Domain generalization (DG) in time series poses significant challenges due to domain shift, particularly under the strict DG setting, where no target domain data are available during training. To address this, we propose AFM-CIR, a unified framework that integrates semantic-similarity-guided Adaptive Fourier Mixing (AFM) with Causality-Inspired Regression (CIR). Specifically, we construct a domain-invariant order-preserving guidance embedding that drives a similarity-based adaptive modulation of amplitude mixing and a bounded shortest-angle phase perturbation, thereby generating label-consistent and causally coherent augmented samples. CIR then enforces invariance and inter-dimensional independence through correlation factorization, while causal sufficiency is encouraged via adversarial masking. We further provide theoretical guarantees of the controllability of phase interventions, supported by mutual information analysis and Lipschitz-spectral norm bounds. Extensive experiments on four widely used benchmark industrial data sets demonstrate that AFM-CIR consistently achieves state-of-the-art performance, outperforming strong ERM, general DG, and task-specific DG baselines.

Cross-domain remaining useful life prediction is a critical task for industrial applications, and existing domain adaptation (DA) techniques typically focus on mitigating domain shift by learning domain-invariant features. These methods help transfer knowledge from a labeled source domain to an unlabeled target domain, thus improving prediction accuracy and robustness. However, the alignment strategies used are often coarse-grained, which can lead to the disruption of inherent temporal dependencies in degradation data. In this article, we introduce the novel order-preserving kernel contrastive (OPKC) learning, which leverages the concept that samples with smaller label differences should exhibit higher kernel similarities in the reproducing Kernel Hilbert space (RKHS), irrespective of their domain origin. This kernel-enhanced, regression-aware contrastive learning technique enables fine-grained instance-level pairwise alignment between the source and target domains, ensuring that label difference information is preserved while maintaining the temporal order information inherent in the data. In addition, OPKC can be seamlessly integrated with adversarial DA methods to further enhance both prediction performance and training stability. Extensive experiments on two widely used industrial benchmark datasets demonstrate that the proposed framework significantly outperforms state-of-the-art transfer learning and contrastive learning methods, achieving relative improvement of 16.9% on the PHM 2012 dataset and 13.2% on the C-MAPSS dataset.

Data-driven methods for predicting remaining useful life (RUL) have received considerable attention in the field of degradation data analysis. The transfer learning (TL) method offers new possibilities for RUL tasks in various operational settings. However, in many engineering applications, challenges in TL arise mainly from the scarcity or high cost of labeled data in the target domain, coupled with incomplete degradation of RUL samples within the target domain. This article proposes an innovative model named deep cross-domain transfer learning for interpretable prediction The model effectively harnesses the advantages of domain adaptation (DA) techniques in mitigating domain distribution disparities and also uses the exceptional visualization capabilities inherent in the variational autoencoder (VAE) model. This method integrates the VAE framework with regression networks and utilizes DA techniques to align feature spaces, achieving cross-domain RUL prediction with unlabeled target domain data and cross-domain visualization of the entire degradation process. The reproducing kernel Hilbert space is considered in domain adaption to control the complexity of hypothesis space. The effectiveness of the proposed method is demonstrated by analyzing the real C-MAPSS dataset.

Predicting remaining useful life (RUL) plays a crucial role in predictive maintenance, improving system reliability, availability, and safety. However, obtaining data from the target domain is often challenging in real-world industrial applications. This article focuses on the domain generalization (DG) problem, where the attention is directed toward adapting algorithms to unseen domains. Building upon the popular algorithm domain adversarial neural network (DANN) for DG, we extend the contrastive adversarial domain adaptation method using a multiple source–source adversarial network to learn domain-invariant features from multiple source domains. In addition, we incorporate the swin-transformer structure into our model to enhance its capability in extracting time–frequency features, leveraging its excellent performance in visual DG problems. Furthermore, to expand the training dataset, we propose a novel augmentation algorithm for time–frequency data. Through predictive experiments in scenarios with unknown domain labels, we validate the contribution of the proposed methods to RUL prediction performance.

The evaluation of long-lifetime and high-reliability products has attracted much attention. Stochastic degradation modeling is one of the most popular methods. The classical stochastic processes are frequently employed to discuss degradation trajectories. Most current work assumes that the underlying probability model of a degradation process is known or fixed in the estimation and model selection procedures. However, the ground-truth degradation model is usually unavailable in engineering applications. This article proposes a feasible parameter estimation and model selection procedure by measuring the distribution divergence among the nonparametric estimated model and some candidate models. In the proposed methods, it is not necessary to assume the availability of a ground-true model, which is replaced by a nonparametric estimated model. The proposed methodologies are suitable for restricted independent and nonidentically distributed samples. We discuss the large sample property of the suggested estimators. We report the Monte Carlo simulation study and practical data analysis to demonstrate our methods.

Kernel Fisher discriminant (KFD) is a popular tool as a nonlinear extension of Fisher's linear discriminant, based on the use of the kernel trick. However, its asymptotic properties are still rarely studied. We first present an operator-theoretical formulation of KFD which elucidates the population target of the estimation problem. Convergence of the KFD solution to its population target is then established. However, the complexity of finding the solution poses significant challenges when n is large and we further propose a sketched estimation approach based on a m×n sketching matrix which possesses the same asymptotic properties (in terms of convergence rate) even when m is much smaller than n. Some numerical results are presented to illustrate the performances of the sketched estimator.

Minimizing the divergence between two probability distributions offers an alternative parameter estimation method. The current literature mainly focuses on minimizing the Kullback-Leibler (K-L) divergence between the true and the proposed models in which the true model is assumed to be known or fixed. In this paper, we propose a parameter estimation method that minimizes the$f$-divergence between two probability distributions. The method is suitable for different situations, no matter the true distribution is known or not. The statistical properties of the estimator, including consistency and asymptotic normality, are established. As an illustration, our method is employed to estimate the degradation model, which is a model frequently used to assess the lifetime of highly reliable products. A simulation study and a real degradation data analysis are presented to illustrate the effectiveness of the proposed estimation method.

Distributed and parallel computing is becoming more important with the availability of extremely large data sets. In this article, we consider this problem for high-dimensional linear quantile regression. We work under the assumption that the coefficients in the regression model are sparse; therefore, a LASSO penalty is naturally used for estimation. We first extend the debiasing procedure, which is previously proposed for smooth parametric regression models to quantile regression. The technical challenges include dealing with the nondifferentiability of the loss function and the estimation of the unknown conditional density. In this article, the main objective is to derive a divide-and-conquer estimation approach using the debiased estimator which is useful under the big data setting. The effectiveness of distributed estimation is demonstrated using some numerical examples.

In this paper, the methods of information geometry are employed to investigate a generalized Bayes rule for prediction. Taking α-divergences as the loss functions, optimality, and asymptotic properties of the generalized Bayesian predictive densities are considered. We show that the Bayesian predictive densities minimize a generalized Bayes risk. We also find that the asymptotic expansions of the densities are related to the coefficients of the α-connections of a statistical manifold. In addition, we discuss the difference between two risk functions of the generalized Bayesian predictions based on different priors. Finally, using the non-informative priors (i.e., Jeffreys and reference priors), uniform prior, and conjugate prior, two examples are presented to illustrate the main results.