time series modelers

Abstract: Learning latent dynamics is central to assessing the current states and forecasting the future trajectories of high-dimensional time series. For locally-stationary latent dynamics $\mathcal{F}(\mathbf{z}_{<t}; \mathbf{c})$ with latent dynamics state $\mathbf{z}_t$ and environment variable $\mathbf{c}$, prior identifiability results have largely focused on $\mathbf{z}_t$ when conditioned on predefined label $u$ of dynamics environments. This leaves two limitations: reliance on predefined labels that hinder generalization to unseen environments, and a limited understanding of the identifiability of $\mathcal{F}$ and $\mathbf{c}$ which -- although offering important structural properties for the identifiability of $\mathbf{z}_t$ -- are learned jointly with $\mathbf{z}_t$. We address these limitations with Meta-iLaD, a novel identifiable latent dynamics framework attained by meta-learning across dynamics environments. Meta-iLaD introduces a novel conditional prior of $\mathbf{c}$, modeled as a feedforward meta-learner to rapidly extract $\mathbf{c}$ from few-shot examples. Meta-iLaD further establishes identifiability for $\mathbf{z}_t$, $\mathbf{c}$, and $\mathcal{F}$ simultaneously, for a general formulation of $\mathcal{F}(\mathbf{z}_{< t};\mathbf{c})$ without restricting the dimension of $\mathbf{c}$ or how it modulates $\mathcal{F}$. On synthetic and real data, we provide strong empirical evidence that 1) conditioning on few-shot examples enables generalization to out-of-distribution environments, and 2) identifiability for $\mathbf{c}$ and $\mathcal{F}$ is critical for accurate forecasting beyond reconstructing observed trajectories. Code is available at https://github.com/GoForIt-YYB/Meta-iLaD.

Abstract: In reservoir computing, performance depends on whether the fixed dynamics of the reservoir suit the task, and the appropriate memory length and frequency response differ across tasks. The reservoir settings are found by a search on each task, and driving the reservoir with the input and collecting its states for every candidate setting dominates the cost of the search. In a reservoir with a linear recurrence, the dynamics decompose into modes, one per eigenvalue, and the eigenvalue distribution becomes a design variable. We propose the Sector-Ring Linear Reservoir (SRLR), in which the eigenvalue angles are fixed at evenly spaced positions around a circle and one decay timescale, chosen from a few values, is assigned to each sector of consecutive angles. To keep the scale of each mode's state equal across candidates, the input weights are scaled using the power spectrum of the training input. To reduce the cost of the search, the states of the modes for each decay-timescale value are computed once and reused across candidates. On prediction and reconstruction benchmarks built from chaotic systems, the SRLR achieved, in all eight cases, a lower mean test error than a reference model in which all modes share one decay timescale, and was competitive with nonlinear echo state network and leaky-integrator echo state network baselines. The selected decay-timescale profiles changed with the prediction target and horizon. The state reuse reduced the number of computed mode trajectories in the search by 99.8% relative to per-candidate recomputation.

Abstract: Latent factors in dynamical systems can play different roles across observations and time. For example, an idea in a group meeting may be expressed by only some participants, inherit information from earlier discussions, and shape what is said next. We formalize this notion as the structural role of a latent factor, describing where it is expressed, whether it receives influence from the past, and whether it transmits influence to the future. The main challenge is correspondence, since the three structures may be identified under different permutations, while a structural role is defined only when they are aligned to the same latent coordinate. We show that, under suitable structural and variability conditions, all three roles are jointly identifiable under a common permutation at each time point. Guided by this theory, we develop RoleTrace, a framework that traces structural roles in a shared latent coordinate system. Synthetic experiments validate structural recovery, while experiments on multiparty meetings examine the identified roles in real sequential interactions.

Abstract: Expectiles are tail-sensitive distributional summaries with important applications in financial risk measurement and forecasting, yet statistical theory for their online estimation from dependent data remains limited. We study a one-pass, stochastic-gradient algorithm with constant learning rate $γ$ for estimating the marginal expectile of a stationary causal process. The recursion admits a unique stationary causal solution and geometrically forgets its initialization. Under short-range dependence, its last iterate is asymptotically Gaussian at the $\sqrtγ$ scale, with variance governed by the population curvature and the long-run variance of the expectile score. For long-memory linear processes the scale changes to $γ^{β-1/2}\ell(1/γ)$, with a Gaussian limit arising from a fractional Brownian motion integral. We further derive the joint long-memory limit for multiple expectile levels and characterize its limiting covariance structure. Various numerical experiments corroborate the predicted Gaussian approximations and regime-specific fluctuation scales.

Abstract: In autoregressive forecasting, long prediction rollouts provide distant supervision, but backpropagation through time (BPTT) carries gradients from those losses through many autoregressive steps. Repeated Jacobian products can make distant gradients dominate the update while amplifying predictable signal and unpredictable noise together; a large distant gradient therefore need not carry reliable learning signal. Motivated by this observation, we introduce Internal Dual-Wiener routing (Internal-DW), a principled backward-only intervention that preserves the full forward rollout and all horizon losses while reliability-weighting internal gradient routes. At each residual block, we derive bounded Wiener gains for the identity and nonlinear routes that balance preserving predictable learning signal against suppressing unpredictable variation, and estimate them from route-level gradient statistics and an explicit noise model. In a controlled system with known gradient signal-to-noise ratio (SNR), we show that distant gradients can grow even as their SNR falls, and that Internal-DW reduces held-out error in recovering predictable gradient signals and improves forecasting. On four history-dominated, weak-drive testbeds, Internal-DW reduces forecast error by 5.2%-13.8% relative to full BPTT, outperforms gradient clipping and Jacobian regularization on all four, and outperforms validation-selected truncated BPTT (TBPTT) on three. It also extends or preserves the fitted optimal training-horizon range across these four testbeds. Across the full benchmark suite, the current Internal-DW estimator has a clear applicability boundary: its benefit diminishes or reverses when usable history is limited or when the selected sampler fails to represent dominant drive-dependent variation. The results show that retaining long-horizon supervision does not require trusting every backward contribution equally.