Daily Papers

Causal discovery, i.e., inferring underlying cause-effect relationships from observations of a scene or system, is an inherent mechanism in human cognition, but has been shown to be highly challenging to automate. The majority of approaches in the literature aiming for this task consider constrained scenarios with fully observed variables or data from stationary time-series. In this work we aim for causal discovery in a more general class of scenarios, scenes with non-stationary behavior over time. For our purposes we here regard a scene as a composition objects interacting with each other over time. Non-stationarity is modeled as stationarity conditioned on an underlying variable, a state, which can be of varying dimension, more or less hidden given observations of the scene, and also depend more or less directly on these observations. We propose a probabilistic deep learning approach called State-Dependent Causal Inference (SDCI) for causal discovery in such conditionally stationary time-series data. Results in two different synthetic scenarios show that this method is able to recover the underlying causal dependencies with high accuracy even in cases with hidden states.

We present a new distribution-free conformal prediction algorithm for sequential data (e.g., time series), called the sequential predictive conformal inference (SPCI). We specifically account for the nature that time series data are non-exchangeable, and thus many existing conformal prediction algorithms are not applicable. The main idea is to adaptively re-estimate the conditional quantile of non-conformity scores (e.g., prediction residuals), upon exploiting the temporal dependence among them. More precisely, we cast the problem of conformal prediction interval as predicting the quantile of a future residual, given a user-specified point prediction algorithm. Theoretically, we establish asymptotic valid conditional coverage upon extending consistency analyses in quantile regression. Using simulation and real-data experiments, we demonstrate a significant reduction in interval width of SPCI compared to other existing methods under the desired empirical coverage.

Statistical inference for non-stationary data is hindered by the failure of classical central limit theorems (CLTs), not least because there is no fixed Gaussian limit to converge to. To resolve this, we introduce relative weak convergence, an extension of weak convergence that compares a statistic or process to a sequence of evolving processes. Relative weak convergence retains the essential consequences of classical weak convergence and coincides with it under stationarity. Crucially, it applies in general non-stationary settings where classical weak convergence fails. We establish concrete relative CLTs for random vectors and empirical processes, along with sequential, weighted, and bootstrap variants, that parallel the state-of-the-art in stationary settings. Our framework and results offer simple, plug-in replacements for classical CLTs whenever stationarity is untenable, as illustrated by applications in nonparametric trend estimation and hypothesis testing.

We present a constraint-based algorithm for learning causal structures from observational time-series data, in the presence of latent confounders. We assume a discrete-time, stationary structural vector autoregressive process, with both temporal and contemporaneous causal relations. One may ask if temporal and contemporaneous relations should be treated differently. The presented algorithm gradually refines a causal graph by learning long-term temporal relations before short-term ones, where contemporaneous relations are learned last. This ordering of causal relations to be learnt leads to a reduction in the required number of statistical tests. We validate this reduction empirically and demonstrate that it leads to higher accuracy for synthetic data and more plausible causal graphs for real-world data compared to state-of-the-art algorithms.

We introduce the concept of programmable feature engineering for time series modeling and propose a feature programming framework. This framework generates large amounts of predictive features for noisy multivariate time series while allowing users to incorporate their inductive bias with minimal effort. The key motivation of our framework is to view any multivariate time series as a cumulative sum of fine-grained trajectory increments, with each increment governed by a novel spin-gas dynamical Ising model. This fine-grained perspective motivates the development of a parsimonious set of operators that summarize multivariate time series in an abstract fashion, serving as the foundation for large-scale automated feature engineering. Numerically, we validate the efficacy of our method on several synthetic and real-world noisy time series datasets.

Recently, channel-independent methods have achieved state-of-the-art performance in multivariate time series (MTS) forecasting. Despite reducing overfitting risks, these methods miss potential opportunities in utilizing channel dependence for accurate predictions. We argue that there exist locally stationary lead-lag relationships between variates, i.e., some lagged variates may follow the leading indicators within a short time period. Exploiting such channel dependence is beneficial since leading indicators offer advance information that can be used to reduce the forecasting difficulty of the lagged variates. In this paper, we propose a new method named LIFT that first efficiently estimates leading indicators and their leading steps at each time step and then judiciously allows the lagged variates to utilize the advance information from leading indicators. LIFT plays as a plugin that can be seamlessly collaborated with arbitrary time series forecasting methods. Extensive experiments on six real-world datasets demonstrate that LIFT improves the state-of-the-art methods by 5.5% in average forecasting performance. Our code is available at https://github.com/SJTU-Quant/LIFT.

We address the problem of learning the dynamics of an unknown non-parametric system linking a target and a feature time series. The feature time series is measured on a sparse and irregular grid, while we have access to only a few points of the target time series. Once learned, we can use these dynamics to predict values of the target from the previous values of the feature time series. We frame this task as learning the solution map of a controlled differential equation (CDE). By leveraging the rich theory of signatures, we are able to cast this non-linear problem as a high-dimensional linear regression. We provide an oracle bound on the prediction error which exhibits explicit dependencies on the individual-specific sampling schemes. Our theoretical results are illustrated by simulations which show that our method outperforms existing algorithms for recovering the full time series while being computationally cheap. We conclude by demonstrating its potential on real-world epidemiological data.

Diffusion models have achieved state-of-the-art performance in generative modeling tasks across various domains. Prior works on time series diffusion models have primarily focused on developing conditional models tailored to specific forecasting or imputation tasks. In this work, we explore the potential of task-agnostic, unconditional diffusion models for several time series applications. We propose TSDiff, an unconditionally trained diffusion model for time series. Our proposed self-guidance mechanism enables conditioning TSDiff for downstream tasks during inference, without requiring auxiliary networks or altering the training procedure. We demonstrate the effectiveness of our method on three different time series tasks: forecasting, refinement, and synthetic data generation. First, we show that TSDiff is competitive with several task-specific conditional forecasting methods (predict). Second, we leverage the learned implicit probability density of TSDiff to iteratively refine the predictions of base forecasters with reduced computational overhead over reverse diffusion (refine). Notably, the generative performance of the model remains intact -- downstream forecasters trained on synthetic samples from TSDiff outperform forecasters that are trained on samples from other state-of-the-art generative time series models, occasionally even outperforming models trained on real data (synthesize).

Temporal Point Processes (TPP) are probabilistic generative frameworks. They model discrete event sequences localized in continuous time. Generally, real-life events reveal descriptive information, known as marks. Marked TPPs model time and marks of the event together for practical relevance. Conditioned on past events, marked TPPs aim to learn the joint distribution of the time and the mark of the next event. For simplicity, conditionally independent TPP models assume time and marks are independent given event history. They factorize the conditional joint distribution of time and mark into the product of individual conditional distributions. This structural limitation in the design of TPP models hurt the predictive performance on entangled time and mark interactions. In this work, we model the conditional inter-dependence of time and mark to overcome the limitations of conditionally independent models. We construct a multivariate TPP conditioning the time distribution on the current event mark in addition to past events. Besides the conventional intensity-based models for conditional joint distribution, we also draw on flexible intensity-free TPP models from the literature. The proposed TPP models outperform conditionally independent and dependent models in standard prediction tasks. Our experimentation on various datasets with multiple evaluation metrics highlights the merit of the proposed approach.

Time series foundation models have demonstrated impressive performance as zero-shot forecasters. However, achieving effectively unified training on time series remains an open challenge. Existing approaches introduce some level of model specialization to account for the highly heterogeneous nature of time series data. For instance, Moirai pursues unified training by employing multiple input/output projection layers, each tailored to handle time series at a specific frequency. Similarly, TimesFM maintains a frequency embedding dictionary for this purpose. We identify two major drawbacks to this human-imposed frequency-level model specialization: (1) Frequency is not a reliable indicator of the underlying patterns in time series. For example, time series with different frequencies can display similar patterns, while those with the same frequency may exhibit varied patterns. (2) Non-stationarity is an inherent property of real-world time series, leading to varied distributions even within a short context window of a single time series. Frequency-level specialization is too coarse-grained to capture this level of diversity. To address these limitations, this paper introduces Moirai-MoE, using a single input/output projection layer while delegating the modeling of diverse time series patterns to the sparse mixture of experts (MoE) within Transformers. With these designs, Moirai-MoE reduces reliance on human-defined heuristics and enables automatic token-level specialization. Extensive experiments on 39 datasets demonstrate the superiority of Moirai-MoE over existing foundation models in both in-distribution and zero-shot scenarios. Furthermore, this study conducts comprehensive model analyses to explore the inner workings of time series MoE foundation models and provides valuable insights for future research.

We present Timer-XL, a generative Transformer for unified time series forecasting. To uniformly predict 1D and 2D time series, we generalize next token prediction, predominantly adopted for causal generation of 1D sequences, to multivariate next token prediction. The proposed paradigm uniformly formulates various forecasting scenarios as a long-context generation problem. We opt for the generative Transformer, which can capture global-range and causal dependencies while providing contextual flexibility, to implement unified forecasting on univariate series characterized by non-stationarity, multivariate time series with complicated dynamics and correlations, and covariate-informed contexts that include both endogenous and exogenous variables. Technically, we propose a universal TimeAttention to facilitate generative Transformers on time series, which can effectively capture fine-grained intra- and inter-series dependencies of flattened time series tokens (patches) and is further strengthened by position embeddings in both temporal and variable dimensions. Timer-XL achieves state-of-the-art performance across challenging forecasting benchmarks through a unified approach. As a large time series model, it demonstrates notable model transferability by large-scale pre-training, as well as contextual flexibility in token lengths, positioning it as a one-for-all forecaster.

Time series foundation models (TSFMs) have recently gained significant attention due to their strong zero-shot capabilities and widespread real-world applications. Such models typically require a computationally costly pretraining on large-scale, carefully curated collections of real-world sequences. To allow for a sample-efficient pretraining of TSFMs, we propose CauKer, a novel algorithm designed to generate diverse, causally coherent synthetic time series with realistic trends, seasonality, and nonlinear interactions. CauKer combines Gaussian Process (GP) kernel composition with Structural Causal Models (SCM) to produce data for sample-efficient pretraining of state-of-the-art classification TSFMs having different architectures and following different pretraining approaches. Additionally, our experiments reveal that CauKer-generated datasets exhibit clear scaling laws for both dataset size (10K to 10M samples) and model capacity (1M to 783M parameters), unlike real-world datasets, which display irregular scaling behavior.

Periodic signals play an important role in daily lives. Although conventional sequential models have shown remarkable success in various fields, they still come short in modeling periodicity; they either collapse, diverge or ignore details. In this paper, we introduce a novel framework inspired by Fourier series to generate periodic signals. We first decompose the given signals into multiple sines and cosines and then conditionally generate periodic signals with the output components. We have shown our model efficacy on three tasks: reconstruction, imputation and conditional generation. Our model outperforms baselines in all tasks and shows more stable and refined results.

Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the denoising diffusion model in the function space which also allows us to naturally handle irregularly-sampled observations. The forward process gradually adds noise to functions, preserving their continuity, while the learned reverse process removes the noise and returns functions as new samples. To this end, we define suitable noise sources and introduce novel denoising and score-matching models. We show how our method can be used for multivariate probabilistic forecasting and imputation, and how our model can be interpreted as a neural process.

Learning accurate predictive models of real-world dynamic phenomena (e.g., climate, biological) remains a challenging task. One key issue is that the data generated by both natural and artificial processes often comprise time series that are irregularly sampled and/or contain missing observations. In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations. NCDSSM employs auxiliary variables to disentangle recognition from dynamics, thus requiring amortized inference only for the auxiliary variables. Leveraging techniques from continuous-discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dynamic states. We propose three flexible parameterizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference. Empirical results on multiple benchmark datasets across various domains show improved imputation and forecasting performance of NCDSSM over existing models.

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect Y can be written as a function of the cause X and a noise source N independent of X, which may be scaled by a positive function g over the cause, i.e., Y = f(X) + g(X)N. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of Y given X as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix -- a canonical representation for discrete-time processes -- which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length ell and state-space size d, we go from O(d ell) na\"ively to O(d + ell). In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR(p) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

Over the past years, foundation models have caused a paradigm shift in machine learning due to their unprecedented capabilities for zero-shot and few-shot generalization. However, despite the success of foundation models in modalities such as natural language processing and computer vision, the development of foundation models for time series forecasting has lagged behind. We present Lag-Llama, a general-purpose foundation model for univariate probabilistic time series forecasting based on a decoder-only transformer architecture that uses lags as covariates. Lag-Llama is pretrained on a large corpus of diverse time series data from several domains, and demonstrates strong zero-shot generalization capabilities compared to a wide range of forecasting models on downstream datasets across domains. Moreover, when fine-tuned on relatively small fractions of such previously unseen datasets, Lag-Llama achieves state-of-the-art performance, outperforming prior deep learning approaches, emerging as the best general-purpose model on average. Lag-Llama serves as a strong contender to the current state-of-art in time series forecasting and paves the way for future advancements in foundation models tailored to time series data.

In this work, we present a novel conformal prediction method for time-series, which we call Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCPI). Specifically, KOWCPI adapts the classic Reweighted Nadaraya-Watson (RNW) estimator for quantile regression on dependent data and learns optimal data-adaptive weights. Theoretically, we tackle the challenge of establishing a conditional coverage guarantee for non-exchangeable data under strong mixing conditions on the non-conformity scores. We demonstrate the superior performance of KOWCPI on real and synthetic time-series data against state-of-the-art methods, where KOWCPI achieves narrower confidence intervals without losing coverage.

Real-world time series often exhibit a non-stationary nature, degrading the performance of pre-trained forecasting models. Test-Time Adaptation (TTA) addresses this by adjusting models during inference, but existing methods typically update the full model, increasing memory and compute costs. We propose PETSA, a parameter-efficient method that adapts forecasters at test time by only updating small calibration modules on the input and output. PETSA uses low-rank adapters and dynamic gating to adjust representations without retraining. To maintain accuracy despite limited adaptation capacity, we introduce a specialized loss combining three components: (1) a robust term, (2) a frequency-domain term to preserve periodicity, and (3) a patch-wise structural term for structural alignment. PETSA improves the adaptability of various forecasting backbones while requiring fewer parameters than baselines. Experimental results on benchmark datasets show that PETSA achieves competitive or better performance across all horizons. Our code is available at: https://github.com/BorealisAI/PETSA

Time series forecasting is widely used in extensive applications, such as traffic planning and weather forecasting. However, real-world time series usually present intricate temporal variations, making forecasting extremely challenging. Going beyond the mainstream paradigms of plain decomposition and multiperiodicity analysis, we analyze temporal variations in a novel view of multiscale-mixing, which is based on an intuitive but important observation that time series present distinct patterns in different sampling scales. The microscopic and the macroscopic information are reflected in fine and coarse scales respectively, and thereby complex variations can be inherently disentangled. Based on this observation, we propose TimeMixer as a fully MLP-based architecture with Past-Decomposable-Mixing (PDM) and Future-Multipredictor-Mixing (FMM) blocks to take full advantage of disentangled multiscale series in both past extraction and future prediction phases. Concretely, PDM applies the decomposition to multiscale series and further mixes the decomposed seasonal and trend components in fine-to-coarse and coarse-to-fine directions separately, which successively aggregates the microscopic seasonal and macroscopic trend information. FMM further ensembles multiple predictors to utilize complementary forecasting capabilities in multiscale observations. Consequently, TimeMixer is able to achieve consistent state-of-the-art performances in both long-term and short-term forecasting tasks with favorable run-time efficiency.

Existing works on general time series forecasting build foundation models with heavy model parameters through large-scale multi-source pre-training. These models achieve superior generalization ability across various datasets at the cost of significant computational burdens and limitations in resource-constrained scenarios. This paper introduces LightGTS, a lightweight general time series forecasting model designed from the perspective of consistent periodical modeling. To handle diverse scales and intrinsic periods in multi-source pre-training, we introduce Periodical Tokenization, which extracts consistent periodic patterns across different datasets with varying scales. To better utilize the periodicity in the decoding process, we further introduce Periodical Parallel Decoding, which leverages historical tokens to improve forecasting. Based on the two techniques above which fully leverage the inductive bias of periods inherent in time series, LightGTS uses a lightweight model to achieve outstanding performance on general time series forecasting. It achieves state-of-the-art forecasting performance on 9 real-world benchmarks in both zero-shot and full-shot settings with much better efficiency compared with existing time series foundation models.

In Online Continual Learning (OCL) a learning system receives a stream of data and sequentially performs prediction and training steps. Important challenges in OCL are concerned with automatic adaptation to the particular non-stationary structure of the data, and with quantification of predictive uncertainty. Motivated by these challenges we introduce a probabilistic Bayesian online learning model by using a (possibly pretrained) neural representation and a state space model over the linear predictor weights. Non-stationarity over the linear predictor weights is modelled using a parameter drift transition density, parametrized by a coefficient that quantifies forgetting. Inference in the model is implemented with efficient Kalman filter recursions which track the posterior distribution over the linear weights, while online SGD updates over the transition dynamics coefficient allows to adapt to the non-stationarity seen in data. While the framework is developed assuming a linear Gaussian model, we also extend it to deal with classification problems and for fine-tuning the deep learning representation. In a set of experiments in multi-class classification using data sets such as CIFAR-100 and CLOC we demonstrate the predictive ability of the model and its flexibility to capture non-stationarity.

Modeling multivariate time series is a well-established problem with a wide range of applications from healthcare to financial markets. Traditional State Space Models (SSMs) are classical approaches for univariate time series modeling due to their simplicity and expressive power to represent linear dependencies. They, however, have fundamentally limited expressive power to capture non-linear dependencies, are slow in practice, and fail to model the inter-variate information flow. Despite recent attempts to improve the expressive power of SSMs by using deep structured SSMs, the existing methods are either limited to univariate time series, fail to model complex patterns (e.g., seasonal patterns), fail to dynamically model the dependencies of variate and time dimensions, and/or are input-independent. We present Chimera that uses two input-dependent 2-D SSM heads with different discretization processes to learn long-term progression and seasonal patterns. To improve the efficiency of complex 2D recurrence, we present a fast training using a new 2-dimensional parallel selective scan. We further present and discuss 2-dimensional Mamba and Mamba-2 as the spacial cases of our 2D SSM. Our experimental evaluation shows the superior performance of Chimera on extensive and diverse benchmarks, including ECG and speech time series classification, long-term and short-term time series forecasting, and time series anomaly detection.

Developing models and algorithms to predict nonstationary time series is a long standing statistical problem. It is crucial for many applications, in particular for fashion or retail industries, to make optimal inventory decisions and avoid massive wastes. By tracking thousands of fashion trends on social media with state-of-the-art computer vision approaches, we propose a new model for fashion time series forecasting. Our contribution is twofold. We first provide publicly a dataset gathering 10000 weekly fashion time series. As influence dynamics are the key of emerging trend detection, we associate with each time series an external weak signal representing behaviours of influencers. Secondly, to leverage such a dataset, we propose a new hybrid forecasting model. Our approach combines per-time-series parametric models with seasonal components and a global recurrent neural network to include sporadic external signals. This hybrid model provides state-of-the-art results on the proposed fashion dataset, on the weekly time series of the M4 competition, and illustrates the benefit of the contribution of external weak signals.

We propose the Fourier Adaptive Lite Diffusion Architecture (FALDA), a novel probabilistic framework for time series forecasting. First, we introduce the Diffusion Model for Residual Regression (DMRR) framework, which unifies diffusion-based probabilistic regression methods. Within this framework, FALDA leverages Fourier-based decomposition to incorporate a component-specific architecture, enabling tailored modeling of individual temporal components. A conditional diffusion model is utilized to estimate the future noise term, while our proposed lightweight denoiser, DEMA (Decomposition MLP with AdaLN), conditions on the historical noise term to enhance denoising performance. Through mathematical analysis and empirical validation, we demonstrate that FALDA effectively reduces epistemic uncertainty, allowing probabilistic learning to primarily focus on aleatoric uncertainty. Experiments on six real-world benchmarks demonstrate that FALDA consistently outperforms existing probabilistic forecasting approaches across most datasets for long-term time series forecasting while achieving enhanced computational efficiency without compromising accuracy. Notably, FALDA also achieves superior overall performance compared to state-of-the-art (SOTA) point forecasting approaches, with improvements of up to 9%.

Time series analysis has witnessed the inspiring development from traditional autoregressive models, deep learning models, to recent Transformers and Large Language Models (LLMs). Efforts in leveraging vision models for time series analysis have also been made along the way but are less visible to the community due to the predominant research on sequence modeling in this domain. However, the discrepancy between continuous time series and the discrete token space of LLMs, and the challenges in explicitly modeling the correlations of variates in multivariate time series have shifted some research attentions to the equally successful Large Vision Models (LVMs) and Vision Language Models (VLMs). To fill the blank in the existing literature, this survey discusses the advantages of vision models over LLMs in time series analysis. It provides a comprehensive and in-depth overview of the existing methods, with dual views of detailed taxonomy that answer the key research questions including how to encode time series as images and how to model the imaged time series for various tasks. Additionally, we address the challenges in the pre- and post-processing steps involved in this framework and outline future directions to further advance time series analysis with vision models.

We incorporate discrete and continuous time Markov processes as building blocks into probabilistic graphical models with latent and observed variables. We introduce the automatic Backward Filtering Forward Guiding (BFFG) paradigm (Mider et al., 2021) for programmable inference on latent states and model parameters. Our starting point is a generative model, a forward description of the probabilistic process dynamics. We backpropagate the information provided by observations through the model to transform the generative (forward) model into a pre-conditional model guided by the data. It approximates the actual conditional model with known likelihood-ratio between the two. The backward filter and the forward change of measure are suitable to be incorporated into a probabilistic programming context because they can be formulated as a set of transformation rules. The guided generative model can be incorporated in different approaches to efficiently sample latent states and parameters conditional on observations. We show applicability in a variety of settings, including Markov chains with discrete state space, interacting particle systems, state space models, branching diffusions and Gamma processes.

Irregularly sampled time series data arise naturally in many application domains including biology, ecology, climate science, astronomy, and health. Such data represent fundamental challenges to many classical models from machine learning and statistics due to the presence of non-uniform intervals between observations. However, there has been significant progress within the machine learning community over the last decade on developing specialized models and architectures for learning from irregularly sampled univariate and multivariate time series data. In this survey, we first describe several axes along which approaches to learning from irregularly sampled time series differ including what data representations they are based on, what modeling primitives they leverage to deal with the fundamental problem of irregular sampling, and what inference tasks they are designed to perform. We then survey the recent literature organized primarily along the axis of modeling primitives. We describe approaches based on temporal discretization, interpolation, recurrence, attention and structural invariance. We discuss similarities and differences between approaches and highlight primary strengths and weaknesses.

Covariates provide valuable information on external factors that influence time series and are critical in many real-world time series forecasting tasks. For example, in retail, covariates may indicate promotions or peak dates such as holiday seasons that heavily influence demand forecasts. Recent advances in pretraining large language model architectures for time series forecasting have led to highly accurate forecasters. However, the majority of these models do not readily use covariates as they are often specific to a certain task or domain. This paper introduces a new method to incorporate covariates into pretrained time series forecasting models. Our proposed approach incorporates covariate information into pretrained forecasting models through modular blocks that inject past and future covariate information, without necessarily modifying the pretrained model in consideration. In order to evaluate our approach, we introduce a benchmark composed of 32 different synthetic datasets with varying dynamics to evaluate the effectivity of forecasting models with covariates. Extensive evaluations on both synthetic and real datasets show that our approach effectively incorporates covariate information into pretrained models, outperforming existing baselines.

Analyzing both temporal and spatial patterns for an accurate forecasting model for financial time series forecasting is a challenge due to the complex nature of temporal-spatial dynamics: time series from different locations often have distinct patterns; and for the same time series, patterns may vary as time goes by. Inspired by the successful applications of deep learning, we propose a new model to resolve the issues of forecasting household leverage in China. Our solution consists of multiple RNN-based layers and an attention layer: each RNN-based layer automatically learns the temporal pattern of a specific series with multivariate exogenous series, and then the attention layer learns the spatial correlative weight and obtains the global representations simultaneously. The results show that the new approach can capture the temporal-spatial dynamics of household leverage well and get more accurate and solid predictive results. More, the simulation also studies show that clustering and choosing correlative series are necessary to obtain accurate forecasting results.

In this paper we present a new framework for time-series modeling that combines the best of traditional statistical models and neural networks. We focus on time-series with long-range dependencies, needed for monitoring fine granularity data (e.g. minutes, seconds, milliseconds), prevalent in operational use-cases. Traditional models, such as auto-regression fitted with least squares (Classic-AR) can model time-series with a concise and interpretable model. When dealing with long-range dependencies, Classic-AR models can become intractably slow to fit for large data. Recently, sequence-to-sequence models, such as Recurrent Neural Networks, which were originally intended for natural language processing, have become popular for time-series. However, they can be overly complex for typical time-series data and lack interpretability. A scalable and interpretable model is needed to bridge the statistical and deep learning-based approaches. As a first step towards this goal, we propose modelling AR-process dynamics using a feed-forward neural network approach, termed AR-Net. We show that AR-Net is as interpretable as Classic-AR but also scales to long-range dependencies. Our results lead to three major conclusions: First, AR-Net learns identical AR-coefficients as Classic-AR, thus being equally interpretable. Second, the computational complexity with respect to the order of the AR process, is linear for AR-Net as compared to a quadratic for Classic-AR. This makes it possible to model long-range dependencies within fine granularity data. Third, by introducing regularization, AR-Net automatically selects and learns sparse AR-coefficients. This eliminates the need to know the exact order of the AR-process and allows to learn sparse weights for a model with long-range dependencies.

In the realm of time series forecasting (TSF), it is imperative for models to adeptly discern and distill hidden patterns within historical time series data to forecast future states. Transformer-based models exhibit formidable efficacy in TSF, primarily attributed to their advantage in apprehending these patterns. However, the quadratic complexity of the Transformer leads to low computational efficiency and high costs, which somewhat hinders the deployment of the TSF model in real-world scenarios. Recently, Mamba, a selective state space model, has gained traction due to its ability to process dependencies in sequences while maintaining near-linear complexity. For TSF tasks, these characteristics enable Mamba to comprehend hidden patterns as the Transformer and reduce computational overhead compared to the Transformer. Therefore, we propose a Mamba-based model named Simple-Mamba (S-Mamba) for TSF. Specifically, we tokenize the time points of each variate autonomously via a linear layer. A bidirectional Mamba layer is utilized to extract inter-variate correlations and a Feed-Forward Network is set to learn temporal dependencies. Finally, the generation of forecast outcomes through a linear mapping layer. Experiments on thirteen public datasets prove that S-Mamba maintains low computational overhead and achieves leading performance. Furthermore, we conduct extensive experiments to explore Mamba's potential in TSF tasks. Our code is available at https://github.com/wzhwzhwzh0921/S-D-Mamba.

In this paper we construct an inferential procedure for Granger causality in high-dimensional non-stationary vector autoregressive (VAR) models. Our method does not require knowledge of the order of integration of the time series under consideration. We augment the VAR with at least as many lags as the suspected maximum order of integration, an approach which has been proven to be robust against the presence of unit roots in low dimensions. We prove that we can restrict the augmentation to only the variables of interest for the testing, thereby making the approach suitable for high dimensions. We combine this lag augmentation with a post-double-selection procedure in which a set of initial penalized regressions is performed to select the relevant variables for both the Granger causing and caused variables. We then establish uniform asymptotic normality of a second-stage regression involving only the selected variables. Finite sample simulations show good performance, an application to investigate the (predictive) causes and effects of economic uncertainty illustrates the need to allow for unknown orders of integration.

Forecasting on sparse multivariate time series (MTS) aims to model the predictors of future values of time series given their incomplete past, which is important for many emerging applications. However, most existing methods process MTS's individually, and do not leverage the dynamic distributions underlying the MTS's, leading to sub-optimal results when the sparsity is high. To address this challenge, we propose a novel generative model, which tracks the transition of latent clusters, instead of isolated feature representations, to achieve robust modeling. It is characterized by a newly designed dynamic Gaussian mixture distribution, which captures the dynamics of clustering structures, and is used for emitting timeseries. The generative model is parameterized by neural networks. A structured inference network is also designed for enabling inductive analysis. A gating mechanism is further introduced to dynamically tune the Gaussian mixture distributions. Extensive experimental results on a variety of real-life datasets demonstrate the effectiveness of our method.

Multivariate time series forecasting is an important machine learning problem across many domains, including predictions of solar plant energy output, electricity consumption, and traffic jam situation. Temporal data arise in these real-world applications often involves a mixture of long-term and short-term patterns, for which traditional approaches such as Autoregressive models and Gaussian Process may fail. In this paper, we proposed a novel deep learning framework, namely Long- and Short-term Time-series network (LSTNet), to address this open challenge. LSTNet uses the Convolution Neural Network (CNN) and the Recurrent Neural Network (RNN) to extract short-term local dependency patterns among variables and to discover long-term patterns for time series trends. Furthermore, we leverage traditional autoregressive model to tackle the scale insensitive problem of the neural network model. In our evaluation on real-world data with complex mixtures of repetitive patterns, LSTNet achieved significant performance improvements over that of several state-of-the-art baseline methods. All the data and experiment codes are available online.

Self-supervised learning has garnered increasing attention in time series analysis for benefiting various downstream tasks and reducing reliance on labeled data. Despite its effectiveness, existing methods often struggle to comprehensively capture both long-term dynamic evolution and subtle local patterns in a unified manner. In this work, we propose TimeDART, a novel self-supervised time series pre-training framework that unifies two powerful generative paradigms to learn more transferable representations. Specifically, we first employ a causal Transformer encoder, accompanied by a patch-based embedding strategy, to model the evolving trends from left to right. Building on this global modeling, we further introduce a denoising diffusion process to capture fine-grained local patterns through forward diffusion and reverse denoising. Finally, we optimize the model in an autoregressive manner. As a result, TimeDART effectively accounts for both global and local sequence features in a coherent way. We conduct extensive experiments on public datasets for time series forecasting and classification. The experimental results demonstrate that TimeDART consistently outperforms previous compared methods, validating the effectiveness of our approach. Our code is available at https://github.com/Melmaphother/TimeDART.

Time series foundation models (TSFMs) have emerged as a powerful paradigm for time series analysis, driven by large-scale pretraining on diverse data corpora. However, time series inherently exhibit heterogeneous information density over time, influenced by system states and signal complexity, presenting significant modeling challenges especially in a zero-shot scenario. Current TSFMs rely on non-adaptive processing pipelines that fail to capture this dynamic nature. For example, common tokenization strategies such as fixed-size patching enforce rigid observational granularity, limiting their ability to adapt to varying information densities. Similarly, conventional positional encodings impose a uniform temporal scale, making it difficult to model diverse periodicities and trends across series. To overcome these limitations, we propose Kairos, a flexible TSFM framework that integrates a dynamic patching tokenizer and an instance-adaptive positional embedding. Kairos adaptively selects tokenization granularity and tailors positional encodings to the unique characteristics of each time series instance. Trained on a large-scale Predictability-Stratified Time Series (PreSTS) corpus comprising over 300 billion time points and adopting a multi-patch prediction strategy in the inference stage, Kairos achieves superior performance with much fewer parameters on two common zero-shot benchmarks, GIFT-Eval and the Time-Series-Library benchmark, consistently outperforming established methods across diverse tasks. The project page is at https://foundation-model-research.github.io/Kairos .

Pre-trained foundation models (FMs) have shown exceptional performance in univariate time series forecasting tasks. However, several practical challenges persist, including managing intricate dependencies among features and quantifying uncertainty in predictions. This study aims to tackle these critical limitations by introducing adapters; feature-space transformations that facilitate the effective use of pre-trained univariate time series FMs for multivariate tasks. Adapters operate by projecting multivariate inputs into a suitable latent space and applying the FM independently to each dimension. Inspired by the literature on representation learning and partially stochastic Bayesian neural networks, we present a range of adapters and optimization/inference strategies. Experiments conducted on both synthetic and real-world datasets confirm the efficacy of adapters, demonstrating substantial enhancements in forecasting accuracy and uncertainty quantification compared to baseline methods. Our framework, AdaPTS, positions adapters as a modular, scalable, and effective solution for leveraging time series FMs in multivariate contexts, thereby promoting their wider adoption in real-world applications. We release the code at https://github.com/abenechehab/AdaPTS.

Time series forecasting plays a crucial role in data mining, driving rapid advancements across numerous industries. With the emergence of large models, time series foundation models (TSFMs) have exhibited remarkable generalization capabilities, such as zero-shot learning, through large-scale pre-training. Meanwhile, Retrieval-Augmented Generation (RAG) methods have been widely employed to enhance the performance of foundation models on unseen data, allowing models to access to external knowledge. In this paper, we introduce TimeRAF, a Retrieval-Augmented Forecasting model that enhance zero-shot time series forecasting through retrieval-augmented techniques. We develop customized time series knowledge bases that are tailored to the specific forecasting tasks. TimeRAF employs an end-to-end learnable retriever to extract valuable information from the knowledge base. Additionally, we propose Channel Prompting for knowledge integration, which effectively extracts relevant information from the retrieved knowledge along the channel dimension. Extensive experiments demonstrate the effectiveness of our model, showing significant improvement across various domains and datasets.

Not all positive pairs are beneficial to time series contrastive learning. In this paper, we study two types of bad positive pairs that can impair the quality of time series representation learned through contrastive learning: the noisy positive pair and the faulty positive pair. We observe that, with the presence of noisy positive pairs, the model tends to simply learn the pattern of noise (Noisy Alignment). Meanwhile, when faulty positive pairs arise, the model wastes considerable amount of effort aligning non-representative patterns (Faulty Alignment). To address this problem, we propose a Dynamic Bad Pair Mining (DBPM) algorithm, which reliably identifies and suppresses bad positive pairs in time series contrastive learning. Specifically, DBPM utilizes a memory module to dynamically track the training behavior of each positive pair along training process. This allows us to identify potential bad positive pairs at each epoch based on their historical training behaviors. The identified bad pairs are subsequently down-weighted through a transformation module, thereby mitigating their negative impact on the representation learning process. DBPM is a simple algorithm designed as a lightweight plug-in without learnable parameters to enhance the performance of existing state-of-the-art methods. Through extensive experiments conducted on four large-scale, real-world time series datasets, we demonstrate DBPM's efficacy in mitigating the adverse effects of bad positive pairs.

In this paper, we present a new approach to time series forecasting. Time series data are prevalent in many scientific and engineering disciplines. Time series forecasting is a crucial task in modeling time series data, and is an important area of machine learning. In this work we developed a novel method that employs Transformer-based machine learning models to forecast time series data. This approach works by leveraging self-attention mechanisms to learn complex patterns and dynamics from time series data. Moreover, it is a generic framework and can be applied to univariate and multivariate time series data, as well as time series embeddings. Using influenza-like illness (ILI) forecasting as a case study, we show that the forecasting results produced by our approach are favorably comparable to the state-of-the-art.

This paper presents OLinear, a linear-based multivariate time series forecasting model that operates in an orthogonally transformed domain. Recent forecasting models typically adopt the temporal forecast (TF) paradigm, which directly encode and decode time series in the time domain. However, the entangled step-wise dependencies in series data can hinder the performance of TF. To address this, some forecasters conduct encoding and decoding in the transformed domain using fixed, dataset-independent bases (e.g., sine and cosine signals in the Fourier transform). In contrast, we utilize OrthoTrans, a data-adaptive transformation based on an orthogonal matrix that diagonalizes the series' temporal Pearson correlation matrix. This approach enables more effective encoding and decoding in the decorrelated feature domain and can serve as a plug-in module to enhance existing forecasters. To enhance the representation learning for multivariate time series, we introduce a customized linear layer, NormLin, which employs a normalized weight matrix to capture multivariate dependencies. Empirically, the NormLin module shows a surprising performance advantage over multi-head self-attention, while requiring nearly half the FLOPs. Extensive experiments on 24 benchmarks and 140 forecasting tasks demonstrate that OLinear consistently achieves state-of-the-art performance with high efficiency. Notably, as a plug-in replacement for self-attention, the NormLin module consistently enhances Transformer-based forecasters. The code and datasets are available at https://anonymous.4open.science/r/OLinear

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

Multivariate probabilistic time series forecasts are commonly evaluated via proper scoring rules, i.e., functions that are minimal in expectation for the ground-truth distribution. However, this property is not sufficient to guarantee good discrimination in the non-asymptotic regime. In this paper, we provide the first systematic finite-sample study of proper scoring rules for time-series forecasting evaluation. Through a power analysis, we identify the "region of reliability" of a scoring rule, i.e., the set of practical conditions where it can be relied on to identify forecasting errors. We carry out our analysis on a comprehensive synthetic benchmark, specifically designed to test several key discrepancies between ground-truth and forecast distributions, and we gauge the generalizability of our findings to real-world tasks with an application to an electricity production problem. Our results reveal critical shortcomings in the evaluation of multivariate probabilistic forecasts as commonly performed in the literature.

We introduce Sundial, a family of native, flexible, and scalable time series foundation models. To predict the next-patch's distribution, we propose a TimeFlow Loss based on flow-matching, which facilitates native pre-training of Transformers on time series without discrete tokenization. Conditioned on arbitrary-length time series, our model is pre-trained without specifying any prior distribution and can generate multiple probable predictions, achieving flexibility in representation learning beyond using parametric densities. Towards time series foundation models, we leverage minimal but crucial adaptations of Transformers and curate TimeBench with 1 trillion time points, comprising mostly real-world datasets and synthetic data. By mitigating mode collapse through TimeFlow Loss, we pre-train a family of Sundial models on TimeBench, which exhibit unprecedented model capacity and generalization performance on zero-shot forecasting. In addition to presenting good scaling behavior, Sundial achieves new state-of-the-art on both point forecasting and probabilistic forecasting benchmarks. We believe that Sundial's pioneering generative paradigm will facilitate a wide variety of forecasting scenarios.

In this paper, we introduce TimeGPT, the first foundation model for time series, capable of generating accurate predictions for diverse datasets not seen during training. We evaluate our pre-trained model against established statistical, machine learning, and deep learning methods, demonstrating that TimeGPT zero-shot inference excels in performance, efficiency, and simplicity. Our study provides compelling evidence that insights from other domains of artificial intelligence can be effectively applied to time series analysis. We conclude that large-scale time series models offer an exciting opportunity to democratize access to precise predictions and reduce uncertainty by leveraging the capabilities of contemporary advancements in deep learning.

In this paper, we introduce FITS, a lightweight yet powerful model for time series analysis. Unlike existing models that directly process raw time-domain data, FITS operates on the principle that time series can be manipulated through interpolation in the complex frequency domain. By discarding high-frequency components with negligible impact on time series data, FITS achieves performance comparable to state-of-the-art models for time series forecasting and anomaly detection tasks, while having a remarkably compact size of only approximately 10k parameters. Such a lightweight model can be easily trained and deployed in edge devices, creating opportunities for various applications. The code is available in: https://github.com/VEWOXIC/FITS

Many businesses and industries nowadays rely on large quantities of time series data making time series forecasting an important research area. Global forecasting models that are trained across sets of time series have shown a huge potential in providing accurate forecasts compared with the traditional univariate forecasting models that work on isolated series. However, there are currently no comprehensive time series archives for forecasting that contain datasets of time series from similar sources available for the research community to evaluate the performance of new global forecasting algorithms over a wide variety of datasets. In this paper, we present such a comprehensive time series forecasting archive containing 20 publicly available time series datasets from varied domains, with different characteristics in terms of frequency, series lengths, and inclusion of missing values. We also characterise the datasets, and identify similarities and differences among them, by conducting a feature analysis. Furthermore, we present the performance of a set of standard baseline forecasting methods over all datasets across eight error metrics, for the benefit of researchers using the archive to benchmark their forecasting algorithms.

Objective: Electroencephalography signals are recorded as a multidimensional dataset. We propose a new framework based on the augmented covariance extracted from an autoregressive model to improve motor imagery classification. Methods: From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to extend the standard approach using these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. Results: The augmented covariance matrix performed noticeably better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. Conclusion: The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information, incorporating nonlinear components of the signal through an embedding procedure, which allows the leveraging of dynamical systems algorithms. Significance: These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Time series anomaly detection forms a very crucial area in several domains but poses substantial challenges. Due to time series data possessing seasonality, trends, noise, and evolving patterns (concept drift), it becomes very difficult to set a general notion of what constitutes normal behavior. Anomalies themselves could be varied, ranging from a single outlier to contextual or collective anomalies, and are normally very rare; hence, the dataset is largely imbalanced. Additional layers of complexities arise due to the problems of increased dimensionality of modern time series, real-time detection criteria, setting up appropriate detection thresholds, and arriving at results that are interpretable. To embrace these multifaceted challenges, very strong, flexible, and interpretable approaches are required. This paper presents THEMIS, a new framework for time series anomaly detection that exploits pretrained knowledge from foundation models. THEMIS extracts embeddings from the encoder of the Chronos time series foundation model and applies outlier detection techniques like Local Outlier Factor and Spectral Decomposition on the self-similarity matrix, to spot anomalies in the data. Our experiments show that this modular method achieves SOTA results on the MSL dataset and performs quite competitively on the SMAP and SWAT^* datasets. Notably, THEMIS exceeds models trained specifically for anomaly detection, presenting hyperparameter robustness and interpretability by default. This paper advocates for pretrained representations from foundation models for performing efficient and adaptable anomaly detection for time series data.

Missing values are common in real-world time series, and multivariate time series forecasting with missing values (MTSF-M) has become a crucial area of research for ensuring reliable predictions. To address the challenge of missing data, current approaches have developed an imputation-then-prediction framework that uses imputation modules to fill in missing values, followed by forecasting on the imputed data. However, this framework overlooks a critical issue: there is no ground truth for the missing values, making the imputation process susceptible to errors that can degrade prediction accuracy. In this paper, we conduct a systematic empirical study and reveal that imputation without direct supervision can corrupt the underlying data distribution and actively degrade prediction accuracy. To address this, we propose a paradigm shift that moves away from imputation and directly predicts from the partially observed time series. We introduce Consistency-Regularized Information Bottleneck (CRIB), a novel framework built on the Information Bottleneck principle. CRIB combines a unified-variate attention mechanism with a consistency regularization scheme to learn robust representations that filter out noise introduced by missing values while preserving essential predictive signals. Comprehensive experiments on four real-world datasets demonstrate the effectiveness of CRIB, which predicts accurately even under high missing rates. Our code is available in https://github.com/Muyiiiii/CRIB.

Multivariate long-term and efficient time series forecasting is a key requirement for a variety of practical applications, and there are complex interleaving time dynamics in time series data that require decomposition modeling. Traditional time series decomposition methods are single and rely on fixed rules, which are insufficient for mining the potential information of the series and adapting to the dynamic characteristics of complex series. On the other hand, the Transformer-based models for time series forecasting struggle to effectively model long sequences and intricate dynamic relationships due to their high computational complexity. To overcome these limitations, we introduce KARMA, with an Adaptive Time Channel Decomposition module (ATCD) to dynamically extract trend and seasonal components. It further integrates a Hybrid Frequency-Time Decomposition module (HFTD) to further decompose Series into frequency-domain and time-domain. These components are coupled with multi-scale Mamba-based KarmaBlock to efficiently process global and local information in a coordinated manner. Experiments on eight real-world datasets from diverse domains well demonstrated that KARMA significantly outperforms mainstream baseline methods in both predictive accuracy and computational efficiency. Code and full results are available at this repository: https://github.com/yedadasd/KARMA

The present work investigates two properties of level crossings of a stationary Gaussian process X(t) with autocorrelation function R_X(tau). We show firstly that if R_X(tau) admits finite second and fourth derivatives at the origin, the length of up-excursions above a large negative level -gamma is asymptotically exponential as -gamma to -infty. Secondly, assuming that R_X(tau) admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two subsequent large levels. The asymptotic results are shown to be effective even for moderate values of crossing level. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.

The goal of the change-point detection is to discover changes of time series distribution. One of the state of the art approaches of the change-point detection are based on direct density ratio estimation. In this work we show how existing algorithms can be generalized using various binary classification and regression models. In particular, we show that the Gradient Boosting over Decision Trees and Neural Networks can be used for this purpose. The algorithms are tested on several synthetic and real-world datasets. The results show that the proposed methods outperform classical RuLSIF algorithm. Discussion of cases where the proposed algorithms have advantages over existing methods are also provided.

Explaining multivariate time series is a compound challenge, as it requires identifying important locations in the time series and matching complex temporal patterns. Although previous saliency-based methods addressed the challenges, their perturbation may not alleviate the distribution shift issue, which is inevitable especially in heterogeneous samples. We present ContraLSP, a locally sparse model that introduces counterfactual samples to build uninformative perturbations but keeps distribution using contrastive learning. Furthermore, we incorporate sample-specific sparse gates to generate more binary-skewed and smooth masks, which easily integrate temporal trends and select the salient features parsimoniously. Empirical studies on both synthetic and real-world datasets show that ContraLSP outperforms state-of-the-art models, demonstrating a substantial improvement in explanation quality for time series data. The source code is available at https://github.com/zichuan-liu/ContraLSP.

Accurate uncertainty measurement is a key step to building robust and reliable machine learning systems. Conformal prediction is a distribution-free uncertainty quantification algorithm popular for its ease of implementation, statistical coverage guarantees, and versatility for underlying forecasters. However, existing conformal prediction algorithms for time series are limited to single-step prediction without considering the temporal dependency. In this paper, we propose a Copula Conformal Prediction algorithm for multivariate, multi-step Time Series forecasting, CopulaCPTS. We prove that CopulaCPTS has finite sample validity guarantee. On several synthetic and real-world multivariate time series datasets, we show that CopulaCPTS produces more calibrated and sharp confidence intervals for multi-step prediction tasks than existing techniques.

Pretrained time series models have enabled inference-only forecasting systems that produce accurate predictions without task-specific training. However, existing approaches largely focus on univariate forecasting, limiting their applicability in real-world scenarios where multivariate data and covariates play a crucial role. We present Chronos-2, a pretrained model capable of handling univariate, multivariate, and covariate-informed forecasting tasks in a zero-shot manner. Chronos-2 employs a group attention mechanism that facilitates in-context learning (ICL) through efficient information sharing across multiple time series within a group, which may represent sets of related series, variates of a multivariate series, or targets and covariates in a forecasting task. These general capabilities are achieved through training on synthetic datasets that impose diverse multivariate structures on univariate series. Chronos-2 delivers state-of-the-art performance across three comprehensive benchmarks: fev-bench, GIFT-Eval, and Chronos Benchmark II. On fev-bench, which emphasizes multivariate and covariate-informed forecasting, Chronos-2's universal ICL capabilities lead to substantial improvements over existing models. On tasks involving covariates, it consistently outperforms baselines by a wide margin. Case studies in the energy and retail domains further highlight its practical advantages. The in-context learning capabilities of Chronos-2 establish it as a general-purpose forecasting model that can be used "as is" in real-world forecasting pipelines.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

We address causal reasoning in multivariate time series data generated by stochastic processes. Existing approaches are largely restricted to static settings, ignoring the continuity and emission of variations across time. In contrast, we propose a learning paradigm that directly establishes causation between events in the course of time. We present two key lemmas to compute causal contributions and frame them as reinforcement learning problems. Our approach offers formal and computational tools for uncovering and quantifying causal relationships in diffusion processes, subsuming various important settings such as discrete-time Markov decision processes. Finally, in fairly intricate experiments and through sheer learning, our framework reveals and quantifies causal links, which otherwise seem inexplicable.

We introduce AutoGluon-TimeSeries - an open-source AutoML library for probabilistic time series forecasting. Focused on ease of use and robustness, AutoGluon-TimeSeries enables users to generate accurate point and quantile forecasts with just 3 lines of Python code. Built on the design philosophy of AutoGluon, AutoGluon-TimeSeries leverages ensembles of diverse forecasting models to deliver high accuracy within a short training time. AutoGluon-TimeSeries combines both conventional statistical models, machine-learning based forecasting approaches, and ensembling techniques. In our evaluation on 29 benchmark datasets, AutoGluon-TimeSeries demonstrates strong empirical performance, outperforming a range of forecasting methods in terms of both point and quantile forecast accuracy, and often even improving upon the best-in-hindsight combination of prior methods.

Extending the forecasting time is a critical demand for real applications, such as extreme weather early warning and long-term energy consumption planning. This paper studies the long-term forecasting problem of time series. Prior Transformer-based models adopt various self-attention mechanisms to discover the long-range dependencies. However, intricate temporal patterns of the long-term future prohibit the model from finding reliable dependencies. Also, Transformers have to adopt the sparse versions of point-wise self-attentions for long series efficiency, resulting in the information utilization bottleneck. Going beyond Transformers, we design Autoformer as a novel decomposition architecture with an Auto-Correlation mechanism. We break with the pre-processing convention of series decomposition and renovate it as a basic inner block of deep models. This design empowers Autoformer with progressive decomposition capacities for complex time series. Further, inspired by the stochastic process theory, we design the Auto-Correlation mechanism based on the series periodicity, which conducts the dependencies discovery and representation aggregation at the sub-series level. Auto-Correlation outperforms self-attention in both efficiency and accuracy. In long-term forecasting, Autoformer yields state-of-the-art accuracy, with a 38% relative improvement on six benchmarks, covering five practical applications: energy, traffic, economics, weather and disease. Code is available at this repository: https://github.com/thuml/Autoformer.

Moment restrictions and their conditional counterparts emerge in many areas of machine learning and statistics ranging from causal inference to reinforcement learning. Estimators for these tasks, generally called methods of moments, include the prominent generalized method of moments (GMM) which has recently gained attention in causal inference. GMM is a special case of the broader family of empirical likelihood estimators which are based on approximating a population distribution by means of minimizing a varphi-divergence to an empirical distribution. However, the use of varphi-divergences effectively limits the candidate distributions to reweightings of the data samples. We lift this long-standing limitation and provide a method of moments that goes beyond data reweighting. This is achieved by defining an empirical likelihood estimator based on maximum mean discrepancy which we term the kernel method of moments (KMM). We provide a variant of our estimator for conditional moment restrictions and show that it is asymptotically first-order optimal for such problems. Finally, we show that our method achieves competitive performance on several conditional moment restriction tasks.

Time series analysis stands as a focal point within the data mining community, serving as a cornerstone for extracting valuable insights crucial to a myriad of real-world applications. Recent advances in Foundation Models (FMs) have fundamentally reshaped the paradigm of model design for time series analysis, boosting various downstream tasks in practice. These innovative approaches often leverage pre-trained or fine-tuned FMs to harness generalized knowledge tailored for time series analysis. This survey aims to furnish a comprehensive and up-to-date overview of FMs for time series analysis. While prior surveys have predominantly focused on either application or pipeline aspects of FMs in time series analysis, they have often lacked an in-depth understanding of the underlying mechanisms that elucidate why and how FMs benefit time series analysis. To address this gap, our survey adopts a methodology-centric classification, delineating various pivotal elements of time-series FMs, including model architectures, pre-training techniques, adaptation methods, and data modalities. Overall, this survey serves to consolidate the latest advancements in FMs pertinent to time series analysis, accentuating their theoretical underpinnings, recent strides in development, and avenues for future exploration.

Long-term time series forecasting (LTSF) is important for various domains but is confronted by challenges in handling the complex temporal-contextual relationships. As multivariate input models underperforming some recent univariate counterparts, we posit that the issue lies in the inefficiency of existing multivariate LTSF Transformers to model series-wise relationships: the characteristic differences between series are often captured incorrectly. To address this, we introduce ARM: a multivariate temporal-contextual adaptive learning method, which is an enhanced architecture specifically designed for multivariate LTSF modelling. ARM employs Adaptive Univariate Effect Learning (AUEL), Random Dropping (RD) training strategy, and Multi-kernel Local Smoothing (MKLS), to better handle individual series temporal patterns and correctly learn inter-series dependencies. ARM demonstrates superior performance on multiple benchmarks without significantly increasing computational costs compared to vanilla Transformer, thereby advancing the state-of-the-art in LTSF. ARM is also generally applicable to other LTSF architecture beyond vanilla Transformer.

Given time series data, how can we answer questions like "what will happen in the future?" and "how did we get here?" These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.

Time series foundation models have shown impressive performance on a variety of tasks, across a wide range of domains, even in zero-shot settings. However, most of these models are designed to handle short univariate time series as an input. This limits their practical use, especially in domains such as healthcare with copious amounts of long and multivariate data with strong temporal and intra-variate dependencies. Our study bridges this gap by cataloging and systematically comparing various context expansion techniques from both language and time series domains, and introducing a novel compressive memory mechanism to allow encoder-only TSFMs to effectively model intra-variate dependencies. We demonstrate the benefits of our approach by imbuing MOMENT, a recent family of multi-task time series foundation models, with the multivariate context.

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs). However, GANs are often unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to these issues, they are (surprisingly) less often considered for time series generation. In this work, we introduce Koopman VAE (KVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leverageing spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stablity of the system can be performed using tools from dynamical systems theory. Our results show that KVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KVAE learns probability density functions that better approximate empirical ground truth distributions.

Conformal prediction is a widely used method to quantify the uncertainty of a classifier under the assumption of exchangeability (e.g., IID data). We generalize conformal prediction to the Hidden Markov Model (HMM) framework where the assumption of exchangeability is not valid. The key idea of the proposed method is to partition the non-exchangeable Markovian data from the HMM into exchangeable blocks by exploiting the de Finetti's Theorem for Markov Chains discovered by Diaconis and Freedman (1980). The permutations of the exchangeable blocks are viewed as randomizations of the observed Markovian data from the HMM. The proposed method provably retains all desirable theoretical guarantees offered by the classical conformal prediction framework in both exchangeable and Markovian settings. In particular, while the lack of exchangeability introduced by Markovian samples constitutes a violation of a crucial assumption for classical conformal prediction, the proposed method views it as an advantage that can be exploited to improve the performance further. Detailed numerical and empirical results that complement the theoretical conclusions are provided to illustrate the practical feasibility of the proposed method.

Time series research has gathered lots of interests in the last decade, especially for Time Series Classification (TSC) and Time Series Forecasting (TSF). Research in TSC has greatly benefited from the University of California Riverside and University of East Anglia (UCR/UEA) Time Series Archives. On the other hand, the advancement in Time Series Forecasting relies on time series forecasting competitions such as the Makridakis competitions, NN3 and NN5 Neural Network competitions, and a few Kaggle competitions. Each year, thousands of papers proposing new algorithms for TSC and TSF have utilized these benchmarking archives. These algorithms are designed for these specific problems, but may not be useful for tasks such as predicting the heart rate of a person using photoplethysmogram (PPG) and accelerometer data. We refer to this problem as Time Series Extrinsic Regression (TSER), where we are interested in a more general methodology of predicting a single continuous value, from univariate or multivariate time series. This prediction can be from the same time series or not directly related to the predictor time series and does not necessarily need to be a future value or depend heavily on recent values. To the best of our knowledge, research into TSER has received much less attention in the time series research community and there are no models developed for general time series extrinsic regression problems. Most models are developed for a specific problem. Therefore, we aim to motivate and support the research into TSER by introducing the first TSER benchmarking archive. This archive contains 19 datasets from different domains, with varying number of dimensions, unequal length dimensions, and missing values. In this paper, we introduce the datasets in this archive and did an initial benchmark on existing models.

Large language models (LLMs) have been increasingly used in time series analysis. However, the potential of multimodal LLMs (MLLMs), particularly vision-language models, for time series remains largely under-explored. One natural way for humans to detect time series anomalies is through visualization and textual description. Motivated by this, we raise a critical and practical research question: Can multimodal LLMs perform time series anomaly detection? To answer this, we propose VisualTimeAnomaly benchmark to evaluate MLLMs in time series anomaly detection (TSAD). Our approach transforms time series numerical data into the image format and feed these images into various MLLMs, including proprietary models (GPT-4o and Gemini-1.5) and open-source models (LLaVA-NeXT and Qwen2-VL), each with one larger and one smaller variant. In total, VisualTimeAnomaly contains 12.4k time series images spanning 3 scenarios and 3 anomaly granularities with 9 anomaly types across 8 MLLMs. Starting with the univariate case (point- and range-wise anomalies), we extend our evaluation to more practical scenarios, including multivariate and irregular time series scenarios, and variate-wise anomalies. Our study reveals several key insights: 1) MLLMs detect range- and variate-wise anomalies more effectively than point-wise anomalies. 2) MLLMs are highly robust to irregular time series, even with 25% of the data missing. 3) Open-source MLLMs perform comparably to proprietary models in TSAD. While open-source MLLMs excel on univariate time series, proprietary MLLMs demonstrate superior effectiveness on multivariate time series. To the best of our knowledge, this is the first work to comprehensively investigate MLLMs for TSAD, particularly for multivariate and irregular time series scenarios. We release our dataset and code at https://github.com/mllm-ts/VisualTimeAnomaly to support future research.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

In recent years, there has been increasing interest in developing foundation models for time series data that can generalize across diverse downstream tasks. While numerous forecasting-oriented foundation models have been introduced, there is a notable scarcity of models tailored for time series classification. To address this gap, we present Mantis, a new open-source foundation model for time series classification based on the Vision Transformer (ViT) architecture that has been pre-trained using a contrastive learning approach. Our experimental results show that Mantis outperforms existing foundation models both when the backbone is frozen and when fine-tuned, while achieving the lowest calibration error. In addition, we propose several adapters to handle the multivariate setting, reducing memory requirements and modeling channel interdependence.

Generating time series data using Generative Adversarial Networks (GANs) presents several prevalent challenges, such as slow convergence, information loss in embedding spaces, instability, and performance variability depending on the series length. To tackle these obstacles, we introduce a robust framework aimed at addressing and mitigating these issues effectively. This advanced framework integrates the benefits of an Autoencoder-generated embedding space with the adversarial training dynamics of GANs. This framework benefits from a time series-based loss function and oversight from a supervisory network, both of which capture the stepwise conditional distributions of the data effectively. The generator functions within the latent space, while the discriminator offers essential feedback based on the feature space. Moreover, we introduce an early generation algorithm and an improved neural network architecture to enhance stability and ensure effective generalization across both short and long time series. Through joint training, our framework consistently outperforms existing benchmarks, generating high-quality time series data across a range of real and synthetic datasets with diverse characteristics.

Conformal prediction offers a powerful framework for building distribution-free prediction intervals for exchangeable data. Existing methods that extend conformal prediction to sequential data rely on fitting a relatively complex model to capture temporal dependencies. However, these methods can fail if the sample size is small and often require expensive retraining when the underlying data distribution changes. To overcome these limitations, we propose Reservoir Conformal Prediction (ResCP), a novel training-free conformal prediction method for time series. Our approach leverages the efficiency and representation learning capabilities of reservoir computing to dynamically reweight conformity scores. In particular, we compute similarity scores among reservoir states and use them to adaptively reweight the observed residuals at each step. With this approach, ResCP enables us to account for local temporal dynamics when modeling the error distribution without compromising computational scalability. We prove that, under reasonable assumptions, ResCP achieves asymptotic conditional coverage, and we empirically demonstrate its effectiveness across diverse forecasting tasks.

Time series data, characterized by its intrinsic long and short-range dependencies, poses a unique challenge across analytical applications. While Transformer-based models excel at capturing long-range dependencies, they face limitations in noise sensitivity, computational efficiency, and overfitting with smaller datasets. In response, we introduce a novel Time Series Lightweight Adaptive Network (TSLANet), as a universal convolutional model for diverse time series tasks. Specifically, we propose an Adaptive Spectral Block, harnessing Fourier analysis to enhance feature representation and to capture both long-term and short-term interactions while mitigating noise via adaptive thresholding. Additionally, we introduce an Interactive Convolution Block and leverage self-supervised learning to refine the capacity of TSLANet for decoding complex temporal patterns and improve its robustness on different datasets. Our comprehensive experiments demonstrate that TSLANet outperforms state-of-the-art models in various tasks spanning classification, forecasting, and anomaly detection, showcasing its resilience and adaptability across a spectrum of noise levels and data sizes. The code is available at https://github.com/emadeldeen24/TSLANet

Deep models have demonstrated remarkable performance in time series forecasting. However, due to the partially-observed nature of real-world applications, solely focusing on the target of interest, so-called endogenous variables, is usually insufficient to guarantee accurate forecasting. Notably, a system is often recorded into multiple variables, where the exogenous variables can provide valuable external information for endogenous variables. Thus, unlike well-established multivariate or univariate forecasting paradigms that either treat all the variables equally or ignore exogenous information, this paper focuses on a more practical setting: time series forecasting with exogenous variables. We propose a novel approach, TimeXer, to ingest external information to enhance the forecasting of endogenous variables. With deftly designed embedding layers, TimeXer empowers the canonical Transformer with the ability to reconcile endogenous and exogenous information, where patch-wise self-attention and variate-wise cross-attention are used simultaneously. Moreover, global endogenous tokens are learned to effectively bridge the causal information underlying exogenous series into endogenous temporal patches. Experimentally, TimeXer achieves consistent state-of-the-art performance on twelve real-world forecasting benchmarks and exhibits notable generality and scalability. Code is available at this repository: https://github.com/thuml/TimeXer.

Time series clustering promises to uncover hidden structural patterns in data with applications across healthcare, finance, industrial systems, and other critical domains. However, without validated ground truth information, researchers cannot objectively assess clustering quality or determine whether poor results stem from absent structures in the data, algorithmic limitations, or inappropriate validation methods, raising the question whether clustering is "more art than science" (Guyon et al., 2009). To address these challenges, we introduce CSTS (Correlation Structures in Time Series), a synthetic benchmark for evaluating the discovery of correlation structures in multivariate time series data. CSTS provides a clean benchmark that enables researchers to isolate and identify specific causes of clustering failures by differentiating between correlation structure deterioration and limitations of clustering algorithms and validation methods. Our contributions are: (1) a comprehensive benchmark for correlation structure discovery with distinct correlation structures, systematically varied data conditions, established performance thresholds, and recommended evaluation protocols; (2) empirical validation of correlation structure preservation showing moderate distortion from downsampling and minimal effects from distribution shifts and sparsification; and (3) an extensible data generation framework enabling structure-first clustering evaluation. A case study demonstrates CSTS's practical utility by identifying an algorithm's previously undocumented sensitivity to non-normal distributions, illustrating how the benchmark enables precise diagnosis of methodological limitations. CSTS advances rigorous evaluation standards for correlation-based time series clustering.

State-space models (SSMs) offer a powerful framework for dynamical system analysis, wherein the temporal dynamics of the system are assumed to be captured through the evolution of the latent states, which govern the values of the observations. This paper provides a selective review of recent advancements in deep neural network-based approaches for SSMs, and presents a unified perspective for discrete time deep state space models and continuous time ones such as latent neural Ordinary Differential and Stochastic Differential Equations. It starts with an overview of the classical maximum likelihood based approach for learning SSMs, reviews variational autoencoder as a general learning pipeline for neural network-based approaches in the presence of latent variables, and discusses in detail representative deep learning models that fall under the SSM framework. Very recent developments, where SSMs are used as standalone architectural modules for improving efficiency in sequence modeling, are also examined. Finally, examples involving mixed frequency and irregularly-spaced time series data are presented to demonstrate the advantage of SSMs in these settings.

Time series forecasting has attracted significant attention in recent decades. Previous studies have demonstrated that the Channel-Independent (CI) strategy improves forecasting performance by treating different channels individually, while it leads to poor generalization on unseen instances and ignores potentially necessary interactions between channels. Conversely, the Channel-Dependent (CD) strategy mixes all channels with even irrelevant and indiscriminate information, which, however, results in oversmoothing issues and limits forecasting accuracy. There is a lack of channel strategy that effectively balances individual channel treatment for improved forecasting performance without overlooking essential interactions between channels. Motivated by our observation of a correlation between the time series model's performance boost against channel mixing and the intrinsic similarity on a pair of channels, we developed a novel and adaptable Channel Clustering Module (CCM). CCM dynamically groups channels characterized by intrinsic similarities and leverages cluster information instead of individual channel identities, combining the best of CD and CI worlds. Extensive experiments on real-world datasets demonstrate that CCM can (1) boost the performance of CI and CD models by an average margin of 2.4% and 7.2% on long-term and short-term forecasting, respectively; (2) enable zero-shot forecasting with mainstream time series forecasting models; (3) uncover intrinsic time series patterns among channels and improve interpretability of complex time series models.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

Time series are the primary data type used to record dynamic system measurements and generated in great volume by both physical sensors and online processes (virtual sensors). Time series analytics is therefore crucial to unlocking the wealth of information implicit in available data. With the recent advancements in graph neural networks (GNNs), there has been a surge in GNN-based approaches for time series analysis. These approaches can explicitly model inter-temporal and inter-variable relationships, which traditional and other deep neural network-based methods struggle to do. In this survey, we provide a comprehensive review of graph neural networks for time series analysis (GNN4TS), encompassing four fundamental dimensions: forecasting, classification, anomaly detection, and imputation. Our aim is to guide designers and practitioners to understand, build applications, and advance research of GNN4TS. At first, we provide a comprehensive task-oriented taxonomy of GNN4TS. Then, we present and discuss representative research works and introduce mainstream applications of GNN4TS. A comprehensive discussion of potential future research directions completes the survey. This survey, for the first time, brings together a vast array of knowledge on GNN-based time series research, highlighting foundations, practical applications, and opportunities of graph neural networks for time series analysis.

The striking fractal geometry of strange attractors underscores the generative nature of chaos: like probability distributions, chaotic systems can be repeatedly measured to produce arbitrarily-detailed information about the underlying attractor. Chaotic systems thus pose a unique challenge to modern statistical learning techniques, while retaining quantifiable mathematical properties that make them controllable and interpretable as benchmarks. Here, we present a growing database currently comprising 131 known chaotic dynamical systems spanning fields such as astrophysics, climatology, and biochemistry. Each system is paired with precomputed multivariate and univariate time series. Our dataset has comparable scale to existing static time series databases; however, our systems can be re-integrated to produce additional datasets of arbitrary length and granularity. Our dataset is annotated with known mathematical properties of each system, and we perform feature analysis to broadly categorize the diverse dynamics present across the collection. Chaotic systems inherently challenge forecasting models, and across extensive benchmarks we correlate forecasting performance with the degree of chaos present. We also exploit the unique generative properties of our dataset in several proof-of-concept experiments: surrogate transfer learning to improve time series classification, importance sampling to accelerate model training, and benchmarking symbolic regression algorithms.

The stable periodic patterns present in time series data serve as the foundation for conducting long-horizon forecasts. In this paper, we pioneer the exploration of explicitly modeling this periodicity to enhance the performance of models in long-term time series forecasting (LTSF) tasks. Specifically, we introduce the Residual Cycle Forecasting (RCF) technique, which utilizes learnable recurrent cycles to model the inherent periodic patterns within sequences, and then performs predictions on the residual components of the modeled cycles. Combining RCF with a Linear layer or a shallow MLP forms the simple yet powerful method proposed in this paper, called CycleNet. CycleNet achieves state-of-the-art prediction accuracy in multiple domains including electricity, weather, and energy, while offering significant efficiency advantages by reducing over 90% of the required parameter quantity. Furthermore, as a novel plug-and-play technique, the RCF can also significantly improve the prediction accuracy of existing models, including PatchTST and iTransformer. The source code is available at: https://github.com/ACAT-SCUT/CycleNet.

Despite the eminent successes of deep neural networks, many architectures are often hard to transfer to irregularly-sampled and asynchronous time series that commonly occur in real-world datasets, especially in healthcare applications. This paper proposes a novel approach for classifying irregularly-sampled time series with unaligned measurements, focusing on high scalability and data efficiency. Our method SeFT (Set Functions for Time Series) is based on recent advances in differentiable set function learning, extremely parallelizable with a beneficial memory footprint, thus scaling well to large datasets of long time series and online monitoring scenarios. Furthermore, our approach permits quantifying per-observation contributions to the classification outcome. We extensively compare our method with existing algorithms on multiple healthcare time series datasets and demonstrate that it performs competitively whilst significantly reducing runtime.

Time-series forecasting is one of the most active research topics in artificial intelligence. Applications in real-world time series should consider two factors for achieving reliable predictions: modeling dynamic dependencies among multiple variables and adjusting the model's intrinsic hyperparameters. A still open gap in that literature is that statistical and ensemble learning approaches systematically present lower predictive performance than deep learning methods. They generally disregard the data sequence aspect entangled with multivariate data represented in more than one time series. Conversely, this work presents a novel neural network architecture for time-series forecasting that combines the power of graph evolution with deep recurrent learning on distinct data distributions; we named our method Recurrent Graph Evolution Neural Network (ReGENN). The idea is to infer multiple multivariate relationships between co-occurring time-series by assuming that the temporal data depends not only on inner variables and intra-temporal relationships (i.e., observations from itself) but also on outer variables and inter-temporal relationships (i.e., observations from other-selves). An extensive set of experiments was conducted comparing ReGENN with dozens of ensemble methods and classical statistical ones, showing sound improvement of up to 64.87% over the competing algorithms. Furthermore, we present an analysis of the intermediate weights arising from ReGENN, showing that by looking at inter and intra-temporal relationships simultaneously, time-series forecasting is majorly improved if paying attention to how multiple multivariate data synchronously evolve.

Recent efforts have been dedicated to enhancing time series forecasting accuracy by introducing advanced network architectures and self-supervised pretraining strategies. Nevertheless, existing approaches still exhibit two critical drawbacks. Firstly, these methods often rely on a single dataset for training, limiting the model's generalizability due to the restricted scale of the training data. Secondly, the one-step generation schema is widely followed, which necessitates a customized forecasting head and overlooks the temporal dependencies in the output series, and also leads to increased training costs under different horizon length settings. To address these issues, we propose a novel generative pretrained hierarchical transformer architecture for forecasting, named GPHT. There are two aspects of key designs in GPHT. On the one hand, we advocate for constructing a mixed dataset for pretraining our model, comprising various datasets from diverse data scenarios. This approach significantly expands the scale of training data, allowing our model to uncover commonalities in time series data and facilitating improved transfer to specific datasets. On the other hand, GPHT employs an auto-regressive forecasting approach under the channel-independent assumption, effectively modeling temporal dependencies in the output series. Importantly, no customized forecasting head is required, enabling a single model to forecast at arbitrary horizon settings. We conduct sufficient experiments on eight datasets with mainstream self-supervised pretraining models and supervised models. The results demonstrated that GPHT surpasses the baseline models across various fine-tuning and zero/few-shot learning settings in the traditional long-term forecasting task, providing support for verifying the feasibility of pretrained time series large models.

Multivariate time series forecasting is essential in domains such as finance, transportation, climate, and energy. However, existing patch-based methods typically adopt fixed-length segmentation, overlooking the heterogeneity of local temporal dynamics and the decoding heterogeneity of forecasting. Such designs lose details in information-dense regions, introduce redundancy in stable segments, and fail to capture the distinct complexities of short-term and long-term horizons. We propose TimeMosaic, a forecasting framework that aims to address temporal heterogeneity. TimeMosaic employs adaptive patch embedding to dynamically adjust granularity according to local information density, balancing motif reuse with structural clarity while preserving temporal continuity. In addition, it introduces segment-wise decoding that treats each prediction horizon as a related subtask and adapts to horizon-specific difficulty and information requirements, rather than applying a single uniform decoder. Extensive evaluations on benchmark datasets demonstrate that TimeMosaic delivers consistent improvements over existing methods, and our model trained on the large-scale corpus with 321 billion observations achieves performance competitive with state-of-the-art TSFMs.

Online updating of time series forecasting models aims to address the concept drifting problem by efficiently updating forecasting models based on streaming data. Many algorithms are designed for online time series forecasting, with some exploiting cross-variable dependency while others assume independence among variables. Given every data assumption has its own pros and cons in online time series modeling, we propose Online ensembling Network (OneNet). It dynamically updates and combines two models, with one focusing on modeling the dependency across the time dimension and the other on cross-variate dependency. Our method incorporates a reinforcement learning-based approach into the traditional online convex programming framework, allowing for the linear combination of the two models with dynamically adjusted weights. OneNet addresses the main shortcoming of classical online learning methods that tend to be slow in adapting to the concept drift. Empirical results show that OneNet reduces online forecasting error by more than 50% compared to the State-Of-The-Art (SOTA) method. The code is available at https://github.com/yfzhang114/OneNet.

We introduce MOMENT, a family of open-source foundation models for general-purpose time-series analysis. Pre-training large models on time-series data is challenging due to (1) the absence of a large and cohesive public time-series repository, and (2) diverse time-series characteristics which make multi-dataset training onerous. Additionally, (3) experimental benchmarks to evaluate these models, especially in scenarios with limited resources, time, and supervision, are still in their nascent stages. To address these challenges, we compile a large and diverse collection of public time-series, called the Time-series Pile, and systematically tackle time-series-specific challenges to unlock large-scale multi-dataset pre-training. Finally, we build on recent work to design a benchmark to evaluate time-series foundation models on diverse tasks and datasets in limited supervision settings. Experiments on this benchmark demonstrate the effectiveness of our pre-trained models with minimal data and task-specific fine-tuning. Finally, we present several interesting empirical observations about large pre-trained time-series models. Our code is available anonymously at anonymous.4open.science/r/BETT-773F/.

Transformer-based foundation models have emerged as a dominant paradigm in time series analysis, offering unprecedented capabilities in tasks such as forecasting, anomaly detection, classification, trend analysis and many more time series analytical tasks. This survey provides a comprehensive overview of the current state of the art pre-trained foundation models, introducing a novel taxonomy to categorize them across several dimensions. Specifically, we classify models by their architecture design, distinguishing between those leveraging patch-based representations and those operating directly on raw sequences. The taxonomy further includes whether the models provide probabilistic or deterministic predictions, and whether they are designed to work with univariate time series or can handle multivariate time series out of the box. Additionally, the taxonomy encompasses model scale and complexity, highlighting differences between lightweight architectures and large-scale foundation models. A unique aspect of this survey is its categorization by the type of objective function employed during training phase. By synthesizing these perspectives, this survey serves as a resource for researchers and practitioners, providing insights into current trends and identifying promising directions for future research in transformer-based time series modeling.

The reliability of deep time series models is often compromised by their tendency to rely on confounding factors, which may lead to misleading results. Our newly recorded, naturally confounded dataset named P2S from a real mechanical production line emphasizes this. To tackle the challenging problem of mitigating confounders in time series data, we introduce Right on Time (RioT). Our method enables interactions with model explanations across both the time and frequency domain. Feedback on explanations in both domains is then used to constrain the model, steering it away from the annotated confounding factors. The dual-domain interaction strategy is crucial for effectively addressing confounders in time series datasets. We empirically demonstrate that RioT can effectively guide models away from the wrong reasons in P2S as well as popular time series classification and forecasting datasets.

Multivariate time series is prevalent in many scientific and industrial domains. Modeling multivariate signals is challenging due to their long-range temporal dependencies and intricate interactions--both direct and indirect. To confront these complexities, we introduce a method of representing multivariate signals as nodes in a graph with edges indicating interdependency between them. Specifically, we leverage graph neural networks (GNN) and attention mechanisms to efficiently learn the underlying relationships within the time series data. Moreover, we suggest employing hierarchical signal decompositions running over the graphs to capture multiple spatial dependencies. The effectiveness of our proposed model is evaluated across various real-world benchmark datasets designed for long-term forecasting tasks. The results consistently showcase the superiority of our model, achieving an average 23\% reduction in mean squared error (MSE) compared to existing models.

Solving Bayesian inverse problems efficiently remains a significant challenge due to the complexity of posterior distributions and the computational cost of traditional sampling methods. Given a series of observations and the forward model, we want to recover the distribution of the parameters, conditioned on observed experimental data. We show, that combining Conditional Flow Mathching (CFM) with transformer-based architecture, we can efficiently sample from such kind of distribution, conditioned on variable number of observations.

We introduce TimeMCL, a method leveraging the Multiple Choice Learning (MCL) paradigm to forecast multiple plausible time series futures. Our approach employs a neural network with multiple heads and utilizes the Winner-Takes-All (WTA) loss to promote diversity among predictions. MCL has recently gained attention due to its simplicity and ability to address ill-posed and ambiguous tasks. We propose an adaptation of this framework for time-series forecasting, presenting it as an efficient method to predict diverse futures, which we relate to its implicit quantization objective. We provide insights into our approach using synthetic data and evaluate it on real-world time series, demonstrating its promising performance at a light computational cost.

Time series forecasting (TSF) is critical in domains like energy, finance, healthcare, and logistics, requiring models that generalize across diverse datasets. Large pre-trained models such as Chronos and Time-MoE show strong zero-shot (ZS) performance but suffer from high computational costs. In this work, We introduce Super-Linear, a lightweight and scalable mixture-of-experts (MoE) model for general forecasting. It replaces deep architectures with simple frequency-specialized linear experts, trained on resampled data across multiple frequency regimes. A lightweight spectral gating mechanism dynamically selects relevant experts, enabling efficient, accurate forecasting. Despite its simplicity, Super-Linear matches state-of-the-art performance while offering superior efficiency, robustness to various sampling rates, and enhanced interpretability. The implementation of Super-Linear is available at https://github.com/azencot-group/SuperLinear{https://github.com/azencot-group/SuperLinear}

Masked time series modeling has recently gained much attention as a self-supervised representation learning strategy for time series. Inspired by masked image modeling in computer vision, recent works first patchify and partially mask out time series, and then train Transformers to capture the dependencies between patches by predicting masked patches from unmasked patches. However, we argue that capturing such patch dependencies might not be an optimal strategy for time series representation learning; rather, learning to embed patches independently results in better time series representations. Specifically, we propose to use 1) the simple patch reconstruction task, which autoencode each patch without looking at other patches, and 2) the simple patch-wise MLP that embeds each patch independently. In addition, we introduce complementary contrastive learning to hierarchically capture adjacent time series information efficiently. Our proposed method improves time series forecasting and classification performance compared to state-of-the-art Transformer-based models, while it is more efficient in terms of the number of parameters and training/inference time. Code is available at this repository: https://github.com/seunghan96/pits.

Spatio-temporal forecasting is essential for real-world applications such as traffic management and urban computing. Although recent methods have shown improved accuracy, they often fail to account for dynamic deviations between current inputs and historical patterns. These deviations contain critical signals that can significantly affect model performance. To fill this gap, we propose ST-SSDL, a Spatio-Temporal time series forecasting framework that incorporates a Self-Supervised Deviation Learning scheme to capture and utilize such deviations. ST-SSDL anchors each input to its historical average and discretizes the latent space using learnable prototypes that represent typical spatio-temporal patterns. Two auxiliary objectives are proposed to refine this structure: a contrastive loss that enhances inter-prototype discriminability and a deviation loss that regularizes the distance consistency between input representations and corresponding prototypes to quantify deviation. Optimized jointly with the forecasting objective, these components guide the model to organize its hidden space and improve generalization across diverse input conditions. Experiments on six benchmark datasets show that ST-SSDL consistently outperforms state-of-the-art baselines across multiple metrics. Visualizations further demonstrate its ability to adaptively respond to varying levels of deviation in complex spatio-temporal scenarios. Our code and datasets are available at https://github.com/Jimmy-7664/ST-SSDL.

Non-linear state-space models, also known as general hidden Markov models, are ubiquitous in statistical machine learning, being the most classical generative models for serial data and sequences in general. The particle-based, rapid incremental smoother PaRIS is a sequential Monte Carlo (SMC) technique allowing for efficient online approximation of expectations of additive functionals under the smoothing distribution in these models. Such expectations appear naturally in several learning contexts, such as likelihood estimation (MLE) and Markov score climbing (MSC). PARIS has linear computational complexity, limited memory requirements and comes with non-asymptotic bounds, convergence results and stability guarantees. Still, being based on self-normalised importance sampling, the PaRIS estimator is biased. Our first contribution is to design a novel additive smoothing algorithm, the Parisian particle Gibbs PPG sampler, which can be viewed as a PaRIS algorithm driven by conditional SMC moves, resulting in bias-reduced estimates of the targeted quantities. We substantiate the PPG algorithm with theoretical results, including new bounds on bias and variance as well as deviation inequalities. Our second contribution is to apply PPG in a learning framework, covering MLE and MSC as special examples. In this context, we establish, under standard assumptions, non-asymptotic bounds highlighting the value of bias reduction and the implicit Rao--Blackwellization of PPG. These are the first non-asymptotic results of this kind in this setting. We illustrate our theoretical results with numerical experiments supporting our claims.

Recent advancements in the collection and analysis of sequential educational data have brought time series analysis to a pivotal position in educational research, highlighting its essential role in facilitating data-driven decision-making. However, there is a lack of comprehensive summaries that consolidate these advancements. To the best of our knowledge, this paper is the first to provide a comprehensive review of time series analysis techniques specifically within the educational context. We begin by exploring the landscape of educational data analytics, categorizing various data sources and types relevant to education. We then review four prominent time series methods-forecasting, classification, clustering, and anomaly detection-illustrating their specific application points in educational settings. Subsequently, we present a range of educational scenarios and applications, focusing on how these methods are employed to address diverse educational tasks, which highlights the practical integration of multiple time series methods to solve complex educational problems. Finally, we conclude with a discussion on future directions, including personalized learning analytics, multimodal data fusion, and the role of large language models (LLMs) in educational time series. The contributions of this paper include a detailed taxonomy of educational data, a synthesis of time series techniques with specific educational applications, and a forward-looking perspective on emerging trends and future research opportunities in educational analysis. The related papers and resources are available and regularly updated at the project page.

Discrete diffusion models with absorbing processes have shown promise in language modeling. The key quantities to be estimated are the ratios between the marginal probabilities of two transitive states at all timesteps, called the concrete score. In this paper, we reveal that the concrete score in absorbing diffusion can be expressed as conditional probabilities of clean data, multiplied by a time-dependent scalar in an analytic form. Motivated by this finding, we propose reparameterized absorbing discrete diffusion (RADD), a dedicated diffusion model without time-condition that characterizes the time-independent conditional probabilities. Besides its simplicity, RADD can reduce the number of function evaluations (NFEs) by caching the output of the time-independent network when the noisy sample remains unchanged in a sampling interval. Empirically, RADD is up to 3.5 times faster while achieving similar performance with the strongest baseline. Built upon the new perspective of conditional distributions, we further unify absorbing discrete diffusion and any-order autoregressive models (AO-ARMs), showing that the upper bound on the negative log-likelihood for the diffusion model can be interpreted as an expected negative log-likelihood for AO-ARMs. Further, our RADD models achieve SOTA performance among diffusion models on 5 zero-shot language modeling benchmarks (measured by perplexity) at the GPT-2 scale. Our code is available at https://github.com/ML-GSAI/RADD.

Analysis and prediction of stock market time series data has attracted considerable interest from the research community over the last decade. Rapid development and evolution of sophisticated algorithms for statistical analysis of time series data, and availability of high-performance hardware has made it possible to process and analyze high volume stock market time series data effectively, in real-time. Among many other important characteristics and behavior of such data, forecasting is an area which has witnessed considerable focus. In this work, we have used time series of the index values of the Auto sector in India during January 2010 to December 2015 for a deeper understanding of the behavior of its three constituent components, e.g., the trend, the seasonal component, and the random component. Based on this structural analysis, we have also designed five approaches for forecasting and also computed their accuracy in prediction using suitably chosen training and test data sets. Extensive results are presented to demonstrate the effectiveness of our proposed decomposition approaches of time series and the efficiency of our forecasting techniques, even in presence of a random component and a sharply changing trend component in the time-series.

Self-supervised learning (SSL) has recently achieved impressive performance on various time series tasks. The most prominent advantage of SSL is that it reduces the dependence on labeled data. Based on the pre-training and fine-tuning strategy, even a small amount of labeled data can achieve high performance. Compared with many published self-supervised surveys on computer vision and natural language processing, a comprehensive survey for time series SSL is still missing. To fill this gap, we review current state-of-the-art SSL methods for time series data in this article. To this end, we first comprehensively review existing surveys related to SSL and time series, and then provide a new taxonomy of existing time series SSL methods by summarizing them from three perspectives: generative-based, contrastive-based, and adversarial-based. These methods are further divided into ten subcategories with detailed reviews and discussions about their key intuitions, main frameworks, advantages and disadvantages. To facilitate the experiments and validation of time series SSL methods, we also summarize datasets commonly used in time series forecasting, classification, anomaly detection, and clustering tasks. Finally, we present the future directions of SSL for time series analysis.

It is commonplace to encounter heterogeneous or nonstationary data, of which the underlying generating process changes across domains or over time. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this paper, we develop a framework for causal discovery from such data, called Constraint-based causal Discovery from heterogeneous/NOnstationary Data (CD-NOD), to find causal skeleton and directions and estimate the properties of mechanism changes. First, we propose an enhanced constraint-based procedure to detect variables whose local mechanisms change and recover the skeleton of the causal structure over observed variables. Second, we present a method to determine causal orientations by making use of independent changes in the data distribution implied by the underlying causal model, benefiting from information carried by changing distributions. After learning the causal structure, next, we investigate how to efficiently estimate the "driving force" of the nonstationarity of a causal mechanism. That is, we aim to extract from data a low-dimensional representation of changes. The proposed methods are nonparametric, with no hard restrictions on data distributions and causal mechanisms, and do not rely on window segmentation. Furthermore, we find that data heterogeneity benefits causal structure identification even with particular types of confounders. Finally, we show the connection between heterogeneity/nonstationarity and soft intervention in causal discovery. Experimental results on various synthetic and real-world data sets (task-fMRI and stock market data) are presented to demonstrate the efficacy of the proposed methods.

We introduce Autoregressive Diffusion Models (ARDMs), a model class encompassing and generalizing order-agnostic autoregressive models (Uria et al., 2014) and absorbing discrete diffusion (Austin et al., 2021), which we show are special cases of ARDMs under mild assumptions. ARDMs are simple to implement and easy to train. Unlike standard ARMs, they do not require causal masking of model representations, and can be trained using an efficient objective similar to modern probabilistic diffusion models that scales favourably to highly-dimensional data. At test time, ARDMs support parallel generation which can be adapted to fit any given generation budget. We find that ARDMs require significantly fewer steps than discrete diffusion models to attain the same performance. Finally, we apply ARDMs to lossless compression, and show that they are uniquely suited to this task. Contrary to existing approaches based on bits-back coding, ARDMs obtain compelling results not only on complete datasets, but also on compressing single data points. Moreover, this can be done using a modest number of network calls for (de)compression due to the model's adaptable parallel generation.

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability eta(x)=P(Y=1mid X=x), we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of 2eta(x)-1 and apply the method to real datasets.

Transformer-based models have significantly advanced time series forecasting, with patch-based input strategies offering efficiency and improved long-horizon modeling. Yet, existing approaches rely on temporally-agnostic patch construction, where arbitrary starting positions and fixed lengths fracture temporal coherence by splitting natural transitions across boundaries. This naive segmentation often disrupts short-term dependencies and weakens representation learning. In response, we propose EntroPE (Entropy-Guided Dynamic Patch Encoder), a novel, temporally informed framework that dynamically detects transition points via conditional entropy and dynamically places patch boundaries. This preserves temporal structure while retaining the computational benefits of patching. EntroPE consists of two key modules, namely an Entropy-based Dynamic Patcher (EDP) that applies information-theoretic criteria to locate natural temporal shifts and determine patch boundaries, and an Adaptive Patch Encoder (APE) that employs pooling and cross-attention to capture intra-patch dependencies and produce fixed-size latent representations. These embeddings are then processed by a global transformer to model inter-patch dynamics. Experiments across long-term forecasting benchmarks demonstrate that EntroPE improves both accuracy and efficiency, establishing entropy-guided dynamic patching as a promising new paradigm for time series modeling. Code is available at: https://github.com/Sachithx/EntroPE.

In this paper, we tackle the important yet under-investigated problem of making long-horizon prediction of event sequences. Existing state-of-the-art models do not perform well at this task due to their autoregressive structure. We propose HYPRO, a hybridly normalized probabilistic model that naturally fits this task: its first part is an autoregressive base model that learns to propose predictions; its second part is an energy function that learns to reweight the proposals such that more realistic predictions end up with higher probabilities. We also propose efficient training and inference algorithms for this model. Experiments on multiple real-world datasets demonstrate that our proposed HYPRO model can significantly outperform previous models at making long-horizon predictions of future events. We also conduct a range of ablation studies to investigate the effectiveness of each component of our proposed methods.

Transformers have achieved superior performances in many tasks in natural language processing and computer vision, which also triggered great interest in the time series community. Among multiple advantages of Transformers, the ability to capture long-range dependencies and interactions is especially attractive for time series modeling, leading to exciting progress in various time series applications. In this paper, we systematically review Transformer schemes for time series modeling by highlighting their strengths as well as limitations. In particular, we examine the development of time series Transformers in two perspectives. From the perspective of network structure, we summarize the adaptations and modifications that have been made to Transformers in order to accommodate the challenges in time series analysis. From the perspective of applications, we categorize time series Transformers based on common tasks including forecasting, anomaly detection, and classification. Empirically, we perform robust analysis, model size analysis, and seasonal-trend decomposition analysis to study how Transformers perform in time series. Finally, we discuss and suggest future directions to provide useful research guidance. To the best of our knowledge, this paper is the first work to comprehensively and systematically summarize the recent advances of Transformers for modeling time series data. We hope this survey will ignite further research interests in time series Transformers.

[This paper was initially published in PHME conference in 2016, selected for further publication in International Journal of Prognostics and Health Management.] This paper describes an Autoregressive Partially-hidden Markov model (ARPHMM) for fault detection and prognostics of equipments based on sensors' data. It is a particular dynamic Bayesian network that allows to represent the dynamics of a system by means of a Hidden Markov Model (HMM) and an autoregressive (AR) process. The Markov chain assumes that the system is switching back and forth between internal states while the AR process ensures a temporal coherence on sensor measurements. A sound learning procedure of standard ARHMM based on maximum likelihood allows to iteratively estimate all parameters simultaneously. This paper suggests a modification of the learning procedure considering that one may have prior knowledge about the structure which becomes partially hidden. The integration of the prior is based on the Theory of Weighted Distributions which is compatible with the Expectation-Maximization algorithm in the sense that the convergence properties are still satisfied. We show how to apply this model to estimate the remaining useful life based on health indicators. The autoregressive parameters can indeed be used for prediction while the latent structure can be used to get information about the degradation level. The interest of the proposed method for prognostics and health assessment is demonstrated on CMAPSS datasets.

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.

A central problem in learning from sequential data is representing cumulative history in an incremental fashion as more data is processed. We introduce a general framework (HiPPO) for the online compression of continuous signals and discrete time series by projection onto polynomial bases. Given a measure that specifies the importance of each time step in the past, HiPPO produces an optimal solution to a natural online function approximation problem. As special cases, our framework yields a short derivation of the recent Legendre Memory Unit (LMU) from first principles, and generalizes the ubiquitous gating mechanism of recurrent neural networks such as GRUs. This formal framework yields a new memory update mechanism (HiPPO-LegS) that scales through time to remember all history, avoiding priors on the timescale. HiPPO-LegS enjoys the theoretical benefits of timescale robustness, fast updates, and bounded gradients. By incorporating the memory dynamics into recurrent neural networks, HiPPO RNNs can empirically capture complex temporal dependencies. On the benchmark permuted MNIST dataset, HiPPO-LegS sets a new state-of-the-art accuracy of 98.3%. Finally, on a novel trajectory classification task testing robustness to out-of-distribution timescales and missing data, HiPPO-LegS outperforms RNN and neural ODE baselines by 25-40% accuracy.

Time series reasoning treats time as a first-class axis and incorporates intermediate evidence directly into the answer. This survey defines the problem and organizes the literature by reasoning topology with three families: direct reasoning in one step, linear chain reasoning with explicit intermediates, and branch-structured reasoning that explores, revises, and aggregates. The topology is crossed with the main objectives of the field, including traditional time series analysis, explanation and understanding, causal inference and decision making, and time series generation, while a compact tag set spans these axes and captures decomposition and verification, ensembling, tool use, knowledge access, multimodality, agent loops, and LLM alignment regimes. Methods and systems are reviewed across domains, showing what each topology enables and where it breaks down in faithfulness or robustness, along with curated datasets, benchmarks, and resources that support study and deployment (https://github.com/blacksnail789521/Time-Series-Reasoning-Survey). Evaluation practices that keep evidence visible and temporally aligned are highlighted, and guidance is distilled on matching topology to uncertainty, grounding with observable artifacts, planning for shift and streaming, and treating cost and latency as design budgets. We emphasize that reasoning structures must balance capacity for grounding and self-correction against computational cost and reproducibility, while future progress will likely depend on benchmarks that tie reasoning quality to utility and on closed-loop testbeds that trade off cost and risk under shift-aware, streaming, and long-horizon settings. Taken together, these directions mark a shift from narrow accuracy toward reliability at scale, enabling systems that not only analyze but also understand, explain, and act on dynamic worlds with traceable evidence and credible outcomes.

Contrastive learning has shown to be effective to learn representations from time series in a self-supervised way. However, contrasting similar time series instances or values from adjacent timestamps within a time series leads to ignore their inherent correlations, which results in deteriorating the quality of learned representations. To address this issue, we propose SoftCLT, a simple yet effective soft contrastive learning strategy for time series. This is achieved by introducing instance-wise and temporal contrastive loss with soft assignments ranging from zero to one. Specifically, we define soft assignments for 1) instance-wise contrastive loss by the distance between time series on the data space, and 2) temporal contrastive loss by the difference of timestamps. SoftCLT is a plug-and-play method for time series contrastive learning that improves the quality of learned representations without bells and whistles. In experiments, we demonstrate that SoftCLT consistently improves the performance in various downstream tasks including classification, semi-supervised learning, transfer learning, and anomaly detection, showing state-of-the-art performance. Code is available at this repository: https://github.com/seunghan96/softclt.

Time series forecasting is an important and forefront task in many real-world applications. However, most of time series forecasting techniques assume that the training data is clean without anomalies. This assumption is unrealistic since the collected time series data can be contaminated in practice. The forecasting model will be inferior if it is directly trained by time series with anomalies. Thus it is essential to develop methods to automatically learn a robust forecasting model from the contaminated data. In this paper, we first statistically define three types of anomalies, then theoretically and experimentally analyze the loss robustness and sample robustness when these anomalies exist. Based on our analyses, we propose a simple and efficient algorithm to learn a robust forecasting model. Extensive experiments show that our method is highly robust and outperforms all existing approaches. The code is available at https://github.com/haochenglouis/RobustTSF.

Recently, denoising diffusion models have led to significant breakthroughs in the generation of images, audio and text. However, it is still an open question on how to adapt their strong modeling ability to model time series. In this paper, we propose TimeDiff, a non-autoregressive diffusion model that achieves high-quality time series prediction with the introduction of two novel conditioning mechanisms: future mixup and autoregressive initialization. Similar to teacher forcing, future mixup allows parts of the ground-truth future predictions for conditioning, while autoregressive initialization helps better initialize the model with basic time series patterns such as short-term trends. Extensive experiments are performed on nine real-world datasets. Results show that TimeDiff consistently outperforms existing time series diffusion models, and also achieves the best overall performance across a variety of the existing strong baselines (including transformers and FiLM).

Predicting the motion of multiple traffic participants has always been one of the most challenging tasks in autonomous driving. The recently proposed occupancy flow field prediction method has shown to be a more effective and scalable representation compared to general trajectory prediction methods. However, in complex multi-agent traffic scenarios, it remains difficult to model the interactions among various factors and the dependencies among prediction outputs at different time steps. In view of this, we propose a transformer-based hierarchical feature guided network (HGNET), which can efficiently extract features of agents and map information from visual and vectorized inputs, modeling multimodal interaction relationships. Second, we design the Feature-Guided Attention (FGAT) module to leverage the potential guiding effects between different prediction targets, thereby improving prediction accuracy. Additionally, to enhance the temporal consistency and causal relationships of the predictions, we propose a Time Series Memory framework to learn the conditional distribution models of the prediction outputs at future time steps from multivariate time series. The results demonstrate that our model exhibits competitive performance, which ranks 3rd in the 2024 Waymo Occupancy and Flow Prediction Challenge.

Recognizing that traditional forecasting models often rely solely on historical demand, this work investigates the potential of data-driven techniques to automatically select and integrate market indicators for improving customer demand predictions. By adopting an exploratory methodology, we integrate macroeconomic time series, such as national GDP growth, from the Eurostat database into Neural Prophet and SARIMAX forecasting models. Suitable time series are automatically identified through different state-of-the-art feature selection methods and applied to sales data from our industrial partner. It could be shown that forecasts can be significantly enhanced by incorporating external information. Notably, the potential of feature selection methods stands out, especially due to their capability for automation without expert knowledge and manual selection effort. In particular, the Forward Feature Selection technique consistently yielded superior forecasting accuracy for both SARIMAX and Neural Prophet across different company sales datasets. In the comparative analysis of the errors of the selected forecasting models, namely Neural Prophet and SARIMAX, it is observed that neither model demonstrates a significant superiority over the other.

Visualizing multiple time series presents fundamental tradeoffs between scalability and visual clarity. Time series capture the behavior of many large-scale real-world processes, from stock market trends to urban activities. Users often gain insights by visualizing them as line charts, juxtaposing or superposing multiple time series to compare them and identify trends and patterns. However, existing representations struggle with scalability: when covering long time spans, leading to visual clutter from too many small multiples or overlapping lines. We propose TiVy, a new algorithm that summarizes time series using sequential patterns. It transforms the series into a set of symbolic sequences based on subsequence visual similarity using Dynamic Time Warping (DTW), then constructs a disjoint grouping of similar subsequences based on the frequent sequential patterns. The grouping result, a visual summary of time series, provides uncluttered superposition with fewer small multiples. Unlike common clustering techniques, TiVy extracts similar subsequences (of varying lengths) aligned in time. We also present an interactive time series visualization that renders large-scale time series in real-time. Our experimental evaluation shows that our algorithm (1) extracts clear and accurate patterns when visualizing time series data, (2) achieves a significant speed-up (1000X) compared to a straightforward DTW clustering. We also demonstrate the efficiency of our approach to explore hidden structures in massive time series data in two usage scenarios.

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population means under the constraint of local differential privacy (LDP). Given bounded observations (X_1, dots, X_n) with mean mu^star that are privatized into (Z_1, dots, Z_n), we present confidence intervals (CI) and time-uniform confidence sequences (CS) for mu^star when only given access to the privatized data. To achieve this, we introduce a nonparametric and sequentially interactive generalization of Warner's famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. For example, our results yield private analogues of Hoeffding's inequality in both fixed-time and time-uniform regimes. We extend these Hoeffding-type CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.

In this study, we propose Feature-aligned N-BEATS as a domain generalization model for univariate time series forecasting problems. The proposed model is an extension of the doubly residual stacking architecture of N-BEATS (Oreshkin et al. [34]) into a representation learning framework. The model is a new structure that involves marginal feature probability measures (i.e., pushforward measures of multiple source domains) induced by the intricate composition of residual operators of N-BEATS in each stack and aligns them stack-wise via an entropic regularized Wasserstein distance referred to as the Sinkhorn divergence (Genevay et al. [14]). The loss function consists of a typical forecasting loss for multiple source domains and an alignment loss calculated with the Sinkhorn divergence, which allows the model to learn invariant features stack-wise across multiple source data sequences while retaining N-BEATS's interpretable design. We conduct a comprehensive experimental evaluation of the proposed approach and the results demonstrate the model's forecasting and generalization capabilities in comparison with methods based on the original N-BEATS.

We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate O(k^{-1/4}). As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is that all algorithms under consideration can be interpreted as approximate descent methods on an implicit smoothing of the problem, given by the Moreau envelope. Specializing to classical circumstances, we obtain the long-sought convergence rate of the stochastic projected gradient method, without batching, for minimizing a smooth function on a closed convex set.

In recent years, generative pre-trained paradigms such as Large Language Models (LLMs) and Large Vision Models (LVMs) have achieved revolutionary advancements and widespread real-world applications. Particularly, the emergence of pre-trained LLMs-based temporal works, compared to previous deep model approaches, has demonstrated superior generalization and robustness, showcasing the potential of generative pre-trained paradigms as foundation models for time series. However, those LLMs-based works mainly focus on cross-modal research, i.e., leveraging the language capabilities of LLMs in time series contexts. Although they have achieved impressive performance, there still exist the issues of concept drift caused by differences in data distribution and inflexibility caused by misalignment of dimensions. To this end, inspired by recent work on LVMs, we reconsider the paradigm of time series modeling. In this paper, we comprehensively explore, for the first time, the effectiveness and superiority of the Generative Pre-trained Diffusion (GPD) paradigm in real-world multivariate time series forecasting (TSF). Specifically, to mitigate performance bias introduced by sophisticated networks, we propose a straightforward MLP diffusion network for unconditional modeling of time series. Then we employ a zero-shot and tuning-free method to predict (generate) future data using historical data as prompts. The GPD paradigm is established on the time series modality, effectively preventing the phenomenon of concept drift, and enabling flexible forecasting of arbitrary lengths. We demonstrate that the GPD paradigm achieves comprehensive performance and generalization comparable to current SOTA LLM-based and deep model paradigms on mainstream benchmarks and various TSF tasks. Extensive experiments validate the potential of the GPD paradigm and its assistance in future related research.

We develop a novel, general and computationally efficient framework, called Divide and Conquer Dynamic Programming (DCDP), for localizing change points in time series data with high-dimensional features. DCDP deploys a class of greedy algorithms that are applicable to a broad variety of high-dimensional statistical models and can enjoy almost linear computational complexity. We investigate the performance of DCDP in three commonly studied change point settings in high dimensions: the mean model, the Gaussian graphical model, and the linear regression model. In all three cases, we derive non-asymptotic bounds for the accuracy of the DCDP change point estimators. We demonstrate that the DCDP procedures consistently estimate the change points with sharp, and in some cases, optimal rates while incurring significantly smaller computational costs than the best available algorithms. Our findings are supported by extensive numerical experiments on both synthetic and real data.

This paper presents TS2Vec, a universal framework for learning representations of time series in an arbitrary semantic level. Unlike existing methods, TS2Vec performs contrastive learning in a hierarchical way over augmented context views, which enables a robust contextual representation for each timestamp. Furthermore, to obtain the representation of an arbitrary sub-sequence in the time series, we can apply a simple aggregation over the representations of corresponding timestamps. We conduct extensive experiments on time series classification tasks to evaluate the quality of time series representations. As a result, TS2Vec achieves significant improvement over existing SOTAs of unsupervised time series representation on 125 UCR datasets and 29 UEA datasets. The learned timestamp-level representations also achieve superior results in time series forecasting and anomaly detection tasks. A linear regression trained on top of the learned representations outperforms previous SOTAs of time series forecasting. Furthermore, we present a simple way to apply the learned representations for unsupervised anomaly detection, which establishes SOTA results in the literature. The source code is publicly available at https://github.com/yuezhihan/ts2vec.

We study the problem of approximating stationary points of Lipschitz and smooth functions under (varepsilon,delta)-differential privacy (DP) in both the finite-sum and stochastic settings. A point w is called an alpha-stationary point of a function F:R^drightarrowR if |nabla F(w)|leq alpha. We provide a new efficient algorithm that finds an Obig(big[sqrt{d}{nvarepsilon}big]^{2/3}big)-stationary point in the finite-sum setting, where n is the number of samples. This improves on the previous best rate of Obig(big[sqrt{d}{nvarepsilon}big]^{1/2}big). We also give a new construction that improves over the existing rates in the stochastic optimization setting, where the goal is to find approximate stationary points of the population risk. Our construction finds a Obig(1{n^{1/3}} + big[sqrt{d}{nvarepsilon}big]^{1/2}big)-stationary point of the population risk in time linear in n. Furthermore, under the additional assumption of convexity, we completely characterize the sample complexity of finding stationary points of the population risk (up to polylog factors) and show that the optimal rate on population stationarity is tilde Thetabig(1{n}+sqrt{d}{nvarepsilon}big). Finally, we show that our methods can be used to provide dimension-independent rates of Obig(1{n}+minbig(big[sqrt{rank}{nvarepsilon}big]^{2/3},1{(nvarepsilon)^{2/5}}big)big) on population stationarity for Generalized Linear Models (GLM), where rank is the rank of the design matrix, which improves upon the previous best known rate.

Time series forecasting is central to decision-making in domains as diverse as energy, finance, climate, and public health. In practice, forecasters face thousands of short, noisy series that vary in frequency, quality, and horizon, where the dominant cost lies not in model fitting, but in the labor-intensive preprocessing, validation, and ensembling required to obtain reliable predictions. Prevailing statistical and deep learning models are tailored to specific datasets or domains and generalize poorly. A general, domain-agnostic framework that minimizes human intervention is urgently in demand. In this paper, we introduce TimeSeriesScientist (TSci), the first LLM-driven agentic framework for general time series forecasting. The framework comprises four specialized agents: Curator performs LLM-guided diagnostics augmented by external tools that reason over data statistics to choose targeted preprocessing; Planner narrows the hypothesis space of model choice by leveraging multi-modal diagnostics and self-planning over the input; Forecaster performs model fitting and validation and, based on the results, adaptively selects the best model configuration as well as ensemble strategy to make final predictions; and Reporter synthesizes the whole process into a comprehensive, transparent report. With transparent natural-language rationales and comprehensive reports, TSci transforms the forecasting workflow into a white-box system that is both interpretable and extensible across tasks. Empirical results on eight established benchmarks demonstrate that TSci consistently outperforms both statistical and LLM-based baselines, reducing forecast error by an average of 10.4% and 38.2%, respectively. Moreover, TSci produces a clear and rigorous report that makes the forecasting workflow more transparent and interpretable.

Many social and environmental phenomena are associated with macroscopic changes in the built environment, captured by satellite imagery on a global scale and with daily temporal resolution. While widely used for prediction, these images and especially image sequences remain underutilized for causal inference, especially in the context of randomized controlled trials (RCTs), where causal identification is established by design. In this paper, we develop and compare a set of general tools for analyzing Conditional Average Treatment Effects (CATEs) from temporal satellite data that can be applied to any RCT where geographical identifiers are available. Through a simulation study, we analyze different modeling strategies for estimating CATE in sequences of satellite images. We find that image sequence representation models with more parameters generally yield a greater ability to detect heterogeneity. To explore the role of model and data choice in practice, we apply the approaches to two influential RCTs -- Banerjee et al. (2015), a poverty study in Cusco, Peru, and Bolsen et al. (2014), a water conservation experiment in Georgia, USA. We benchmark our image sequence models against image-only, tabular-only, and combined image-tabular data sources, summarizing practical implications for investigators in a multivariate analysis. Land cover classifications over satellite images facilitate interpretation of what image features drive heterogeneity. We also show robustness to data and model choice of satellite-based generalization of the RCT results to larger geographical areas outside the original. Overall, this paper shows how satellite sequence data can be incorporated into the analysis of RCTs, and provides evidence about the implications of data, model, and evaluation metric choice for causal analysis.

This paper studies sequence modeling for prediction tasks with long range dependencies. We propose a new formulation for state space models (SSMs) based on learning linear dynamical systems with the spectral filtering algorithm (Hazan et al. (2017)). This gives rise to a novel sequence prediction architecture we call a spectral state space model. Spectral state space models have two primary advantages. First, they have provable robustness properties as their performance depends on neither the spectrum of the underlying dynamics nor the dimensionality of the problem. Second, these models are constructed with fixed convolutional filters that do not require learning while still outperforming SSMs in both theory and practice. The resulting models are evaluated on synthetic dynamical systems and long-range prediction tasks of various modalities. These evaluations support the theoretical benefits of spectral filtering for tasks requiring very long range memory.

Time series data in real-world applications such as healthcare, climate modeling, and finance are often irregular, multimodal, and messy, with varying sampling rates, asynchronous modalities, and pervasive missingness. However, existing benchmarks typically assume clean, regularly sampled, unimodal data, creating a significant gap between research and real-world deployment. We introduce Time-IMM, a dataset specifically designed to capture cause-driven irregularity in multimodal multivariate time series. Time-IMM represents nine distinct types of time series irregularity, categorized into trigger-based, constraint-based, and artifact-based mechanisms. Complementing the dataset, we introduce IMM-TSF, a benchmark library for forecasting on irregular multimodal time series, enabling asynchronous integration and realistic evaluation. IMM-TSF includes specialized fusion modules, including a timestamp-to-text fusion module and a multimodality fusion module, which support both recency-aware averaging and attention-based integration strategies. Empirical results demonstrate that explicitly modeling multimodality on irregular time series data leads to substantial gains in forecasting performance. Time-IMM and IMM-TSF provide a foundation for advancing time series analysis under real-world conditions. The dataset is publicly available at https://github.com/blacksnail789521/Time-IMM, and the benchmark library can be accessed at https://github.com/blacksnail789521/IMM-TSF. Project page: https://blacksnail789521.github.io/time-imm-project-page/

Data scarcity in the brain-computer interface field can be alleviated through the use of generative models, specifically diffusion models. While diffusion models have previously been successfully applied to electroencephalogram (EEG) data, existing models lack flexibility w.r.t.~sampling or require alternative representations of the EEG data. To overcome these limitations, we introduce a novel approach to conditional diffusion models that utilizes classifier-free guidance to directly generate subject-, session-, and class-specific EEG data. In addition to commonly used metrics, domain-specific metrics are employed to evaluate the specificity of the generated samples. The results indicate that the proposed model can generate EEG data that resembles real data for each subject, session, and class.

Recent advances in deep learning have driven rapid progress in time series forecasting, yet many state-of-the-art models continue to struggle with robust performance in real-world applications, even when they achieve strong results on standard benchmark datasets. This persistent gap can be attributed to the black-box nature of deep learning architectures and the inherent limitations of current evaluation frameworks, which frequently lack the capacity to provide clear, quantitative insights into the specific strengths and weaknesses of different models, thereby complicating the selection of appropriate models for particular forecasting scenarios. To address these issues, we propose a synthetic data-driven evaluation paradigm, SynTSBench, that systematically assesses fundamental modeling capabilities of time series forecasting models through programmable feature configuration. Our framework isolates confounding factors and establishes an interpretable evaluation system with three core analytical dimensions: (1) temporal feature decomposition and capability mapping, which enables systematic evaluation of model capacities to learn specific pattern types; (2) robustness analysis under data irregularities, which quantifies noise tolerance thresholds and anomaly recovery capabilities; and (3) theoretical optimum benchmarking, which establishes performance boundaries for each pattern type-enabling direct comparison between model predictions and mathematical optima. Our experiments show that current deep learning models do not universally approach optimal baselines across all types of temporal features.The code is available at https://github.com/TanQitai/SynTSBench

Moments when a time series changes its behaviour are called change points. Detection of such points is a well-known problem, which can be found in many applications: quality monitoring of industrial processes, failure detection in complex systems, health monitoring, speech recognition and video analysis. Occurrence of change point implies that the state of the system is altered and its timely detection might help to prevent unwanted consequences. In this paper, we present two online change-point detection approaches based on neural networks. These algorithms demonstrate linear computational complexity and are suitable for change-point detection in large time series. We compare them with the best known algorithms on various synthetic and real world data sets. Experiments show that the proposed methods outperform known approaches.

In this work, we present a variety of novel information-theoretic generalization bounds for learning algorithms, from the supersample setting of Steinke & Zakynthinou (2020)-the setting of the "conditional mutual information" framework. Our development exploits projecting the loss pair (obtained from a training instance and a testing instance) down to a single number and correlating loss values with a Rademacher sequence (and its shifted variants). The presented bounds include square-root bounds, fast-rate bounds, including those based on variance and sharpness, and bounds for interpolating algorithms etc. We show theoretically or empirically that these bounds are tighter than all information-theoretic bounds known to date on the same supersample setting.

Time series classification (TSC) on multivariate time series is a critical problem. We propose a novel multi-view approach integrating frequency-domain and time-domain features to provide complementary contexts for TSC. Our method fuses continuous wavelet transform spectral features with temporal convolutional or multilayer perceptron features. We leverage the Mamba state space model for efficient and scalable sequence modeling. We also introduce a novel tango scanning scheme to better model sequence relationships. Experiments on 10 standard benchmark datasets demonstrate our approach achieves an average 6.45% accuracy improvement over state-of-the-art TSC models.

We study nonparametric density estimation in non-stationary drift settings. Given a sequence of independent samples taken from a distribution that gradually changes in time, the goal is to compute the best estimate for the current distribution. We prove tight minimax risk bounds for both discrete and continuous smooth densities, where the minimum is over all possible estimates and the maximum is over all possible distributions that satisfy the drift constraints. Our technique handles a broad class of drift models, and generalizes previous results on agnostic learning under drift.

Multivariate time series classification is a crucial task in data mining, attracting growing research interest due to its broad applications. While many existing methods focus on discovering discriminative patterns in time series, real-world data does not always present such patterns, and sometimes raw numerical values can also serve as discriminative features. Additionally, the recent success of Transformer models has inspired many studies. However, when applying to time series classification, the self-attention mechanisms in Transformer models could introduce classification-irrelevant features, thereby compromising accuracy. To address these challenges, we propose a novel method, VSFormer, that incorporates both discriminative patterns (shape) and numerical information (value). In addition, we extract class-specific prior information derived from supervised information to enrich the positional encoding and provide classification-oriented self-attention learning, thereby enhancing its effectiveness. Extensive experiments on all 30 UEA archived datasets demonstrate the superior performance of our method compared to SOTA models. Through ablation studies, we demonstrate the effectiveness of the improved encoding layer and the proposed self-attention mechanism. Finally, We provide a case study on a real-world time series dataset without discriminative patterns to interpret our model.

The recent boom of linear forecasting models questions the ongoing passion for architectural modifications of Transformer-based forecasters. These forecasters leverage Transformers to model the global dependencies over temporal tokens of time series, with each token formed by multiple variates of the same timestamp. However, Transformers are challenged in forecasting series with larger lookback windows due to performance degradation and computation explosion. Besides, the embedding for each temporal token fuses multiple variates that represent potential delayed events and distinct physical measurements, which may fail in learning variate-centric representations and result in meaningless attention maps. In this work, we reflect on the competent duties of Transformer components and repurpose the Transformer architecture without any modification to the basic components. We propose iTransformer that simply applies the attention and feed-forward network on the inverted dimensions. Specifically, the time points of individual series are embedded into variate tokens which are utilized by the attention mechanism to capture multivariate correlations; meanwhile, the feed-forward network is applied for each variate token to learn nonlinear representations. The iTransformer model achieves state-of-the-art on challenging real-world datasets, which further empowers the Transformer family with promoted performance, generalization ability across different variates, and better utilization of arbitrary lookback windows, making it a nice alternative as the fundamental backbone of time series forecasting. Code is available at this repository: https://github.com/thuml/iTransformer.

Generative modeling of time series is a central challenge in time series analysis, particularly under data-scarce conditions. Despite recent advances in generative modeling, a comprehensive understanding of how state-of-the-art generative models perform under limited supervision remains lacking. In this work, we conduct the first large-scale study evaluating leading generative models in data-scarce settings, revealing a substantial performance gap between full-data and data-scarce regimes. To close this gap, we propose a unified diffusion-based generative framework that can synthesize high-fidelity time series across diverse domains using just a few examples. Our model is pre-trained on a large, heterogeneous collection of time series datasets, enabling it to learn generalizable temporal representations. It further incorporates architectural innovations such as dynamic convolutional layers for flexible channel adaptation and dataset token conditioning for domain-aware generation. Without requiring abundant supervision, our unified model achieves state-of-the-art performance in few-shot settings-outperforming domain-specific baselines across a wide range of subset sizes. Remarkably, it also surpasses all baselines even when tested on full datasets benchmarks, highlighting the strength of pre-training and cross-domain generalization. We hope this work encourages the community to revisit few-shot generative modeling as a key problem in time series research and pursue unified solutions that scale efficiently across domains. Code is available at https://github.com/azencot-group/ImagenFew.

The SIML (abbreviation of Separating Information Maximal Likelihood) method, has been introduced by N. Kunitomo and S. Sato and their collaborators to estimate the integrated volatility of high-frequency data that is assumed to be an It\^o process but with so-called microstructure noise. The SIML estimator turned out to share many properties with the estimator introduced by P. Malliavin and M.E. Mancino. The present paper establishes the consistency and the asymptotic normality under a general sampling scheme but without microstructure noise. Specifically, a fast convergence shown for Malliavin--Mancino estimator by E. Clement and A. Gloter is also established for the SIML estimator.

This article presents Individual Conditional Expectation (ICE) plots, a tool for visualizing the model estimated by any supervised learning algorithm. Classical partial dependence plots (PDPs) help visualize the average partial relationship between the predicted response and one or more features. In the presence of substantial interaction effects, the partial response relationship can be heterogeneous. Thus, an average curve, such as the PDP, can obfuscate the complexity of the modeled relationship. Accordingly, ICE plots refine the partial dependence plot by graphing the functional relationship between the predicted response and the feature for individual observations. Specifically, ICE plots highlight the variation in the fitted values across the range of a covariate, suggesting where and to what extent heterogeneities might exist. In addition to providing a plotting suite for exploratory analysis, we include a visual test for additive structure in the data generating model. Through simulated examples and real data sets, we demonstrate how ICE plots can shed light on estimated models in ways PDPs cannot. Procedures outlined are available in the R package ICEbox.

Multivariate time series data for real-world applications typically contain a significant amount of missing values. The dominant approach for classification with such missing values is to impute them heuristically with specific values (zero, mean, values of adjacent time-steps) or learnable parameters. However, these simple strategies do not take the data generative process into account, and more importantly, do not effectively capture the uncertainty in prediction due to the multiple possibilities for the missing values. In this paper, we propose a novel probabilistic framework for classification with multivariate time series data with missing values. Our model consists of two parts; a deep generative model for missing value imputation and a classifier. Extending the existing deep generative models to better capture structures of time-series data, our deep generative model part is trained to impute the missing values in multiple plausible ways, effectively modeling the uncertainty of the imputation. The classifier part takes the time series data along with the imputed missing values and classifies signals, and is trained to capture the predictive uncertainty due to the multiple possibilities of imputations. Importantly, we show that na\"ively combining the generative model and the classifier could result in trivial solutions where the generative model does not produce meaningful imputations. To resolve this, we present a novel regularization technique that can promote the model to produce useful imputation values that help classification. Through extensive experiments on real-world time series data with missing values, we demonstrate the effectiveness of our method.

Spatiotemporal forecasting techniques are significant for various domains such as transportation, energy, and weather. Accurate prediction of spatiotemporal series remains challenging due to the complex spatiotemporal heterogeneity. In particular, current end-to-end models are limited by input length and thus often fall into spatiotemporal mirage, i.e., similar input time series followed by dissimilar future values and vice versa. To address these problems, we propose a novel self-supervised pre-training framework Spatial-Temporal-Decoupled Masked Pre-training (STD-MAE) that employs two decoupled masked autoencoders to reconstruct spatiotemporal series along the spatial and temporal dimensions. Rich-context representations learned through such reconstruction could be seamlessly integrated by downstream predictors with arbitrary architectures to augment their performances. A series of quantitative and qualitative evaluations on six widely used benchmarks (PEMS03, PEMS04, PEMS07, PEMS08, METR-LA, and PEMS-BAY) are conducted to validate the state-of-the-art performance of STD-MAE. Codes are available at https://github.com/Jimmy-7664/STD-MAE.