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UniSTOK: Uniform Inductive Spatio-Temporal Kriging

Lewei Xie, Haoyu Zhang, Juan Yuan, Liangjun You, Yulong Chen, Yifan Zhang

TL;DR

UniSTOK is a plug-and-play framework that enhances existing inductive kriging backbones under missing observation that forms a dual-branch input consisting of the original observations and a jigsaw-augmented counterpart that synthesizes proxy signals only at missing entries.

Abstract

Spatio-temporal kriging aims to infer signals at unobserved locations from observed sensors and is critical to applications such as transportation and environmental monitoring. In practice, however, observed sensors themselves often exhibit heterogeneous missingness, forcing inductive kriging models to rely on crudely imputed inputs. This setting brings three key challenges: (1) it is unclear whether an value is a true signal or a missingness-induced artifact; (2) missingness is highly heterogeneous across sensors and time; (3) missing observations distort the local spatio-temporal structure. To address these issues, we propose Uniform Inductive Spatio-Temporal Kriging (UniSTOK), a plug-and-play framework that enhances existing inductive kriging backbones under missing observation. Our framework forms a dual-branch input consisting of the original observations and a jigsaw-augmented counterpart that synthesizes proxy signals only at missing entries. The two branches are then processed in parallel by a shared spatio-temporal backbone with explicit missingness mask modulation. Their outputs are finally adaptively fused via dual-channel attention. Experiments on multiple real-world datasets under diverse missing patterns demonstrate consistent and significant improvements.

UniSTOK: Uniform Inductive Spatio-Temporal Kriging

TL;DR

UniSTOK is a plug-and-play framework that enhances existing inductive kriging backbones under missing observation that forms a dual-branch input consisting of the original observations and a jigsaw-augmented counterpart that synthesizes proxy signals only at missing entries.

Abstract

Spatio-temporal kriging aims to infer signals at unobserved locations from observed sensors and is critical to applications such as transportation and environmental monitoring. In practice, however, observed sensors themselves often exhibit heterogeneous missingness, forcing inductive kriging models to rely on crudely imputed inputs. This setting brings three key challenges: (1) it is unclear whether an value is a true signal or a missingness-induced artifact; (2) missingness is highly heterogeneous across sensors and time; (3) missing observations distort the local spatio-temporal structure. To address these issues, we propose Uniform Inductive Spatio-Temporal Kriging (UniSTOK), a plug-and-play framework that enhances existing inductive kriging backbones under missing observation. Our framework forms a dual-branch input consisting of the original observations and a jigsaw-augmented counterpart that synthesizes proxy signals only at missing entries. The two branches are then processed in parallel by a shared spatio-temporal backbone with explicit missingness mask modulation. Their outputs are finally adaptively fused via dual-channel attention. Experiments on multiple real-world datasets under diverse missing patterns demonstrate consistent and significant improvements.
Paper Structure (39 sections, 3 theorems, 28 equations, 13 figures, 3 tables)

This paper contains 39 sections, 3 theorems, 28 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. In-Sample Imputation
  4. Out-of-Sample Imputation
  5. Problem Definition
  6. Spatio temporal Kriging
  7. Inductive Spatio temporal Kriging
  8. Method
  9. Overview
  10. Virtual-Node Jigsaw Mechanism
  11. Window Encoding Modeling
  12. Node Encoding Modeling
  13. Selection of Replaced and Replacement Sets
  14. Jigsaw Construction and Replacement
  15. Missingness Mask Modulation
  16. ...and 24 more sections

Key Result

proposition 1

Let $\mathcal{M} \subset \mathbb{R}^n$ be the manifold of sensor observations equipped with the Euclidean metric. Under random missingness with rate $p \in (0,1)$ and deterministic imputation $z \in \mathbb{R}^n$, the observed manifold $\tilde{\mathcal{M}}$ satisfies:

Figures (13)

  • Figure 1: A introdution for ISK under observed missing.
  • Figure 2: Framework of UniSTOK
  • Figure 3: Performance under random, mixed, and block missingness, and the relative gains brought by Ours.
  • Figure 4: Relative error reduction (%) brought by UniSTOK when applied to different ISK backbones on METR-LA, PEMS-BAY, and NREL.
  • Figure 5: Ablation study (INCREASE backbone, mixed missingness). Removing Jigsaw or Mask modulation causes the largest degradation; Attention fusion and the auxiliary loss provide additional improvements.
  • ...and 8 more figures

Theorems & Definitions (3)

  • proposition 1: Geometric Distortion under Missing Data
  • proposition 2: Optimal Transport Reconstruction Bound
  • proposition 3: Information Gain from Mask Modeling