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.

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.

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.

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.

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.

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.

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.

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.

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.