PHM will be introduced in detail.

2.1 Baseline hazard function

The baseline hazard function $h_0\left(t\right)$ can be modeled in various forms, including constant [3, 4], linear form [5], quadratic polynomial [6, 7], lognormal distribution [8, 9], and Weibull distribution [10, 11, 12]. Alternatively, following Cox’s strategy, a distribution-free approach is employed to directly estimate the baseline hazard rate from historical failure event data [13, 14].

PHM can be classified into two types based on the form of the baseline hazard function: semi-parametric PHM and full-parametric PHM. When the baseline hazard function is not specified, the model in Eq. (1) is often referred to as a semi-parametric PHM. In fact, one of the major advantages of semi-parametric PHM is that there is no need to define a specific form for the baseline hazard function, which makes semi-parametric PHM more flexible. The original semi-parametric PHM can be fully parameterized by defining a specific form of the baseline hazard function. The Weibull distribution is often used to describe the baseline hazard function of PHM, because it covers various types of failure rates (increasing failure rate, constant failure rate, and decreasing failure rate) and can better fit the equipment degradation data. In the form of Weibull distribution, Eq. (1) can be further deduced as follows:

$$\mathrm{h}\left(\mathrm{t},{\mathrm{Z}}_{\mathrm{t}}\right)=\frac{\mathrm{\beta }}{\mathrm{\alpha }}{\left(\frac{\mathrm{t}}{\mathrm{\alpha }}\right)}^{\mathrm{\beta }-1}\mathrm{\psi }\left(\mathrm{\gamma }{\mathrm{Z}}_{\mathrm{t}}\right)$$E5

where $\alpha >0$ is the scale parameter, and $\beta >0$ is the shape parameter. The Weibull distSribution function is used as an example to demonstrate the fully parameterized method of semi-parametric PHM. The baseline hazard function of PHM can be simply extended to any other function form in addition to the Weibull distribution function.

2.2 Link function

The form of the link function $\psi \left(\gamma Z_t\right)$ depends on the given failure event data and must satisfy the condition $\psi \left(\gamma Z_t\right)>0$. Cox proposed three link function forms, namely linear form $1+\mathit{\gamma Z}$, inverse linear form $1/\left(1+\mathit{\gamma Z}\right)$, and exponential function form $exp\left(\mathit{\gamma Z}\right)$. However, for all possible values of $Z$, it is difficult to choose the coefficient $\gamma$ in the linear and inverse linear forms to satisfy the above conditions. This criterion is better satisfied by the exponential function form. Moreover, an exponential function can also approximates the experimental data well [10]. Expressing $\psi \left(\gamma Z_t\right)$ in the form of an exponential function, Eq. (1) can be rewritten as the following form:

$$h\left(t,Z\left(t\right)\right)=h_0\left(t\right)exp\left[\sum _{i=1}^n{\gamma }_i{Z}_i\left(t\right)\right]$$E6

where $h\left(t,Z\left(t\right)\right)$ represents the failure rate at time $t$ under the influence of the covariate vector $Z\left(t\right)$. The symbol ${\gamma }_i,i=1,2,\dots ,n$, is the coefficient corresponding to the covariate $Z_i\left(t\right)$, indicating the degree of influence of each covariate on the failure rate, and $n$ represents the number of covariates. Therefore, determining the covariate process and parameters of PHM is crucial for assessing equipment failure rates.

2.3 Covariate process

In reliability modeling, the environmental factors or self-degradation characteristics that affect the system failure rate are commonly referred to as covariates. Covariates are important factors in PHM, and the choice of covariates has a direct impact on the accuracy of reliability and life prediction. According to the internal and external factors affecting equipment failure, it can be divided into internal covariates and external covariates. Internal covariates include the system’s own structural design [15], materials [16], degradation state characteristics [17], and so on. The current state of the system can be reflected according to the degradation characteristics. Whereas external covariates can usually be regarded as “risk factors” that can affect the failure time of the system, such as temperature [18, 19], humidity [20], weather conditions [21, 22], and other external operating environment.

During PHM modeling, a preliminary analysis of covariates should be performed to identify state indicators that have a significant impact on equipment failure rates. The covariates of PHM were determined by Vlok et al. [23] and Ghodrati et al. [24] based on the experience of maintenance technicians, which is extremely subjective. This could lead to the omission of other significant factors, as well as a high correlation between them. There are many methods to test the influence of covariates on the system failure rate. Most of the existing studies use methods such as P-value [25, 26], Wald test [27], likelihood ratio test [28], and score test [21]. When determining covariates, it is usually required that the correlation coefficient between the covariates be as small as possible. Therefore, Lin et al. [29], Carr et al. [30], and Chen et al. [31] employed principal component analysis (PCA) to analyze condition monitoring data and built PHM using principal components rather than the original covariates. This method is helpful to eliminate the collinearity between the original covariates and reduce the number of covariates in PHM. Makis et al. [32] used dynamic principal component analysis to reduce the dimensionality of the transmission oil data. Dynamic principal component analysis is an extension of the original PCA, which can achieve dimensionality reduction when the data have autocorrelation. Mazidi et al. [33] used several statistical techniques to decrease the dimension of the original monitoring data and select parameters, including PCA, Pearson, Spearman and Kendall correlation, mutual information, regression ReliefF, and decision trees. Ahmad et al. [34] used Failure Mode Effect and Criticality Analysis (FMECA) to identify external covariates that may affect the failure rate of transmission belts in cutting process system. Another key reason they use FMECA is that it can classify censored and uncensored data. Then, a statistical analysis of censored and uncensored time-to-failure data was performed by applying Failure Time Modeling (FTM) based on PHM considering the effects of external covariates. In order to investigate the impact of different covariates, Kabir et al. [35] and Kabir et al. [36] stratified the data according to the material type of the water mains and whether they had previously failed to establish distinct PHMs. They identified significant covariates in different models using the Bayesian model averaging (BMA) method. Based on the assumption that the heavier the operational use of components, the higher the probability of component failure, Verhagen et al. [37] used Extreme Value Analysis (EVA) and Maximum Difference Analysis (MDA) techniques to identify the operational factors that lead to the high failure rate of aircraft components. Wu et al. [28] used the Z test to investigate the effect of time-varying environmental covariates on the failure rate of wind turbine components, and the likelihood ratio test was used to find the best combination of covariates. Li et al. [38] and Thijssens et al. [39] used the Akaike Information Criterion (AIC) to choose covariates. In addition, the software SPSS [40, 41] and EXAKT [42, 43, 44] can also be applied to identify critical covariates affecting the equipment failure rate.

2.4 Parameter estimation

Parameter estimation is an important step in PHM modeling, including baseline hazard function parameter estimation and link function parameter estimation. The covariate most closely connected to system failure should have a higher weight in the link function, and the corresponding covariate coefficient ${\gamma }_i$ should be greater. The covariates having a weak link to failure should be given less weight, and the corresponding covariate coefficient ${\gamma }_i$ should be smaller. The model will be fairly close to reality if only covariates related to system failure are included. Therefore, the accuracy of the model parameter estimation has a considerable influence on the calculation result of the total failure rate of the objective equipment. Generally, the parameters of PHM are estimated by partial likelihood function [45, 46] or maximum likelihood function [8, 47], or related software programs, such as SPSS [40, 41], EXAKT [48, 49], SYSTAT [3], survival package for R [50], coxphfit function of Matlab [25], and so on. The likelihood estimation function formula of PHM parameter estimation is simple, and the maximization process is robust. However, the method based on classical likelihood estimation may suffer from slow convergence. Moreover, the maximum likelihood method cannot quantify the uncertainty in model predictions and field data. Uncertainties can be found in the whole process of calculation and modeling, including those resulting from test data and model parameters. For the lack of sufficient experimental or field data, Zuashkiani et al. [51] used expert knowledge to compensate and developed a method combining expert knowledge and statistical data to estimate the parameters of PHM. Considering the uncertainties in model predictions and field data, Jiang et al. [52] built a Bayesian network to represent nonlinear PHM based on historical inspection data. When updating the distribution function to estimate the model parameters in Bayesian networks, it is necessary to calculate the marginal function, which usually requires the high-dimensional integration problem on the prior distribution. Therefore, they used the Markov Chain Monte Carlo technique to solve the difficulties in parameter estimation.

The framework for equipment prognostics and health management based on PHM is shown in Figure 2, which includes data acquisition, PHM modeling, prognostics, and health management. Firstly, state indications connected to equipment failure, such as temperature, current, or vibration signals of the objective equipment, are collected using manual operation, sensors, or specific test tools. Then, P-value, principal component analysis or expert experience is used to determine the covariates that have a significant impact on the equipment failure rate. At the same time, maximum likelihood estimation, Bayesian update or SPSS, EXAKT, and other software are used to estimate the parameters for the PHM modeling. Finally, the reliability indicators such as equipment reliability or RUL are estimated according to the established PHM to achieve the purpose of state prediction. The basic structural form of PHM (containing the baseline hazard function and link function), as well as the covariate determination method and commonly used parameter estimation methods, is all introduced in this section. Next, as shown in Figure 2, we will concentrate on the prediction and evaluation of PHM in the domains of cutting tools, bearings, water supply pipes, and high-reliability devices and primarily review the covariate indicators selected in various literatures, as well as reliability estimation and RUL prediction based on PHM.

Figure 2.

Framework diagram of equipment prognostics and health management based on PHM.

3. Research progress of PHM in prognostics

In equipment failure prediction, reliability and RUL are two key health indicators. Reliability refers to the ability of a product to accomplish a specific function under specified conditions and within a specified time, indicating the probability that the product will fail to occur within a certain period of time. RUL refers to the continuous operating time of the equipment from the present moment to the occurrence of a potential failure. A potential risk to an effective forecasting system lies in accurately assessing reliability, RUL, and other relevant reliability indicators. Therefore, in order to avoid failures, accurate prediction of reliability and RUL through quantitative methods based on the current state of the machine and operating history is crucial for making preventive maintenance (PM) decisions. In PHM, the RUL of the equipment can be derived from the relevant reliability function, as shown in Eq. (4). Initially, Bendell [53] pointed out that PHM offers a lot of potential in the field of reliability assessment. According to the existing research, PHM has been widely employed in failure data analysis, reliability assessment, and life prediction in various fields, including hardware and software [54, 55]. For example, valves [56, 57], aircraft cargo doors [58], mining loader cables [59, 60, 61], distribution network cables [41, 62], printed circuit boards [63], mobile handsets [64], electrical appliances [65], automotive air-conditioning compressors [66], and so on. In addition, Barker et al. [67] applied PHM to describe the instantaneous rate of recovery of an electric power system after an outage and the likelihood of recovery occurs prior to a given point in time. Mohammad et al. [68] used PHM and Markov chains to analyze the reliability of load-sharing systems with a k-out-of-n structure. Zhao et al. [69] described a task reliability modeling method based on the Quality State Task Network (QSTN), used the WPHM to estimate the reliability of the cylinder head manufacturing system, and then described the overall operation state of the system. In various application examples, PHM is widely used in reliability assessment and life prediction in the domains of cutting tools, bearings, water supply pipes, and high-reliability devices. Table 1 gives an introductory summary of PHM in various application domains, regarding issues and failures, common measures, common covariates, and example data.

Application domains	Issues and failures	Common measures	Common covariates	Example data
Cutting tools	Tool wear,; tool fracture,; poor surface finish,; blade cracking	Vibration,; tool wear value,; cutting parameters	Cutting speed, feed rate and depth of cut [70],; root mean square and peak [8, 9],; cutting speed and the feed rate [49, 71],; tool wear, cutting speed and feed rate [72],; cutting speed [50],; the logarithm of cutting speed [73]	A CNC lathe FTC-20, FAIR FRIEND Group Taiwan [8],; a six-axis Boehringer NG 200 [49],; Gamma process simulation [50],; a CNC SOMAB “UNIMAB 450” lathe [73]
Bearings	Outer-race, inner-race, roller, and cage failures	Vibration,; oil debris,; acoustic emission	Natural logarithm of root mean square and kurtosis [74, 75],; kurtosis factor and crest factor [76],; kurtosis [77],; standard deviation, root mean square, and root amplitude sequences [78]	Experimental data of bearings 6205-2RS (SKF) from Case Western Reserve University [76],; the prognostic data repository contributed by Intelligent Maintenance System, University of Cincinnati [77, 79]
Water supply pipes	Corrosion,; leaking of joints, main barrel and line valves,; blockage,; break	Physical parameters of pipes, pressure, surrounding environment	Diameter, length, corrosivity, soil stability, internal pressure and the percentage of the pipe covered with low development land [80], material, diameter, length, vintage, soil type and the number of previous failures [81], length, diameter, pipe material, soil resistivity, soil resistivity, freezing index, rain deficit [82]	A pipe database collected in Laramie, Wyoming [83], the water distribution system serving the western part of the province of Ferrara [84], a dataset on pipe breakage from the city of Limassol (Cyprus) [85], the failure database of water distribution network in the City of Calgary, Alberta, Canada [35, 36, 82]

Table 1.

Introductory summarization of PHM in various application domains.

3.1 Cutting tools

In industrial manufacturing, the cost of consumables such as cutting tools cannot be ignored. The estimation error of cutting tool reliability or residual life may result in a large amount of production loss. On the one hand, overestimating tool reliability or residual life can lead to substandard parts being produced, as well as poor surface quality and machine damage. On the other hand, the underestimate of tool reliability or residual life may reduce the overall productivity and raise production cost, since insufficient tool use and shutdown loss result from needless frequent tool replacement. Therefore, effectively predicting tool reliability and RUL can help production managers to develop better tool replacement strategies, improve production planning, and increase production efficiency. At present, some scholars have used PHM to combine machining time and different working conditions to estimate tool reliability or RUL. Mazzuchi et al. [70] used PHM to evaluate the reliability of machine tools and used a full Bayesian method to compare the prior and posterior distributions of the parameters involved in the model to reflect tool aging and the importance of each covariate. In an automated manufacturing system, it is usually necessary for the same tool to cut multiple parts from different materials with different cutting parameters in order to save the space of tool magazine and avoid frequent tool changes while increasing production efficiency. In this situation, Liu et al. [86] derived a formula for calculating tool reliability under various cutting conditions with random machining time. Ding et al. [8] and Ding et al. [9] used PHM to analyze tool wear reliability by extracting the root mean square and peak of time domain indicators from the tool vibration signal as covariates. Cutting speed was chosen as a covariate of PHM by Equeter et al. [50], and the Mean Up Time of cutting tools was calculated using the integral of the reliability function. However, they only investigated the effect of cutting speed as a covariate on tool life. Shaban et al. [49] presented the reliability and RUL curves of tool cutting titanium metal matrix composites (Ti-MMCs) under various cutting speeds and feed rates. In the work of Equeter et al. [73], the authors converted the cutting speed of the tool to logarithmic form and used it as a covariate of PHM to predict the average available time of the tool. The logarithmic conversion of cutting speed data can provide more accurate prediction results, as demonstrated by a numerical case. Aramesh et al. [71] proposed a cutting tool life prediction model that took into account the influence of cutting parameters, machining time, and different tool wear stages (initial wear zone, steady wear zone, and rapid wear zone). The model provides very good estimates of tool life and critical points at which changes of states take place, as well as can calculate each between-states transition time. Aramesh et al. [72] developed a model for estimating the RUL of the worn tool under various cutting situations purely based on the actual wear of the tool, regardless of its usage history. This is a significant advantage of this model over other models in practical applications.

3.2 Bearings

Bearings are widely employed in a variety of areas as the essential components of rotating machinery. Due to the severe operating environment, bearing failure is one of the most common causes of machine failure, so it is necessary to perform reliability assessment and RUL prediction on bearings to prevent unexpected failures or accidents. Ding et al. [87] extracted the kurtosis and the root mean square in the bearing vibration signal as covariates reflecting the bearing operating state to evaluate the reliability of the bearing. To evaluate the reliability of bearing on site, Ding et al. [76] extracted the kurtosis factor and the crest factor as the covariates of PHM. The evaluation results can reflect the trend of failure occurrence and development. In some cases, it is very difficult to collect data from the actual system. Therefore, Leturiondo et al. [25] employed a physical model to generate synthetic data related to bearing degradation in order to fit the PHM and then estimate the bearing reliability further. The PHM presented by Liao et al. [74] takes into account both hard failure and multiple degradation features. The model is able to predict the mean RUL of a component based on online degradation information. Liao et al. [75] further compared the approach of Liao et al. [74] with the logistic regression model, demonstrating that the estimated RUL value based on PHM is closer to the actual life through a bearing test.

In recent research work, some scholars have combined PHM with artificial intelligence algorithms to estimate bearing reliability and predict RUL. Caesarendra et al. [77] used reliability theory and PHM to estimate the failure degradation of bearings and regarded it as a target vector. At the same time, combined with the kurtosis in the bearing vibration signal, they trained the support vector machine and established the life prediction model. The trained support vector machine is then utilized to predict the failure time of an individual bearing. Combining a neural network and PHM, Wang et al. [78] proposed a three-phase prognostic algorithm for bearings reliability evaluation and life prediction, which included feature selection, feature prediction, and RUL prediction. To begin, the most useful time-dependent features of vibration signals were extracted. Then, the feed-forward neural network is established as an identification model to predict the future features trends. Finally, PHM is used to estimate the reliability and RUL of the bearing. Qiu et al. [79] proposed an ensemble RUL prediction model by combining feature extraction, genetic algorithm, support vector regression, and WPHM. In this approach, genetic algorithm and signal feature extraction techniques are used to construct an effective health indicator. Secondly, support vector regression is used to predict the future development of the system operation behavior. Finally, RUL prediction is implemented using the WPHM prediction function.

3.3 Water supply pipes

The failure of water supply pipes usually affects other nearby infrastructure, which can lead to catastrophic consequences, so a large number of articles have been published to study the break risk process of water supply pipes. Kleiner et al. [88] outlined the application of various statistical models to water mains degradation, of which PHM is one of the most commonly used statistical models for estimating break failure of water mains. PHM was initially used by Jeffrey [7] to model the failure rate of water distribution system. Andreou et al. [89] and Andreou et al. [90] introduced the concept of early and late stages of water distribution system failures and used PHM to predict the deterioration of water distribution system in early stages with fewer breaks. They distinguished the different stages of pipe breaks based on a fixed number of failures, which only applies to the specific scenario considered, and did not explicitly describe the method used to identify the different stages. Park [91] and Park [92] developed a methodology to assess and track changes in the hazard functions between water main breaks by using PHM. As the number of pipe breaks increases, the critical points when the hazard function changes into different functional forms can be obtained to distinguish different stages of pipe failure. Park et al. [93] and Park et al. [94] divided cast iron 6-inch pipes into seven groups according to the break history of the water distribution system and constructed different PHMs for each group to estimate the reliability of the pipes. When there are only brief maintenance records, Le Gat et al. [95] discussed the efficiency of WPHM in fault prediction of water networks. Alvisi et al. [84] further investigated the model proposed by Le Gat et al. [95], pointing out that WPHM can exploit the information available on both the characteristics of the pipes in which breakages occur and their age to make the prediction results more stable and reliable. Instead of calculating the expected number of failures for a group of pipes, Clark et al. [96] and Karaa et al. [80] used PHM to calculate the probability of a pipe breaking or leaking for each pipe. Vanrenterghem-Raven et al. [97] created a simple prioritization index based on the ratio of pipe failure rates to determine which pipes should be replaced first. PHM was used by Fuchs-Hanusch et al. [81] to estimate the years when the failures occur with a defined probability. Moreover, they proposed a whole of life cost calculation method due to the long lifetime of water supply pipes. Christodoulou [85] used a 5-year dataset to study the impact of several risk factors on pipe failure, such as pipe material, diameter, and accident type. Regardless of the quality and quantity of data utilized in the model, there is inherent uncertainty when predicting the failure of water pipes. In order to explain the variability of these unknown factors, Clark et al. [83] incorporated a shared frailty into the PHM to account for the unspecified variability affecting the pipe breaks. Kabir et al. [35] and Kabir et al. [36] developed a Bayesian framework for predicting water main failure in the face of uncertainties. The proposed Bayesian Weibull proportional hazards model (BWPHM) is applied in this study to develop survival curves and predict water main failure rates. The results of their case indicated that the predicted 95% uncertainty bounds of the proposed BWPHMs capture effectively the observed water main failures. Applying the receiver operating characteristics curve, Debón et al. [98] compared PHM and generalized linear model for evaluating the risk of failure in water supply networks. Kimutai et al. [82] compared the predicted effects of Cox PHM, WPHM, and Poisson model in the break of cast iron, ductile iron, and plastic water pipes. The results recommended that a combined model should be used according to the rate of degradation and material type of the system. Xie et al. [99] used PHM to study the blockage risk of vitrified clay wastewater pipes and identified the pipes with the highest risk of failure due to blockage. In a cost-constrained environment, targeted inspection, plan maintenance, and replacement programs can be carried out to reduce the serious consequences caused by blockage.

3.4 High-reliability device

The PHM is a very popular tool in reliability theory and applications, which can be used to simulate the impact of another environment on the reliability of a baseline environment. For long-life and high-reliability devices (such as some electronic components [100, 101], etc.), it is difficult to obtain their failure data in a short time. Accelerated testing is a method for decreasing the life of high reliability devices or accelerating their performance degradation. The results of the tests are obtained in a shorter period of time under an accelerated stress environment, and PHM is then utilized to predict the failure behavior of the device under various operating conditions [102, 103]. There are generally two approaches to comprehensively utilize the failure data at various stress levels. The first is to convert data collected under high stress into data collected under normal operating conditions in order to expand the sample size and improve the accuracy of parameter estimation, reliability assessment, and life prediction [104]. Another approach is to establish the relationship between stress environment and lifetime by using acceleration models such as Arrhenius models, power law models, and exponential models [5]. Comparing PHM with the accelerated failure time model, Newby [105] pointed out the greatest advantage of PHM is that it does not need to specify a baseline failure rate, and it can quantitatively analyze the impact of each covariate on the total failure rate. Elsayed et al. [6] and Finkelstein [106] generalized the covariates of a single stress type to two stress types, allowing for the collection of a large number of failure time data in a short period of time. PHM was introduced into dealing with accelerated degradation test data by Chen et al. [107], who proposed a model based on the proportional degradation hazards model. They plotted the reliability curves of carbon-film resistors at normal stress condition based on the proposed model. To explore the reliability trend of Metal-Oxide-Semiconductor Field-Effect Transistors, Zheng et al. [108] conducted an accelerated degradation test with temperature as the accelerated stress. They established a PHM with the degradation trend and temperature as covariates, in which the degradation trend is defined by the Wiener process, because the degradation trend contains more information than the degradation state. The reliability results predicted in this reference are closer to the real scenario than the PHM with only temperature or only the deterioration state as covariate.

3.5 Others

In addition to the above domains, PHM can also be used in many other industries such as batteries, pumps, wind turbines, and so on. Some scholars also use PHM to describe the probability of hard failure of equipment. Hard failure generally refers to the sudden failure of equipment due to hidden manufacturing defects, excessive loads, or other stresses. When a hard failure occurs, the degradation signal tends to exhibit different values, such as the resistance value of a lead-acid battery in an automobile. Zhou et al. [109] proposed a two-stage approach, with an offline modeling stage based on historical data and an online prediction stage based on the degradation signal of each individual unit. This method is suitable for predicting the RUL of batteries without a powerful computing platform such as vehicle microcontrollers; however, the prediction results are relatively conservative. The variance of resistance between different batteries becomes larger as time increases, and there are noticeable individual differences among units. Therefore, in the RUL prediction framework of Man et al. [110], the authors applied the Wiener process with drift to characterize the degradation path of the battery resistance. However, because of the Markov nature of the Wiener process, their prediction methods rely on the most recent observational information. In comparison to Zhou et al. [109], the prediction accuracy is relatively low when current observations deviate from the degenerate path.

In the hard failure prediction method, the above studies do not consider potential change points in condition monitoring signals. However, change point detection and equipment degradation modeling are interrelated, which directly affects the accuracy of residual life prediction. You et al. [111] detected the change point in advance through the statistical process control method and divided the life cycle of the equipment into two zones: the stable zone and the degradation zone. This study was limited to the assumption that the change point was fixed and did not consider the impact of other factors on the change point during equipment operation. Son et al. [112] extended the method of Zhou et al. [109] to predict the RUL for individual units with considering the change point in condition monitoring signals, where the change point is captured based on the concordance correlation coefficient (CCC). Although this method improves the accuracy of RUL prediction, it also increases the complexity and computational burden of the model.

Because of the complexity of modern mechanical systems and the diversity of failure modes, it is necessary to consider the competition and interaction between different failure modes when analyzing the failure of the whole system. Zhang et al. [113] proposed a mixture Weibull proportional hazards model (MWPHM) to predict the failure of a high-pressure water descaling pump with two failure modes of sealing ring wear and thrust bearing damage. The system failure probability density is obtained by proportionally accumulating the probability density of multiple failure modes. Compared with traditional WPHM, MWPHM can provide more detailed life information, and its failure probability distribution is closer to the actual distribution. In this model, the prediction of failure time depends on the choice of reliability threshold. However, they assumed the reliability threshold was fixed, ignoring the fact that the reliability of the repaired system may change at the moment of failure.

In recent years, machine learning methods have been continuously developed. These theories and methods have been widely employed in engineering applications and scientific fields to address complex problems. In reliability engineering, some scholars compared the reliability and RUL prediction results based on PHM with the prediction results of neural network [103] and random forests method [114]. Li et al. [114] investigated the effect of failure time data with heavy-tailed behavior on the RUL prediction error. The results showed that the RUL prediction method based on PHM can make more accurate mechanical failure predictions than random forests. Izquierdo et al. [115] proposed a reliability model based on a dynamic artificial neural network by combining the neural network model and dynamic PHM concept. The model combines the benefits of neural networks for analyzing unknown interactions between environmental variables with the benefits of PHM for integrating dynamic operational environments. Mazidi et al. [116] created three neural network models to simulate the normal behavior of three features of wind turbine rotor speed, gearbox temperature, and generator winding temperature. Deviation signals are defined and calculated as accumulated time series of differences between neural network predictions and actual measurements. These signals are then used to develop a health condition model for each considered feature of wind turbine in order to perform anomaly detection. By combining autoregressive moving average model, PHM, and support vector machine, Tran et al. [117] proposed a three-stage method for estimating low methane compressor performance degradation and RUL. The method only uses the normal operation condition of the machine to create an identification model to recognize the dynamic system behavior and does not need data of whole machine life. Chen et al. [118] designed a deep learning structure called merged-long-short term memory (M-LSTM) network for health index modeling, which they subsequently integrated with PHM to predict the RUL of an automobile. However, since the repaired automobile cannot be recovered to its original state, the authors only consider the first maintenance record of automobiles, which makes it difficult to construct the health index of the automobile.

PHM has a certain application in prognostics and has achieved considerable results, but there is still a lack of a lot of research work to extend the prognostics scheme based on PHM.

4. Conclusions

As one of the most commonly used statistical models, PHM has received extensive attention and has been applied in a variety of domains. This chapter has summarized the research progress of prognostics based on PHM, focusing on the baseline hazard function, link function, covariate process, and parameter estimation methods. Data analysis, reliability estimation, and RUL prediction based on PHM are systematically discussed.

According to the review and research of PHM in this chapter, the advantages of PHM in the field of prognostics are summarized. Compared with other statistical methods, the main advantages of PHM are as follows.

1. PHM does not require assumptions about the nature or form of the baseline hazard function, and any type of distribution function can be used as the baseline hazard function. Therefore, PHM can be applied to reliability analysis of equipment in a variety of engineering domains, and it has the characteristics of universality, flexibility, and simplicity.

2. PHM directly models the failure rate, which has strong interpretability. The relationship between failure rate and condition monitoring information has been established, allowing the condition monitoring information to be used more effectively to update the equipment state. Furthermore, the influence of various covariates on the total failure rate may be easily assessed.

3. PHM can better simulate the influence of multiple internal and external degradation information on equipment failure, including environmental factors, aging factors, and degradation factors. Therefore, PHM is applicable when the failure of equipment is related to multiple influencing factors.

4. Compared with other data-driven methods, PHM can achieve good modeling results with a limited amount of degraded data. The accuracy of equipment condition prediction will improve as the amount of gathered event data and condition monitoring data grows.

Although PHM has made significant development in the field of prognostics and health management and has many advantages mentioned above, there are still a few aspects that need to be studied further. The current challenges facing PHM are listed below in order to point out the development direction for researchers.

1. Proportionality assumption. The application of PHM needs to satisfy the proportionality assumption, which has a fixed model form. It can only be used if the influence of the degradation process on the failure rate satisfies the link function, which is a fairly severe requirement.

2. Determine covariates. PHM can take into account the effects of multiple covariates on the failure rate of equipment at the same time. However, the results of parameter estimation will be biased if a relevant covariate is omitted or the accuracy of covariate measurement varies. Furthermore, because multiple covariates are associated with the same equipment degradation process, there may be correlations between them, which might influence the accuracy of the prediction results if not treated properly.

3. Data fusion. It is a challenge for data-specific fusion methods, such as the fusion between vibration signals, current signals, and oil signals. Complex systems usually involve multidimensional covariate processes. To assure the accuracy of system state prediction, more research into how to properly integrate diverse forms of data and adopt a more reasonable combination structure deserves further study.

4. Calculation difficulty. For a covariate process affected by stochastic degradation, a stochastic process must be used to describe the degradation process of covariates, which increases the calculation burden. The calculation of PHM becomes extremely difficult when the degradation process of the system incorporates multiple covariates. Calculating high-dimensional data presents a number of challenges.

5. Data problems. It is difficult to obtain event data and condition monitoring data simultaneously in practical applications. Especially for high-reliability systems and crucial equipment, they are not allowed to run to failure. This leads to a small number of failed samples, posing a major barrier to data-driven methods.

Combining the above advantages and disadvantages, PHM is suitable for imperfect observation systems with small degraded data samples and can make better use of condition monitoring information in various dimensions. The data fusion and model calculation problems of PHM can be greatly solved when combined with other methods (such as Bayesian iteration, artificial intelligence, and so on), offering it irreplaceable theoretical value and application prospect in the field of prognostics and health management in complex systems.