Synthetic Time Series Forecasting with Transformer Architectures: Extensive Simulation Benchmarks

Time series forecasting plays a critical role in domains such as energy, finance, and healthcare, where accurate predictions inform decision-making under uncertainty. Although Transformer-based models have demonstrated success in sequential modeling, their adoption for time series remains limited by challenges such as noise sensitivity, long-range dependencies, and a lack of inductive bias for temporal structure. In this work, we present a unified and principled framework for benchmarking three prominent Transformer forecasting architectures-Autoformer, Informer, and Patchtst-each evaluated through three architectural variants: Minimal, Standard, and Full, representing increasing levels of complexity and modeling capacity. We conduct over 1500 controlled experiments on a suite of ten synthetic signals, spanning five patch lengths and five forecast horizons under both clean and noisy conditions. Our analysis reveals consistent patterns across model families. To advance this landscape further, we introduce the Koopman-enhanced Transformer framework, Deep Koopformer, which integrates operator-theoretic latent state modeling to improve stability and interpretability. We demonstrate its efficacy on nonlinear and chaotic dynamical systems. Our results highlight Koopman based Transformer as a promising hybrid approach for robust, interpretable, and theoretically grounded time series forecasting in noisy and complex real-world conditions.

Hierarchical Representations for Evolving Acyclic Vector Autoregressions (HEAVe)

Causal networks offer an intuitive framework to understand influence structures within time series systems. However, the presence of cycles can obscure dynamic relationships and hinder hierarchical analysis. These networks are typically identified through multivariate predictive modelling, but enforcing acyclic constraints significantly increases computational and analytical complexity. Despite recent advances, there remains a lack of simple, flexible approaches that are easily tailorable to specific problem instances. We propose an evolutionary approach to fitting acyclic vector autoregressive processes and introduces a novel hierarchical representation that directly models structural elements within a time series system. On simulated datasets, our model retains most of the predictive accuracy of unconstrained models and outperforms permutation-based alternatives. When applied to a dataset of 100 cryptocurrency return series, our method generates acyclic causal networks capturing key structural properties of the unconstrained model. The acyclic networks are approximately sub-graphs of the unconstrained networks, and most of the removed links originate from low-influence nodes. Given the high levels of feature preservation, we conclude that this cryptocurrency price system functions largely hierarchically. Our findings demonstrate a flexible, intuitive approach for identifying hierarchical causal networks in time series systems, with broad applications to fields like econometrics and social network analysis.

Degree-Optimized Cumulative Polynomial Kolmogorov-Arnold Networks

We introduce cumulative polynomial Kolmogorov-Arnold networks (CP-KAN), a neural architecture combining Chebyshev polynomial basis functions and quadratic unconstrained binary optimization (QUBO). Our primary contribution involves reformulating the degree selection problem as a QUBO task, reducing the complexity from $O(D^N)$ to a single optimization step per layer. This approach enables efficient degree selection across neurons while maintaining computational tractability. The architecture performs well in regression tasks with limited data, showing good robustness to input scales and natural regularization properties from its polynomial basis. Additionally, theoretical analysis establishes connections between CP-KAN's performance and properties of financial time series. Our empirical validation across multiple domains demonstrates competitive performance compared to several traditional architectures tested, especially in scenarios where data efficiency and numerical stability are important. Our implementation, including strategies for managing computational overhead in larger networks is available in Ref.~\citep{cpkan_implementation}.

FCPCA: Fuzzy clustering of high-dimensional time series based on common principal component analysis

Clustering multivariate time series data is a crucial task in many domains, as it enables the identification of meaningful patterns and groups in time-evolving data. Traditional approaches, such as crisp clustering, rely on the assumption that clusters are sufficiently separated with little overlap. However, real-world data often defy this assumption, exhibiting overlapping distributions or overlapping clouds of points and blurred boundaries between clusters. Fuzzy clustering offers a compelling alternative by allowing partial membership in multiple clusters, making it well-suited for these ambiguous scenarios. Despite its advantages, current fuzzy clustering methods primarily focus on univariate time series, and for multivariate cases, even datasets of moderate dimensionality become computationally prohibitive. This challenge is further exacerbated when dealing with time series of varying lengths, leaving a clear gap in addressing the complexities of modern datasets. This work introduces a novel fuzzy clustering approach based on common principal component analysis to address the aforementioned shortcomings. Our method has the advantage of efficiently handling high-dimensional multivariate time series by reducing dimensionality while preserving critical temporal features. Extensive numerical results show that our proposed clustering method outperforms several existing approaches in the literature. An interesting application involving brain signals from different drivers recorded from a simulated driving experiment illustrates the potential of the approach.

Non-Stationary Time Series Forecasting Based on Fourier Analysis and Cross Attention Mechanism

Time series forecasting has important applications in financial analysis, weather forecasting, and traffic management. However, existing deep learning models are limited in processing non-stationary time series data because they cannot effectively capture the statistical characteristics that change over time. To address this problem, this paper proposes a new framework, AEFIN, which enhances the information sharing ability between stable and unstable components by introducing a cross-attention mechanism, and combines Fourier analysis networks with MLP to deeply explore the seasonal patterns and trend characteristics in unstable components. In addition, we design a new loss function that combines time-domain stability constraints, time-domain instability constraints, and frequency-domain stability constraints to improve the accuracy and robustness of forecasting. Experimental results show that AEFIN outperforms the most common models in terms of mean square error and mean absolute error, especially under non-stationary data conditions, and shows excellent forecasting capabilities. This paper provides an innovative solution for the modeling and forecasting of non-stationary time series data, and contributes to the research of deep learning for complex time series.

TransPL: VQ-Code Transition Matrices for Pseudo-Labeling of Time Series Unsupervised Domain Adaptation

Unsupervised domain adaptation (UDA) for time series data remains a critical challenge in deep learning, with traditional pseudo-labeling strategies failing to capture temporal patterns and channel-wise shifts between domains, producing sub-optimal pseudo-labels. As such, we introduce TransPL, a novel approach that addresses these limitations by modeling the joint distribution $P(\mathbf{X}, y)$ of the source domain through code transition matrices, where the codes are derived from vector quantization (VQ) of time series patches. Our method constructs class- and channel-wise code transition matrices from the source domain and employs Bayes' rule for target domain adaptation, generating pseudo-labels based on channel-wise weighted class-conditional likelihoods. TransPL offers three key advantages: explicit modeling of temporal transitions and channel-wise shifts between different domains, versatility towards different UDA scenarios (e.g., weakly-supervised UDA), and explainable pseudo-label generation. We validate TransPL's effectiveness through extensive analysis on four time series UDA benchmarks and confirm that it consistently outperforms state-of-the-art pseudo-labeling methods by a strong margin (6.1% accuracy improvement, 4.9% F1 improvement), while providing interpretable insights into the domain adaptation process through its learned code transition matrices.

Context-Aware Probabilistic Modeling with LLM for Multimodal Time Series Forecasting

Time series forecasting is important for applications spanning energy markets, climate analysis, and traffic management. However, existing methods struggle to effectively integrate exogenous texts and align them with the probabilistic nature of large language models (LLMs). Current approaches either employ shallow text-time series fusion via basic prompts or rely on deterministic numerical decoding that conflict with LLMs' token-generation paradigm, which limits contextual awareness and distribution modeling. To address these limitations, we propose CAPTime, a context-aware probabilistic multimodal time series forecasting method that leverages text-informed abstraction and autoregressive LLM decoding. Our method first encodes temporal patterns using a pretrained time series encoder, then aligns them with textual contexts via learnable interactions to produce joint multimodal representations. By combining a mixture of distribution experts with frozen LLMs, we enable context-aware probabilistic forecasting while preserving LLMs' inherent distribution modeling capabilities. Experiments on diverse time series forecasting tasks demonstrate the superior accuracy and generalization of CAPTime, particularly in multimodal scenarios. Additional analysis highlights its robustness in data-scarce scenarios through hybrid probabilistic decoding.

MSDformer: Multi-scale Discrete Transformer For Time Series Generation

Discrete Token Modeling (DTM), which employs vector quantization techniques, has demonstrated remarkable success in modeling non-natural language modalities, particularly in time series generation. While our prior work SDformer established the first DTM-based framework to achieve state-of-the-art performance in this domain, two critical limitations persist in existing DTM approaches: 1) their inability to capture multi-scale temporal patterns inherent to complex time series data, and 2) the absence of theoretical foundations to guide model optimization. To address these challenges, we proposes a novel multi-scale DTM-based time series generation method, called Multi-Scale Discrete Transformer (MSDformer). MSDformer employs a multi-scale time series tokenizer to learn discrete token representations at multiple scales, which jointly characterize the complex nature of time series data. Subsequently, MSDformer applies a multi-scale autoregressive token modeling technique to capture the multi-scale patterns of time series within the discrete latent space. Theoretically, we validate the effectiveness of the DTM method and the rationality of MSDformer through the rate-distortion theorem. Comprehensive experiments demonstrate that MSDformer significantly outperforms state-of-the-art methods. Both theoretical analysis and experimental results demonstrate that incorporating multi-scale information and modeling multi-scale patterns can substantially enhance the quality of generated time series in DTM-based approaches. The code will be released upon acceptance.

A Study of Data-driven Methods for Inventory Optimization

This paper shows a comprehensive analysis of three algorithms (Time Series, Random Forest (RF) and Deep Reinforcement Learning) into three inventory models (the Lost Sales, Dual-Sourcing and Multi-Echelon Inventory Model). These methodologies are applied in the supermarket context. The main purpose is to analyse efficient methods for the data-driven. Their possibility, potential and current challenges are taken into consideration in this report. By comparing the results in each model, the effectiveness of each algorithm is evaluated based on several key performance indicators, including forecast accuracy, adaptability to market changes, and overall impact on inventory costs and customer satisfaction levels. The data visualization tools and statistical metrics are the indicators for the comparisons and show some obvious trends and patterns that can guide decision-making in inventory management. These tools enable managers to not only track the performance of different algorithms in real-time but also to drill down into specific data points to understand the underlying causes of inventory fluctuations. This level of detail is crucial for pinpointing inefficiencies and areas for improvement within the supply chain.

Heart2Mind: Human-Centered Contestable Psychiatric Disorder Diagnosis System using Wearable ECG Monitors

Psychiatric disorders affect millions globally, yet their diagnosis faces significant challenges in clinical practice due to subjective assessments and accessibility concerns, leading to potential delays in treatment. To help address this issue, we present Heart2Mind, a human-centered contestable psychiatric disorder diagnosis system using wearable electrocardiogram (ECG) monitors. Our approach leverages cardiac biomarkers, particularly heart rate variability (HRV) and R-R intervals (RRI) time series, as objective indicators of autonomic dysfunction in psychiatric conditions. The system comprises three key components: (1) a Cardiac Monitoring Interface (CMI) for real-time data acquisition from Polar H9/H10 devices; (2) a Multi-Scale Temporal-Frequency Transformer (MSTFT) that processes RRI time series through integrated time-frequency domain analysis; (3) a Contestable Diagnosis Interface (CDI) combining Self-Adversarial Explanations (SAEs) with contestable Large Language Models (LLMs). Our MSTFT achieves 91.7% accuracy on the HRV-ACC dataset using leave-one-out cross-validation, outperforming state-of-the-art methods. SAEs successfully detect inconsistencies in model predictions by comparing attention-based and gradient-based explanations, while LLMs enable clinicians to validate correct predictions and contest erroneous ones. This work demonstrates the feasibility of combining wearable technology with Explainable Artificial Intelligence (XAI) and contestable LLMs to create a transparent, contestable system for psychiatric diagnosis that maintains clinical oversight while leveraging advanced AI capabilities. Our implementation is publicly available at: https://github.com/Analytics-Everywhere-Lab/heart2mind.