Trend-Adjusted Time Series Models with an Application to Gold Price Forecasting

Time series data play a critical role in various fields, including finance, healthcare, marketing, and engineering. A wide range of techniques (from classical statistical models to neural network-based approaches such as Long Short-Term Memory (LSTM)) have been employed to address time series forecasting challenges. In this paper, we reframe time series forecasting as a two-part task: (1) predicting the trend (directional movement) of the time series at the next time step, and (2) forecasting the quantitative value at the next time step. The trend can be predicted using a binary classifier, while quantitative values can be forecasted using models such as LSTM and Bidirectional Long Short-Term Memory (Bi-LSTM). Building on this reframing, we propose the Trend-Adjusted Time Series (TATS) model, which adjusts the forecasted values based on the predicted trend provided by the binary classifier. We validate the proposed approach through both theoretical analysis and empirical evaluation. The TATS model is applied to a volatile financial time series (the daily gold price) with the objective of forecasting the next days price. Experimental results demonstrate that TATS consistently outperforms standard LSTM and Bi-LSTM models by achieving significantly lower forecasting error. In addition, our results indicate that commonly used metrics such as MSE and MAE are insufficient for fully assessing time series model performance. Therefore, we also incorporate trend detection accuracy, which measures how effectively a model captures trends in a time series.

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.

A Dataset of Dengue Hospitalizations in Brazil (1999 to 2021) with Weekly Disaggregation from Monthly Counts

This data paper describes and publicly releases this dataset (v1.0.0), published on Zenodo under DOI 10.5281/zenodo.18189192. Motivated by the need to increase the temporal granularity of originally monthly data to enable more effective training of AI models for epidemiological forecasting, the dataset harmonizes municipal-level dengue hospitalization time series across Brazil and disaggregates them to weekly resolution (epidemiological weeks) through an interpolation protocol with a correction step that preserves monthly totals. The statistical and temporal validity of this disaggregation was assessed using a high-resolution reference dataset from the state of Sao Paulo (2024), which simultaneously provides monthly and epidemiological-week counts, enabling a direct comparison of three strategies: linear interpolation, jittering, and cubic spline. Results indicated that cubic spline interpolation achieved the highest adherence to the reference data, and this strategy was therefore adopted to generate weekly series for the 1999 to 2021 period. In addition to hospitalization time series, the dataset includes a comprehensive set of explanatory variables commonly used in epidemiological and environmental modeling, such as demographic density, CH4, CO2, and NO2 emissions, poverty and urbanization indices, maximum temperature, mean monthly precipitation, minimum relative humidity, and municipal latitude and longitude, following the same temporal disaggregation scheme to ensure multivariate compatibility. The paper documents the datasets provenance, structure, formats, licenses, limitations, and quality metrics (MAE, RMSE, R2, KL, JSD, DTW, and the KS test), and provides usage recommendations for multivariate time-series analysis, environmental health studies, and the development of machine learning and deep learning models for outbreak forecasting.

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.

Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation

Inverse parallel schemes remain indispensable tools for computing the roots of nonlinear systems, yet their dynamical behavior can be unexpectedly rich, ranging from strong contraction to oscillatory or chaotic transients depending on the choice of algorithmic parameters and initial states. A unified analytical-data-driven methodology for identifying, measuring, and reducing such instabilities in a family of uni-parametric inverse parallel solvers is presented in this study. On the theoretical side, we derive stability and bifurcation characterizations of the underlying iterative maps, identifying parameter regions associated with periodic or chaotic behavior. On the computational side, we introduce a micro-series pipeline based on kNN-driven estimation of the local largest Lyapunov exponent (LLE), applied to scalar time series derived from solver trajectories. The resulting sliding-window Lyapunov profiles provide fine-grained, real-time diagnostics of contractive or unstable phases and reveal transient behaviors not captured by coarse linearized analysis. Leveraging this correspondence, we introduce a Lyapunov-informed parameter selection strategy that identifies solver settings associated with stable behavior, particularly when the estimated LLE indicates persistent instability. Comprehensive experiments on ensembles of perturbed initial guesses demonstrate close agreement between the theoretical stability diagrams and empirical Lyapunov profiles, and show that the proposed adaptive mechanism significantly improves robustness. The study establishes micro-series Lyapunov analysis as a practical, interpretable tool for constructing self-stabilizing root-finding schemes and opens avenues for extending such diagnostics to higher-dimensional or noise-contaminated problems.

EvoMorph: Counterfactual Explanations for Continuous Time-Series Extrinsic Regression Applied to Photoplethysmography

Wearable devices enable continuous, population-scale monitoring of physiological signals, such as photoplethysmography (PPG), creating new opportunities for data-driven clinical assessment. Time-series extrinsic regression (TSER) models increasingly leverage PPG signals to estimate clinically relevant outcomes, including heart rate, respiratory rate, and oxygen saturation. For clinical reasoning and trust, however, single point estimates alone are insufficient: clinicians must also understand whether predictions are stable under physiologically plausible variations and to what extent realistic, attainable changes in physiological signals would meaningfully alter a model's prediction. Counterfactual explanations (CFE) address these "what-if" questions, yet existing time series CFE generation methods are largely restricted to classification, overlook waveform morphology, and often produce physiologically implausible signals, limiting their applicability to continuous biomedical time series. To address these limitations, we introduce EvoMorph, a multi-objective evolutionary framework for generating physiologically plausible and diverse CFE for TSER applications. EvoMorph optimizes morphology-aware objectives defined on interpretable signal descriptors and applies transformations to preserve the waveform structure. We evaluated EvoMorph on three PPG datasets (heart rate, respiratory rate, and oxygen saturation) against a nearest-unlike-neighbor baseline. In addition, in a case study, we evaluated EvoMorph as a tool for uncertainty quantification by relating counterfactual sensitivity to bootstrap-ensemble uncertainty and data-density measures. Overall, EvoMorph enables the generation of physiologically-aware counterfactuals for continuous biomedical signals and supports uncertainty-aware interpretability, advancing trustworthy model analysis for clinical time-series applications.

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.

MemKD: Memory-Discrepancy Knowledge Distillation for Efficient Time Series Classification

Deep learning models, particularly recurrent neural networks and their variants, such as long short-term memory, have significantly advanced time series data analysis. These models capture complex, sequential patterns in time series, enabling real-time assessments. However, their high computational complexity and large model sizes pose challenges for deployment in resource-constrained environments, such as wearable devices and edge computing platforms. Knowledge Distillation (KD) offers a solution by transferring knowledge from a large, complex model (teacher) to a smaller, more efficient model (student), thereby retaining high performance while reducing computational demands. Current KD methods, originally designed for computer vision tasks, neglect the unique temporal dependencies and memory retention characteristics of time series models. To this end, we propose a novel KD framework termed Memory-Discrepancy Knowledge Distillation (MemKD). MemKD leverages a specialized loss function to capture memory retention discrepancies between the teacher and student models across subsequences within time series data, ensuring that the student model effectively mimics the teacher model's behaviour. This approach facilitates the development of compact, high-performing recurrent neural networks suitable for real-time, time series analysis tasks. Our extensive experiments demonstrate that MemKD significantly outperforms state-of-the-art KD methods. It reduces parameter size and memory usage by approximately 500 times while maintaining comparable performance to the teacher model.

Learning to Reason: Temporal Saliency Distillation for Interpretable Knowledge Transfer

Knowledge distillation has proven effective for model compression by transferring knowledge from a larger network called the teacher to a smaller network called the student. Current knowledge distillation in time series is predominantly based on logit and feature aligning techniques originally developed for computer vision tasks. These methods do not explicitly account for temporal data and fall short in two key aspects. First, the mechanisms by which the transferred knowledge helps the student model learning process remain unclear due to uninterpretability of logits and features. Second, these methods transfer only limited knowledge, primarily replicating the teacher predictive accuracy. As a result, student models often produce predictive distributions that differ significantly from those of their teachers, hindering their safe substitution for teacher models. In this work, we propose transferring interpretable knowledge by extending conventional logit transfer to convey not just the right prediction but also the right reasoning of the teacher. Specifically, we induce other useful knowledge from the teacher logits termed temporal saliency which captures the importance of each input timestep to the teacher prediction. By training the student with Temporal Saliency Distillation we encourage it to make predictions based on the same input features as the teacher. Temporal Saliency Distillation requires no additional parameters or architecture specific assumptions. We demonstrate that Temporal Saliency Distillation effectively improves the performance of baseline methods while also achieving desirable properties beyond predictive accuracy. We hope our work establishes a new paradigm for interpretable knowledge distillation in time series analysis.

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.