Many failure mechanisms of machinery are closely related to the behavior of condition monitoring (CM) signals. To achieve a cost-effective preventive maintenance strategy, accurate remaining useful life (RUL) prediction based on the signals is of paramount importance. However, the CM signals are often recorded at different factories and production lines, with limited amounts of data. Unfortunately, these datasets have rarely been shared between the sites due to data confidentiality and ownership issues, a lack of computing and storage power, and high communication costs associated with data transfer between sites and a data center. Another challenge in real applications is that the CM signals are often not explicitly specified \textit{a priori}, meaning that existing methods, which often usually a parametric form, may not be applicable. To address these challenges, we propose a new prognostic framework for RUL prediction using the joint modeling of nonlinear degradation signals and time-to-failure data within a federated learning scheme. The proposed method constructs a nonparametric degradation model using a federated multi-output Gaussian process and then employs a federated survival model to predict failure times and probabilities for in-service machinery. The superiority of the proposed method over other alternatives is demonstrated through comprehensive simulation studies and a case study using turbofan engine degradation signal data that include run-to-failure events.

Industrial prognostics aims to develop data-driven methods that leverage high-dimensional degradation signals from assets to predict their failure times. The success of these models largely depends on the availability of substantial historical data for training. However, in practice, individual organizations often lack sufficient data to independently train reliable prognostic models, and privacy concerns prevent data sharing between organizations for collaborative model training. To overcome these challenges, this article proposes a statistical learning-based federated model that enables multiple organizations to jointly train a prognostic model while keeping their data local and secure. The proposed approach involves two key stages: federated dimension reduction and federated (log)-location-scale regression. In the first stage, we develop a federated randomized singular value decomposition algorithm for multivariate functional principal component analysis, which efficiently reduces the dimensionality of degradation signals while maintaining data privacy. The second stage proposes a federated parameter estimation algorithm for (log)-location-scale regression, allowing organizations to collaboratively estimate failure time distributions without sharing raw data. The proposed approach addresses the limitations of existing federated prognostic methods by using statistical learning techniques that perform well with smaller datasets and provide comprehensive failure time distributions. The effectiveness and practicality of the proposed model are validated using simulated data and a dataset from the NASA repository.

Industrial prognostics focuses on utilizing degradation signals to forecast and continually update the residual useful life of complex engineering systems. However, existing prognostic models for systems with multiple failure modes face several challenges in real-world applications, including overlapping degradation signals from multiple components, the presence of unlabeled historical data, and the similarity of signals across different failure modes. To tackle these issues, this research introduces two prognostic models that integrate the mixture (log)-location-scale distribution with deep learning. This integration facilitates the modeling of overlapping degradation signals, eliminates the need for explicit failure mode identification, and utilizes deep learning to capture complex nonlinear relationships between degradation signals and residual useful lifetimes. Numerical studies validate the superior performance of these proposed models compared to existing methods.

The use of tensors is progressively widespread in the realms of data analytics and machine learning. As an extension of vectors and matrices, a tensor is a multi-dimensional array of numbers that provides a means to represent data across multiple dimensions. As an illustration, Figure 1 shows an image stream that can be seen as a three-dimensional tensor, where the first two dimensions denote the pixels within each image, while the third dimension represents the distinct images in the sequence. One of the advantages of representing data as a tensor, as opposed to reshaping it into a vector or matrix, lies in its ability to capture intricate relationships within the data, especially when interactions occur across multiple dimensions. For instance, the image stream depicted in Figure 1 exhibits a spatiotemporal correlation structure. Specifically, pixels within each image have spatial correlation, and pixels at the same location across multiple images are temporally correlated. Transforming the image stream into a vector or matrix would disrupt the spatiotemporal correlation structure, whereas representing it as a three-dimensional tensor preserves this correlation. In addition to capturing intricate relationships, other benefits of using tensors include compatibility with multi-modal data (i.e., accommodating diverse types of data in a unified structure) and facilitating parallel processing (i.e., enabling the parallelization of operations), etc. As a result, the volume of research in tensor-based data analytics has been rapidly increasing in recent years (Shen et al., 2022; Gahrooei et al., 2021; Yan et al., 2019; Hu et al., 2023; Zhen et al., 2023; Zhang et al., 2023).

Industrial prognostic aims to predict the failure time of machines by utilizing their degradation signals. This is typically achieved by establishing a statistical learning model that maps the degradation signals of machines to their time-to-failure (TTFs) [1, 2]. Similar to that of many other statistical learning models, the implementation of prognostic models usually consists of two steps: model training and real-time monitoring (also known as model testing or deployment). Model training focuses on using a historical dataset that comprises the degradation signals and TTFs of some failed machines to estimate the parameters of the prognostic model; real-time monitoring feeds the real-time degradation signals from a partially degraded onsite machine into the prognostic model trained earlier to predict its TTF or TTF distribution. Most existing prognostic models assume that a historical dataset from a decent number of failed machines is available for model training [3, 4, 5, 6, 7]. In reality, however, the amount of historical data owned by a single organization (e.g., a company, a university lab, a factory, etc.) might be small or not large enough to train a reliable prognostic model.

Prognostics is concerned with predicting the future health of the equipment and any potential failures. With the advances in the Internet of Things (IoT), data-driven approaches for prognostics that leverage the power of machine learning models are gaining popularity. One of the most important categories of data-driven approaches relies on a predefined or learned health indicator to characterize the equipment condition up to the present time and make inference on how it is likely to evolve in the future. In these approaches, health indicator forecasting that constructs the health indicator curve over the lifespan using partially observed measurements (i.e., health indicator values within an initial period) plays a key role. Existing health indicator forecasting algorithms, such as the functional Empirical Bayesian approach, the regression-based formulation, a naive scenario matching based on the nearest neighbor, have certain limitations. In this paper, we propose a new `generative + scenario matching' algorithm for health indicator forecasting. The key idea behind the proposed approach is to first non-parametrically fit the underlying health indicator curve with a continuous Gaussian Process using a sample of run-to-failure health indicator curves. The proposed approach then generates a rich set of random curves from the learned distribution, attempting to obtain all possible variations of the target health condition evolution process over the system's lifespan. The health indicator extrapolation for a piece of functioning equipment is inferred as the generated curve that has the highest matching level within the observed period. Our experimental results show the superiority of our algorithm over the other state-of-the-art methods.