DRAN: A Distribution and Relation Adaptive Network for Spatio-temporal Forecasting

Xiaobei Zou; Luolin Xiong; Kexuan Zhang; Cesare Alippi; Yang Tang

cs.LGApr 2, 2025

DRAN: A Distribution and Relation Adaptive Network for Spatio-temporal Forecasting

Authors: Xiaobei Zou, Luolin Xiong, Kexuan Zhang, Cesare Alippi, Yang Tang

Organizations: Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China · Faculty of Informatics, Università della Svizzera italiana, 69000 Lugano, Switzerland · Department of Electronics, Information and Bioengineering, Politecnico di Milano, 20133 Milan, Italy

Abstract

Spatio-temporal forecasting remains challenging under non-stationary environments because both data distributions and spatial relations evolve over time. Temporal normalization and de-normalization are widely used to mitigate distribution shifts, but they may distort inter-node relationships and thereby impair spatial dependency modeling. To address these issues, we propose the Distribution and Relation Adaptive Network (DRAN) for spatio-temporal forecasting. DRAN incorporates a Spatial Factor Learner (SFL) module, which enables effective normalization and de-normalization while preserving spatial dependencies in spatio-temporal systems. To model evolving spatial interactions, DRAN further proposes the Dynamic-Static Fusion Learner (DSFL) module. DSFL decomposes features into static and dynamic components and adaptively fuses them according to input variability. Experiments on six benchmark datasets show that DRAN outperforms state-of-the-art baselines. Additional analyses demonstrate that SFL consistently reduces spatial-relation distortion across multiple normalization schemes, whereas DSFL captures complementary static and dynamic dependencies and adjusts their contributions according to temporal variability.

Figures & tables

Fig. 1: Time variance in spatio-temporal systems and corresponding distribution adaptation strategies. (a) Temporal variations of a spatio-temporal system, including shifts in temporal distributions and changes in node connectivity. The color of each node represents the mean value of the node variables (consistent with the colorbar), while the vertical size of each ellipse indicates the variance. (b) Comparison between conventional normalization and de-normalization strategies for distribution adaptation, which may disrupt the effectiveness of spatial relational learning, and the proposed SFL module for preserving spatial distribution structures. “Norm.” and “De-norm.” denote normalization and de-normalization, respectively.

Fig. 2: The overall architecture of DRAN. DRAN follows a temporal-spatial learning paradigm. SFL facilitates the normalization and de-normalization processes, denoted as “Norm.” and “De-norm.”, respectively. DSFL decomposes the representations into dynamic and static components, and models spatial dependencies through attention mechanisms together with the adaptive adjacency matrix ${\bm{A}}_{\rm St}$ constructed from learnable node embeddings.

Symbol	Description
$L,~H$	Lookback and forecasting length.
$N,~C$	Number of nodes and input variables.
$C^{\prime}$	Latent feature dimension.
$\bm{X}$ , $\hat{\bm{X}}$	Observed and predicted spatio-temporal sequence.
$\bm{\mu}_{X},\bm{\sigma}_{X}$	Temporal mean and standard deviation.
$\bm{H}_{\rm raw}$	Features extracted from unnormalized inputs.

TABLE I: Summary of main notation and hyperparameters.

Hyper -parameter	Weather	NYCBike1	NYCBike2	NYCTaxi	PeMS04	PeMS08
$\alpha$	0.05	0.05	0.05	1	0.001	0.1
$\beta$	0.5	0.5	0.5	0.5	5	0.05

TABLE II: Balanced hyperparameter selection

Attributes	Duration time	Freq.	Node number	Length (In $\rightarrow$ Out)
Weather	01/01/2012 $\sim$ 31/12/2022	1 h	263	24 $\;\rightarrow\;$	12
NYCBike1	01/04/2014 $\sim$ 30/09/2014	30 min	128	19 $\;\rightarrow\;$	1
NYCBike2	01/07/2016 $\sim$ 29/08/2016	30 min	200	35 $\;\rightarrow\;$	1
NYCTaxi	01/01/2015 $\sim$ 01/03/2015	30 min	200	35 $\;\rightarrow\;$	1
PeMS04	01/01/2018 $\sim$ 28/02/2018	5 min	307	12 $\;\rightarrow\;$	12
PeMS08	01/07/2016 $\sim$ 31/08/2016	5 min	170	12 $\;\rightarrow\;$	12

TABLE III: Dataset details

Task type	Methods	Task adaptive	Dynamic adaptive
Time series forecasting	DA-RNN [ 27 ]	✗	✗
	InfoTS [ 54 ]	✓	✓
	AutoTCL [ 55 ]	✓	✓
Spatio-temporal forecasting	TGCN [ 15 ]	✗	✗
	STGCN [ 16 ]	✗	✗
	GCGRU [ 17 ]	✗	✗

TABLE IV: Baseline methods.

Model	Weather			NYCBike1			NYCBike2
Model	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$
DA-RNN [ 27 ]	5.492 ( $\pm$ 0.725)	1.874 ( $\pm$ 0.276)	3.758 ( $\pm$ 0.670)	15.773 ( $\pm$ 2.406)	61.948 ( $\pm$ 7.968)	5.970 ( $\pm$ 0.955)	15.159 ( $\pm$ 3.591)	63.687 ( $\pm$ 9.591)	3.567 ( $\pm$ 1.106)
InfoTS [ 54 ]	1.274 ( $\pm$ 0.137)	0.435 ( $\pm$ 0.053)	0.492 ( $\pm$ 0.035)	6.526 ( $\pm$ 0.336)	33.681 ( $\pm$ 1.831)	0.923 ( $\pm$ 0.044)	6.259 ( $\pm$ 0.368)	30.628 ( $\pm$ 1.626)	0.455 ( $\pm$ 0.044)
AutoTCL [ 55 ]	1.194 ( $\pm$ 0.022)	0.408 ( $\pm$ 0.008)	0.420 ( $\pm$ 0.019)	6.213 ( $\pm$ 0.213)	28.824 ( $\pm$ 0.808)	0.978 ( $\pm$ 0.024)	5.772 ( $\pm$ 0.246)	28.639 ( $\pm$ 0.993)	0.476 ( $\pm$ 0.026)
STGCN [ 16 ]	2.074 ( $\pm$ 1.004)	0.709 ( $\pm$ 0.396)	1.960 ( $\pm$ 0.918)	17.141 ( $\pm$ 0.142)	58.498 ( $\pm$ 1.610)	11.634 ( $\pm$ 1.827)	17.297 ( $\pm$ 0.244)	55.595 ( $\pm$ 0.805)	7.990 ( $\pm$ 0.046)
TGCN [ 15 ]	1.864 ( $\pm$ 0.797)	0.635 ( $\pm$ 0.311)	1.769 ( $\pm$ 0.704)	7.544 ( $\pm$ 0.311)	34.848 ( $\pm$ 1.287)	3.166 ( $\pm$ 0.713)	11.488 ( $\pm$ 8.575)	36.789 ( $\pm$ 7.720)	3.402 ( $\pm$ 3.165)

TABLE V: The prediction results on weather, NYCBike1 and NYCBike2 datasets.

Model	NYCTaxi			PeMS04			PeMS08
Model	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$	MAE $\downarrow$	MAPE(%) $\downarrow$	WD $\downarrow$
DA-RNN [ 27 ]	26.682 ( $\pm$ 8.176)	69.028 ( $\pm$ 12.085)	11.912 ( $\pm$ 9.342)	138.741 ( $\pm$ 19.896)	192.640 ( $\pm$ 37.494)	80.935 ( $\pm$ 12.140)	109.276 ( $\pm$ 13.097)	123.719 ( $\pm$ 33.187)	69.984 ( $\pm$ 11.873)
InfoTS [ 54 ]	13.286 ( $\pm$ 1.695)	21.505 ( $\pm$ 0.562)	1.670 ( $\pm$ 0.038)	25.801 ( $\pm$ 0.408)	20.058 ( $\pm$ 1.193)	11.188 ( $\pm$ 0.644)	23.604 ( $\pm$ 1.243)	14.642 ( $\pm$ 0.583)	12.469 ( $\pm$ 1.371)
AutoTCL [ 55 ]	13.119 ( $\pm$ 1.729)	21.601 ( $\pm$ 0.509)	1.768 ( $\pm$ 0.060)	23.814 ( $\pm$ 0.043)	17.355 ( $\pm$ 0.200)	10.374 ( $\pm$ 0.180)	20.879 ( $\pm$ 0.088)	13.006 ( $\pm$ 0.121)	10.074 ( $\pm$ 0.094)
STGCN [ 16 ]	25.227 ( $\pm$ 4.222)	29.533 ( $\pm$ 5.694)	8.563 ( $\pm$ 10.891)	29.114 ( $\pm$ 5.555)	23.466 ( $\pm$ 7.692)	26.608 ( $\pm$ 4.168)	27.173 ( $\pm$ 10.017)	16.585 ( $\pm$ 7.589)	13.060 ( $\pm$ 11.726)
TGCN [ 15 ]	23.227 ( $\pm$ 10.239)	44.356 ( $\pm$ 10.694)	5.563 ( $\pm$ 6.891)	34.853 ( $\pm$ 0.263)	28.078 ( $\pm$ 0.743)	30.670 ( $\pm$ 0.337)	37.969 ( $\pm$ 0.324)	30.431 ( $\pm$ 0.690)	23.658 ( $\pm$ 0.523)

TABLE VI: The prediction results on NYCTaxi, Pems04 and Pems08 datasets.

Fig. 3: Horizon-wise MAE comparison of DRAN and two strong baselines on Weather, PeMS04, and PeMS08. The curves and shaded regions denote the mean and standard deviation over five random seeds, respectively.

Fig. 4: Visualization of prediction results. (a)–(c) Absolute prediction errors of DRAN, RGSL, and DST-Mamba on the Weather dataset, respectively. (d)–(h) Prediction results on NYCBike1, where (d) shows the ground truth, (e) and (f) show the predictions of DRAN and ST-SSL, respectively, and (g) and (h) present their corresponding absolute prediction errors.

Fig. 5: Computational cost analysis on the NYCTaxi dataset. The x-axis denotes inference time (in seconds), and the y-axis represents MAE. Bubble size indicates the number of model parameters. DRAN achieves an accuracy-efficiency trade-off, obtaining lower MAE with competitive inference time compared to existing methods.

Dataset	$\bm{H}_{\rm tem}$ vs. $\bm{X}$		$\bm{H}_{\rm spa}$ vs. $\bm{X}$
Dataset	PDD $\downarrow$	${\rm WD}_{\rm rel}$ $\downarrow$	PDD $\downarrow$	${\rm WD}_{\rm rel}$ $\downarrow$
Weather	0.491 ( $\pm$ 0.205)	0.177 ( $\pm$ 0.056)	0.180 ( $\pm$ 0.054)	0.086 ( $\pm$ 0.029)
NYCBike1	0.481 ( $\pm$ 0.075)	0.470 ( $\pm$ 0.087)	0.215 ( $\pm$ 0.071)	0.129 ( $\pm$ 0.078)
NYCBike2	0.177 ( $\pm$ 0.034)	0.143 ( $\pm$ 0.036)	0.168 ( $\pm$ 0.071)	0.103 ( $\pm$ 0.072)
NYCTaxi	0.130 ( $\pm$ 0.024)	0.109 ( $\pm$ 0.029)	0.118 ( $\pm$ 0.040)	0.079 ( $\pm$ 0.035)
PeMS04	0.198 ( $\pm$ 0.020)	0.437 ( $\pm$ 0.233)	0.171 ( $\pm$ 0.202)	0.137 ( $\pm$ 0.024)

TABLE VII: Quantitative comparison of spatial-distribution preservation.

Dataset	Model	Clean	Gaussian Noise			Missing Sensors			High-dynamic
Dataset	Model	Clean	$\sigma=0.1$	$\sigma=0.2$	$\sigma=0.3$	$\rho=10\%$	$\rho=20\%$	$\rho=30\%$	Top 10%
Weather	DRAN	0.676 ( $\pm$ 0.005)	1.369 ( $\pm$ 0.060)	2.039 ( $\pm$ 0.037)	2.557 ( $\pm$ 0.031)	1.227 ( $\pm$ 0.032)	1.328 ( $\pm$ 0.062)	1.397 ( $\pm$ 0.051)	0.948 ( $\pm$ 0.009)
Weather	RGSL	0.727 ( $\pm$ 0.003)	1.481 ( $\pm$ 0.004)	2.161 ( $\pm$ 0.006)	2.871 ( $\pm$ 0.004)	1.315 ( $\pm$ 0.004)	1.385 ( $\pm$ 0.016)	1.415 ( $\pm$ 0.046)	0.973 ( $\pm$ 0.003)
PeMS08	DRAN	13.366 ( $\pm$ 0.117)	14.366 ( $\pm$ 0.040)	16.061 ( $\pm$ 0.061)	18.483 ( $\pm$ 0.162)	21.919 ( $\pm$ 1.719)	29.591 ( $\pm$ 2.030)	39.017 ( $\pm$ 1.771)	19.325 ( $\pm$ 0.035)
PeMS08	STAEformer	13.538 ( $\pm$ 0.034)	14.854 ( $\pm$ 0.110)	17.168 ( $\pm$ 0.442)	20.344 ( $\pm$ 0.961)	22.412 ( $\pm$ 0.707)	32.739 ( $\pm$ 1.328)	41.851 ( $\pm$ 1.805)	19.434 ( $\pm$ 0.095)

TABLE VIII: Robustness comparison under test-time perturbations and naturally high-dynamic periods.

Strategies	Weather	NYCBike1	NYCBike2	NYCTaxi	PeMS04	PeMS08
Strategies	MAE $\downarrow$	MAE $\downarrow$	MAE $\downarrow$	MAE $\downarrow$	MAE $\downarrow$	MAE $\downarrow$
+RevIN	0.844 ( $\pm$ 0.029)	5.419 ( $\pm$ 0.178)	5.386 ( $\pm$ 0.187)	11.656 ( $\pm$ 1.486)	18.875 ( $\pm$ 0.196)	13.602 ( $\pm$ 0.282)
+DAIN	1.004 ( $\pm$ 0.453)	5.541 ( $\pm$ 0.189)	5.424 ( $\pm$ 0.174)	11.867 ( $\pm$ 1.401)	18.695 ( $\pm$ 0.231)	13.704 ( $\pm$ 0.070)
+DAIN+SFL	0.840 ( $\pm$ 0.069)	5.291 ( $\pm$ 0.182)	5.276 ( $\pm$ 0.017)	11.517 ( $\pm$ 1.307)	18.305 ( $\pm$ 0.346)	13.366 ( $\pm$ 0.116)
+Non-st	1.194 ( $\pm$ 0.002)	5.502 ( $\pm$ 0.163)	5.266 ( $\pm$ 0.229)	12.426 ( $\pm$ 1.016)	18.642 ( $\pm$ 0.064)	13.995 ( $\pm$ 0.555)
+Non-st+SFL	0.676 ( $\pm$ 0.005)	5.046 ( $\pm$ 0.141)	4.845 ( $\pm$ 0.203)	10.721 ( $\pm$ 0.980)	18.132 ( $\pm$ 0.008)	13.366 ( $\pm$ 0.117)

TABLE IX: The effectiveness of SFL on various temporal normalization methods.

Fig. 6: Comparison of spatial distributions before and after SFL. Panels (a)–(d) correspond to the Weather, NYCBike1, NYCBike2, and NYCTaxi datasets, respectively. Each panel shows Gaussian kernel density estimates [ 62 ] for the raw inputs $\bm{X}$ , the latent representations before SFL $\bm{H}_{\rm tem}$ , and the representations after SFL $\bm{H}_{\rm spa}$ , denoted as “Lookback”, “Before SFL”, and “After SFL”, respectively.

Fig. 7: Visualization of static and dynamic relations learned by DSFL on the Weather dataset. Panels (a) and (b) are the adjacency matrices of static and dynamic branches. Panels (c)–(h) show the relation strengths for three nodes located in different areas of Weather dataset. Panels (c), (e), and (g) show the static relations of the nodes, while panels (d), (f) and (h) display the dynamic relations. The orange triangles represent the selected target nodes. The darker color indicates a closer relationship between nodes.

Strategies

Weather

NYCBike1

NYCBike2

NYCTaxi

PeMS04

PeMS08

MAE

\downarrow

\downarrow

MAE

\downarrow

\downarrow

MAE

\downarrow

\downarrow

MAE

\downarrow

\downarrow

MAE

\downarrow

\downarrow

MAE

\downarrow

\downarrow

DRAN

0.676 (

\pm

0.005)

0.392 (

\pm

0.011)

5.046 (

\pm

0.141)

0.415 (

\pm

0.148)

4.845 (

\pm

0.203)

0.437 (

\pm

0.039)

10.721 (

\pm

0.980)

0.750 (

\pm

0.146)

18.132 (

\pm

0.008)

4.687 (

\pm

0.187)

13.366 (

\pm

0.117)

4.180 (

\pm

0.075)

w/o SFL &

\bm{\mathcal{L}_{\rm spa}}

1.194 (

\pm

0.002)

2.000 (

\pm

0.045)

5.502 (

\pm

0.163)

0.720 (

\pm

0.061)

5.266 (

\pm

0.229)

1.495 (

\pm

0.021)

12.426 (

\pm

1.016)

1.520 (

\pm

0.076)

18.642 (

\pm

0.064)

5.712 (

\pm

0.099)

13.995 (

\pm

0.555)

4.822 (

\pm

0.265)

w/o

\bm{\mathcal{L}_{\rm spa}}

0.887 (

\pm

0.004)

1.168 (

\pm

0.069)

5.403 (

\pm

0.185)

0.688 (

\pm

0.080)

5.148 (

\pm

0.182)

0.696 (

\pm

0.007)

11.451 (

\pm

1.231)

1.371 (

\pm

0.118)

18.281 (

\pm

0.046)

5.485 (

\pm

0.186)

13.719 (

\pm

0.301)

4.536 (

\pm

0.076)

w/o DSFL

1.193 (

\pm

0.002)

1.542 (

\pm

0.072)

5.529 (

\pm

0.170)

0.730 (

\pm

0.080)

5.536 (

\pm

0.221)

1.577 (

\pm

0.058)

12.276 (

\pm

1.404)

1.612 (

\pm

0.143)

18.695 (

\pm

0.131)

5.586 (

\pm

0.157)

13.663 (

\pm

0.142)

4.613 (

\pm

0.126)

w/o Decomposition

0.794 (

\pm

0.013)

0.669 (

\pm

0.038)

5.404 (

\pm

0.170)

0.655 (

\pm

0.071)

5.258 (

\pm

0.193)

0.600 (

\pm

0.058)

11.375 (

\pm

1.348)

1.345 (

\pm

0.098)

18.275 (

\pm

0.070)

4.908 (

\pm

0.145)

13.537 (

\pm

0.084)

4.534 (

\pm

0.069)

TABLE X: The ablation results.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Attributes	Training set	Test set	Validation set	Node number	Feature number
Weather	5,623	1,607	803	263	1
NYCBike1	3,023	864	431	128	2
NYCBike2	1,912	546	274	200	2
NYCTaxi	1,912	546	274	200	2
PeMS04	10,181	3,394	3,394	307	1
PeMS08	10,700	3,566	3,566	170	1

Appendix

TABLE A.1: Additional datasets details

Weather		NYCBike1		NYCBike2		NYCTaxi		PeMS04		PeMS08
$\alpha$	MAE	$\alpha$	MAE	$\alpha$	MAE	$\alpha$	MAE	$\alpha$	MAE	$\alpha$	MAE
0.001	0.856	0.001	5.102	0.001	4.958	0.001	11.375	0.001	18.186	0.001	13.604
0.010	0.895	0.010	5.075	0.010	4.942	0.010	11.423	0.010	18.249	0.010	13.682
0.050	0.667	0.050	5.037	0.050	4.855	0.050	11.191	0.050	18.343	0.050	13.560
0.100	0.726	0.100	5.082	0.100	4.972	0.100	11.085	0.100	18.240	0.100	13.499
0.500	0.758	0.500	5.101	0.500	5.040	0.500	10.892	0.500	18.299	0.500	13.526

Appendix

TABLE A.2: The selection of balanced hyperparameters.

Methods	Hyperparameters selection
DA-RNN [ 27 ]	hidden dimension {64, 128,256}
InfoTS [ 54 ]	$\alpha$ {0.01, 0.1, 0.5, 1, 5, 10}; $beta$ : {0.01, 0.1, 0.5, 1, 5, 10}
AutoTCL [ 55 ]	$\beta$ {0.0003, 0.001, 0.01, 0.1,0.3}; $\lambda$ {0.0003, 0.001, 0.01, 0.1,0.3}; local loss {0.0003, 0.001, 0.01, 0.1, 0.5}
TGCN [ 15 ]	recurrent layer {1, 2, 3}; hidden dimension {32, 64, 128}
GCGRU [ 17 ]	recurrent layer {1, 2, 3}; hidden dimension {32, 64, 128}
AGCRN [ 33 ]	recurrent layer {1, 2, 3}; hidden dimension {32, 64, 128}

Appendix

TABLE A.3: Hyperparameter candidate ranges

Methods	Parameter settings	Weather	NYCBike1	NYCBike2	NYCTaxi	PeMS04	PeMS08
DA-RNN [ 27 ]	hidden dimension	64	64	128	128	128	128
InfoTS [ 54 ]	$\alpha$	0.5	5	10	5	5	1
InfoTS [ 54 ]	$\beta$	10	0.5	10	0.5	0.5	5
AutoTCL [ 55 ]	$\beta$	0.01	0.1	0.1	0.1	0.3	0.3
	$\lambda$	0.1	0.01	0.001	0.01	0.3	0.1
	Local loss	0.1	0.0003	0.0003	0.0003	0.01	0.001

Appendix

TABLE A.4: Final tuned hyperparameters

Fig. 9: Additional node-level temporal prediction results. Panels (a) and (b) show temperature forecasts for two Weather nodes, while panels (c) and (d) show inflow and outflow forecasts for one NYCBike1 node.

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