Uniform Inductive Spatio-Temporal Kriging

Lewei Xie; Haoyu Zhang; Yulong Chen; Liangjun You; Zongxian Yang; Yifan Zhang

cs.AIMar 5, 2026

Uniform Inductive Spatio-Temporal Kriging

Authors: Lewei Xie, Haoyu Zhang, Yulong Chen, Liangjun You, Zongxian Yang, Yifan Zhang

Organizations: City University of Hong Kong (Dongguan) · City University of Hong Kong

Abstract

Inductive spatio-temporal kriging infers signals at unobserved locations from observed sensors, but real-world observations are often incomplete and exhibit block-wise missingness caused by failures, interruptions, or maintenance. A common impute-then-krige pipeline suffers from objective mismatch: better reconstruction on observed sensors does not necessarily improve downstream kriging, and value-dependent imputation bias can be propagated to unobserved nodes. We propose UniSTOK, a plug-and-play framework for inductive spatio-temporal kriging under incomplete observations. We first introduce Reliability-guided Signal Regulation (RSR), which estimates entry-wise reliability from temporal continuity and spatial support, and uses it to regulate the input signals so that reliable observations are emphasized while long-gap or weakly supported entries are suppressed before spatial propagation. We further introduce Residual Bias Calibration (RBC), which estimates value-conditioned residual prototypes after the main predictor converges and learns context-correction amplitudes to adaptively calibrate systematic over- or under-estimation in final kriging predictions. Extensive experiments on real-world datasets show that UniSTOK consistently improves multiple kriging backbones.

Figures & tables

Figure 1: Overview of the studied setting, key challenge, and proposed solution.

Figure 2: Overview of UniStok , which consists of three components: Reliability-guided Signal Regulation (RSR), the shared-backbone main predictor, and Residual Bias Calibration (RBC).

Models		METR-LA			PEMS-BAY			NREL-AL		Avg. Improv.(%)
Models		MAE	RMSE	MAPE	MAE	RMSE	MAPE	MAE	RMSE	Avg. Improv.(%)
DCRNN ICLR’18	Base	6.20	9.89	17.30	3.70	6.76	8.86	2.11	4.23	–
DCRNN ICLR’18	+Ours	5.93 $\downarrow$ 4.4	9.48 $\downarrow$ 4.1	16.34 $\downarrow$ 5.5	3.50 $\downarrow$ 5.4	6.70 $\downarrow$ 0.9	8.71 $\downarrow$ 1.7	2.08 $\downarrow$ 1.4	4.06 $\downarrow$ 4.0	3.4
STGCN IJCAI’18	Base	6.28	10.36	17.42	3.68	6.62	8.72	2.25	4.34	–
STGCN IJCAI’18	+Ours	5.76 $\downarrow$ 8.3	9.40 $\downarrow$ 9.3	16.30 $\downarrow$ 6.4	3.40 $\downarrow$ 7.6	6.59 $\downarrow$ 0.5	8.43 $\downarrow$ 3.3	1.97 $\downarrow$ 12.4	3.85 $\downarrow$ 11.3	7.4
GMAN AAAI’20	Base	6.44	11.27	17.98	4.01	7.58	9.12	1.71	3.24	–

Table 1: Performance comparison of plug-in enhancement across backbones and datasets. Lower is better. Gray : base model results; Black : results with our plug-in ( +Ours ).

\downarrow

x : relative improvement (%) of +Ours over Base.

Model		METR-LA	PEMS-BAY	NREL-AL
ImputeFormer+STA-GANN	MAE	$6.04\pm 0.15$	$3.84\pm 0.05$	$1.91\pm 0.05$
	RMSE	$9.34\pm 0.23$	$6.76\pm 0.09$	$3.95\pm 0.19$
	MAPE	$18.11\pm 0.88$	$9.14\pm 0.05$	–
UniStok	MAE	5.84 $\pm$ 0.05	3.37 $\pm$ 0.00	1.76 $\pm$ 0.01
	RMSE	9.22 $\pm$ 0.05	6.09 $\pm$ 0.03	3.41 $\pm$ 0.03
	MAPE	16.86 $\pm$ 0.13	8.32 $\pm$ 0.04	–

Table 2: Comparison with a two-stage pre-imputation pipeline.

Figure 3: Ablation study of the core modules on three datasets. Full denotes the complete UniStok framework; w/o RSR removes reliability-guide signal regulation; w/o RBC removes residual bias calibration; Original denotes the backbone without our framework.

Figure 4: Hyperparameter sensitivity of UniStok with IGNNK on METR-LA, varying the number of residual bins $K$ , the residual correction bound $\eta$ , and the reliability regulation strength $\alpha$ .

Figure 5: Parameter–accuracy trade-off on METR-LA. Empty and filled circles denote base backbones and their UniStok -enhanced variants, respectively.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Notation	Definition
Data & Task
$\mathcal{G}=(\mathcal{V},\mathcal{E})$	Sensor graph with node set $\mathcal{V}$ and edge set $\mathcal{E}$
$\mathcal{V}_{o}$ / $\mathcal{V}_{u}$	Observed / unobserved node sets, with $\|\mathcal{V}_{o}\|=N$ and $\|\mathcal{V}_{u}\|=U$
$\mathcal{X}\in\mathbb{R}^{N\times T}$	Historical measurements at observed nodes over the full time horizon
$\mathcal{Y}\in\mathbb{R}^{U\times T}$	Ground-truth signals at unobserved nodes
$\mathcal{A}\in[0,1]^{(N+U)\times(N+U)}$	Weighted adjacency matrix of the whole sensor graph

Appendix

Table 3: Notations used in UniStok .

Figure 6: Relative MAE degradation under random and block missingness.

Figure 7: Stage-1 imputation MAE on observed sensors under block missingness.

Dataset	Backbone	Best Method	Best MAE	Worst Method	Worst MAE
METR-LA	IGNNK	Linear	6.230	KNN	6.434
	KITS	Zero	6.548	ImputeFormer	6.781
	STGCN	Zero	6.340	Mean	6.545
	STA-GANN	Zero	6.229	ImputeFormer	6.641
	DARKFARSEER	Mean	6.375	Zero	6.793
	INCREASE-SP	SAITS	6.201	Zero	6.392

Appendix

Table 4: Stage-2 kriging performance of the imputation-first pipeline under block missingness. We report the best and worst imputation method for each dataset-backbone pair.

Figure 8: Comparison between Stage-1 reconstruction ranking and Stage-2 kriging ranking. For each dataset, rows list the imputation methods ordered by reconstruction MAE, and each cell shows the downstream kriging rank under a specific ISK backbone.

Method	GME	MAE	Low CB	Mid CB	High CB
Zero	+0.37	9.92	+15.33	-4.81	-9.40
Mean	-0.18	7.74	+9.60	-4.51	-5.63
Linear	-0.38	8.08	+7.49	-4.25	-4.38
KNN	+0.67	7.84	+9.28	-2.05	-5.21
GRIN	+1.80	5.51	+7.97	-0.40	-2.17
SAITS	+1.38	4.80	+7.08	-1.04	-1.89

Appendix

Table 5: Stage-1 imputation bias on simulated missing entries of observed nodes. Positive values indicate overestimation, and negative values indicate underestimation.

Backbone	Setting	GME	Low-value CB	High-value CB
DARKFARSEER	No missing	2.10	6.85	-2.10
DARKFARSEER	Imputation	[4.40, 4.60]	[12.16, 13.42]	[-4.25, -4.12]
IGNNK	No missing	1.20	5.35	-1.85
IGNNK	Imputation	[2.49, 2.86]	[11.57, 12.20]	[-3.72, -3.42]
INCREASE-SP	No missing	1.35	5.80	-1.90
INCREASE-SP	Imputation	[2.47, 2.81]	[10.94, 12.39]	[-3.85, -3.40]

Appendix

Table 6: Kriging bias under the no-missing reference and block-wise incomplete observations. The no-missing setting uses original observed values without imputation, while the imputation setting reports the range across seven imputation methods. Positive values indicate overestimation, and negative values indicate underestimation.

Models		METR-LA			PEMS-BAY			NREL-AL		Avg. Improv.(%)
Models		MAE	RMSE	MAPE	MAE	RMSE	MAPE	MAE	RMSE	Avg. Improv.(%)
DCRNN ICLR’18	Base	6.03	9.51	17.09	3.58	6.64	8.75	1.98	3.87	–
DCRNN ICLR’18	+Ours	5.82 $\downarrow$ 3.5	9.44 $\downarrow$ 0.7	16.21 $\downarrow$ 5.1	3.41 $\downarrow$ 4.7	6.57 $\downarrow$ 1.1	8.35 $\downarrow$ 4.6	1.76 $\downarrow$ 11.1	3.25 $\downarrow$ 16.0	5.9
STGCN IJCAI’18	Base	6.04	9.65	17.32	3.59	6.70	8.83	2.01	3.93	–
STGCN IJCAI’18	+Ours	5.72 $\downarrow$ 5.3	9.34 $\downarrow$ 3.2	16.20 $\downarrow$ 6.5	3.34 $\downarrow$ 7.0	6.46 $\downarrow$ 3.6	8.22 $\downarrow$ 6.9	1.66 $\downarrow$ 17.4	3.30 $\downarrow$ 16.0	8.2
GMAN AAAI’20	Base	6.37	11.10	17.62	3.92	7.34	8.99	1.69	3.13	–

Appendix

Table 7: Performance comparison under random missingness across backbones and datasets. Lower is better. Gray : base model results; Black : results with our plug-in ( +Ours ).

\downarrow

x : relative improvement (%) of +Ours over Base.

Models		METR-LA			PEMS-BAY			NREL-AL		Avg. Improv.(%)
Models		MAE	RMSE	MAPE	MAE	RMSE	MAPE	MAE	RMSE	Avg. Improv.(%)
DCRNN ICLR’18	Base	6.01	9.47	17.02	3.54	6.61	8.67	1.90	3.72	–
DCRNN ICLR’18	+Ours	5.75 $\downarrow$ 4.3	9.42 $\downarrow$ 0.5	16.13 $\downarrow$ 5.2	3.38 $\downarrow$ 4.5	6.53 $\downarrow$ 1.2	8.36 $\downarrow$ 3.6	1.74 $\downarrow$ 8.4	3.24 $\downarrow$ 12.9	5.1
STGCN IJCAI’18	Base	6.07	9.63	17.29	3.59	6.71	8.86	1.93	3.88	–
STGCN IJCAI’18	+Ours	5.69 $\downarrow$ 6.3	9.26 $\downarrow$ 3.8	16.11 $\downarrow$ 6.8	3.32 $\downarrow$ 7.5	6.44 $\downarrow$ 4.0	8.13 $\downarrow$ 8.2	1.62 $\downarrow$ 16.1	3.29 $\downarrow$ 15.2	8.5
GMAN AAAI’20	Base	6.32	10.74	17.55	3.94	7.40	8.95	1.65	3.08	–

Appendix

Table 8: Performance comparison under the no-missing setting across backbones and datasets. Lower is better. Gray : base model results; Black : results with our plug-in ( +Ours ).

\downarrow

x : relative improvement (%) of +Ours over Base.

Metric	Setting	DCRNN	STGCN	GMAN	IGNNK	SATCN	INCREASE	KITS	STA-GANN	DarkFarseer
MAE	Base	74.23	78.52	97.37	78.55	79.36	76.28	85.09	76.07	99.23
MAE	+Ours	72.60 $\downarrow$ 2.2	76.99 $\downarrow$ 1.9	93.12 $\downarrow$ 4.4	77.60 $\downarrow$ 1.2	78.27 $\downarrow$ 1.4	74.85 $\downarrow$ 1.9	78.16 $\downarrow$ 8.1	75.85 $\downarrow$ 0.3	97.86 $\downarrow$ 1.4
RMSE	Base	112.59	114.85	143.81	115.29	115.38	115.37	130.74	113.17	146.18
RMSE	+Ours	111.46 $\downarrow$ 1.0	113.40 $\downarrow$ 1.3	138.04 $\downarrow$ 4.0	114.50 $\downarrow$ 0.7	113.96 $\downarrow$ 1.2	110.71 $\downarrow$ 4.0	117.22 $\downarrow$ 10.3	110.63 $\downarrow$ 2.2	143.67 $\downarrow$ 1.7

Appendix

Table 9: Performance comparison on NZ-Highway with naturally occurring block-wise missing observations. Lower is better. Gray : base model results; Black : results with our plug-in ( +Ours ).

\downarrow

x : relative improvement (%) of +Ours over Base.

Panel of Figure 9 — Figure 9: Module-level ablation results based on the average relative degradation over the full model. The RSR variants include w/o temporal reliability , w/o spatial reliability , and w/o residual compensation . The fusion variants include regulated-only , M as second branch , and w/o gate . The RBC variants include w/o context amplitude , w/o bin-balanced loss , and hard-bin RBC .

Panel of Figure 10 — Figure 10: Additional hyperparameter sensitivity results of UniStok with IGNNK as the backbone. We analyze the number of residual bins $K$ in RBC, the residual correction bound $\eta$ , and the reliability regulation strength $\alpha$ in RSR. Lower MAE and RMSE indicate better performance.

Figure 11: Additional visualization of RSR and adaptive fusion on four representative samples. Each column corresponds to one sample. From top to bottom, the three rows show the observation mask $M$ , the RSR-induced input difference $\widetilde{X}-X$ , and the adaptive fusion gate $G$ . Larger $G$ indicates stronger reliance on the RSR-enhanced branch.

Figure 12: From left to right: ground-truth values, predictions before RBC, predictions after RBC, and learned correction amplitudes.

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