Constructing robust health indicators from complex engineered systems via anticausal learning

Abstract

In prognostics and health management (PHM), the task of constructing comprehensive health indicators (HI) from huge amounts of condition monitoring data plays a crucial role. HIs may influence both the accuracy and reliability of remaining useful life (RUL) prediction, and ultimately the assessment of system’s degradation status. Most of the existing methods assume apriori an oversimplified degradation law of the investigated machinery, which in practice may not appropriately reflect the reality. Especially for safety–critical engineered systems with a high level of complexity that operate under time-varying external conditions, degradation labels are not available, and hence, supervised approaches are not applicable. To address the above-mentioned challenges for extrapolating HI values, we propose a novel anticausal-based framework with reduced model complexity, by predicting the cause from the causal models’ effects. Two heuristic methods are presented for inferring the structural causal models. First, the causal driver is identified from complexity estimate of the time series, and second, the set of the effect measuring parameters is inferred via Granger Causality. Once the causal models are known, off-line anticausal learning only with few healthy cycles ensures strong generalization capabilities that helps obtaining robust online predictions of HIs. We validate and compare our framework on the NASA’s N-CMAPSS dataset with real-world operating conditions as recorded on board of a commercial jet, which are utilized to further enhance the CMAPSS simulation model. The proposed framework with anticausal learning outperforms existing deep learning architectures by reducing the average root-mean-square error (RMSE) across all investigated units by nearly 65%.

Introduction

All modern engineered systems inevitably go through a continuously evolving health degradation process, which finally may lead to their replacement, usually via conventional preventive maintenance policies (Shafiee, 2015). It is extremely important to obtain a transparent knowledge of the machinery’s degradation levels so that unscheduled maintenance activities with great operational cost and possible reputational damage can be prevented (Rodrigues et al., 2012). According to Groenenboom (Groenenboom, 2018) in the domain of aviation, handling such issues by deploying intelligent condition monitoring approaches may enable airlines to save about $3 bn. per year. For developing such sophisticated solutions, it may be either impractical or infeasible to measure the exact health status of the machinery (Lei et al., 2018), let alone to unveil its hidden degradation trends. In this regard, Prognostics and Health Management (PHM) (Goebel et al., 2017) aims for the prediction of the remaining useful life (RUL) of the investigated machinery from comprehensive health indicator (HI) values that hopefully reflect the true health status. Such PHM schemes may not only improve the system’s reliability and cost-efficiency, but further prevent major accidents with potential loss of human lives. Hence, the accurate estimation of HIs, in particular under time-varying operating conditions with different degradation effects, plays a critical role to the final assessment of the RUL prediction, and ultimately to the effectiveness of the entire PHM framework.

Generally, two main categories of HIs are widely known based on the direct association of the information carrier with the gradual deterioration of the machinery (Hu et al., 2012, Lei et al., 2018). First, Physical Health Indicators (PHIs) mostly utilize domain-driven features that characterize the system’s degradation condition in a straightforward manner. In such PHI approaches (Javed et al., 2014, Medjaher et al., 2013, Benkedjouh et al., 2013), signal processing methods (e.g, wavelet transform) are usually employed with statistical-based ones so that the physical characteristics of the system are captured. PHIs are typically extracted from univariate raw vibration signals sampled at high frequencies, and measured in single mechanical components, such as bearings and gears (Ali et al., 2015, Hu et al., 2016). In more complex multi-component systems (e.g, jet engines) is much more difficult for PHIs to accurately extract the overall health status. Alternatively, Virtual Health Indicators (VHIs) are generally extracted by applying fusion and dimensionality reduction techniques either on multidimensional sensor readings or on individual physical features (Yang et al., 2016, Baraldi et al., 2018, Wang et al., 2008, Lei et al., 2018). In general, VHIs are one-dimensional unitless agents and they should clearly depict the health status of the machine regardless of any variations in the operating conditions. In this work, we focus on the later category of HIs, which is considered more challenging, since multifaceted degradation from complex systems must be extracted and summarized into a single highly representative and robust HI.

VHIs can be constructed by supervised or unsupervised learning methods, depending on the availability of the degradation (RUL) labels. For instance, Guo et al. (2018) proposed a supervised method to construct HIs via deep convolutional neural networks (CNN) by utilizing the cumulative service life in percentage from rolling element bearings as the target label. Although the authors used a non-linear mapping to construct the HI, they assumed a linear degradation trajectory for RUL labeling that adds a major restriction to the method, as different operating conditions may vary within the service life of the asset, and accordingly, they might accelerate or distort the degradation process. Similarly, the authors in Chen et al. (2020) employed in a non-linear way CNNs with long–short term memory neural networks (LSTMs) to capture long term dependencies in the time series which they ultimately used for bearing HI construction. Even though deep learning architectures are designed to overcome the tedious hand-crafting effort of manual feature extraction (Qin et al., 2016), the former study require sufficiently large amounts of run-to-failure vibration data to map the automatically extracted features to the target value that represents the health status of the investigated bearings. Both previous methods are basically supervised and might work well for individual components. However, in complex mechanical modules or systems, such as jet engines, it is often extremely costly or even impossible to quantitatively capture the holistic degradation state with a high accuracy that might be further used as the label at a specific service time. Finally, the dependence of these methods on run-to-failure training data is obviously an additional factor of limited applicability in real-operating conditions of safety–critical systems. Such limitations pave the way for unsupervised learning methods, in which only data from healthy conditions are utilized to learn the prediction algorithms.

Data-driven approaches that employ machine learning models, and especially deep neural network architectures, are currently proposed for PHM applications under nonlinear and multidimensional settings (Fink et al., 2020, Thoppil et al., 2021). Besides these fundamental challenges, changing operating conditions in engineered systems may rapidly deteriorate the model’s HI predictions due to sensitivity issues to perturbations of the input independent variables, which is attributed to the lack of robustness (Khan et al., 2021). Uncertainties that emanate either from measurement errors or from any stochasticity of the degradation process further impact the robustness of the model (Lei et al., 2018) in such a way that it is infeasible for HIs to be later used for RUL prediction purposes. On the other hand, another source of error might originate from model complexity itself. For example, deep neural network architectures that are trained with vast amounts of data, they consist of millions of parameters in order to learn the mapping functions between the input variables and the output. Such models usually suffer from increased complexity, and at the end they might yield poor generalization performance on future unseen data. Developing data-driven solutions with low model complexity that are able to robustly account for data variations is the key for ensuring the safety and reliability of engineered systems.

Most of the existing works rely on methods that learn horizontal dependencies in the data without the underlying causal structure under consideration. Such approaches usually lack generalization abilities and robustness, since they may collapse in non-i.i.d. (independent and identically distributed) regime, in which the probability distribution may strongly vary between the source and the target domain (e.g., different operating conditions). Since causal relationships are inherently invariant and stable over different domains (Bühlmann, 2020), models that are built on this principle can effectively address the challenges from non-i.i.d. settings. Such useful properties are actually entailed in structural causal models (Pearl, 2009, Peters et al., 2017) that mathematically describe the underlying causal relationships between cause and effect via deterministic functions, and noise variables to account any randomness in the model. Intuitively, SCMs represent the mechanism that is responsible for the data generation, and they represent a trade-off between physical and statistical models, as it is summarized in Table 1. In the seminal work of Schölkopf et al. (2012), the notion of structural causal models is utilized to investigate their implications on machine learning problems, like covariate shift and efficient usage of data in semi-supervised learning. In particular, the authors showed that in case of alignment of the causal direction $X\to Y$ with the predictive one (predicting $Y$ from $X$), where the input of the model $X$ is the cause and the target $Y$ is the effect, robustness to covariate shift is easier to achieve. On the other hand, when the causal direction $X\leftarrow Y$ is the opposite with the predictive one, where we are trying to predict the cause $Y$ from the effect $X$, this is said to be anticausal prediction and semi-supervised learning can interestingly work. Experiments in Schölkopf et al. (2012) with linear models for semi-supervised learning demonstrate the effectiveness of the approach, which we also adopt in the proposed framework due to availability of few data from healthy state.

Recently, the authors in Khan et al. (2021) highlight the importance of integrating causal models towards achieving robust AI-based solutions for PHM applications. Furthermore, they assert that the black-box problem in deep learning hinders any transparency of how input variables are interrelated with each other and with the outcome. This problem can be eliminated by the inherent interpretability of the structural causal models. We employ causality to seamlessly model the degradation process of the system and build multiple structural causal models from the time series of the monitoring signals and the inferred causal driver. Hence, we exploit the remarkable invariance properties of the causal mechanisms (Bühlmann, 2020, Schölkopf, 2019) and learn our models upon these mechanisms for better generalization and robustness.

The main contributions of this work are summarized as follows:

• A complexity-estimate metric for time series data to rank operational and environmental parameters (potential causes) by computing the largest variability in relation with the time scale. The intuition behind this metric is that within same time periods having more, and larger peaks and valleys will yield higher complexity estimate. In addition to limited background knowledge from the application domain, we finally select the causal driver from the overall parameter set.

• A causal feature selection for time series is developed based on non-linear Granger Causality to capture complex dependencies between the inferred causal driver and the measuring parameters. The computed causal indices are then used for selecting the dependent measuring parameters, from which a set of structural causal models is yielded.

• A novel HI is proposed by introducing anticausal learning for robust predictions of the holistic health status of complex engineered systems under time-varying operating conditions. To the best of our knowledge, the proposed framework is the first that integrates such powerful techniques from the field of causal inference for PHM applications.

• Our anticausal-based framework is capable of employing any kind of regression method. However, we show empirically that indeed the linear regression outperforms all other methods as it inherently disentangles the derived structural causal models in the anticausal direction. Last but not least, the proposed approach requires no assumption (e.g., bilinear) regarding the underlying degradation law, which most of methods do.

The rest of the paper is organized as follows: Section 2 reviews the related work of data-driven methods employed for VHI construction in different PHM applications. Section 3 presents the required theoretical background. In Section 4, the framework and details of the proposed anticausal learning approach are briefly introduced. In Section 5, details of the experimental results with the analysis and discussion on the effectiveness of the proposed method is presented. Finally, Section 6 concludes the paper and presents the potential of future work.