Non-Stationary Time Series

Time-series anomaly detection (TSAD) identifies deviations from patterns learned from historical data. In non-stationary settings, distribution drift and true anomalies can cause similar local changes, making it difficult to tell whether a deviation reflects abnormality or evolving context. Existing methods typically adapt to detected shifts or learn drift-insensitive representations, but do not resolve this ambiguity. We define this problem as \emph{temporal change disambiguation}: determining whether a local deviation is explained by broader temporal evolution. We introduce MORA, a drift-robust TSAD framework that reconstructs the same local target from paired short- and long-term views. The reconstruction gap measures contextual support for a local deviation, and a data-dependent correction mechanism conservatively adjusts the primary local anomaly score. Context can only reduce the score when it improves reconstruction of the same target. MORA needs neither drift annotations nor online adaptation. Experiments on four TSAD benchmarks show strong robustness to non-stationarity while preserving sensitivity to genuine anomalies.

Reducing nonstationarity in a persistent time series entails deciding how much of its temporal dependence to remove. In finance, fractional differencing is often tuned using the Augmented Dickey--Fuller (ADF) test, limiting the search to a one-parameter family of lag profiles and addressing input preservation only indirectly. We propose StaFIR, a causal finite-impulse-response filter with a learned nonnegative mixture of exponential lag profiles. Its convex learning objective balances empirical stationarity with similarity to the input. We evaluate StaFIR on ARFIMA--GARCH controlled settings and rolling financial series, including a realized-volatility forecasting task. The experiments show that StaFIR adjusts its filtering strength to persistence while limiting unnecessary transformation in stationary regimes. In downstream forecasting, there is no clear accuracy difference from fixed half-order differencing, while StaFIR achieves higher measured similarity to the raw signal. A complementary direct forecasting experiment finds that greater input similarity is associated with smaller forecasting penalties, although the raw representation remains stronger.

Time series anomaly detection (TSAD) is increasingly deployed in streaming settings, where data arrive sequentially and may exhibit non-stationarity. As a result, several works from the recent literature propose streaming anomaly detection methods that rely on incremental updates to adapt over time. However, most of these approaches originate from the streaming outlier detection literature and largely ignore core characteristics of time series anomalies. Moreover, their empirical evaluation is typically conducted on synthetic or small-scale benchmarks with limited diversity, making it unclear whether streaming methods are truly advantageous in realistic TSAD scenarios. In this work, we carry out the first large-scale experimental study comparing streaming and static TSAD methods under a unified streaming evaluation benchmark. We consider a realistic setting in which an initial batch of data is available for model training, followed by online evaluation of both detection accuracy and computational efficiency. In addition, we propose a distribution-drift dataset of real time series, called TSB- drift, to isolate scenarios where streaming updates are theoretically justified. Our results show that, contrary to common assumptions, static TSAD methods significantly outperform streaming approaches in most streaming settings. Such finding highlights a critical gap between the design of existing streaming methods and the requirements of modern TSAD, and calls for a rethinking of how streaming capabilities should be integrated into TSAD.

Many physiological time series, such as cardiac and brain recordings, exhibit cyclostationarity: their statistics vary periodically with an underlying cycle phase. Corruption from motion, poor contact, and physiological interference obscures morphology needed for diagnosis, making signal restoration essential. Existing diffusion approaches condition on corrupted observations alone and must learn cyclic structure implicitly. We instead propose two inductive biases which encode cyclostationarity: a shift-covariant wavelet representation and dense per-sample phase conditioning inferred from the corrupted input. We further introduce a training-free cyclostationarity index that quantifies phase structure and predicts when phase conditioning will help. Finally, we propose antithetic coupling of reverse trajectories to reduce sampling variance while achieving comparable performance with fivefold fewer network evaluations. Across modalities, our results show that explicitly encoding measurable cyclic structure improves physiological time-series restoration.

Real-world time series contain evolving underlying dynamics with irregular variations that lack stable temporal patterns and are often referred to as noise. Existing methods address this mixture by filtering frequencies or suppressing noisy observations. They either miss temporal evolution or risk suppressing useful dynamics. We propose DrafTS, a model-agnostic framework that aims to reduce noise while preserving evolving dynamics through time-aware Decomposition with ResiduAl correction For Time Series. DrafTS uses features derived from instantaneous amplitude and frequency to guide decomposition into a primary component intended to capture underlying dynamics. A task-specific backbone models the primary component, while a lightweight correction module uses residual information to correct the backbone output. Across four time series modeling tasks, DrafTS improves six diverse backbones, demonstrating its effectiveness. Code is at https://github.com/Autumn61q/DrafTS

Temporal systems often exhibit non-stationary behaviour, such as seasonal climate variation or glucose fluctuations in patients with type-1 diabetes. One way to model non-stationarity is through discrete latent regimes, i.e., stationary segments of time. Such systems induce a Markov Switching Model (MSM), a class of Hidden Markov Models with autoregressive dependencies among latent regimes and observed variables. Identifying latent regimes is challenging in the presence of frequent regime switches and nonlinear and non-Gaussian dynamics, particularly when there are instantaneous effects between the variables, e.g., due to slow rates of measurements. In this work, we establish the identifiability of both latent regimes and regime-dependent causal structures under temporal regime dependencies, nonlinear lagged and instantaneous effects, and independent noise from the exponential family. Our identifiability theory subsumes non-temporal mixtures of causal models. Furthermore, we introduce FlowMSM, a regime detection framework that can be paired with any stationary causal discovery method to recover regime-dependent causal structures. Experiments on synthetic benchmarks and a financial economics dataset demonstrate the effectiveness of our approach to detect latent regimes and discover causal structures from non-stationary time series.

Medical time series are central to healthcare, enabling continuous monitoring and supporting timely clinical decisions. Despite recent progress, existing methods struggle to jointly model local-global dynamics and handle nonstationarities like baseline drift, while often failing to capture latent channel interactions. To address these challenges, we propose MedMamba, an end-to-end architecture that integrates state space models with domain-specific inductive biases. Specifically, MedMamba first employs multi-scale convolutional embeddings to capture discriminative local morphology. Second, to mitigate nonstationarity, we introduce a tri-branch differential state space encoder that processes raw, temporal-difference, and frequency-domain views, fusing them to emphasize informative patterns while suppressing drift. Furthermore, to uncover latent channel correlations, we design a spatial graph Mamba module that learns a directed dependency structure regularized toward sparsity and acyclicity, which obviates the need for predefined graphs. Extensive experiments on five real-world datasets demonstrate that MedMamba achieves state-of-the-art performance while maintaining linear computational complexity, and ablation studies validate each component's contribution.Code is available at https://github.com/zhangda1018/MedMamba.

In dynamic acoustic environments with time-varying interferers, effective beamforming requires identifying stationary regions over time. The Capon beamformer, a whitened matched filter constrained to maintain unity gain in the desired direction, theoretically relies on the instantaneous ensemble covariance matrix. Practical implementations rely on the batch Capon (or Sample Matrix Inversion), which estimates the sample covariance matrix (SCM) by averaging over a block of snapshots. This practical approach implicitly assumes that the data within the batch window is stationary and can be coherently combined. In non-stationary settings, a batch approach that averages over fixed or excessively long windows fails, as moving interferers smear the SCM and degrade the beamformer's nulling capabilities. To address this, this paper introduces a temporally segmented distortionless response beamformer. Inspired by the segmented least squares method, which fits piecewise polynomials to data while penalizing excessive segmentation to prevent overfitting, the framework extends practical Capon beamforming by incorporating data-driven temporal segmentation. This formulation minimizes output power while dynamically adapting the SCM estimation windows to local stationarity, offering a principled approach to tracking time-varying interferers.

Modeling the dynamics of non-stationary stochastic systems requires balancing the representational power of deep learning with the mathematical transparency of classical models. While classical Markov transition operators provide explicit, theoretically grounded rules for system evolution, their empirical estimation collapses due to severe data sparsity when applied to high-resolution, high-noise environments. We explore this statistical barrier using financial time series as a canonical, real-world testbed. To overcome the degeneracy of empirical counting, we introduce a framework that utilizes neural networks strictly as parameterization engines to generate explicit, time-varying Markov transition matrices. By constraining the neural network to output its predictions as a formal stochastic operator, we maintain complete structural interpretability. We demonstrate that these learned operators successfully capture complex regime shifts: the state-conditioned model achieves mean row heterogeneity $\barρ = 0.0073$ while the state-free ablation collapses to exactly zero, and operator row entropy correlates with realized variance at $r = -0.62$ ($p \approx 10^{-251}$), revealing that high-volatility regimes homogenize transition dynamics rather than diversify them. Furthermore, rather than enforcing the Chapman-Kolmogorov equations as a rigid structural requirement, we repurpose them as a localized diagnostic tool to pinpoint specific temporal windows where first-order memory assumptions break down. Ultimately, this framework demonstrates how neural networks can be constrained to make rigorous, classical operator analysis viable for complex real-world time series.

Time-varying dependence is often modeled with dynamic correlations or Gaussian graphical models, but multivariate systems can change through tail behavior, asymmetry, or conditional structure even when correlations are nearly stable. We introduce Dynamic Vine Copulas (DVC), a temporal vine-copula framework for estimating and diagnosing sequence-wide non-Gaussian dependence. DVC fixes a chosen vine factorization for comparability; the framework applies to C-, D-, and R-vines, and our experiments use fixed-root-order C-vines. Pair-copula states evolve through smooth parameter trajectories or temporally regularized family-switching paths. The main diagnostic is a held-out comparison between a full vine and its matched 1-truncated version, which separates flexible first-tree pairwise dependence from evidence contributed by higher-tree conditional terms. At the population level, under a correct fixed vine and the simplifying assumption, this contrast equals the higher-tree component of a vine total-correlation decomposition; in finite samples, it is a predictive diagnostic. In controlled benchmarks, DVC detects Student-t degrees-of-freedom changes, Clayton-to-Gumbel switches, and recurrent conditional-interaction episodes missed or conflated by Gaussian dynamic baselines. The higher-tree score remains near zero in pairwise-only regimes and rises during conditional-interaction regimes. On Allen Visual Behavior Neuropixels data, DVC identifies a reproducible time-indexed higher-tree signal that is positive across held-out splits and vanishes under a decorrelated null, indicating simultaneous cross-area dependence. DVC therefore provides a flexible temporal copula model and an interpretable test of whether temporal dependence changes are pairwise or conditional.

Real-world time series forecasting faces the fundamental challenge of non-stationary statistical properties, including shifts in mean and variance over time. While reversible instance normalization (RevIN) has shown promise by stationarizing inputs and denormalizing outputs, it relies on the strong assumption that historical and future distributions remain identical. We observe that in many practical applications, distribution shifts follow cyclical patterns that correlate with periodic positions (e.g., seasonal and holiday volatility). To this end, we propose PAMod, a lightweight yet powerful framework that models cyclical distribution shifts via Phase-Amplitude Modulation in the normalized feature space. PAMod learns periodic embeddings to modulate representations: phase modulation captures mean shifts, while amplitude modulation adapts to variance changes. Crucially, we prove mathematically that modulating in normalized space is equivalent to applying dynamic denormalization, offering an elegant unification of distribution adaptation and representation learning. Extensive experiments on twelve real-world benchmarks demonstrate that PAMod achieves state-of-the-art performance with fewer computational resources. Furthermore, our modulation mechanism, as a novel plug-and-play technique, can improve existing time-series forecasting methods with simple integration.

The widespread availability of complex time series data in various domains such as environmental science, epidemiology, and economics demands robust causal discovery methods that can identify intricate contemporaneous and lagged relationships in non-stationary, nonlinear, and noisy settings. Existing constraint-based methods often rely heavily on conditional independence tests that degrade for limited data samples and complex distributions, while score-based methods impose strong statistical assumptions. Recent methods address special cases such as change point detection or distribution shifts, but struggle to provide a unified solution. We propose the Transformer Integrated Temporal Causal Discovery (TTCD) Framework, a novel end-to-end approach that learns contemporaneous and lagged causal relations from non-stationary time series. TTCD introduces a Non-Stationary Feature Learner integrating temporal and frequency-domain attention with dynamic non-stationarity profiling, and a custom Causal Structure Learner. A key innovation is reconstruction-guided causal signal distillation, to distill essential causal signals through the reconstruction process of the transformer decoder, which mitigates noise and spurious correlations while preserving meaningful dependencies. The Causal Structure Learner operates on distilled reconstructed signals to infer the underlying causal graph without restrictive assumptions on noise distributions or data generation processes. Experiments on synthetic, benchmark, and real world datasets show that TTCD consistently outperforms state-of-the-art baselines in both accuracy and consistency with domain knowledge, demonstrating the approach's effectiveness for causal discovery in challenging real world contexts.

We consider streaming principal component analysis when the stochastic data generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world datasets.

Conventional acoustic beamformers typically assume short-time stationarity and process frequency bins independently, ignoring inter-frequency correlations. This is suboptimal for almost-periodic noise sources such as engines, fans, and musical instruments: these signals are better modeled as (almost) cyclostationary (ACS) processes with statistically correlated spectral components. This paper introduces the cyclic minimum power distortionless response (cMPDR) beamformer, which extends the conventional MPDR to jointly exploit spatial and spectral correlations. Building on frequency-shifted (FRESH) filtering, it suppresses noise components that are coherent across harmonically related frequencies, reducing residual noise beyond what spatial filtering alone can achieve. To address inharmonicity, where partials deviate from exact integer multiples of a fundamental frequency, we estimate resonant frequencies from a periodogram and derive frequency shifts from their pairwise spacing. Theoretical analysis yields closed-form expressions for residual noise and proves that output power decreases monotonically with the number of cyclic components. Experiments on synthetic harmonic noise and real UAV motor recordings confirm these findings: in low-SNR scenarios, the cMPDR achieves up to 5 dB improvement in SI-SDR over the MPDR, yields consistent STOI gains, and remains effective with a single microphone. When spectral correlation is absent, the method reduces to conventional MPDR and does not degrade performance. These results suggest that cyclic processing is a viable direction for acoustic noise reduction that deserves further investigation. Code and audio samples are available at https://github.com/Screeen/cMPDR.