In prognostics and health management (PHM), different authors frame the prognostics problem using different methods [1,2]^[1][2]. Typically, the goal is to create a regression model that can provide at each moment a numerical estimate or forecast of the residual time to the end of life of the equipment ^[3]. The equipment can be either a battery, valve, aircraft structure, or other engineering studied valves’ different fault stages (corresponding to intervals of residual lives) with sensor data to predict the remaining useful life (RUL). The concept of RUL is defined as the residual life of a system or component, measured in usage units (e.g., calendar time and number of cycles) at a given instant in time.

2. Prognostics Modeling Approaches

It is not simple to produce a precise definition of prognostics. Different authors have stated slightly different definitions, often with distinct meanings. However, at its simplest, the concept conveys the original meaning of the Greek words pro “before” and gnosis “knowledge”. It signifies knowing beforehand what the probable outcome, forecast, or prediction will be. This definition is in agreement with the perception of ^[4] of prognostics as the science of making predictions of engineering systems. This notion also concurs with ISO 13381-1:2004 [9]^[5], where prognostics is stated to be the “estimation of time to failure and risk for one or more existing and future failure modes”. Despite the generality of these definitions, other authors tend to equate prognostics to remaining useful life (RUL) estimation. RUL estimation is concerned with the operational performance of the system that falls outside a specific region of acceptable behavior [10]^[6]. It relates to the concept of end of life (EoL), which Daigle and Goebel [10]^[6] defined as the earliest time point at which one or more performance constraints are violated. The remaining useful life (RUL), at time instant 𝑡_𝑃, is then defined as Although the overall notion equates prognostics to RUL estimation, there are many approaches to prognostics, and not all of them necessarily focus on directly estimating residual life [11]^[7]. A few publications have focused on binary classification and multiclass classification rather than remaining useful life estimation. 

3. Hidden Markov Modeling

Hidden Markov models (HMMs) are statistical methods where the observed data can be modeled as the resulting output of hidden internal states. This kind of methodology utilizes inference to calculate the likelihood of each hidden state for each data instance. In HMM, the hidden states define how the data are generated. HMM has been used in some prognostics and health management (PHM) studies to classify sequences of health monitoring data. Ramasso [12]^[8] proposed an evidential HMM to address the problem of fault diagnostics and prognostics based on C-MAPSS data [13]^[9]. The model signaled a fault whenever a sequence of four hidden states was detected: steady, transition, up, and faulty. The authors reported a global accuracy of 75% against 68% with a classical HMM. Other works that have focused on applying classification methods based on HMM are by Ramasso and Denoeux [14]^[10] and Ramasso and Gouriveau [15]^[11], who further extended the work of Ramasso [12]^[8]. The underlying idea and assumptions were essentially the same as those of Ramasso [12]^[8]. Despite the positive results of these works, HMMs have their set of limitations. The Viterbi algorithm is computationally expensive, both in memory and time, possibly making the training slow. Additionally, if the observed sequence is too short, HMM models may not be suitable. Furthermore, HMMs require the configuration of many parameters.

4. Health Stage Classification Approaches

Other authors have studied the application of classification methods in prognostics. For example, Ref. [16]^[12] proposed the use of a pre-classification of health monitoring data before a suitable neural network is selected to perform the remaining useful life estimation. Instead of using a single RUL ESN model, the authors applied several ESN sub-models, chosen according to the outcome of a classification scheme. The authors reported that the combined method achieved better estimation performance on C-MAPSS data [13]^[9] compared to the approaches of classical ESN and ESN trained by the Kalman filter. Despite the positive results, the authors acknowledged that the main limitation of their approach was the absence of an actual method to match a specific engine unit to an ESN sub-model. In their work, they assumed that the classification scheme was always correct. In their recommendations, Ref. [16]^[12] reinforced the need to investigate how to establish the relationship between the different data classes and the RUL submodels. Ref. [17]^[13] followed similar reasoning to [16]^[12] using different RUL models over different prediction horizons according to a classification scheme of feature predictability. The authors argued that it is critical to build a model for prognostics and define the appropriate set of features over different horizons. In their work, feature sets were determined by the Fuzzy-C-means clustering algorithm. However, the authors did not disclose the details about this classification step. Other work focusing on machine prognostics based on classification is that of [18]^[14]. The authors used a multi-classification scheme to identify the different discrete degradation health stages that a system goes through. Five different classifiers were tested to capture the health stages of different systems in several simulated and industrial case studies. The study results suggest that accuracy depends on the selected classification technique and that support vector machines (SVMs) can produce the best results out of the five tested classifiers. Additionally, the authors argued that the optimal selection of the number of health stages is a vital aspect of these approaches. The best performing model in this study, the SVM, has some limitations, the most critical being its difficulty to process large data sets effectively. Additionally, the SVM classification algorithm typically does not perform well when the data have noise. Furthermore, the SVM does not provide natural probability estimates, as they need to be computed using a time-consuming procedure. Another important work that used the SVM algorithm for classification is by [19]^[15]. The authors proposed a binary SVM that classified each flight according to their degradation stage: healthy or faulty. The method was applied to the prognostics of engine bleed valves, with promising results. Other authors that used SVM approaches to prognostics include [20,21]^[16][17]. Another important work is by Tamilselvan and Wang [5]^[18] who proposed a deep belief network to classify the health state of multi-sensor condition monitoring data. The model is used for diagnostics purposes on two applications: aircraft engine health and electric power transformer condition. Ref. [8]^[19] also worked in health stage classification using the locally weighted linear regression (LWR) method. Interestingly, the authors noted the trade-off between the number of health stages and prediction accuracy. The more degradation stages, the more accurate the predictive model but the longer the training time. In the work of Allegorico and Mantini [22]^[20], the authors proposed an anomaly detection method based on machine learning classification methods to detect fault patterns in the engine exhaust gas temperature (EGT) profile data of E-class gas turbines. The authors tested logistic regression and artificial neural networks techniques on a real-world gas turbine case. A proprietary baseline algorithm based on the monitoring of the EGT spread was subjected to testing. The authors reported that the logistic regression classifier showed better performance in precision and recall than the other two algorithms. The authors mentioned the high training accuracy of the neural network in contrast with the poor testing performance. This finding may indicate that the network algorithm over-fitted the data. Despite the positive results of the logistic regression approach, the assumption of linearity between the dependent features and the target variables can hinder the utilization of this technique in more complex engineering scenarios. Linearly separable data are rarely found in health monitoring applications. Although traditionally developed for two-dimensional image data, convolutional neural networks (CNNs) can model univariate time series forecasting problems. A work using CNNs with promising results is by [23]^[21], where the authors proposed a multivariate convolutional neural network for time series classification. The method was evaluated on the prognostics and health management 2015 challenge data. The prediction problem was framed as a binary classification problem, with several models, one per plant and fault mode, being trained and used to detect abnormal data patterns. The proposed neural approach outperformed other deep learning methods (vanilla convolutional neural network and vanilla neural network), ensemble methods (random forests and xgboost), and simple linear regression. Another work that explored deep learning classifiers in prognostics and health management is by [24]^[22]. Their paper presented a hierarchical multiclass classification method using deep neural networks and a weighted support vector machine to discriminate spacecraft data. The deep network was used to reduce the dimensionality of the original spacecraft data. The multiclass weighted support vector machine method was used for classification. The results suggested that the proposed neural network with weighted support vector machines was more accurate and faster than the K-nearest neighbors, traditional SVM, and naive Bayes method. One interesting finding of the authors was that classification algorithms achieved high classification accuracy when the number of classes was small. The results, however, suggested that the technique had some difficulties in dealing with large datasets. Ref. [25]^[23] proposed a bearing fault detection strategy that used singular value decomposition for feature extraction and transfer learning for K-nearest neighbors classification. The authors concluded that performance was driven mainly by the volume of data and the ratio between target and auxiliary data. Despite the study’s utility, the focus was placed on the transfer learning methodology rather than the classification approach. Other notable work was performed by [26]^[24], who also classified system health states based on multidimensional sensor signals. The authors proposed a set of classifiers whose predictions were weighted according to an accuracy-based weighting scheme based on k-fold cross-validation. Five algorithms were selected as member classifiers: back-propagation neural network, support vector machines, deep belief networks, self-organizing maps, and Mahalanobis distance classifier. Naturally, classifiers with higher classification accuracy had larger weights (importance or influence) on the final fusion results. The authors used a relatively complex voting system, designated as weighted majority voting with dominance, to fuse the classifier outputs and predict system health conditions. The integrated fusion system was demonstrated on C-MAPSS data [13]^[9] and on rolling bearing data, with positive results. However, the offline training process was computationally expensive, focusing on the fusion methodology rather than the classification approaches.

5. Failure Mode Classification Approaches

Classification techniques are not only used to categorize health stages. For example, Ref. [6]^[25] proposed the use of a classifier based on multivariate sensor data to assign the system to different fault modes. Similarly, Ref. [27]^[26] tested three classification schemes to identify electronic failure modes. The authors used Euclidean, Mahalanobis, and Bayesian distance classifiers based on a feature extraction technique in the joint time–frequency analysis to classify different fault modes using pre-failure feature space. Their results suggest that it is possible to identify the regions and dominant progression directions of different failure modes of electronic equipment subjected to mechanical shock and vibration. The classification of failure modes is vital to forecast impending failure and to support prognostics. The main limitations of Bayesian classification techniques, such as the ones employed by [27]^[26], are the assumptions of Gaussian distributed data in each one of the classes, and the existence of equiprobable classes. In most cases, sensor features do not exhibit these properties, which impedes the widespread utilization of these approaches. Ref. [7]^[27] also used classification techniques to detect operational conditions. A recent interesting contribution is by ^[28]. The authors proposed a deep learning cross-domain fault classification method for rotating machinery. Experiments on different rotating machinery datasets suggest the superiority of the method. The authors reported positive results but recognize the issue of the large network size. The authors also acknowledged the need to optimize the network architecture.