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).