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.

Prediction models inform important clinical decisions, aiding in diagnosis, prognosis, and treatment planning. The predictive performance of these models is typically assessed through discrimination and calibration. However, changes in the distribution of the data impact model performance. In health-care, a typical change is a shift in case-mix: for example, for cardiovascular risk management, a general practitioner sees a different mix of patients than a specialist in a tertiary hospital. This work introduces a novel framework that differentiates the effects of case-mix shifts on discrimination and calibration based on the causal direction of the prediction task. When prediction is in the causal direction (often the case for prognosis predictions), calibration remains stable under case-mix shifts, while discrimination does not. Conversely, when predicting in the anti-causal direction (often with diagnosis predictions), discrimination remains stable, but calibration does not. A simulation study and empirical validation using cardiovascular disease prediction models demonstrate the implications of this framework. This framework provides critical insights for evaluating and deploying prediction models across different clinical settings, emphasizing the importance of understanding the causal structure of the prediction task.

In medical imaging, particularly in early disease detection and prognosis tasks, discerning the rationale behind an AI model's predictions is crucial for evaluating the reliability of its decisions. Conventional explanation methods face challenges in identifying discernible decisive features in medical image classifications, where discriminative features are subtle or not immediately apparent. To bridge this gap, we propose an explainable model that is equipped with both decision reasoning and feature identification capabilities. Our approach not only detects influential image patterns but also uncovers the decisive features that drive the model's final predictions. By implementing our method, we can efficiently identify and visualise class-specific features leveraged by the data-driven model, providing insights into the decision-making processes of deep learning models. We validated our model in the demanding realm of medical prognosis task, demonstrating its efficacy and potential in enhancing the reliability of AI in healthcare and in discovering new knowledge in diseases where prognostic understanding is limited.

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.

Operating units often experience various failure modes in complex systems, leading to distinct degradation paths. Relying on a prognostic model trained on a single failure mode may lead to poor generalization performance across multiple failure modes. Therefore, accurately identifying the failure mode is of critical importance. Current prognostic approaches either ignore failure modes during degradation or assume known failure mode labels, which can be challenging to acquire in practice. Moreover, the high dimensionality and complex relations of sensor signals make it challenging to identify the failure modes accurately. To address these issues, we propose a novel failure mode diagnosis method that leverages a dimension reduction technique called UMAP (Uniform Manifold Approximation and Projection) to project and visualize each unit's degradation trajectory into a lower dimension. Then, using these degradation trajectories, we develop a time series-based clustering method to identify the training units' failure modes. Finally, we introduce a monotonically constrained prognostic model to predict the failure mode labels and RUL of the test units simultaneously using the obtained failure modes of the training units. The proposed prognostic model provides failure mode-specific RUL predictions while preserving the monotonic property of the RUL predictions across consecutive time steps. We evaluate the proposed model using a case study with the aircraft gas turbine engine dataset.

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.

This paper proposes a supervised dimension reduction methodology for tensor data which has two advantages over most image-based prognostic models. First, the model does not require tensor data to be complete which expands its application to incomplete data. Second, it utilizes time-to-failure (TTF) to supervise the extraction of low-dimensional features which makes the extracted features more effective for the subsequent prognostic. Besides, an optimization algorithm is proposed for parameter estimation and closed-form solutions are derived under certain distributions.

Data-driven industrial health prognostics require rich training data to develop accurate and reliable predictive models. However, stringent data privacy laws and the abundance of edge industrial data necessitate decentralized data utilization. Thus, the industrial health prognostics field is well suited to significantly benefit from federated learning (FL), a decentralized and privacy-preserving learning technique. However, FL-based health prognostics tasks have hardly been investigated due to the complexities of meaningfully aggregating model parameters trained from heterogeneous data to form a high performing federated model. Specifically, data heterogeneity among edge devices, stemming from dissimilar degradation mechanisms and unequal dataset sizes, poses a critical statistical challenge for developing accurate federated models. We propose a pioneering FL-based health prognostic model with a feature similarity-matched parameter aggregation algorithm to discriminatingly learn from heterogeneous edge data. The algorithm searches across the heterogeneous locally trained models and matches neurons with probabilistically similar feature extraction functions first, before selectively averaging them to form the federated model parameters. As the algorithm only averages similar neurons, as opposed to conventional naive averaging of coordinate-wise neurons, the distinct feature extractors of local models are carried over with less dilution to the resultant federated model. Using both cyclic degradation data of Li-ion batteries and non-cyclic data of turbofan engines, we demonstrate that the proposed method yields accuracy improvements as high as 44.5\% and 39.3\% for state-of-health estimation and remaining useful life estimation, respectively.

Automatic segmentation of head and neck cancer (HNC) tumors and lymph nodes plays a crucial role in the optimization treatment strategy and prognosis analysis. This study aims to employ nnU-Net for automatic segmentation and radiomics for recurrence-free survival (RFS) prediction using pretreatment PET/CT images in multi-center HNC cohort. A multi-center HNC dataset with 883 patients (524 patients for training, 359 for testing) was provided in HECKTOR 2022. A bounding box of the extended oropharyngeal region was retrieved for each patient with fixed size of 224 x 224 x 224 $mm^{3}$. Then 3D nnU-Net architecture was adopted to automatic segmentation of primary tumor and lymph nodes synchronously.Based on predicted segmentation, ten conventional features and 346 standardized radiomics features were extracted for each patient. Three prognostic models were constructed containing conventional and radiomics features alone, and their combinations by multivariate CoxPH modelling. The statistical harmonization method, ComBat, was explored towards reducing multicenter variations. Dice score and C-index were used as evaluation metrics for segmentation and prognosis task, respectively. For segmentation task, we achieved mean dice score around 0.701 for primary tumor and lymph nodes by 3D nnU-Net. For prognostic task, conventional and radiomics models obtained the C-index of 0.658 and 0.645 in the test set, respectively, while the combined model did not improve the prognostic performance with the C-index of 0.648.