Many forms of sensitive data, such as web traffic, mobility data, or hospital occupancy, are inherently sequential. The standard method for training machine learning models while ensuring privacy for units of sensitive information, such as individual hospital visits, is differentially private stochastic gradient descent (DP-SGD). However, we observe in this work that the formal guarantees of DP-SGD are incompatible with time series specific tasks like forecasting, since they rely on the *privacy amplification* attained by training on small, unstructured batches sampled from an unstructured dataset. In contrast, batches for forecasting are generated by (1) sampling sequentially structured time series from a dataset, (2) sampling contiguous subsequences from these series, and (3) partitioning them into context and ground-truth forecast windows. We theoretically analyze the privacy amplification attained by this *structured subsampling* to enable the training of forecasting models with sound and tight event- and user-level privacy guarantees. Towards more private models, we additionally prove how data augmentation amplifies privacy in self-supervised training of sequence models. Our empirical evaluation demonstrates that amplification by structured subsampling enables the training of forecasting models with strong formal privacy guarantees.

Multi-modal time series analysis has recently emerged as a prominent research area, driven by the increasing availability of diverse data modalities, such as text, images, and structured tabular data from real-world sources. However, effective analysis of multi-modal time series is hindered by data heterogeneity, modality gap, misalignment, and inherent noise. Recent advancements in multi-modal time series methods have exploited the multi-modal context via cross-modal interactions based on deep learning methods, significantly enhancing various downstream tasks. In this tutorial and survey, we present a systematic and up-to-date overview of multi-modal time series datasets and methods. We first state the existing challenges of multi-modal time series analysis and our motivations, with a brief introduction of preliminaries. Then, we summarize the general pipeline and categorize existing methods through a unified cross-modal interaction framework encompassing fusion, alignment, and transference at different levels (i.e., input, intermediate, output), where key concepts and ideas are highlighted. We also discuss the real-world applications of multi-modal analysis for both standard and spatial time series, tailored to general and specific domains. Finally, we discuss future research directions to help practitioners explore and exploit multi-modal time series. The up-to-date resources are provided in the GitHub repository. https://github.com/UConn-DSIS/Multi-modal-Time-Series-Analysis.

Large Language Models (LLMs) and Foundation Models (FMs) have recently become prevalent for time series forecasting tasks. While fine-tuning LLMs enables domain adaptation, they often struggle to generalize across diverse and unseen datasets. Moreover, existing Time Series Foundation Models (TSFMs) still face challenges in handling non-stationary dynamics and distribution shifts, largely due to the lack of effective mechanisms for adaptation. To this end, we present TS-RAG, a retrieval-augmented generation framework for time series forecasting that enhances the generalization and interpretability of TSFMs. Specifically, TS-RAG leverages pre-trained time series encoders to retrieve semantically relevant segments from a dedicated knowledge base, enriching the contextual representation of the input query. Furthermore, we propose an Adaptive Retrieval Mixer (ARM) module that dynamically fuses the retrieved patterns with the TSFM's internal representation, improving forecasting accuracy without requiring task-specific fine-tuning. Thorough empirical studies on seven public benchmark datasets demonstrate that TS-RAG achieves state-of-the-art zero-shot forecasting performance, outperforming the existing TSFMs by up to 6.84\% across diverse domains while also providing desirable interpretability. Our code and data are available at: https://github.com/UConn-DSIS/TS-RAG.

The attention mechanism is a core component of the Transformer architecture. Various methods have been developed to compute attention scores, including multi-head attention (MHA), multi-query attention, group-query attention and so on. We further analyze the MHA and observe that its performance improves as the number of attention heads increases, provided the hidden size per head remains sufficiently large. Therefore, increasing both the head count and hidden size per head with minimal parameter overhead can lead to significant performance gains at a low cost. Motivated by this insight, we introduce Simulated Attention Score (SAS), which **maintains a compact model size while simulating a larger number of attention heads and hidden feature dimension per head.** This is achieved by projecting a low-dimensional head representation into a higher-dimensional space, effectively increasing attention capacity without increasing parameter count. Beyond the head representations, we further extend the simulation approach to feature dimension of the key and query embeddings, enhancing expressiveness by mimicking the behavior of a larger model while preserving the original model size. **To control the parameter cost, we also propose Parameter-Efficient Attention Aggregation (PEAA).** Comprehensive experiments on a variety of datasets and tasks demonstrate the effectiveness of the proposed SAS method, achieving significant improvements over different attention variants.

Probabilistic time series forecasting is a challenging problem due to the long sequences involved, the large number of samples needed for accurate probabilistic inference, and the need for real-time inference in many applications. These challenges necessitate methods that are not only accurate but computationally efficient. Unfortunately, most current state-of-the-art methods for time series forecasting are based on Transformers, which scale poorly due to quadratic complexity in sequence length, and are therefore needlessly computationally inefficient. Moreover, with a few exceptions, these methods have only been evaluated for non-probabilistic point estimation. In this work, we address these two shortcomings. For the first, we introduce VQ-TR, which maps large sequences to a discrete set of latent representations as part of the Attention module. This not only allows us to attend over larger context windows with linear complexity in sequence length but also allows for effective regularization to avoid overfitting. For the second, we provide what is to the best of our knowledge the first systematic comparison of modern Transformer-based time series forecasting methods for probabilistic forecasting. In this comparison, we find that VQ-TR performs better or comparably to all other methods while being computationally efficient.

Recently, there has been a growing interest in leveraging pre-trained large language models (LLMs) for various time series applications. However, the semantic space of LLMs, established through the pre-training, is still underexplored and may help yield more distinctive and informative representations to facilitate time series forecasting. To this end, we propose Semantic Space Informed Prompt learning with LLM ($S^2$IP-LLM) to align the pre-trained semantic space with time series embedding space and perform time series forecasting based on learned prompts from the joint space. We first design a tokenization module tailored for cross-modality alignment, which explicitly concatenates patches of decomposed time series components to create embeddings that effectively encode the temporal dynamics. Next, we leverage the pre-trained word token embeddings to derive semantic anchors and align selected anchors with time series embeddings by maximizing the cosine similarity in the joint space. This way, $S^2$IP-LLM can retrieve relevant semantic anchors as prompts to provide strong indicators (context) for time series that exhibit different temporal dynamics. With thorough empirical studies on multiple benchmark datasets, we demonstrate that the proposed $S^2$IP-LLM can achieve superior forecasting performance over state-of-the-art baselines. Furthermore, our ablation studies and visualizations verify the necessity of prompt learning informed by semantic space.

Quantifying aleatoric uncertainty is a challenging task in machine learning. It is important for decision making associated with data-dependent uncertainty in model outcomes. Recently, many empirical studies in modeling aleatoric uncertainty under regression settings primarily rely on either a Gaussian likelihood or moment matching. However, the performance of these methods varies for different datasets whereas discussions on their theoretical guarantees are lacking. In this work, we investigate theoretical aspects of these approaches and establish risk bounds for their estimates. We provide conditions that are sufficient to guarantee the PAC-learnablility of the aleatoric uncertainty. The study suggests that the likelihood and moment matching-based methods enjoy different types of guarantee in their risk bounds, i.e., they calibrate different aspects of the uncertainty and thus exhibit distinct properties in different regimes of the parameter space. Finally, we conduct empirical study which shows promising results and supports our theorems.

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.

The Schrödinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence analysis of the Schrödinger bridge algorithm based on approximated projections. As for its practical applications, we apply SBP to probabilistic time series imputation by generating missing values conditioned on observed data. We show that optimizing the transport cost improves the performance and the proposed algorithm achieves the state-of-the-art result in healthcare and environmental data while exhibiting the advantage of exploring both temporal and feature patterns in probabilistic time series imputation.

Many kinds of simultaneously-observed event sequences exhibit mutually exciting or inhibiting patterns. Reliable detection of such temporal dependencies is crucial for scientific investigation. A common model is the Multivariate Hawkes Process (MHP), whose impact function naturally encodes a causal structure in Granger causality. However, the vast majority of existing methods use a transformed standard MHP intensity with a constant baseline, which may be inconsistent with real-world data. On the other hand, modeling irregular and unknown background dynamics directly is a challenge, as one struggles to distinguish the effect of mutual interaction from that of fluctuations in background dynamics. In this paper, we address the short-term temporal dependency detection issue. We show that maximum likelihood estimation (MLE) for cross-impact from MHP has an error that can not be eliminated, but may be reduced by an order of magnitude using a heterogeneous intensity not for the target HP but for the interacting HP. Then we propose a robust and computationally-efficient modification of MLE that does not rely on the prior estimation of the heterogeneous intensity and is thus applicable in a data-limited regime (e.g., few-shot, unrepeated observations). Extensive experiments on various datasets show that our method outperforms existing ones by notable margins, with highlighted novel applications in neuroscience.

Detecting out-of-distribution (OOD) samples is a problem of practical importance for a reliable use of deep neural networks (DNNs) in production settings. The corollary to this problem is the detection in-distribution (ID) samples, which is applicable to domain adaptation scenarios for augmenting a train set with ID samples from other data sets, or to continual learning for replay from the past. For both ID or OOD detection, we propose a principled yet simple approach of (empirically) estimating KL-Divergence, in its dual form, for a given test set w.r.t. a known set of ID samples in order to quantify the contribution of each test sample individually towards the divergence measure and accordingly detect it as OOD or ID. Our approach is compute-efficient and enjoys strong theoretical guarantees. For WideResnet101 and ViT-L-16, by considering ImageNet-1k dataset as the ID benchmark, we evaluate the proposed OOD detector on 51 test (OOD) datasets, and observe drastically and consistently lower false positive rates w.r.t. all the competitive methods. Moreover, the proposed ID detector is evaluated, using ECG and stock price datasets, for the task of data augmentation in domain adaptation and continual learning settings, and we observe higher efficacy compared to relevant baselines.

Information-theoretic clustering is one of the most promising and principled approaches to finding clusters with minimal apriori assumptions. The key criterion therein is to maximize the mutual information between the data points and their cluster labels. Such an approach, however, does not explicitly promote any type of inter-cluster behavior. We instead propose to maximize the Kullback-Leibler divergence between the underlying data distributions associated to clusters (referred to as cluster distributions). We show it to entail the mutual information criterion along with maximizing cross entropy between the cluster distributions. For practical efficiency, we propose to empirically estimate the objective of KL-D between clusters in its dual form leveraging deep neural nets as a dual function approximator. Remarkably, our theoretical analysis establishes that estimating the divergence measure in its dual form simplifies the problem of clustering to one of optimally finding k-1 cut points for k clusters in the 1-D dual functional space. Overall, our approach enables linear-time clustering algorithms with theoretical guarantees of near-optimality, owing to the submodularity of the objective. We show the empirical superiority of our approach w.r.t. current state-of-the-art methods on the challenging task of clustering noisy timeseries as observed in domains such as neuroscience, healthcare, financial markets, spatio-temporal environmental dynamics, etc.

Point processes often have a natural interpretation with respect to a continuous process. We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. In this framework, we relate the return times of a diffusion in a continuous path space to new arrivals of the point process. This leads to a continuous sample path that is used to describe the underlying mechanism generating the arrival distribution. These models arise in many disciplines, such as financial settings where actions in a market are determined by a hidden continuous price or in neuroscience where a latent stimulus generates spike trains. Based on the developments in Itô’s excursion theory, we propose methods for inferring and sampling from the point process derived from the latent diffusion process. We illustrate the approach with numerical examples using both simulated and real data. The proposed methods and framework provide a basis for interpreting point processes through the lens of diffusions.

Estimating Transfer Entropy (TE) between time series is a highly impactful problem in fields such as finance and neuroscience. The well-known nearest neighbor estimator of TE potentially fails if temporal dependencies are noisy and long ranged, primarily because it estimates TE indirectly relying on the estimation of joint entropy terms in high dimensions, which is a hard problem in itself. Other estimators, such as those based on Copula entropy or conditional mutual information have similar limitations. Leveraging the successes of modern discriminative models that operate in high dimensional (noisy) feature spaces, we express TE as a difference of two conditional entropy terms, which we directly estimate from conditional likelihoods computed in-sample from any discriminator (timeseries forecaster) trained per maximum likelihood principle. To ensure that the in-sample log likelihood estimates are not overfit to the data, we propose a novel perturbation model based on locality sensitive hash (LSH) functions, which regularizes a discriminative model to have smooth functional outputs within local neighborhoods of the input space. Our estimator is consistent, and its variance reduces linearly in sample size. We also demonstrate its superiority w.r.t. state-of-the-art estimators through empirical evaluations on a synthetic as well as real world datasets from the neuroscience and finance domains.