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Learning with Calibration: Exploring Test-Time Computing of Spatio-Temporal Forecasting

Wei Chen, Yuxuan Liang

TL;DR

The paper addresses non-stationary shifts in spatio-temporal forecasting by introducing ST-TTC, a lightweight test-time calibration framework. It combines a Spectral Domain Calibrator that applies per-band amplitude and phase corrections in the Fourier domain after a backbone predictor, with a Flash Gradient Update that updates calibration parameters online using a streaming memory queue and a single-step gradient rule. The approach leverages frequency-domain representations to capture periodic biases efficiently, using a grouped parameterization over $M=\tfrac{T}{2}+1$ frequency bins divided into $G$ groups, and updates parameters with a small learning rate $\eta$ to ensure stable descent. Extensive experiments across 14 real-world STF datasets and diverse backbones show consistent gains in MAE, RMSE, and MAPE, improved robustness to distribution shifts, and favorable efficiency compared with baselines, highlighting the method's universality and practicality for real-time forecasting tasks. Mathematical notation is kept in $...$ delimiters to ensure precise interpretation of FFT-based calibration and update dynamics.

Abstract

Spatio-temporal forecasting is crucial in many domains, such as transportation, meteorology, and energy. However, real-world scenarios frequently present challenges such as signal anomalies, noise, and distributional shifts. Existing solutions primarily enhance robustness by modifying network architectures or training procedures. Nevertheless, these approaches are computationally intensive and resource-demanding, especially for large-scale applications. In this paper, we explore a novel test-time computing paradigm, namely learning with calibration, ST-TTC, for spatio-temporal forecasting. Through learning with calibration, we aim to capture periodic structural biases arising from non-stationarity during the testing phase and perform real-time bias correction on predictions to improve accuracy. Specifically, we first introduce a spectral-domain calibrator with phase-amplitude modulation to mitigate periodic shift and then propose a flash updating mechanism with a streaming memory queue for efficient test-time computation. ST-TTC effectively bypasses complex training-stage techniques, offering an efficient and generalizable paradigm. Extensive experiments on real-world datasets demonstrate the effectiveness, universality, flexibility and efficiency of our proposed method.

Learning with Calibration: Exploring Test-Time Computing of Spatio-Temporal Forecasting

TL;DR

The paper addresses non-stationary shifts in spatio-temporal forecasting by introducing ST-TTC, a lightweight test-time calibration framework. It combines a Spectral Domain Calibrator that applies per-band amplitude and phase corrections in the Fourier domain after a backbone predictor, with a Flash Gradient Update that updates calibration parameters online using a streaming memory queue and a single-step gradient rule. The approach leverages frequency-domain representations to capture periodic biases efficiently, using a grouped parameterization over frequency bins divided into groups, and updates parameters with a small learning rate to ensure stable descent. Extensive experiments across 14 real-world STF datasets and diverse backbones show consistent gains in MAE, RMSE, and MAPE, improved robustness to distribution shifts, and favorable efficiency compared with baselines, highlighting the method's universality and practicality for real-time forecasting tasks. Mathematical notation is kept in delimiters to ensure precise interpretation of FFT-based calibration and update dynamics.

Abstract

Spatio-temporal forecasting is crucial in many domains, such as transportation, meteorology, and energy. However, real-world scenarios frequently present challenges such as signal anomalies, noise, and distributional shifts. Existing solutions primarily enhance robustness by modifying network architectures or training procedures. Nevertheless, these approaches are computationally intensive and resource-demanding, especially for large-scale applications. In this paper, we explore a novel test-time computing paradigm, namely learning with calibration, ST-TTC, for spatio-temporal forecasting. Through learning with calibration, we aim to capture periodic structural biases arising from non-stationarity during the testing phase and perform real-time bias correction on predictions to improve accuracy. Specifically, we first introduce a spectral-domain calibrator with phase-amplitude modulation to mitigate periodic shift and then propose a flash updating mechanism with a streaming memory queue for efficient test-time computation. ST-TTC effectively bypasses complex training-stage techniques, offering an efficient and generalizable paradigm. Extensive experiments on real-world datasets demonstrate the effectiveness, universality, flexibility and efficiency of our proposed method.

Paper Structure

This paper contains 36 sections, 2 theorems, 20 equations, 9 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Methodology
  5. What to Compute $?$ Spectral Domain Calibrator with Phase-Amplitude Modulation
  6. How to Compute $?$ Flash Gradient Update with Streaming Memory Queue
  7. Experiments
  8. Experimental Setup
  9. Effectiveness Study (RQ1)
  10. Universality Study (RQ2)
  11. Flexibility Study (RQ3)
  12. Mechanism & Robustness Study (RQ4)
  13. Efficiency & Lightweight Study (RQ5)
  14. Conclusion
  15. More Related Work
  16. ...and 21 more sections

Key Result

Theorem 1

Let $Y \in \mathbb{C}^{B \times N \times M}$ be the original frequency-domain representation of the backbone’s prediction $y \in \mathbb{R}^{B \times N \times T}$, and $y' \in \mathbb{R}^{B \times N \times T}$ be the calibrated output. Suppose the amplitude and phase modulation parameters satisfy $| where $\|Y\|_2$ is the $\ell_2$-norm of $Y$.

Figures (9)

  • Figure 1: Conceptual visualization comparison of different spatio-temporal learning paradigms under test environment. (a) Test-Time Training requires the use of additional pretext tasks in the training and test phases to optimize the self-supervision head or the overall model parameters $f_{\theta}$. (b) Online Continual Learning, by optimizing some internal parameters $f_{\theta(\delta)}$ of the model, requires additional modifications to the internal architecture of the network. Our (c) Test-Time Computing method only requires a lightweight calibrator $g_{\theta}$, which is a seamless and lightweight plug-and-play module.
  • Figure 2: Relative improvements of different models w/ ST-TTC in the few-shot setting.
  • Figure 3: Left: relative improvement of long-term setting. Right: visualization study of PEMS-08.
  • Figure 4: Performance on LargeST.
  • Figure 5: Relative improvement using our ST-TTC in the OOD learning setting.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Theorem 1: Approximate Bound on Output Perturbation
  • proof
  • Remark 1
  • Proposition 2: Controlled Descent on Streaming Memory Queues
  • proof
  • Remark 2