Nonlinear Dynamic Factor Analysis With a Transformer Network

The paper develops a Transformer architecture for estimating dynamic factors from multivariate time series data under flexible identification assumptions. Performance on small datasets is improved substantially by using a conventional factor model as prior information via a regularization term in the training objective. The results are interpreted with Attention matrices that quantify the relative importance of variables and their lags for the factor estimate. Time variation in Attention patterns can help detect regime switches and evaluate narratives. Monte Carlo experiments suggest that the Transformer is more accurate than the linear factor model, when the data deviate from linear-Gaussian assumptions. An empirical application uses the Transformer to construct a coincident index of U.S. real economic activity.

Learnable Koopman-Enhanced Transformer-Based Time Series Forecasting with Spectral Control

This paper proposes a unified family of learnable Koopman operator parameterizations that integrate linear dynamical systems theory with modern deep learning forecasting architectures. We introduce four learnable Koopman variants-scalar-gated, per-mode gated, MLP-shaped spectral mapping, and low-rank Koopman operators which generalize and interpolate between strictly stable Koopman operators and unconstrained linear latent dynamics. Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator while retaining compatibility with expressive nonlinear backbones such as Patchtst, Autoformer, and Informer. We evaluate the proposed operators in a large-scale benchmark that also includes LSTM, DLinear, and simple diagonal State-Space Models (SSMs), as well as lightweight transformer variants. Experiments across multiple horizons and patch lengths show that learnable Koopman models provide a favorable bias-variance trade-off, improved conditioning, and more interpretable latent dynamics. We provide a full spectral analysis, including eigenvalue trajectories, stability envelopes, and learned spectral distributions. Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.

From Observations to States: Latent Time Series Forecasting

Deep learning has achieved strong performance in Time Series Forecasting (TSF). However, we identify a critical representation paradox, termed Latent Chaos: models with accurate predictions often learn latent representations that are temporally disordered and lack continuity. We attribute this phenomenon to the dominant observation-space forecasting paradigm. Most TSF models minimize point-wise errors on noisy and partially observed data, which encourages shortcut solutions instead of the recovery of underlying system dynamics. To address this issue, we propose Latent Time Series Forecasting (LatentTSF), a novel paradigm that shifts TSF from observation regression to latent state prediction. Specifically, LatentTSF employs an AutoEncoder to project observations at each time step into a higher-dimensional latent state space. This expanded representation aims to capture underlying system variables and impose a smoother temporal structure. Forecasting is then performed entirely in the latent space, allowing the model to focus on learning structured temporal dynamics. Theoretical analysis demonstrates that our proposed latent objectives implicitly maximize mutual information between predicted latent states and ground-truth states and observations. Extensive experiments on widely-used benchmarks confirm that LatentTSF effectively mitigates latent chaos, achieving superior performance. Our code is available in https://github.com/Muyiiiii/LatentTSF.

ACFormer: Mitigating Non-linearity with Auto Convolutional Encoder for Time Series Forecasting

Time series forecasting (TSF) faces challenges in modeling complex intra-channel temporal dependencies and inter-channel correlations. Although recent research has highlighted the efficiency of linear architectures in capturing global trends, these models often struggle with non-linear signals. To address this gap, we conducted a systematic receptive field analysis of convolutional neural network (CNN) TSF models. We introduce the "individual receptive field" to uncover granular structural dependencies, revealing that convolutional layers act as feature extractors that mirror channel-wise attention while exhibiting superior robustness to non-linear fluctuations. Based on these insights, we propose ACFormer, an architecture designed to reconcile the efficiency of linear projections with the non-linear feature-extraction power of convolutions. ACFormer captures fine-grained information through a shared compression module, preserves temporal locality via gated attention, and reconstructs variable-specific temporal patterns using an independent patch expansion layer. Extensive experiments on multiple benchmark datasets demonstrate that ACFormer consistently achieves state-of-the-art performance, effectively mitigating the inherent drawbacks of linear models in capturing high-frequency components.

Explainable AI to Improve Machine Learning Reliability for Industrial Cyber-Physical Systems

Industrial Cyber-Physical Systems (CPS) are sensitive infrastructure from both safety and economics perspectives, making their reliability critically important. Machine Learning (ML), specifically deep learning, is increasingly integrated in industrial CPS, but the inherent complexity of ML models results in non-transparent operation. Rigorous evaluation is needed to prevent models from exhibiting unexpected behaviour on future, unseen data. Explainable AI (XAI) can be used to uncover model reasoning, allowing a more extensive analysis of behaviour. We apply XAI to to improve predictive performance of ML models intended for industrial CPS. We analyse the effects of components from time-series data decomposition on model predictions using SHAP values. Through this method, we observe evidence on the lack of sufficient contextual information during model training. By increasing the window size of data instances, informed by the XAI findings, we are able to improve model performance.

Sparse Tucker Decomposition and Graph Regularization for High-Dimensional Time Series Forecasting

Existing methods of vector autoregressive model for multivariate time series analysis make use of low-rank matrix approximation or Tucker decomposition to reduce the dimension of the over-parameterization issue. In this paper, we propose a sparse Tucker decomposition method with graph regularization for high-dimensional vector autoregressive time series. By stacking the time-series transition matrices into a third-order tensor, the sparse Tucker decomposition is employed to characterize important interactions within the transition third-order tensor and reduce the number of parameters. Moreover, the graph regularization is employed to measure the local consistency of the response, predictor and temporal factor matrices in the vector autoregressive model.The two proposed regularization techniques can be shown to more accurate parameters estimation. A non-asymptotic error bound of the estimator of the proposed method is established, which is lower than those of the existing matrix or tensor based methods. A proximal alternating linearized minimization algorithm is designed to solve the resulting model and its global convergence is established under very mild conditions. Extensive numerical experiments on synthetic data and real-world datasets are carried out to verify the superior performance of the proposed method over existing state-of-the-art methods.

Split-and-Conquer: Distributed Factor Modeling for High-Dimensional Matrix-Variate Time Series

In this paper, we propose a distributed framework for reducing the dimensionality of high-dimensional, large-scale, heterogeneous matrix-variate time series data using a factor model. The data are first partitioned column-wise (or row-wise) and allocated to node servers, where each node estimates the row (or column) loading matrix via two-dimensional tensor PCA. These local estimates are then transmitted to a central server and aggregated, followed by a final PCA step to obtain the global row (or column) loading matrix estimator. Given the estimated loading matrices, the corresponding factor matrices are subsequently computed. Unlike existing distributed approaches, our framework preserves the latent matrix structure, thereby improving computational efficiency and enhancing information utilization. We also discuss row- and column-wise clustering procedures for settings in which the group memberships are unknown. Furthermore, we extend the analysis to unit-root nonstationary matrix-variate time series. Asymptotic properties of the proposed method are derived for the diverging dimension of the data in each computing unit and the sample size $T$. Simulation results assess the computational efficiency and estimation accuracy of the proposed framework, and real data applications further validate its predictive performance.

Wavelet-Driven Masked Multiscale Reconstruction for PPG Foundation Models

Wearable foundation models have the potential to transform digital health by learning transferable representations from large-scale biosignals collected in everyday settings. While recent progress has been made in large-scale pretraining, most approaches overlook the spectral structure of photoplethysmography (PPG) signals, wherein physiological rhythms unfold across multiple frequency bands. Motivated by the insight that many downstream health-related tasks depend on multi-resolution features spanning fine-grained waveform morphology to global rhythmic dynamics, we introduce Masked Multiscale Reconstruction (MMR) for PPG representation learning - a self-supervised pretraining framework that explicitly learns from hierarchical time-frequency scales of PPG data. The pretraining task is designed to reconstruct randomly masked out coefficients obtained from a wavelet-based multiresolution decomposition of PPG signals, forcing the transformer encoder to integrate information across temporal and spectral scales. We pretrain our model with MMR using ~17 million unlabeled 10-second PPG segments from ~32,000 smartwatch users. On 17 of 19 diverse health-related tasks, MMR trained on large-scale wearable PPG data improves over or matches state-of-the-art open-source PPG foundation models, time-series foundation models, and other self-supervised baselines. Extensive analysis of our learned embeddings and systematic ablations underscores the value of wavelet-based representations, showing that they capture robust and physiologically-grounded features. Together, these results highlight the potential of MMR as a step toward generalizable PPG foundation models.

Chain-of-thought Reviewing and Correction for Time Series Question Answering

With the advancement of large language models (LLMs), diverse time series analysis tasks are reformulated as time series question answering (TSQA) through a unified natural language interface. However, existing LLM-based approaches largely adopt general natural language processing techniques and are prone to reasoning errors when handling complex numerical sequences. Different from purely textual tasks, time series data are inherently verifiable, enabling consistency checking between reasoning steps and the original input. Motivated by this property, we propose T3LLM, which performs multi-step reasoning with an explicit correction mechanism for time series question answering. The T3LLM framework consists of three LLMs, namely, a worker, a reviewer, and a student, that are responsible for generation, review, and reasoning learning, respectively. Within this framework, the worker generates step-wise chains of thought (CoT) under structured prompts, while the reviewer inspects the reasoning, identifies erroneous steps, and provides corrective comments. The collaboratively generated corrected CoT are used to fine-tune the student model, internalizing multi-step reasoning and self-correction into its parameters. Experiments on multiple real-world TSQA benchmarks demonstrate that T3LLM achieves state-of-the-art performance over strong LLM-based baselines.

TFEC: Multivariate Time-Series Clustering via Temporal-Frequency Enhanced Contrastive Learning

Multivariate Time-Series (MTS) clustering is crucial for signal processing and data analysis. Although deep learning approaches, particularly those leveraging Contrastive Learning (CL), are prominent for MTS representation, existing CL-based models face two key limitations: 1) neglecting clustering information during positive/negative sample pair construction, and 2) introducing unreasonable inductive biases, e.g., destroying time dependence and periodicity through augmentation strategies, compromising representation quality. This paper, therefore, proposes a Temporal-Frequency Enhanced Contrastive (TFEC) learning framework. To preserve temporal structure while generating low-distortion representations, a temporal-frequency Co-EnHancement (CoEH) mechanism is introduced. Accordingly, a synergistic dual-path representation and cluster distribution learning framework is designed to jointly optimize cluster structure and representation fidelity. Experiments on six real-world benchmark datasets demonstrate TFEC's superiority, achieving 4.48% average NMI gains over SOTA methods, with ablation studies validating the design. The code of the paper is available at: https://github.com/yueliangy/TFEC.