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Applications · Time Series

Benjamin Walker, Andrew McLeod, Tiexin QIN, Yichuan Cheng, Haoliang Li, Terry Lyons

The vector field of a controlled differential equation (CDE) describes the relationship between a *control* path and the evolution of a *solution* path. Neural CDEs (NCDEs) treat time series data as observations from a control path, parameterise a CDE's vector field using a neural network, and use the solution path as a continuously evolving hidden state. As their formulation makes them robust to irregular sampling rates, NCDEs are a powerful approach for modelling real-world data. Building on neural rough differential equations (NRDEs), we introduce Log-NCDEs, a novel, effective, and efficient method for training NCDEs. The core component of Log-NCDEs is the Log-ODE method, a tool from the study of rough paths for approximating a CDE's solution. Log-NCDEs are shown to outperform NCDEs, NRDEs, the linear recurrent unit, S5, and MAMBA on a range of multivariate time series datasets with up to $50{,}000$ observations.

Applications · Time Series

M. Saquib Sarfraz, Mei-Yen Chen, Lukas Layer, Kunyu Peng, Marios Koulakis

The current state of machine learning scholarship in Timeseries Anomaly Detection (TAD) is plagued by the persistent use of flawed evaluation metrics, inconsistent benchmarking practices, and a lack of proper justification for the choices made in novel deep learning-based model designs. Our paper presents a critical analysis of the status quo in TAD, revealing the misleading track of current research and highlighting problematic methods, and evaluation practices. ***Our position advocates for a shift in focus from solely pursuing novel model designs to improving benchmarking practices, creating non-trivial datasets, and critically evaluating the utility of complex methods against simpler baselines***. Our findings demonstrate the need for rigorous evaluation protocols, the creation of simple baselines, and the revelation that state-of-the-art deep anomaly detection models effectively learn linear mappings. These findings suggest the need for more exploration and development of simple and interpretable TAD methods. The increment of model complexity in the state-of-the-art deep-learning based models unfortunately offers very little improvement. We offer insights and suggestions for the field to move forward.

Applications · Time Series

Yanfei Zhou, Lars Lindemann, Matteo Sesia

This paper presents a new conformal method for generating *simultaneous* forecasting bands guaranteed to cover the *entire path* of a new random trajectory with sufficiently high probability. Prompted by the need for dependable uncertainty estimates in motion planning applications where the behavior of diverse objects may be more or less unpredictable, we blend different techniques from online conformal prediction of single and multiple time series, as well as ideas for addressing heteroscedasticity in regression. This solution is both principled, providing precise finite-sample guarantees, and effective, often leading to more informative predictions than prior methods.

Deep Learning · Sequential Models, Time series

Andrea Cini, Danilo Mandic, Cesare Alippi

Relationships among time series can be exploited as inductive biases in learning effective forecasting models. In hierarchical time series, relationships among subsets of sequences induce hard constraints (hierarchical inductive biases) on the predicted values. In this paper, we propose a graph-based methodology to unify relational and hierarchical inductive biases in the context of deep learning for time series forecasting. In particular, we model both types of relationships as dependencies in a pyramidal graph structure, with each pyramidal layer corresponding to a level of the hierarchy. By exploiting modern - trainable - graph pooling operators we show that the hierarchical structure, if not available as a prior, can be learned directly from data, thus obtaining cluster assignments aligned with the forecasting objective. A differentiable reconciliation stage is incorporated into the processing architecture, allowing hierarchical constraints to act both as an architectural bias as well as a regularization element for predictions. Simulation results on representative datasets show that the proposed method compares favorably against the state of the art.

Applications · Neuroscience, Cognitive Science

Ali Behrouz, Parsa Delavari, Farnoosh Hashemi

Effective brain representation learning is a key step toward the understanding of cognitive processes and diagnosis of neurological diseases/disorders. Existing studies have focused on either (1) voxel-level activity, where only a single weight relating the voxel activity to the task (i.e., aggregation of voxel activity over a time window) is considered, missing their temporal dynamics, or (2) functional connectivity of the brain in the level of region of interests, missing voxel-level activities. We bridge this gap and design BrainMixer, an unsupervised learning framework that effectively utilizes both functional connectivity and associated time series of voxels to learn voxel-level representation in an unsupervised manner. BrainMixer employs two simple yet effective MLP-based encoders to simultaneously learn the dynamics of voxel-level signals and their functional correlations. To encode voxel activity, BrainMixer fuses information across both time and voxel dimensions via a dynamic attention mechanism. To learn the structure of the functional connectivity, BrainMixer presents a temporal graph patching and encodes each patch by combining its nodes' features via a new adaptive temporal pooling. Our experiments show that BrainMixer attains outstanding performance and outperforms 14 baselines in different downstream tasks and setups.

Deep Learning · Sequential Models, Time series

Shuai Zhang, Chuan Zhou, Yang Liu, PENG ZHANG, Xixun Lin, Zhiming Ma

We present a novel perspective on temporal point processes (TPPs) by reformulating their intensity processes as solutions to stochastic differential equations (SDEs). In particular, we first prove the equivalent SDE formulations of several classical TPPs, including Poisson processes, Hawkes processes, and self-correcting processes. Based on these proofs, we introduce a unified TPP framework called Neural Jump-Diffusion Temporal Point Process (NJDTPP), whose intensity process is governed by a neural jump-diffusion SDE (NJDSDE) where the drift, diffusion, and jump coefficient functions are parameterized by neural networks. Compared to previous works, NJDTPP exhibits model flexibility in capturing intensity dynamics without relying on any specific functional form, and provides theoretical guarantees regarding the existence and uniqueness of the solution to the proposed NJDSDE. Experiments on both synthetic and real-world datasets demonstrate that NJDTPP is capable of capturing the dynamics of intensity processes in different scenarios and significantly outperforms the state-of-the-art TPP models in prediction tasks.

Weizhong Zhang, Zhiwei Zhang, Renjie Pi, Zhongming Jin, Yuan Gao, Jieping Ye, Kani Chen

Diffusion models have exhibited remarkable advancements in generating high-quality data. However, a critical drawback is their computationally intensive inference process, which requires a large number of timesteps to generate a single sample. Existing methods address this challenge by decoupling the forward and reverse processes, and they rely on handcrafted rules for sampling acceleration, leading to the risk of discarding important steps. In this paper, we propose an Efficient Denoising Diffusion method via Probabilistic Masking (EDDPM) that can identify and skip the redundant steps during training. To determine whether a timestep should be skipped or not, we employ probabilistic reparameterization to continualize the binary determination mask. The mask distribution parameters are learned jointly with model weights. By incorporating a real-time sparse constraint, our method can effectively identify and eliminate unnecessary steps during the training iterations, thereby improving inference efficiency. Notably, as the model becomes fully trained, the random masks converge to a sparse and deterministic one, retaining only a small number of essential steps. Empirical results demonstrate the superiority of our proposed EDDPM over the state-of-the-art sampling acceleration methods across various domains. EDDPM can generate high-quality samples with only 20% of the steps for time series imputation and achieve 4.89 FID with 5 steps for CIFAR-10. Moreover, when starting from a pretrained model, our method efficiently identifies the most informative timesteps within a single epoch, which demonstrates the potential of EDDPM to be a practical tool to explore large diffusion models with limited resources.

Miscellaneous Aspects of Machine Learning · Causality

Kun Zhang, Shaoan Xie, Ignavier Ng, Yujia Zheng

In many problems, the measured variables (e.g., image pixels) are just mathematical functions of the latent causal variables (e.g., the underlying concepts or objects). For the purpose of making predictions in changing environments or making proper changes to the system, it is helpful to recover the latent causal variables $Z_i$ and their causal relations represented by graph $\mathcal{G}_Z$. This problem has recently been known as causal representation learning. This paper is concerned with a general, completely nonparametric setting of causal representation learning from multiple distributions (arising from heterogeneous data or nonstationary time series), without assuming hard interventions behind distribution changes. We aim to develop general solutions in this fundamental case; as a by product, this helps see the unique benefit offered by other assumptions such as parametric causal models or hard interventions. We show that under the sparsity constraint on the recovered graph over the latent variables and suitable sufficient change conditions on the causal influences, interestingly, one can recover the moralized graph of the underlying directed acyclic graph, and the recovered latent variables and their relations are related to the underlying causal model in a specific, nontrivial way. In some cases, most latent variables can even be recovered up to component-wise transformations. Experimental results verify our theoretical claims.

Applications · Time Series

Yunhao Zhang, Liu Minghao, Shengyang Zhou, Junchi Yan

Despite the success of self-supervised pre-training in texts and images, applying it to multivariate time series (MTS) falls behind tailored methods for tasks like forecasting, imputation and anomaly detection. We propose a general-purpose framework, named UP2ME (**U**nivariate **P**re-training to **M**ultivariate Fin**e**-tuning). It conducts task-agnostic pre-training when downstream tasks are unspecified. Once the task and setting (e.g. forecasting length) are determined, it gives sensible solutions with frozen pre-trained parameters, which has not been achieved before. UP2ME is further refined by fine-tuning. A univariate-to-multivariate paradigm is devised to address the heterogeneity of temporal and cross-channel dependencies. In univariate pre-training, univariate instances with diverse lengths are generated for Masked AutoEncoder (MAE) pre-training, discarding cross-channel dependency. The pre-trained model handles downstream tasks by formulating them into specific mask-reconstruction problems. In multivariate fine-tuning, it constructs a dependency graph among channels using the pre-trained encoder to enhance cross-channel dependency capture. Experiments on eight real-world datasets show its SOTA performance in forecasting and imputation, approaching task-specific performance in anomaly detection. Our code is available at https://github.com/Thinklab-SJTU/UP2ME.

Deep Learning · Sequential Models, Time series

Yang Yang, Chao Yang, Boyang Li, Yinghao Fu, Shuang Li

Our goal is to $\textit{efficiently}$ discover a compact set of temporal logic rules to explain irregular events of interest. We introduce a neural-symbolic rule induction framework within the temporal point process model. The negative log-likelihood is the loss that guides the learning, where the explanatory logic rules and their weights are learned end-to-end in a $\textit{differentiable}$ way. Specifically, predicates and logic rules are represented as $\textit{vector embeddings}$, where the predicate embeddings are fixed and the rule embeddings are trained via gradient descent to obtain the most appropriate compositional representations of the predicate embeddings. To make the rule learning process more efficient and flexible, we adopt a $\textit{sequential covering algorithm}$, which progressively adds rules to the model and removes the event sequences that have been explained until all event sequences have been covered. All the found rules will be fed back to the models for a final rule embedding and weight refinement. Our approach showcases notable efficiency and accuracy across synthetic and real datasets, surpassing state-of-the-art baselines by a wide margin in terms of efficiency.

Linxiao Yang, Yunze Tong, Xinyue Gu, Liang Sun

How to explain temporal models is a significant challenge due to the inherent characteristics of time series data, notably the strong temporal dependencies and interactions between observations. Unlike ordinary tabular data, data at different time steps in time series usually interact dynamically, forming influential patterns that shape the model’s predictions, rather than only acting in isolation. Existing explanatory approaches for time series often overlook these crucial temporal interactions by treating time steps as separate entities, leading to a superficial understanding of model behavior. To address this challenge, we introduce FDTempExplainer, an innovative model-agnostic explanation method based on functional decomposition, tailored to unravel the complex interplay within black-box time series models. Our approach disentangles the individual contributions from each time step, as well as the aggregated influence of their interactions, in a rigorous framework. FDTempExplainer accurately measures the strength of interactions, yielding insights that surpass those from baseline models. We demonstrate the effectiveness of our approach in a wide range of time series applications, including anomaly detection, classification, and forecasting, showing its superior performance to the state-of-the-art algorithms.

Applications · Time Series

Tijin Yan, Hengheng Gong, Yongping He, Yufeng Zhan, Yuanqing Xia

Probabilistic time series modeling based on generative models has attracted lots of attention because of its wide applications and excellent performance. However, existing state-of-the-art models, based on stochastic differential equation, not only struggle to determine the drift and diffusion coefficients during the design process but also have slow generation speed. To tackle this challenge, we firstly propose decomposable denoising diffusion model ($\text{D}^3\text{M}$) and prove it is a general framework unifying denoising diffusion models and continuous flow models. Based on the new framework, we propose some simple but efficient probability paths with high generation speed. Furthermore, we design a module that combines a special state space model with linear gated attention modules for sequence modeling. It preserves inductive bias and simultaneously models both local and global dependencies. Experimental results on 8 real-world datasets show that $\text{D}^3\text{M}$ reduces RMSE and CRPS by up to 4.6% and 4.3% compared with state-of-the-arts on imputation tasks, and achieves comparable results with state-of-the-arts on forecasting tasks with only 10 steps.

Miscellaneous Aspects of Machine Learning · Causality

Anpeng Wu, Haoxuan Li, Kun Kuang, zhang keli, Fei Wu

This paper studies the causal relations from subsampled time series, in which measurements are sparse and sampled at a coarser timescale than the causal timescale of the underlying system. In such data, because there are numerous missing time-slices (i.e., cross-sections at each time point) between two consecutive measurements, conventional causal discovery methods designed for standard time series data would produce significant errors. To learn causal relations from subsampled time series, a typical solution is to conduct different interventions and then make a comparison. However, full interventions are often expensive, unethical, or even infeasible, particularly in fields such as health and social science. In this paper, we first explore how readily available two-time-slices data can replace intervention data to improve causal ordering, and propose a novel Descendant Hierarchical Topology algorithm with Conditional Independence Test (DHT-CIT) to learn causal relations from subsampled time series using only two time-slices. Specifically, we develop a conditional independence criterion that can be applied iteratively to test each node from time series and identify all of its descendant nodes. Empirical results on both synthetic and real-world datasets demonstrate the superiority of our DHT-CIT algorithm.

Deep Learning · Sequential Models, Time series

Gerald Woo, Chenghao Liu, Akshat Kumar, Caiming Xiong, Silvio Savarese, Doyen Sahoo

Deep learning for time series forecasting has traditionally operated within a one-model-per-dataset framework, limiting its potential to leverage the game-changing impact of large pre-trained models. The concept of *universal forecasting*, emerging from pre-training on a vast collection of time series datasets, envisions a single Large Time Series Model capable of addressing diverse downstream forecasting tasks. However, constructing such a model poses unique challenges specific to time series data: (i) cross-frequency learning, (ii) accommodating an arbitrary number of variates for multivariate time series, and (iii) addressing the varying distributional properties inherent in large-scale data. To address these challenges, we present novel enhancements to the conventional time series Transformer architecture, resulting in our proposed **M**asked Enc**o**der-based Un**i**ve**r**s**a**l T**i**me Series Forecasting Transformer (**Moirai**). Trained on our newly introduced Large-scale Open Time Series Archive (LOTSA) featuring over 27B observations across nine domains, Moirai achieves competitive or superior performance as a zero-shot forecaster when compared to full-shot models. Code, data, and model weights can be found at https://github.com/SalesforceAIResearch/uni2ts.

Deep Learning · Sequential Models, Time series

Theodor Westny, Arman Mohammadi, Daniel Jung, Erik Frisk

This paper addresses the training of Neural Ordinary Differential Equations (neural ODEs), and in particular explores the interplay between numerical integration techniques, stability regions, step size, and initialization techniques. It is shown how the choice of integration technique implicitly regularizes the learned model, and how the solver's corresponding stability region affects training and prediction performance. From this analysis, a stability-informed parameter initialization technique is introduced. The effectiveness of the initialization method is displayed across several learning benchmarks and industrial applications.

Theory · Optimization

Yafei Wang, Bo Pan, Mei Li, Jianya Lu, Lingchen Kong, Bei Jiang, Linglong Kong

Conditional Stochastic Optimization (CSO) is a powerful modelling paradigm for optimization under uncertainty. The existing literature on CSO is mainly based on the independence assumption of data, which shows that the solution of CSO is asymptotically consistent and enjoys a finite sample guarantee. The independence assumption, however, does not typically hold in many important applications with dependence patterns, such as time series analysis, operational control, and reinforcement learning. In this paper, we aim to fill this gap and consider a Sample Average Approximation (SAA) for CSO with dependent data. Leveraging covariance inequalities and independent block sampling technique, we provide theoretical guarantees of SAA for CSO with dependent data. In particular, we show that SAA for CSO retains asymptotic consistency and a finite sample guarantee under mild conditions. In addition, we establish the sample complexity $O(d / \varepsilon^4)$ of SAA for CSO, which is shown to be of the same order as independent cases. Through experiments on several applications, we verify the theoretical results and demonstrate that dependence does not degrade the performance of the SAA approach in real data applications.

Deep Learning · Sequential Models, Time series

Shida Wang, Qianxiao Li

In this paper, we investigate the long-term memory learning capabilities of state-space models (SSMs) from the perspective of parameterization. We prove that state-space models without any reparameterization exhibit a memory limitation similar to that of traditional RNNs: the target relationships that can be stably approximated by state-space models must have an exponential decaying memory. Our analysis identifies this ``curse of memory'' as a result of the recurrent weights converging to a stability boundary, suggesting that a reparameterization technique can be effective. To this end, we introduce a class of reparameterization techniques for SSMs that effectively lift its memory limitations. Besides improving approximation capabilities, we further illustrate that a principled choice of reparameterization scheme can also enhance optimization stability. We validate our findings using synthetic datasets, language models and image classifications.

Sumanth Varambally, Yian Ma, Rose Yu

Discovering causal relationships from time series data is significant in fields such as finance, climate science, and neuroscience. However, contemporary techniques rely on the simplifying assumption that data originates from the same causal model, while in practice, data is heterogeneous and can stem from different causal models. In this work, we relax this assumption and perform causal discovery from time series data originating from *a mixture of causal models*. We propose a general variational inference-based framework called MCD to infer the underlying causal models as well as the mixing probability of each sample. Our approach employs an end-to-end training process that maximizes an evidence-lower bound for the data likelihood. We present two variants: MCD-Linear for linear relationships and independent noise, and MCD-Nonlinear for nonlinear causal relationships and history-dependent noise. We demonstrate that our method surpasses state-of-the-art benchmarks in causal discovery tasks through extensive experimentation on synthetic and real-world datasets, particularly when the data emanates from diverse underlying causal graphs. Theoretically, we prove the identifiability of such a model under some mild assumptions. Implementation is available at [https://github.com/Rose-STL-Lab/MCD](https://github.com/Rose-STL-Lab/MCD).

Deep Learning · Sequential Models, Time series

Maksim Sinelnikov, Manuel Haussmann, Harri Lähdesmäki

Longitudinal data are important in numerous fields, such as healthcare, sociology and seismology, but real-world datasets present notable challenges for practitioners because they can be high-dimensional, contain structured missingness patterns, and measurement time points can be governed by an unknown stochastic process. While various solutions have been suggested, the majority of them have been designed to account for only one of these challenges. In this work, we propose a flexible and efficient latent-variable model that is capable of addressing all these limitations. Our approach utilizes Gaussian processes to capture correlations between samples and their associated missingness masks as well as to model the underlying point process. We construct our model as a variational autoencoder together with deep neural network parameterised decoder and encoder models, and develop a scalable amortised variational inference approach for efficient model training. We demonstrate competitive performance using both simulated and real datasets.

Miscellaneous Aspects of Machine Learning · Sequential, Network, and Time Series Modeling

Shubhanshu Shekhar, Aaditya Ramdas

We consider the problem of sequential change detection under minimal assumptions on the distribution generating the stream of observations. Formally, our goal is to design a scheme for detecting any changes in a parameter or functional $\theta$ of the data stream distribution that has small detection delay, but guarantees control on the frequency of false alarms in the absence of changes. We describe a simple reduction from sequential change detection to sequential estimation using confidence sequences (CSs): begin a new level-$(1-\alpha)$ CS at each time step, and proclaim a change as soon as the intersection of all active CSs becomes empty. We prove that the average run length of our scheme is at least $1/\alpha$, resulting in a change detection scheme with minimal structural assumptions (thus allowing for possibly dependent observations, and nonparametric distribution classes), but strong guarantees. We also describe an interesting parallel with Lorden's reduction from change detection to sequential testing and connections to the recent ''e-detector'' framework.