SMart: A Multi-source Multi-phase Time Series Representation Transfer Framework

Time series representation learning (TSRL) has attracted growing research interests in recent years. Two recent explorations in TSRL are: i) exploiting a transformer-based framework to learn time series; ii) instead of using only the targeted dataset, borrowing time series from other datasets to to facilitate representation transfer. While these two explorations are shown effective, the self-supervised time series recovery task in (i) and the single-source dataset used in (ii) are technically simple and thus can be enhanced with new ideas. In this work, we propose a new TSRL framework, namely multi-source multi-phase time series representation transfer (SMart), which has two novel mechanisms to address the aforementioned deficiencies: 1) a multi-phase recurrence plots recovery task, in three alternative modes, for guiding the encoder to embed time series dynamics into the time series representation; and 2) a source dataset selector to select multiple suitable source datasets to supplement the original target dataset for pre-training the TSRL encoder. Experimental results show that SMart outperforms several state-of-the-art models for time series representation learning, classification and regression on both uni-variate and multi-variate time series datasets, reducing mean absolute error up to 19.5% for time series regression, and increasing average accuracy up to 1.34\% for time series classification.

Poisson-Gamma Dynamical Systems with Time-varying Transition Dynamics

Bayesian methodologies for handling count-valued time series have gained prominence due to their ability to infer interpretable latent structures and to estimate uncertainties. Among these Bayesian models, Poisson-Gamma Dynamical Systems (PGDSs) are proven to be effective in capturing the evolving dynamics underlying observed count sequences. However, the state-of-the-art PGDS still falls short in capturing the transition dynamics that are commonly observed in real-world count time series. To mitigate this limitation, a PGDS with time-varying transition kernel (TV-PGDS), is proposed to allow the underlying transition matrices to evolve over time. Three specifically-designed Dirichlet Markov chains (Dir-Dir, Dir-Gam-Dir, PR-Gam-Dir) are constructed to accommodate heterogeneous structural mutations within these dependencies. Leveraging Dirichlet-Multinomial-Beta data augmentation techniques, a fully-conjugate and efficient Gibbs sampler is developed to perform posterior simulation. Experiments show that, in comparison with related models, the proposed PGDS achieves improved predictive performance due to its capacity to learn time-varying dependency structure captured by the time-evolving transition matrices.

A convolutional framework for detecting event-driven dynamics in energy price series

This paper develops a general convolutional neural network (CNN) framework for detecting heterogeneous event-driven dynamics in univariate time series windows. We show that the induced CNN class exactly represents classifiers based on range, maximum drawup, maximum drawdown and slope change, and uniformly approximates realised volatility and autoregressive explosiveness on compact domains. We further establish error bounds for representative rules in finite samples and an oracle inequality for learning across them. Simulations show that the proposed model can match or outperform classifiers based on individual statistics as the training sample grows. In an application to six daily energy price series, a hierarchical CNN distinguishes event windows and event families. Applied without retraining to observations withheld after 20 February 2026, the fitted model identifies predominantly geopolitical dynamics in several oil and refined product series around the outbreak of the 2026 Iran war, while distinguishing a contemporaneous natural gas spike associated with weather.

Liquid Gated Attention

Real-world time series often exhibit irregular sampling and extended temporal horizons, requiring models to capture continuous-time dynamics across arbitrary intervals without prohibitive scaling costs. Discrete-time methods collapse variable time intervals into static positional steps; solver-dependent continuous-time models preserve temporal structure but rely on sequential integration, precluding parallelization; and solver-free approximations avoid this cost yet none couples observed time intervals with input-driven state modulation. We propose Liquid Gated Attention (LGA), a solver-free parallel temporal operator. By parameterizing an input-driven gating mechanism with observed time intervals, LGA introduces a continuous-time inductive bias and formulates hidden state evolution as a fast-weight associative memory, enabling parallel computation across the temporal dimension. Using matrix associativity in non-causal encoding and a prefix scan in causal encoding, LGA attains linear temporal complexity in sequence length in both modes. A sequence-level normalization bounds cumulative temporal decay for stable long-horizon optimization. Building on LGA, we instantiate LFormer, a modular backbone for continuous-time representation learning. Across six tasks and sixteen datasets spanning up to 17,984 steps, LFormer demonstrates long-range dependency modeling, fine-grained state tracking, and trajectory reconstruction from sparse and noisy observations, while delivering competitive performance against state-of-the-art discrete-time and continuous-time baselines with linear scaling efficiency.

When the Martingale Never Stops Firing: Anytime-Valid Gating on Real Forecast Streams

Machine learning systems are increasingly corrected while they run, and the decision of when to intervene is increasingly delegated to statistical monitors. Anytime-valid inference promises evidence that can be acted on at any moment, exactly the guarantee this setting needs, and it is moving from theory into deployed monitoring. Conformal test martingales are the change-detection instrument, and Ville's inequality caps their false-alarm probability on exchangeable data. The guarantee is conditional. A deployment inherits it only if the stream it monitors behaves exchangeably. The premise is hardest to satisfy where these monitors are most useful, on dependent data and inside loops where the monitor modifies the learner whose scores it reads. It is also rarely measured. We measure it in a pre-specified case study, where such a monitor gates the online updates of a Kalman adapter correcting frozen time-series foundation models on five forecasting streams. On exchangeable synthetic streams, the same implementation fires in at most 1 of 60 runs. On the real streams, at alpha = 0.05, 135 of 135 clean-stream runs fired. The construction does not explain the firing; the failure comes from the deployed score stream itself. Repeated fires hold the gate's drift response active, and the gated filter amplifies the very transient it was designed to prevent. The component worth keeping makes no validity claim. Huber-style gating of the filter's own updates cuts isolated-spike degradation by an order of magnitude with no dataset specific tuning. Anytime-valid methods proposed for dependent data should therefore be accompanied by null-calibration controls and mechanism traces.

Generalized Gibbs Ensemble Weighting for Forecast Combination

Forecast combination is a reliable way to improve predictive performance when several forecasting models are available. Simple aggregation rules such as the mean, median, trimmed mean, inverse-loss weighting, and exponential weighting are often strong baselines, but their relative performance can vary across datasets, forecast horizons, deployment settings, and levels of disagreement among base forecasters. We develop Generalized Gibbs Ensemble Weighting (GGEW), a probabilistic framework that treats forecasting models as experts and assigns ensemble weights using a Gibbs-style exponential transformation of normalized predictive loss. The framework extends this basic weighting rule through numerical stabilization, diversity-aware score corrections, and online hyperparameter adaptation. GGEW produces a family of related methods, including Stable Gibbs weighting, Directional Gibbs-NCL, and Symmetric Gibbs-NCL. These variants share one core algorithm and differ only in the score used inside the exponential weighting rule. For sequential deployment, we adopt a UCB-style bandit mechanism, called online Local-UCB, to adapt the learning rate, diversity strength, and Gibbs variant without evaluating the full hyperparameter grid at every prediction step. We evaluate GGEW on official M4 competition forecast submissions and external rolling-origin deployment experiments using Monash Traffic Hourly, Electricity Hourly, and Solar Weekly datasets. Results suggest that Gibbs-style adaptive weighting is a useful and competitive tool across several benchmark settings, although its relative performance varies across datasets, forecast horizons, deployment protocols, and forecast disagreement groups. The contribution is not a universal dominance claim, but a framework and empirical study motivating further investigation of when adaptive Gibbs-style forecast combination is useful.

Forecasting Multiple Observables with SCROLL: Score-Trained Uncertainty for Stochastic Dynamics

Forecasting a stochastic dynamical system rarely means a single number: one wants several observables---future state, threshold event, regime label---each with its own likelihood. Standard multi-task recipes balance per-task losses, tuned or learned. We instead compose the observables' likelihoods in per-task free-routed last-layer beliefs on a shared backbone; this absorbs unit-dependent loss scaling into likelihood parameters learned in the same gradient pass. Stochastic dynamics supply what static benchmarks cannot: computable ground truth for the predictive variance. Results land where theory puts them: on the well-specified, homoscedastic Ornstein--Uhlenbeck process the learned predictive law recovers the analytic kernel and correctly specified baselines tie. On heteroscedastic systems (stochastic Lorenz-63, real air-quality data) the belief's input-dependent variance separates: best single-run NLL on the state and regime tasks, calibration matched only by arms whose NLL it beats, at a fraction of the tuned grids' cost. On the real series the state margin holds across five rolling origins.

Towards A Unified Information Bottleneck Framework for Time Series Explanations

Explaining deep learning models operating on time series data is crucial in various applications that require transparent and interpretable insights into model behavior. {Existing explanation methods generally fall into two categories: attribution-based explanations, which identify the temporal regions most responsible for a prediction, and counterfactual explanations, which reveal how an input should be modified to alter the model's decision.} {Despite valuable insights, these two fields are largely studied independently. This disconnect leaves attribution methods lacking causal validation, while counterfactual methods suffer from severe instability, producing adversarial-like noise instead of meaningful explanations.} In this work, we revisit time-series explainability from an information-theoretic perspective and show that existing explainers are vulnerable to trivial solutions and distributional shifts. To address these limitations, we propose a unified objective function for explainable time series learning that bridges attribution and counterfactual reasoning within a single framework. Building upon the Information Bottleneck principle, our formulation explicitly prevents trivial explanations and out-of-distribution counterfactuals. {Based on this objective function, we introduce {\modelname}, a novel explanation framework that learns a parametric transformation network to construct explanation-embedded instances, where preserved information yields attribution explanations and controlled information removal produces stable counterfactual explanations.} We evaluate {\modelname} on synthetic and real-world benchmarks against state-of-the-art baselines. Extensive quantitative and qualitative results show that {\modelname} consistently outperforms competing methods, yielding faithful attributions and stable counterfactual explanations.

NVExplain: Explaining Time Series Forecasting with Latent Trajectory Analysis and Structure-Preserving Surrogates

Time series forecasting models are widely used in high-stakes settings, yet their predictions remain difficult to interpret because existing post-hoc methods often ignore temporal dependence and fail to provide horizon-specific explanations. We propose a model-agnostic explainability framework that explains forecasting predictions by attributing each forecast horizon to temporally relevant historical lags. The framework models forecasting as a latent trajectory and introduces semantic flow to quantify how information evolves across time in the model's internal representations. By aggregating semantic flow, it constructs a lag-horizon attribution matrix that captures horizon-resolved temporal influence. To improve explainability, we further generate structure-preserving perturbations and fit sparse local surrogate models, producing human-readable and temporally coherent explanations. We evaluate the method using faithfulness and stability diagnostics across multiple benchmark datasets. Results show that the semantic-flow variant achieves competitive or superior faithfulness compared to standard post-hoc baselines, while being substantially more computationally efficient. Stability analysis further demonstrates that the explanations are robust and identifies regimes where interpretation should be applied with caution.

Single State Update Predictive Coding training for Time Series Forecasting and Anomaly Detection

Predictive Coding (PC) is a neural learning paradigm that enables parallelizable neural network layer updates. However, the main bottleneck of PC Networks (PCN) is the sequential backwards error propagation. To tackle this, we introduce a training technique that pairs a Generative PCN with a support Encoding PCN. The two PCNs are trained in parallel to match their neural activations, without sequential propagation. We apply this to time series anomaly detection and show that our approach results in more stable, continuous, online learning.

Causal Analysis for Time Series Foundation Models

Transitioning from bespoke time series models towards time series foundation models changes the relationship of model and application from one-to-one to one-to-many. This shift introduces concentration risk as many, potentially high-risk, forecasting applications are exposed to the same biases and failure modes of a single time series foundation model. At the same time, this centralization allows for economies of scale in model development and validation. In this study we investigate how biases and failure modes of time series foundation models can be identified before deployment. We propose a causal analysis framework to investigate the ability of a time series foundation model to preserve time series patterns. To achieve this, we intervene on parameterized synthetic time series generators and measure the corresponding change in model output under ceteris paribus conditions. We apply our causal analysis framework to Chronos-2 and TimesFM-2.5 and test them across six distinct time series patterns. We find safe configurations for trend and harmonic oscillation patterns. The results also indicate a bias in both models towards overestimating persistence, sudden failures for both models against the regime switch pattern and failure for TimesFM-2.5 against the energy-release pattern. Our review of the original works for both models indicates that the findings might be explained by the data used for pretraining. We conclude our study with suggestions for further model development, recommendations for application-specific model selection, and a discussion of limitations and further research directions.

ALPHABET: A Laplace-Pole History Aggregator with Banked Exponential Transport

Can a sequence model remain competitive with only a few thousand parameters and an explicitly auditable prediction interface? We introduce ALPHABET, a compact linear-time model that compresses temporal history into stable complex pole modes: a direct bank synthesizes its modal states back into the feature trajectory, an independent cascaded bank analyzes the transformed trajectory without resynthesis, and an affine head reads only modal energies and lag moments from both banks. We characterize the temporal information this descriptor retains: for a stationary, fully observed feature process, each mode energy is a frequency-localized measurement of the second-order spectrum, the continuum of such measurements identifies the spectrum, and almost every mode separates any fixed finite set of spectrally distinct classes. On a Gaussian control with matched low-lag statistics, the learned descriptor approaches the Bayes oracle where raw autocovariances remain at chance. Across the fixed 82-task registry, ALPHABET attains mean rank 3.97 in the complete ten-family comparison. At the common-width D=64 runtime anchor, its 6,437 parameters deliver 5.02 times faster inference and 3.93 times faster complete training steps than the nine baselines on average.

Counterfactual Transition Graphs: Evaluating Cross-Class Transition Quality

Counterfactual (CF) explanations for time-series classifiers are usually evaluated one example at a time: what minimal edit flips this single window's prediction? We argue that the more informative question for diagnostic interpretability is structural: how does the classifier connect its own classes to each other? We propose a counterfactual transition graph (CGT) in which each node is a class and each edge weight is the CF reliability of the transition from one prototype to another under a proximity aware retrieval sweep. On a six-class hand-movement task, we induce a CGT that reveals a non-trivial topology, which is not predicted by the binary confusion matrix: it shows that counterfactual reachability does not align with classifier accuracy and even runs counter to it (Spearman $ρ=-0.37$ over the 15 pairs), i.e. the boundaries the classifier separates most confidently are among those an in-distribution edit can least often cross. Our framework is method agnostic, i.e. any CF-explainers can be used. Presently, we use it to juxtapose replacement-based CFs with gradient-based CFs; gradient-based methods reach almost any class by stepping off the data manifold, while replacement-based methods stay on it and fail on precisely the rigid boundaries.

FreKoo++: Learning Continuous Spectral Dynamics for Temporal Domain Generalization

Temporal Domain Generalization (TDG) aims to learn from historical domains and generalize to unseen future distributions under concept drift. Nevertheless, prevailing TDG methods struggle with complex real-world streaming scenarios involving both multi-scale drift patterns (e.g., long-term periodicity intertwined with short-term incremental changes) and local uncertainties, especially in continuous settings where observations arrive irregularly. To address this limitation, we propose FreKoo++, a novel continuous spectral-dynamical framework that pioneers the unification of continuous Koopman modal dynamics with adaptive spectral disentanglement. Specifically, FreKoo++ maps source-domain parameters into a compact latent space, modeling their evolution as a superposition of learnable continuous modes where complex eigenvalues jointly encode oscillatory frequency and temporal growth or decay. This formulation naturally accommodates irregular timestamps and supports arbitrary horizon extrapolation without rigid discrete stepping. Furthermore, we propose a new adaptive soft spectral weighting mechanism backed by stability and spectral regularization, which automatically isolates persistent dominant dynamics from transient noise without relying on manual frequency thresholds. We derive modal approximation and generalization bounds that characterize how amplitude and eigenvalue estimation errors propagate with the prediction horizon. Extensive experiments on both discrete and continuous TDG benchmarks demonstrate that FreKoo++ achieves state-of-the-art performance under complex multi-scale drifts and irregular sampling.

Counterfactual Evaluation of Temporal Observation Protocols

We study counterfactual protocol evaluation: whether data collected under a realised observation protocol determine the predictive value of alternatives that were never deployed. Protocol value is the population $R^2$ of the Bayes-optimal predictor of a fixed trajectory-level target from the measurements an alternative would collect. We show that even infinite benchmark data need not determine this value: distinct latent covariance structures can induce the same benchmark measurement--target law while assigning different values to the same alternative. We develop a value-specific identification theory in which only latent ambiguity that changes the alternative's value matters. For linear targets, invisible covariance directions certify non-identification, while targeted measurements can restore identification without recovering the full latent covariance; an exact permutation construction extends the result to nonlinear aggregate targets. With finite dense calibration data, uniform error bounds control protocol-selection regret and distinguishable value gaps. Exact marginal gains then support cost-constrained, target-aware observation design. Simulations and retrospective analyses of Sleep-EDF and Long-Term AF show that broad temporal-layout differences can be more reliably distinguished than fine placements selected from finite data. Together, these results connect identification, calibration resolution and observation design for undeployed protocols.

Efficient Regression Models for Scan Statistics

We introduce a new class of regression models for scan statistics on real-valued signals. These allow for improved fitting of non-stationary signals to contrast with the interval anomalies identified by the scan statistics. Our models can represent generalized likelihood ratio statistics. While these methods naively require $O(n^4)$ for a length $n$ signal, we provide algorithmic improvements which lead to linear time algorithms (with assumptions on max interval width). Our methods, especially ones based on Nadaraya-Watson kernel regression, are demonstrated as especially effective in detecting both synthetically planted anomalies, and for identifying a real ``platforming'' issue in interferometric astronomy.

Structured Learning on Mapper Representations

Modern machine learning (ML) methods are highly effective for prediction tasks, but many commonly used representations reduce complex data to fixed dimensional embeddings that may suppress multiscale structural organization. The Mapper algorithm from topological data analysis (TDA) provides a different perspective by decomposing data into overlapping local regions connected through a nerve construction, producing a structured representation that captures geometric organization, local statistical behavior, and relational connectivity simultaneously. In this work, we develop a framework for learning over Mapper induced structured representations. Rather than treating Mapper as a preprocessing step that produces a graph for downstream learning, we treat the full Mapper construction as part of the representation itself. We study mathematical properties of these representations, including invariance under relabeling, a distance functional on the space of Mapper representations, structural complexity of multiscale decompositions, and learning oriented stability under representation perturbations. Experiments on time series and graph classification datasets validate the proposed framework through controlled studies of representation ablation, Mapper parameter sensitivity, and the geometry of the induced representation space. Together, these results demonstrate how the proposed mathematical framework enables systematic comparison, interpretation, and analysis of Mapper representations, providing practical tools for studying representation geometry, structural complexity, and learning stability in learning tasks.

Spectral Pre-Filtering for Context-Adaptive Sensor Fusion: A Four-Role FFT-GDCB Integration for High-Stakes Decision Systems

Context-adaptive Kalman filters calibrate their noise covariance matrices Q and R from innovation residuals via online regression. When the underlying sensor or signal carries periodic structure -- mechanical LiDAR rotation harmonics, engine vibration, ground multipath, weekly and annual demand cycles, dosing-interval rhythms, weekly media-buying cadence -- the regression input is contaminated and the fitted covariance models structural modes rather than genuine state uncertainty. We introduce a four-role FFT pre-filter that solves this problem at $O(N\log N)$ cost and serves three additional roles "for free": (i) it whitens coloured noise before the Kalman update, restoring the optimality assumption; (ii) it cleans innovations before covariance regression, preventing periodic contamination of $\hat{R}$ and $\hat{Q}$; (iii) it generates spectral context features that enrich the downstream bandit's regime-selection state; (iv) it deseasonalises the input feature vector before any supervised regression that produces a sensitivity coefficient (beta, dose offset, bid modifier). We position the algorithm inside the Gated Decoupled Compositional Bandits (GDCB) family, where it acts as a preprocessing layer for the supervised scaler. The single $O(N\log N)$ FFT call thereby serves four downstream consumers, fits in <0.1% of the sensor-fusion or pricing-pipeline compute budget, and is a drop-in addition with no changes to the Kalman filter, bandit, or runtime composition operator. We summarise empirical validation across six independent domains (rocket descent, autonomous-vehicle tracking, short-term rental pricing, clinical drug dosing, airline fare distribution, and ad-operations bid calibration), all returning a PROVES verdict under a pre-registered evaluation protocol.

More Experts, Worse Dynamics: Inverse Scaling and Spectral Bias in Mixture-of-Experts State-Space Models

Mixture-of-Experts (MoE) architectures are commonly motivated as a way to increase expressivity by decomposing complex systems into simpler local dynamics. This intuition has recently been extended to spectral state-space models, where mixing stable operators is assumed to enable adaptation to heterogeneous or regime-switching time series. We critically evaluate this assumption in a controlled synthetic setting designed to isolate dynamical rather than representational challenges. We study a next-step prediction task on sequences composed of three regimes: chaotic dynamics generated by the Mackey-Glass system, a stable oscillatory regime, and a noise-dominated autoregressive regime. Across extensive ablations including capacity scaling, oracle routing, frozen-expert variants, and comparisons to output-level MoE baselines, operator-level mixture models consistently fail to outperform a single-expert baseline. Increasing the number of experts leads to inverse scaling, routing collapses or fails to induce meaningful specialization, and even perfect regime supervision does not prevent degradation in global performance. Furthermore, we show that apparent improvements in mean squared error on chaotic trajectories can be misleading. Phase-space analysis reveals that lower error often arises from temporal smoothing that destroys the geometry of the underlying attractor rather than from faithful modeling of the dynamics. These results identify a likely limitation of operator interpolation under the studied parameterization and training protocol, and underscore the need for geometry-aware evaluation when assessing regime-switching dynamical systems.

Reaching the Tail: Calibration Diversity Drives Conformal Coverage under Data Scarcity

Multi-horizon rare-event forecasting is hard under long macroeconomic series' data constraints: labeled events are scarce, and standard uncertainty quantification assumes an exchangeability that autocorrelation violates. A controlled ablation shows an apparent rare-event threshold for Adaptive Conformal Inference instead reflects calibration-set size. Across 200 random calibration sets, support width of the nonconformity-score distribution explains up to 85% of coverage variance versus 2% for rare-event count; the same, not the same magnitude, replicates across synthetic conditions and five countries (five-country Spearman $ρ$ 0.45-0.66 vs. 0.02-0.23). A diversity-maximizing selector built on this is the only strategy tested that improves long-horizon coverage (67.8% to 81.4% at six months); Mondrian, shift-robust, and extreme-value alternatives fail to close it. Mondrian even worsens coverage under oracle labels. A compact proposition explains why: coverage deficit reflects how closely the calibration set's upper quantile reaches the test distribution's. Diversity is necessary, not sufficient. Demonstrated on a two-stage U.S. recession-forecasting framework with RegressorChain, whether six-month coverage reaches 90% under honest scoring remains open, a question this paper quantifies rather than resolves.

Root cause analysis via difference graph discovery from linear time-series data

Root cause analysis aims to identify the mechanisms responsible for anomalies in complex dynamical systems. In this paper, we study root cause analysis in linear time-series through the lens of difference graph discovery. We focus on effect-defying root causes, corresponding to variables whose causal coefficients change between a normal and an anomalous regime. We formalize this problem using linear discrete-time dynamic structural causal models and adapt several methods originally introduced for discovering difference graphs between two populations to the time-series setting, where the two populations are replaced by a normal and an anomalous regime. We first evaluate the proposed approaches on simulated data, and then demonstrate their practical relevance on real-world datasets from IT monitoring and intensive care monitoring. Our results show how difference graph discovery can help localize causal mechanisms responsible for anomalous behavior.

Free-Probability Kernels for Zero-Rollout Hyperparameter Selection in Reservoir Computing

Reservoir computing (RC) couples a fixed recurrent dynamical system with a trained lightweight readout, but this efficiency is partly lost during hyperparameter selection: the recurrent gain, input scale, and leakage rate determine the reservoir's stability and temporal processing regime and are usually tuned through many rollouts. We introduce a deterministic, pilot-informed selector for leaky linear reservoirs followed by coordinate-wise nonlinear features. Free probability yields cross-lag propagation coefficients that summarize how the reservoir mixes past inputs. In the large-width limit, these coefficients define a deterministic temporal kernel that approximates the finite-reservoir feature geometry. Kernel ridge regression on a short labelled pilot sequence therefore ranks candidate operating regimes without instantiating or rolling out a reservoir, and the selected configuration transfers across widths. Across ten synthetic temporal benchmarks, zero-rollout selection obtains a mean deployment score of $0.772$, compared with $0.774$ for exhaustive simulation-based search, while avoiding $156\,600$ selection rollouts. With a small rollout budget, the proposed ranking provides the strongest mean performance at every tested budget and reaches the exhaustive reference using $4.8\%$ of its rollout cost. On four public electricity-transformer-temperature (ETT) forecasting datasets, five retained candidates recover the exhaustive operating point on three datasets. On multivariate cellular-traffic forecasting, 15 rollouts per cell reach the 462-rollout exhaustive reference and outperform random search and Bayesian optimization at low budgets. These results position free-probability kernels as deterministic surrogates for selecting reservoir operating regimes when validation rollouts are scarce.

A Critical Audit of Spatiotemporal Forecasting Benchmark Datasets and Baselines

Graph neural networks (GNNs) are routinely employed for short-range forecasting on multivariate time series with a spatial graph structure. Despite the availability of many alternative datasets, method innovations within this domain are predominantly assessed against a rather limited set of benchmark datasets, most notably Chickenpox, PedalMe, WikiMaths, METR-LA, and PEMS-BAY. The evaluation protocols contain baselines spanning from historical averages to classical machine learning approaches. These baselines often show competitive performance compared to GNNs. In the present work, we take a step back and analyse the benchmark datasets via classical time series methods to uncover why spatially-unaware linear models pose a stronger competitor than previously reported, casting further doubt on the discriminative reliability of the aforementioned widely adopted datasets. Our statistical analysis provides a toolset for identifying significant spatial and temporal correlations, while revealing a structural bias introduced by first-order differenced datasets. We therefore recommend reducing the over-reliance on such datasets for method comparison, and instead advocate for more rigorous statistical evaluation. By applying the results of our analysis to a simple hybrid model, we show how our methodology can lead to novel ways of developing GNN models

Class-Conditioned Gaussian Mixture Modeling for Imbalanced Time Series Quantification

Quantification, estimating class prevalences in bags of unlabeled instances is vital in domains where aggregate statistics are more important than individual instance labels, such as biosignal monitoring, fall detection, and activity recognition. We investigate this issue in the challenging setting of imbalanced time series data and develop CC-GMNet-TS, a class-conditioned Gaussian mixture quantifier that combines a Transformer-based feature extractor with per-class latent mixtures. Unlike previous mixture-based quantifiers, which use a single Gaussian mixture shared by all classes, CC-GMNet-TS assigns each class its own compact mixture in a bounded latent space and scores segment embeddings against these class-specific components to create bag-level representations that emphasize rare but informative patterns. Bags are constructed from labeled pools using the Artificial Prevalence Protocol (APP) and prior shift bag sampling (PShift) to cover a wide range of class prevalence scenarios, and the model is trained end-to-end with a quantification-oriented loss. Experiments on three benchmarks: EMG Data for Gestures, SmartFallMM, and UCI-HAR show that CC-GMNet-TS achieves lower error across the three benchmarks compared to traditional aggregators and recent deep quantifiers, while ablations confirm the contributions of both the Transformer backbone and class-conditioned mixtures during PShift.

Let Time Tell: Identification and Gaussian Process Estimation for Interrupted Time Series

We study causal inference in interrupted time series designs where a treatment affects every unit simultaneously, so that the contemporaneous controls used by difference-in-differences and synthetic control are unavailable and the counterfactual must be extrapolated from a unit's own pre-treatment history. We establish identification within the potential outcomes framework and estimate the counterfactual by Gaussian process regression. Rather than committing to a single best-fitting trend, the estimator retains the functions consistent with the pre-treatment series and widens its intervals where extrapolation magnifies their divergence. Connecting it to reproducing kernel Hilbert space theory, we derive a bias decomposition that isolates the component extrapolation inflates and a worst-case bound on that component, justifying the Gaussian process estimator's posterior variance as extrapolation-aware uncertainty quantification. In closed form, the band equals the worst-case divergence the model class permits among functions consistent with the pre-treatment data. The method is illustrated with calibrated simulations and an analysis of handgun purchases after the Supreme Court's Heller decision, a universal treatment whose practical effect concentrates in a single jurisdiction. An R package, gpss, implements the approach.

End-to-end Early Classification of Time Series in Non-Stationary Environments

Early Classification of Time Series (ECTS) requires making accurate decisions as early as possible in inherently online and evolving environments. Yet, most existing methods assume stationarity and rely on separable designs, where classification and triggering are optimized independently, an assumption that fundamentally limits their adaptability under drift. In this work, we challenge this paradigm and study ECTS under non-stationary conditions. We provide the first systematic comparison between separable and end-to-end approaches across controlled drifting scenarios. Building on Reinforcement Learning, we introduce DQeND, a unified architecture that jointly learns representation, classification, and triggering decisions, while remaining directly comparable to state-of-the-art separable baselines. Across a wide range of drifts, DQeND demonstrates strong robustness across various non-stationary scenarios, consistently outperforming separable baselines. An ablation study further highlights that jointly updating representation and decision modules is critical to these gains. Overall, our results indicate that end-to-end learning can offer improved adaptation capabilities for ECTS in dynamic environments, and motivate further investigation of alternatives to separable designs.

Scale-Aware Pretraining of Time Series Foundation Models via Multi-Patch Token Alignment and Hybrid Masking

Pretraining time series foundation models across heterogeneous datasets necessitates effective handling of varying sampling frequencies. Current methods either employ dataset-specific patch sizes and separate FFNs, leading to fragmented representations, or enforce a fixed patch size that neglects inherent temporal variations. To address this, we propose SATS, featuring a scale-aware token alignment mechanism that treats patch size as an explicit notion of scale. By incorporating a contrastive-inspired alignment regularizer, SATS aligns representation spaces across scales while preserving distinct modeling capacities. Furthermore, a hybrid masking strategy combining random and contiguous masking is introduced to capture multi-scale temporal structures. Experimental results on LSTF benchmarks demonstrate that SATS achieves a 9.2% improvement in MSE and an 8.3% gain in GIFT-Eval MASE compared to competitive baselines. Notably, SATS consistently delivers SOTA performance while achieving a 65.6% increase in model efficiency over advanced baselines, highlighting its effectiveness and scalability in time series pretraining.

skchange: Fast and Flexible Algorithms for Changepoint Detection

Skchange is an open-source Python library for detecting structural changes in time series. It implements modern change detection algorithms within a unified and extensible framework. The algorithms are modular and composable, and they include changepoint search methods based on both cost minimisation and statistical tests. Key features include the detection of anomalous segments in addition to changepoints; theoretically well-founded fast and approximate search methods; theoretically well-founded algorithms for high-dimensional data, covering settings where either few or many features change simultaneously; utilities for automatic and data-driven penalty calibration, which balances false alarms against missed detections; and a large collection of built-in costs and statistical tests. The design follows established scikit-learn conventions to streamline both user and contributor experience, and Numba is used extensively to achieve high computational performance. Source code and documentation are available at https://github.com/NorskRegnesentral/skchange.

Lévy Attention: Single-Pass Predictive Uncertainty for Continuous-Time Attention

Deep models for irregularly-sampled time series answer queries at arbitrary continuous timestamps, yet report nothing about how far each answer should be trusted. We show the attention layer itself can close that gap: with the right stochastic formulation, the pass that makes each prediction also reports, in closed form and at no extra cost, how far it should be trusted. We introduce Lévy Attention, a cross-attention operator whose output is a stochastic integral against an inhomogeneous Poisson random measure: query-key compatibilities assemble an intensity over a continuous (time x channel) index space, the measure scatters atoms under it, and the output averages an interpolated value field at those atoms. In expectation it reduces to a mollified cosine-kernel attention, so it replaces a softmax layer and trains with exact gradients. What softmax discards, the Poisson construction preserves in closed form: the evidence $Λ_q$ (total compatibility mass) and the disagreement $\mathrm{tr}\,Σ_V(q)$ (value spread). An exact variance identity makes their combination $\hatσ(q)=\sqrt{\mathrm{tr}\,Σ_V(q)\,\varphi(Λ_q)}$ the root-mean-square deviation of the sampled operator, emitted by the deterministic pass with no trained head. Empirically, disagreement carries the signal, while the evidence factor swings from uninformative on dense data to strongly informative on sparse. On t-PatchGNN the operator swap costs at most 5.6% accuracy against a matched control and nothing on the sparsest dataset. The free disagreement signal improves on 20-pass MC dropout across matched five-seed suites, and $\hatσ$ scales a calibrated Gaussian whose zero-sample CRPS beats a fifty-draw sampler; a split-conformal wrapper reaches nominal coverage at every level, and one pass ranks 3,383 unseen patients by trust in 1.4 seconds.

Causal Local States: Scalable Simultaneous Causal Network Inference and Forecasting for Dynamical Systems

Machine learning methods predict many real-world systems with remarkable accuracy, but they are typically treated as black boxes that offer no insight into which interactions drive the dynamics. Causal discovery methods reconstruct the interaction network from observational data, but without regard to whether the inferred structure supports prediction. Existing approaches combining both tasks rely on a single global hyperparameter, such as a causal threshold or a fixed neighborhood size, which cannot recover the structure of heterogeneous systems. Here we introduce causal local states (CLS), a framework that simultaneously infers an approximate Granger-causal interaction network and forecasts the system dynamics. For each node independently, we select the smallest set of neighbors that allows a predictive model to forecast the node near-optimally, and the resulting neighborhoods are then combined for a forecast of the full system. On three benchmarks of increasing difficulty, we achieve reconstruction of the underlying networks with high fidelity and forecasts on par with a model that is supplied with the true network, providing a step toward explainable and scalable forecasting of complex systems.