Information Sciences 648 (2023) 119560

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Information Sciences

journal homepage: www.elsevier.com/locate/ins

Learning and integration of adaptive hybrid graph structures for 

multivariate time series forecasting

Ting Guo a,b, Feng Hou b,∗, Yan Pang a, Xiaoyun Jia c, Zhongwei Wang a, 
Ruili Wang a,b,∗

a School of Logistics and Transportation, Central South University of Forestry and Technology, Changsha, Hunan, China
b School of Mathematical and Computational Sciences, Massey University, Auckland, New Zealand
c Institute of Governance & School of Politics and Public Administration, Shandong University, Qingdao, Shandong, China

A R T I C L E I N F O A B S T R A C T

Keywords:

Multivariate time series forecasting

Global graph

Local graph

Graph structure learning

Information fusion

Recent status-of-the-art methods for multivariate time series forecasting can be categorized 
into graph-based approach and global-local approach. The former approach uses graphs to 
represent the dependencies among variables and apply graph neural networks to the forecasting 
problem. The latter approach decomposes the matrix of multivariate time series into global 
components and local components to capture the shared information across variables. However, 
both approaches cannot capture the propagation delay of the dependencies among individual 
variables of a multivariate time series, for example, the congestion at intersection A has delayed 
effects on the neighboring intersection B. In addition, graph-based forecasting methods cannot 
capture the shared global tendency across the variables of a multivariate time series; and global-

local forecasting methods cannot reflect the nonlinear inter-dependencies among variables of a 
multivariate time series. In this paper, we propose to combine the advantages of both approaches 
by integrating Adaptive Global-Local Graph Structure Learning with Gated Recurrent Units 
(AGLG-GRU). We learn a global graph to represent the shared information across variables. 
And we learn dynamic local graphs to capture the local randomness and nonlinear dependencies 
among variables. We apply diffusion convolution and graph convolution operations to global and 
dynamic local graphs to integrate the information of graphs and update gated recurrent unit for 
multivariate time series forecasting. The experimental results on seven representative real-world 
datasets demonstrate that our approach outperforms various existing methods.

1. Introduction

Precisely forecasting the future states of systems and environment is vital for engineering, management, and decision (policy) 
making. Thus, time series forecasting, a task of predicting the future values of variables based on historical data, has been studied for 
over five decades. With the availability of massive amounts of real-time records of multiple variables collected by networked low-

cost sensors and devices, time series forecasting has evolved into multivariate time series forecasting. Each variable in a multivariate 
time series is dependent not only on the past values of its own but also depends on the past or current values of other variables. 

* Corresponding authors.
Available online 21 August 2023
0020-0255/© 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).

E-mail address: f.hou@massey.ac.nz (F. Hou).

https://doi.org/10.1016/j.ins.2023.119560

Received 22 April 2023; Received in revised form 26 June 2023; Accepted 13 August 2023

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Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Multivariate time series forecasting [24] can exploit such complex dependencies between variables and has been applied in electrical 
grid management, traffic congestion reduction for intelligent transportation systems, and financial services.

Traditional statistical forecasting methods, such as Auto-regressive (AR), Auto-regressive Integrated Moving Average (ARIMA) 
[45], and Vector Auto-regressive (VAR), are based on the linear dependency assumption between variables, which is not valid for 
most cases. Moreover, the complexity grows quadratically with the number of variables.

With the success of deep learning in computer vision, speech [31], and natural language processing, the RNN-based Seq2Seq 
model for language generation is adopted as a popular architecture for multi-step multivariate time series forecasting [15]. To capture 
the non-linearity of multivariate time series, LSTNet [19] uses convolutional neural networks to encode the short-term patterns of 
a local variable. Convolutional neural networks are also used by TPA-LSTM [29] to compute attention scores for multiple steps. 
However, these RNN-based Seq2Seq models can capture seasonality only when the dataset possesses homogeneous seasonal patterns 
[14]. Moreover, such models focus on the local forecasting of individual time series based on historical data and cannot efficiently 
capture the pair-wise interdependencies between variables. LSTM-FC [48] use a fully connected neural networks to integrate the 
predictions of variables, this can capture the pair-wise dependencies to some degree but is incompetent for data in non-Euclidean 
space [1].

Graph is very effective for modeling the pair-wise dependencies between variables, including the variables in non-Euclidean space. 
Recently, with the advancement in graph neural networks, many deep learning methods for multivariate time series forecasting have 
used deep learning on graphs to model the complex dependencies among variables. For example, DCRNN [34] proposes diffusion 
convolutional recurrent neural network for traffic forecasting; ASTGCN [13] uses attention-based graph convolutional networks 
for multivariate time series forecasting. For these graph-based multivariate time series forecasting methods, the nodes of a graph 
represent the variables of a multivariate time series; the edges represent the degree of dependencies between a pair of variables; and 
the graph is denoted by an adjacency matrix. The graph for multivariate time series forecasting usually is pre-defined or hand-crafted 
to model the inter-dependencies [34]. Such pre-defined graph cannot express the genuine interdependencies. Thus, researchers 
propose to learn a static adjacency matrix [38] or dynamic adjacency matrices [20] for multi-step forecasting. The former cannot 
reflect the dynamic interdependencies and the latter cannot capture the global components of a multivariate time series. Graph-based 
methods are good at modeling the interdependencies yet cannot identify the common behaviors of all the variables.

Global-local methods are proposed to capture and exploit the common behaviors of all variables of a multivariate time series. 
The key notion of global-local methods is to identify the shared global pattern of multiple time series and use it to facilitate the 
forecasting of local time series (variables). The global components of a multivariate time series characterize the general tendencies 
and dependencies of all the variables of a multivariate time series, for example the seasonality of electricity consumption and the 
congestion on roads during rush hours. Global-local forecasting models [24], [27] decompose the matrix of multivariate time series 
into an addition of 𝑘 latent global components (𝑘 < 𝑛, 𝑛 is the number of variables) and a local component; and the future values of 
each variable are predicted based on these components. These global-local models can capture the similar trends of multiple variables 
as shared information to accelerate the forecasting of local (individual) time series. However, due to the linear addition form of local 
and global components, global-local methods cannot efficiently capture the nonlinearity of time series.

The limitations of existing graph-based and global-local forecasting methods can be summarized as follows: (i) Graph-based 
forecasting methods cannot capture the shared global tendency across the variables of a multivariate time series; (ii) Global-local 
forecasting methods cannot reflect the nonlinear interdependencies among variables of a multivariate time series; (iii) Existing 
methods cannot capture the propagation delay between the local variables of a multivariate time series, for example, the congestion 
at intersection A has delayed effects on the next intersection B.

In this paper, we propose to combine the advantages of graph-based forecasting methods and global-local forecasting methods 
by learning a global graph and multiple dynamic graphs for multi-step multivariate time series forecasting. The global graph is to 
capture the global information of a multivariate time series. The dynamic graphs are for capturing duo-local interdependencies and 
propagation delays between variables of the same multivariate time series. We propose a novel method to integrate and incorporate 
the global and local information by a joint convolution operation on the global and dynamic local graphs. We introduce a regular-

ization term for model training to increase forecasting precision. We name our method as Adaptive Global-Local Graph Structure 
Learning and Integration with Gated Recurrent Units (AGLG-GRU) for multivariate time series forecasting. Our contributions can be 
summarized as follows:

1. We learn a global graph by sampling from a differentiable probabilistic graphical model to capture the shared information and 
global dependencies among variables of a multivariate time series.

2. We use a hyper-network to learn dynamic local graphs to capture the local randomness and correlations between pairs of 
variables; and multi-scale historical data are input to the hyper-network to capture the delayed temporal dependencies of variables.

3. We devise a mechanism to integrate the information of global and dynamic local graphs by a joint graph convolution operation, 
and the integrated information is used to update the gated recurrent units for multi-step forecasting.

The remainder of this manuscript is organized as follows. In Section 2, related work is reviewed. The proposed methods are 
described in detail in Section 3. Section 4 presents the results of extensive experiments and the analysis of the results. This paper 
concludes with conclusions and future work in Section 5.

2. Related work

In recent decades, data-driven methods for multivariate time series forecasting have shown significant improvements over tradi-
2

tional knowledge-driven methods. With the success of deep learning, data-driven methods have predominantly adopted variants of 



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

deep learning methods to improve forecasting performance. Although deep recurrent neural networks (RNN) can be directly used on 
multivariate time series data [10], simple RNN cannot efficiently model the complex dependencies among variables. Recent methods 
use graph or decomposing time series data into global and local components to model the dependencies of variables. These methods 
can be categorized into graph-based methods and global-local methods.

2.1. Graph-based methods for multivariate time series forecasting

Graph-based methods use graphs to represent the complex dependencies between variables and apply graph neural networks 
(GNN) to the graphs to forecast the future values of variables. Deep learning operations on graphs are usually performed in the spatial 
domain or spectral domain. Spatial-domain GNNs are usually applied to Euclidean data (e.g., images) with shareable parameters. 
Spectral-domain GNNs are performed on the eigenvectors of matrices associated with a graph, such as its adjacency matrix or 
Laplacian matrix. For graph-based multivariate time series forecasting, GNN operations are performed in spectral domain based the 
adjacency matrix of a graph.

Initially, the adjacency matrices are pre-defined and fixed for the model training. For instance, DCRNN [22] computes the pairwise 
road network distances between sensors and builds the adjacency matrix using thresholded Gaussian kernel; STGCN [42] also builds 
the distance-based graph similarly; some other methods use grid-based graph [41]. However, these pre-defined adjacency matrices 
are built using the prior geo-information, which is not available for some multivariate time series. Moreover, such static graphs 
cannot reflect the complex dependencies and dynamic properties of multivariate time series. AGC-Seq2Seq [47] proposes to learn 
a Laplacian matrix during the model training stage, however the Laplacian matrix is still fixed during the inference stage. ASTGCN 
[13] accompanies the adjacency matrix with a learned spatial attention matrix to dynamically adjust the impacting weights between 
nodes (variables); however still uses the empirical Laplacian matrix as a mask matrix [12].

Recently, inspired by the research in generic graph structure learning [9], adaptive graph [37], [46], [2], [11], [33] is proposed 
to learn unseen graph structures automatically from multivariate time series without prior knowledge. Among these studies, some 
proposed methods directly learn the adjacency matrix as model parameters [38], [13]; some methods use a latent network to extract 
features for constructing the graph [46], [12]; others calculate node similarities from node embeddings [37], [2]. However, such 
adaptively learned adjacency matrices are fixed for multi-step forecasting, thus cannot reflect the dynamic evolution of time series. 
To tackle this issue, some researchers propose to learn a series of adjacency matrices in a recurrent manner [20].

Many deep learning operations are implemented on the pre-defined graphs or learned adaptive graphs to forecast the future values 
based on the captured dependencies of variables. Graph convolutional networks are widely used for graph-based multivariate time 
series forecasting [42], [49], [13], [2], [39]. DCRNN [22] models the dynamics of the traffic flow as a diffusion process and proposes 
the diffusion convolution operation to capture the pair-wise spatial correlations using bidirectional random walks on a directed 
graph. DCRNN uses an encoder-decoder architecture to capture temporal dependency. Rank influence learning is introduced to the 
diffusion convolutional networks to adjust the importance of neighboring nodes [16]. A common practice of graph-based forecasting 
methods is to use the so-called spatial-temporal graph neural networks to model the spatial and temporal correlations separately [22], 
[42], [16], [49], [39], [46], [13], [2], i.e., they use graph convolutional networks or diffusion convolutional networks to capture the 
spatial dependencies and use recurrent neural networks to capture the temporal correlations. However, recurrent neural networks 
employed in these methods cannot capture long-range temporal sequences. Graph waveNet [37], [25] is proposed to handle long 
sequences. Inspired by the wavenet architecture for audio generation, the spatial-temporal layers in graph wavenet are connected by 
residual connections and the output of each spatial-temporal layer is skip connected to the output layer. Multi Scale Graph Wavenet 
[25] model is able to capture the global trends of change of variables with a learnable adjacency matrix.

2.2. Global-local methods for multivariate time series forecasting

Global-local methods [27], [36], [26], [24] assume that multivariate time series are composed of latent global components and 
local components. The global components are designed to capture the global trends and dependencies among variables of the entire 
multivariate time series. The local components are considered to handle the randomness of individual variables.

TRMF (Temporal Regularized Matrix Factorization) [43] employs matrix factorization to extract the global components of multi-

variate time series. Unlike the graph-based approaches, TRMF uses well-studied time series models to describe temporal dependencies 
explicitly; and designs a new regularization term to incorporate the temporal dependency into the matrix factorization model. TRMF 
can not only be used for forecasting of time series but also used for missing-value imputation of time series.

DeepGLO [27] extends the TRMF method to non-linear settings by imposing the temporal regularization with a temporal convo-

lution network (TCN). DeepGLO is composed of two components: a matrix factorization model regularized by a TCN (TCN-MF) for 
global patterns and a temporal convolution network for local patterns of individual time series. TCN-MF extracts basis time series as 
global components and the predictions from TCN-MF are used as covariates for the local temporal convolution network. DeepGLO 
can capture the global features from a whole dataset as well as the local patterns of individual time series while both training and 
prediction.

DFMRE (Deep Factor Models with Random Effects) [36] is a global-local forecasting method based on a global DNN backbone 
and local probabilistic graphical models. They assume that each time series is determined by a global component (non-random) and a 
local random component. The global components are a linear combination of 𝑘 latent global factors modeled by RNNs. Three models 
are proposed to model the local random components based on three typical local choices: white noise processes, linear dynamical 
3

systems (LDS) and Gaussian processes (GPs).



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

DeepAR [26] is a global-local forecasting method based on autoregressive recurrent neural networks. DeepAR learns a global 
RNN model from historical data of all variables, while the local components are obtained by probabilistic sampling based on prior 
distribution.

The aforementioned methods only focus on global patterns for forecasting and ignore the local patterns of the time series.

DeepGate [24] decomposes multivariate time series into the underlying global and local series and predicts the future value of 
each series based on these global and local series. They adopt a pre-training-based transfer training scheme to maintain global–local 
decomposition in a joint learning scenario. Unlike other global–local methods which use global factors as latent features, DeepGate 
explicitly model global and local features in an additive manner to support disentanglement.

3. Methodology

In this section, we first introduce some preliminaries and the formulation of multivariate time series forecasting in SubSection 3.1. 
We present our method, Adaptive Global-Local Graph Structure Learning and Integration with Gated Recurrent Units (AGLG-GRU) 
for multivariate time series forecasting, in SubSection 3.2. Finally, SubSection 3.5 describes how our multivariate forecasting model 
is trained and regularized.

3.1. Preliminaries and problem formulation

Multivariate time series forecasting is to predict the future values of correlated multi-variables based on the historical data of 
these variables. The historical data of multivariate time series with auxiliary prior knowledge can be denoted as a tensor 𝑿𝑇×𝑁×𝐹 , 
where 𝑇 is the length of temporal sequence, 𝑁 represents the number of different variables (e.g., nodes in sensors network), 𝐹 is 
the dimension of features of each node. Concatenating input signals with auxiliary features, we assume the inputs for model learning 
𝑿𝑇×𝑁×𝐹 =

{
𝑿𝑇×𝑁×𝑓1 ,𝑿𝑇×𝑁×𝑓2 ,⋯ ,𝑿𝑇×𝑁×𝑓𝑖 ,⋯

}
and the first column 𝑿𝑇×𝑁×𝑓1 is the target value for forecasting. For example, we 

usually use traffic volume as target for traffic flow forecasting, traffic speed for highway speed forecasting, and daily electricity 
consumption observations for city electricity forecasting. The primary feature of input signals can be coupled with some auxiliary 
features 𝑿𝑇×𝑁×𝑓𝑖 , such as holidays information, historical weather observations, road network distance, point of interest (PoI) 
similarities [4].

For variables with geospatial properties, such as variables of a traffic network, multivariate time series can be denoted as a 
weighted graph 

(
 ,,𝑨𝑃

)
. Here,  is the set of vertices that represent the different sensors (e.g., energy sensors or traffic segments) 

on networks; 𝑁 = | | denotes the number of nodes (variables) in a graph and  is a set of connected edges; 𝑨𝑃 is a pre-defined 
adjacency matrix with 𝑨𝑃

𝑖𝑗
= 𝑐 > 0 if 

(
𝑣𝑖, 𝑣𝑗

)
∈  and 𝑨𝑃

𝑖𝑗
= 0 if 

(
𝑣𝑖, 𝑣𝑗

)
∉ , which is a mathematical structure created by computing 

distances and up-down relationships between the nodes of a graph.

Multivariate time series forecasting can be formulated as follows: Given the historical 𝑇 steps data 𝑿(𝑡−𝑇 )∶(𝑡−1) of 𝑁 variables 
(nodes) before the current time step 𝑡 and the corresponding graph (𝑁, , 𝑨𝑃 ), the objective is to learn a function  that is able to 
predict the target value of the next 𝑆 steps �̃�𝑡∶(𝑡+𝑆) as represented by Equation (1).

�̃�𝑡∶(𝑡+𝑆) = 
(
𝑿(𝑡−𝑇 )∶(𝑡−1),(𝑁,,𝑨𝑃 )

)
(1)

where �̃�𝑡∶(𝑡+𝑆) =
(
�̃�𝑡,𝑁×𝑓1 , �̃�(𝑡+1),𝑁×𝑓1 ,⋯ , �̃�(𝑡+𝑆),𝑁×𝑓1

)
for multi-step forecasting (𝑆 > 1).

Multivariate time series forecasting is challenging due to:

• Complex spatial dependency and underlying global trends of change. The variables in a multivariate time series are intrinsically 
dependent with each other, and have similar trends of change. Existing graph-based multivariate time series forecasting methods 
focus on modeling the spatial dependency but ignore the identification of the global trends of changes of all variables, while 
global-local forecasting methods do not consider the spatial dependency among variables. Thus, it is necessary to develop a 
method to capture both the spatial dependency and the global trends of change.

• Dynamic non-linear temporal dependency and randomness of local variables. The current value of a variable depends on the history 
of itself and many other variables, and variables show independent randomness. Existing graph-based methods focus on cap-

turing temporal dependency but ignore the randomness of local variables. Global-local methods ignore the dynamic temporal 
dependency of variables. Thus, it is essential to consider both the temporal dependency and the randomness of local variables.

3.2. The architecture of AGLG-GRU

As shown in Fig. 1, our multivariate time series forecasting framework AGLG-GRU consists of four components: (1) a global 
graph structure learning module, (2) a local graph structure learning module, (3) a GRU module for hierarchical weighted feature 
integration and (4) a GRU-based encoder-predictor module. First, in the global graph structure learning module, a global proba-

bilistic graphical model is employed to extract the global (shared) trends of variables at the whole observation scale. Second, in 
the local graph structure learning module, a local hyper-network is used to extract the local features of time series and capture 
the dynamic dependencies among variables at dynamic observation scale. Third, the GRU module for hierarchical weighted feature 
4

fusion adaptively merges the global-local feature embedded in the graphs. Finally, the fused feature vector is fed to a GRU-based 



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 1. The overall architecture of AGLG-GRU. The Encoder and Predictor Consist of a module of Hierarchical Feature Integration with GRU. An Adaptive Global-Local 
Graph Structure Learning Module adaptively generates global and local graphs for the encoder and predictor. The global graph learning module is used to capture 
the global trends and shared information of variables (nodes). The local graph learning module is used to learn the local randomness of variables and the correlation 
between variables.

encoder-predictor for single-step and multi-step ahead forecasting. ⟨𝐺𝑂⟩ indicates the first of the predictor inputs, and the remaining 
predictor inputs are derived from the previous outputs.

3.3. Adaptive global-local graph structure learning

In this section, we present our method for generating the global graph 𝑨𝐺 and local graphs 𝑨𝐿𝑖 . The global graph is used to 
capture the shared global trends of all variables. The local graphs are generated dynamically to capture the local randomness and 
dynamic dependencies among variables. To improve the sensitivity to random changes in non-periodic time series, we fuse the 
learned global features with the learned local features in the subsequent stage.

3.3.1. Global graph structure learning

We aim to use a global graph to capture the global trends and shared information of variables. By measuring the global similarity 
of variables, we can capture the delayed dependencies between a pair of variables. As shown in Fig. 2, the probabilistic graph 
model for learning the global graph structure is composed of three components: a feature extractor, a link predictor, and a sampling 
component.

The feature extractor uses a shared convolution kernel to extract the shared features (e.g., periodic fluctuations, seasonality) 
of all variables from the whole time series and auxiliary data. In particular, we first feed time series 𝑿𝑇×𝑁×𝐹 into two layers of 
convolutional neural network 𝑪𝒐𝒏𝒗𝟐𝒅 and then use a fully-connected layer 𝔽ℂ to reduce the dimension of node feature in the 
feature extractor. The generated node vector 𝒗𝒊 represents the temporal characteristics of the 𝑖𝑡ℎ variable. Two variables with similar 
temporal characteristics will have similar node vectors. The feature of 𝑖𝑡ℎ node 𝑣𝑖 is generated as Equation (2).

𝒗𝑖 = 𝔽ℂ
(
𝑪𝒐𝒏𝒗𝟐𝒅

(
𝑿0∶𝑇 ,𝑖,0∶𝐹

))
(2)

As shown in Equation (3), we concatenate a pair of node vectors 
(
𝒗𝑖,𝒗𝑗

)
and implement a fully-connected layer to generate an 

edge scalar 𝜖𝑖𝑗 in the link predictor.

𝜖𝑖𝑗 = 𝔽ℂ
(
𝒗𝑖‖𝒗𝑗) (3)

As part of the sampling procedure for global graph structure learning, we employ the Gumbel reparameterization trick [28] to 
determine the parameters of the global adjacency matrix 𝑨𝐺 based on sampling variables from discrete distributions. Equation (4)

shows the formula for generating 𝑨𝐺 .[ ( ( )) ( )]

5

𝑨𝐺
𝑖𝑗
= sigmoid log 𝜖𝑖𝑗∕ 1 − 𝜖𝑖𝑗 + 𝑔1

𝑖𝑗
− 𝑔2

𝑖𝑗
(4)



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 2. The framework of adaptive global-local graph structure learning.

where 𝑔1
𝑖𝑗
, 𝑔2
𝑖𝑗
∼Gumbel(0, 1) for all 𝑖, 𝑗.

3.3.2. Local graph structure learning

We aim to use local graphs to capture the local randomness of variables and their dynamic correlations. The local randomness 
and dynamic correlations are more prominent when variables are observed at a smaller observation scale. Thus, we split the given 
historical data into smaller segments, and learn a series of dynamic local graphs 

(
𝑨𝐿0 ,𝑨𝐿1 ,⋯ ,𝑨𝐿𝑚 ,⋯ ,𝑨𝐿𝑀

)
from these segments. 

This process is denoted in the left part of Fig. 2.

Specifically, we first divide the given historical data 𝑿𝑇×𝑁×𝐹 into independent segments along the time dimension, and use an 
aggregator to aggregate the features of each segment along the time dimension. We implement the aggregator by performing an 
averaging operation on each segment. This process will generate a sequence of new features 

[
𝝃(1),𝝃(2),… ,𝝃(𝑚),… ,𝝃(𝑀)] as follows:

𝝃(𝑚) =𝔸𝕍𝔾
(
𝑿((𝑚−1)𝐵+1∶𝑚𝐵)) ∈𝑅𝑁×𝐹 (5)

where 𝑀 is the total number of segments, 𝐵 is the size of segments, 𝔸𝕍𝔾 indicates an averaging aggregator. According to Equation 
(1), our model requires 𝑇 step historical data to predict future time series variables at step 𝑆. Therefore, the length of the input in 
the model is dimensionally consistent with the historical data provided in 𝑇 steps.

We then use a feature segment 𝝃(𝑚) to perform graph convolution on the pre-refined graph 𝑨𝑃 and the global graph 𝑨𝐺 respec-

tively, as follows.

𝑷 𝑚 =𝑮𝒄𝒐𝒗𝒏
(
𝝃(𝑚),𝑨𝑃

)
𝑮𝑚 =𝑮𝒄𝒐𝒗𝒏

(
𝝃(𝑚),𝑨𝐺

) (6)

where 𝑮𝒄𝒐𝒗𝒏 represents the graph convolution with learnable parameters. In case of missing, incomplete or incorrect 𝑨𝑃 data in 
the dataset, we will use 𝑨𝐺 instead of 𝑨𝑃 or use different weights in the subsequent steps to adjust the importance of 𝑨𝑃 data in the 
feature construction. In the adjacency matrices 𝑨𝑃 and 𝑨𝐺 , there are static distance-based relations between nodes, as well as global 
features, that can be used in the message-passing process for handling dynamic node status information.

Based on the outputs in Equation (6), we compute a dynamic node embedding 𝑬𝑚
𝑖

for each variable. The dynamic node embed-

dings are computed based on the element-wise multiplication between 𝑷 𝑚, 𝑮𝑚 and the static node embeddings 𝑽 𝒆𝒄𝑖 as follows.

𝑬𝑚
𝑖
= tanh

(
𝜑𝑝

(
𝑷 𝑚
𝑖
⊙ 𝑽 𝒆𝒄𝑖

)
+𝜑𝑔

(
𝑮𝑚
𝑖
⊙ 𝑽 𝒆𝒄𝑖

))
(7)

where 𝜑𝑝 and 𝜑𝑔 are hyper-parameters that control the weights of different components, ⊙ denotes the element-wise multiplication. 
By adjusting weights in case of missing, incomplete, or incorrect data in the dataset, we are able to optimize prediction accuracy 
based on experimental results.

The adjacency matrix of the 𝑚th local graph 𝑨𝐿𝑚 is derived by computing the pairwise similarity between variables. The elements 
of 𝑨𝐿𝑚 are computed as follows:

𝑨
𝐿𝑚
𝑖𝑗

= ReLU
(
𝜏𝑬𝑚

𝑖
𝑬𝑚𝑇

𝑗

)
(8)
6

where 𝜏 controls the saturation rate of the activation function.



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 3. Structure Of the hierarchical weighted feature fusion GRU with 𝐾 feature fusion layers.

3.4. Hierarchical feature integration with GRUs

The adaptive global and local graphs capture different aspects of the features of multivariate time series. In this section, we 
describe how to integrate the information captured by the global graph and a local graph. The details of global-local graph integration 
using a GRU are shown in Fig. 3. It should be noted that the GRUs in the encoder and predictor (as shown in Fig. 1) are the same 
except the initialization of the hidden state.

3.4.1. Hierarchical weighted feature integration

As shown in Fig. 3, we recurrently perform 𝐾-hop feature fusion to integrate the information captured by the global and local 
graphs; then we update an GRU with the integrated information. The operations of each hop are illustrated in the right part of Fig. 3. 
The operations include: (1) a graph diffusion convolution on the global graph 𝑨𝐺 ; (2) a graph convolution on the dynamic local 
graph 𝑨𝐿𝑡 ; (3) a linear weighted addition operation.

For the global graph 𝑨𝐺 , we perform the diffusion convolution operation from DCRNN [34] defined using a bidirectional random 
graph walk. The diffusion convolution operation calculates bidirectional transition matrices that represent inflows and outflows 
during diffusion. As multi-step diffusion produces homogeneous nodes, complicating parametric training and confusing node features, 
we only use one-step diffusion convolution. In the 𝑘-th feature fusiong layer, our one-step graph diffusion convolution on the global 
graph is defined in Equation (9).

𝑯 (𝑘−1) ⋆𝑨𝐺 =
[
𝜔0 +𝜔1

(
𝑫−1
𝑂
𝑨𝐺

)
+𝜔2

(
𝑫−1
𝐼

(
𝑨𝐺

)𝑇)]
𝑯 (𝑘−1) (9)

where 𝑯 (𝑘−1) denotes the output of the previous feature fusion layer; ⋆ denotes our one-step graph diffusion convolution; 𝑫𝑂 and 
𝑫𝐼 are the out-degree and in-degree matrices of 𝐴𝐺 respectively, 𝜔0, 𝜔1, 𝜔2 are model hyperparameters.

Based on the graph diffusion convolution defined in Equation (9), the information propagation step is defined in Equation (10), 
where the 𝑡-th feature segment 𝝃(𝑡) and hidden state of the previous time step 𝑯 𝑡−1 are concatenated as the input for the 𝐾-hop 
feature fusion.

𝑯 (0) = 𝝃(𝑡)‖𝑯 𝑡−1

𝑯 (𝑘) = 𝛼
(
𝑯 (𝑘−1) ⋆𝑨𝐺

)
+ 𝛽

(
𝑨𝐿𝑡𝑯 (𝑘−1))+ 𝛾 (𝝃(𝑡)‖𝑯 𝑡−1

) (10)

where ∗ denotes a tensor-tensor multiplication, ⋆ denotes the graph diffusion convolution, 𝛼, 𝛽 and 𝛾 are hyper-parameters that 
control the weights of different components.

We learn different integrated feature representations for updating different gates of GRU. The information selection step Θ𝑞 for 
the gate 𝑞 of GRU is defined in Equation (11).

Θ𝑞 =
∑𝐾

𝑗=0𝑯
(𝑘)𝜔𝑗𝑞 (11)

where 𝑞 denotes that the integrated feature representation is used to update the 𝑞 gate of GRU, 𝜔𝑗𝑞 are learnable parameters, 𝐾 is the 
depth of information propagation. Finally, the outputs of our weighted feature fusion unit are fed into a gated recurrent unit.

3.4.2. Updating GRUs for the GRU-based encoder-predictor model

The GRUs updating process is defined in the following form, utilizing the hierarchically integrated feature representation.

( )

7

𝑯 𝑡 =𝑼 𝑡 ⊙𝑯 𝑡−1 + 1 −𝑼 𝑡 ⊙𝑪 𝑡 (12)



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Table 1

Statistics of the seven datasets used for experiments.

Datasets Sample Sample Rate [34] Nodes Input length Output length

METR-LA 34272 5 minutes 207 12 12

PEMS04 16992 5 minutes 307 12 12

PEMS08 17856 5 minutes 170 12 12

TRAFFIC 17,544 1 hour 862 168 1

SOLAR-ENERGY 52,560 10 minutes 137 168 1

ELECTRICITY 26,304 1 hour 321 168 1

EXCHANGE-RATE 7588 1 day 8 168 1

𝑪 𝑡 = tanh
(
Θ𝑐

[
𝝃(𝑡)‖(𝑹𝑡 ⊙𝑯 𝑡−1

)]
+ 𝑏𝑐

)
(13)

and the update gate 𝑼 𝑡 and reset gate 𝑹𝑡 can be respectively given by

𝑹𝑡 = 𝜎
(
Θ𝑟

[
𝝃(𝑡)‖𝑯 𝑡−1

]
+ 𝑏𝑟

)
(14)

𝑼 𝑡 = 𝜎
(
Θ𝑢

[
𝝃(𝑡)‖𝑯 𝑡−1

]
+ 𝑏𝑢

)
(15)

where 𝜎(⋅) denotes the sigmoid activation function, 𝑹𝑡, 𝑼 𝑡 are reset gate and update gate at the current time step 𝑡, Θ𝑟, Θ𝑢, Θ𝑐 are 
different K-hop feature fusion layers with learnable parameters for the corresponding graph convolution modules.

As shown in Fig. 1, when our model makes multi-step predictions, it is an sequence-to-sequence model with an encoder and a 
decoder (predictor). We use the final hidden state of the encoder to initialize the decoder. The decoder will be trained with the 
teacher forcing mode, i.e., the previous ground truth observations are provided for training. During testing time, the predictions by 
the model will be used for predicting values of the next time step. To alleviate the performance degradation caused by the discrepancy 
of distribution between the training and testing mode, we use scheduled sampling [3] during the training. More details about our 
model training are provided in the next section.

3.5. Training strategy

In our research, we consider the time series forecasting problem as a regression problem. We use the Mean Absolute Error (MAE) 
between the predicting result �̂�𝑡∶(𝑡+𝑆−1) and the ground truth data 𝑿𝑡∶(𝑡+𝑆−1) =

(
𝑿𝑡,𝑁×𝑓1 ,𝑿(𝑡+1),𝑁×𝑓1 ,⋯ ,𝑿(𝑡+𝑆−1),𝑁×𝑓1

)
∈𝑅𝑆×𝑁×𝑓1 as 

the base training loss. The base training loss 𝑳𝑡
base

is defined as Equation (16).

𝑳𝑡
base

(
�̂�𝑡∶(𝑡+𝑆−1),𝑿𝑡∶(𝑡+𝑆−1)

)
= 1
𝑆

𝑡+𝑆−1∑
𝑡′=𝑡

|||�̂�𝑡′ ,𝑁×𝑓1 −𝑿𝑡′ ,𝑁×𝑓1
||| (16)

We also propose a cross-entropy loss 𝑳𝐶𝐸 between the edge scalar 𝜖𝑖𝑗 of the learned global graph (Equation (3)) and the pre-defined 
adjacency matrix values 𝐴𝑃

𝑖𝑗
. By introducing the pre-defined graph in the loss function, we increase the robustness of our model to 

noises of the training data. Thus, we can use 𝑳𝐶𝐸 as a regularization term to alleviate over-fitting. 𝑳𝐶𝐸 is defined in Equation (17):

𝑳𝐶𝐸 =
∑
𝑖𝑗

[
𝑨𝑃
𝑖𝑗
log 𝜖𝑖𝑗 −

(
1 −𝑨𝑃

𝑖𝑗

)
log

(
1 − 𝜖𝑖𝑗

)]
(17)

For our method, the overall training loss is defined as 𝑳 =
∑
𝑡𝑳

𝑡

base
+ 𝛿𝑳𝐶𝐸 , where 𝛿 > 0 is the regularization magnitude. Different 

weights 𝛿 > 0 are applied depending on whether 𝑨𝑃 data are missing, incomplete, or incorrect in the dataset. We demonstrate 
experimentally that prior knowledge 𝑨𝑃 can help substantially improve the effectiveness and quality of the learned global graph 
by incorporating 𝑳𝐶𝐸 as a regularization term. During training and testing, forecasting accuracy can be significantly improved by 
learning the appropriate graph.

4. Experiments

We conduct extensive experiments to evaluate our proposed AGLG-GRU model on several real-world datasets for multivariate 
time series forecasting. The extensive experiments include both single-step and multi-step forecasting. We compare our model with 
several competitive baselines and conduct ablation studies to assess the effectiveness of key components of our model.

4.1. Datasets

We use seven datasets to evaluate our method. These seven datasets include data from traffic, energy consumption and finance, 
and they have been widely used by research in multivariate time series forecasting. Thus, we can conveniently compare our method 
8

with other methods. The statistics of the seven datasets are listed in Table 1. More details about the datasets are as follows:



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

• METR-LA [18] This dataset on public transportation is derived from sensors installed at intersections and loops in Los Angeles, 
California [34]. There are 207 sensors in this dataset created based on pairwise distances from a demarcation point. During 
March 1 through June 30, 2012, this sensor network recorded traffic speeds in miles per hour (mph) at five-minute intervals. 
The METR-LA dataset also presents adjacency data in a graphical format through a 207 × 207 matrix representation.

• PEMS04 [8] This data collection represents actual public transit speeds recorded by the performance measurement system of the 
California Transportation Agency (CalTrans). PEMS04 includes traffic data from 307 detectors in the Bay Area during January 
and February 2018.

• PEMS08 [8] This is a collection of public transit speed data collected by the performance measurement system of the California 
Transportation Agency. PEMS08 contains 170 detectors, and the time range selected is July to August 2016. The time-frequency 
and velocity units are identical to the METR-LA data set.

• TRAFFIC [21] This traffic dataset is provided by the California Department of Transportation, 862 sensors collected measure-

ments during 2015 and 2016 and reported their findings. These figures represent the highway occupancy rates in the San 
Francisco Bay region.

• SOLAR-ENERGY [21] This is a dataset on solar energy generated by the National Renewable Energy Laboratory based on 
information received from 137 solar panels in Alabama State in 2007.

• ELECTRICITY [21] This is a dataset on electricity based on the information provided by the UCI Machine Learning Repository. 
It contains information regarding the energy use of 321 clients between 2012 and 2014.

• EXCHANGE-RATE [21] This dataset consists of the daily exchange rates for eight nations from 1990 to 2016. The list has eight 
countries: Australia, Great Britain, Canada, Switzerland, China, Japan, New Zealand, and Singapore.

4.2. Experimental settings

The time granularity of the three data sets that use speed as an informative characteristic is five minutes. For multi-step fore-

casting, the length of the input sequence is twelve, whereas the size of the target sequence is twelve. For single-step forecasting, the 
sequence length of the input data is 168 and the sequence length of the output data is one. We adjust the velocity data by zero-mean 
normalization, and any missing values are filled in by near interpolation. We take the road sensors to be vertex points and use their 
midpoints to calculate the location of the road segment. The pre-defined adjacency matrix 𝑨𝑃 is generated by applying a Gaussian 
threshold kernel to the Euclidean distance between the two road segments [30].

Hyper-parameter values are optimized via grid search. All databases have a batch size of 64, while AGLG-GRU has a depth of 1 
for its gated recurrent unit. Following the convolution layer, the learning rate is set to 0.1, 0.01, and 0.001, while the dropout rate 
is set to 0.1, 0.2, and 0.3. We adopt a variable number of hidden units from [16,32,64,96,128] since the dimension of the gated 
recurrent unit hidden state can significantly impact the accuracy of its forecasting. Given the preliminary results, the hidden unit 
dimension has been set to 64 for METR-LA and PEMS04 and 32 for PEMS08. For the purpose of preventing overfitting, the Adam 
optimizer uses the early stop mechanism to train the model. The specific values of the hyper-parameters, such as 𝛼, 𝛽, and hidden 
cell dimension, are adjusted based on the experimental results with different data sets. We compare the experimental results of some 
parameters at different values in the subsection 4.7.

Following [19], we divide the four sets of single-step predictive datasets into three sets in chronological order: the training set 
(60%) the validation set (20%), and the test set (20%) and we train independent models to predict the target future step (horizon) 3, 
6, 12, and 24. We divide these multi-step forecasting datasets into three sets in chronological order - training data (70%), validation 
data (20%), and test data (10%) and our multi-steps forecasting are based on a random optimizer 15 minutes, 30 minutes, 45 minutes, 
and an hour in advance.

4.3. Evaluation metrics

We use four evaluation metrics to evaluate the performance of single-step and multi-step forecasting, namely Relative Squared 
Error (RSE) [21], Empirical Correlation Coefficient (CORR) [21], Root Mean Squared Error (RMSE) [8], and Mean Absolute Error 
(MAE) [18]. For RMSE, MAE, and RSE, lower values are better. For CORR, higher values are better. The formal definition of the 
evaluation metrics is introduced as Equations (18).

RSE(𝑥, �̂�) =

√∑
(𝑖,𝑡)∈ΩTest

(
𝑥𝑖𝑡−�̂�𝑖𝑡

)2
√∑

(𝑖,𝑡)∈ΩTest
(
𝑥𝑖𝑡−mean

(
𝑥𝑖
))2 ,

CORR(𝑥, �̂�) = 1
𝑛

∑𝑛

𝑖=1

∑
𝑡

(
𝑥𝑖𝑡−mean

(
𝑥𝑖
))(

�̂�𝑖𝑡−mean
(
�̂�𝑖
))

√∑
𝑡

(
𝑥𝑖𝑡−mean

(
𝑥𝑖
))2(

�̂�𝑖𝑡−mean
(
�̂�𝑖
))2 ,

RMSE(𝑥, �̂�) =
√

1
𝑛

∑𝑛

𝑖=1
(
�̂�𝑖 − 𝑥𝑖

)2
,

MAE(𝑥, �̂�) = 1|𝑛|
∑𝑛

𝑖=1
||�̂�𝑖 − 𝑥𝑖||

(18)
9

where 𝑥𝑖 denotes the ground truth and the �̂�𝑖 represents the predicted time series.



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 4. The multi-step forecasting accuracy of different methods.

4.4. Baselines

We compare our proposed model, AGLG-GRU, with models that reflect recent improvements in multivariate time series forecast-

ing. Baseline methods are summarized as follows:

4.4.1. Single-step forecasting baselines

• AR [5]: A prediction model using auto-regression.

• VAR-MLP [50]: A hybrid model of vector auto-regressive model and multilayer perceptron.

• RNN-GRU [44]: A prediction model based on recurrent neural network with fully connected gated recurrent unit.

• TPA-LSTM [29]: A forecasting method based on recurrent neural network with attention mechanism.

• MTGNN [38]: A graph neural network framework for multivariate time series forecasting. It consists of a graph learning module, 
a graph convolution module and a temporal convolution module.

4.4.2. Multi-step forecasting baselines

• ARIMA [45]: A widely used time series analysis technique based on the combination of autoregression and moving average.

• VAR [16]: A method based on vector auto-regression.

• FC-LSTM [48]: A method based on recurrent neural networks with fully connected Long Short-term Memory hidden unit.

• Graph WaveNet [37]: A method consists of an adaptive dependency graph learning module and a stacked dilated 1D convolution 
module.

• STGCN [42]: A method uses spatial-temporal graph convolution network and incorporates graph convolutions with 1D convo-

lutions.

• DCRNN [22]: A method based on diffusion convolution recurrent neural network. It incorporates diffusion graph convolution 
with recurrent neural network in an encoder-decoder architecture.

• AGCRN [2]: A GNN-based and RNN-based model, which employs adaptive graph learning and integrates gated recurrent unit 
with graph convolutions.

• ASTGCN [13]: A model uses a devised spatial-temporal attention mechanism to capture the spatial-temporal correlations through 
calculating the attention matrix.

4.5. Main results

This section presents experimental results of single-step and multi-step forecasting. Table 2 and 3 provide the experimental results 
for AGLG-GRU, demonstrating that AGLG-GRU achieves state-of-the-art results on the majority of the tasks.

4.5.1. Single-step forecasting performance

In this experiment, we compare AGLG-GRU with other time series prediction models. Table 2 shows the experimental results 
for the single-step forecasting task. The AGLG-GRU routinely achieves the best results across nearly all data horizons pertaining 
to solar energy, traffic, and electricity. Particularly for traffic data, AGLG-GRU has demonstrated considerable RSE improvement. 
As a result of the more natural nature of traffic data, AGLG-GRU improves the results of our model, which aligns better with our 
assumptions regarding spatio-temporal dependence. The future occupancy rate of a road is inherently affected by its past and its 
related highways. Due to the reduced graph size and the lack of available exchange-rate training examples, AGLG-GRU does not 
enhance the exchange-rate data.

4.5.2. Multi-step forecasting performance

Table 3 lists the performances of our model and baselines for predicting 15 minutes, 30 minutes, 45 minutes, and one hour on 
the METR-LA, PEMS04, and PEMS08 datasets. The results can be summarized as follows:

1) The underlying properties of highly complex traffic flow data are easier to capture using deep learning algorithms than with 
10

typical time series methods (such as ARIMA and VAR).



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Table 2

Comparative results of single-step forecasting.

Data Method 3 horizon 6 horizon 12 horizon 24 horizon

RSE CORR RSE CORR RSE CORR RSE CORR

Solar-Energy AR 0.2435 0.971 0.379 0.9263 0.5911 0.8107 0.8699 0.5314

VAR-MLP 0.1922 0.9829 0.2679 0.9655 0.4244 0.9508 0.6841 0.7149

RNN-GRU 0.1932 0.9823 0.2628 0.9675 0.4163 0.915 0.4852 0.8823

TPA-LSTM 0.1803 0.985 0.2347 0.9742 0.3234 0.9487 0.4389 0.9081

MTGNN 0.1778 0.9852 0.2348 0.9726 0.3109 0.9509 0.427 0.9031

AGLG-GRU 0.1762 0.9842 0.2302 0.9682 0.3021 0.9532 0.413 0.9084

Traffic AR 0.5991 0.7752 0.6218 0.7568 0.6252 0.7544 0.63 0.7519

VAR-MLP 0.5582 0.8245 0.6579 0.7695 0.6023 0.7929 0.6146 0.7891

RNN-GRU 0.5358 0.8511 0.5522 0.8405 0.5562 0.8345 0.5633 0.83

TPA-LSTM 0.4487 0.8812 0.4658 0.8717 0.4641 0.8717 0.4765 0.8625

MTGNN 0.4162 0.8963 0.4754 0.8667 0.4461 0.8794 0.4535 0.8810

AGLG-GRU 0.4173 0.8958 0.4722 0.8541 0.4427 0.8755 0.4526 0.8842

Electricity AR 0.0995 0.8845 0.1035 0.8632 0.105 0.8591 0.1054 0.8595

VAR-MLP 0.1393 0.8708 0.1620 0.8389 0.1557 0.8192 0.1274 0.8679

RNN-GRU 0.1102 0.8597 0.1144 0.8623 0.1183 0.8472 0.1295 0.8651

TPA-LSTM 0.0823 0.9439 0.0916 0.9337 0.0964 0.925 0.1006 0.9133

MTGNN 0.0745 0.9474 0.0878 0.9316 0.0916 0.9278 0.0953 0.9234

AGLG-GRU 0.0738 0.9434 0.0864 0.9302 0.0912 0.9283 0.0947 0.9274

Exchange-Rate AR 0.0228 0.9734 0.0279 0.9656 0.0353 0.9526 0.0445 0.9357

VAR-MLP 0.0265 0.8609 0.0394 0.8725 0.0407 0.8280 0.0578 0.7675

RNN-GRU 0.0192 0.9786 0.0264 0.9712 0.0408 0.9531 0.0626 0.9223

TPA-LSTM 0.0174 0.979 0.0241 0.9709 0.0341 0.9564 0.0444 0.9381

MTGNN 0.0194 0.9786 0.0259 0.9708 0.0349 0.9551 0.0456 0.9372

AGLG-GRU 0.0191 0.9792 0.0233 0.9701 0.0328 0.9548 0.0449 0.9372

Table 3

Comparison with baselines on multi-step traffic flow forecasting.

Data Method 15 min 30 min 45 min 60 min

MAE RMSE MAE RMSE MAE RMSE MAE RMSE

METR-LA ARIMA 3.9 8.21 5.15 10.45 5.82 11.83 6.92 13.23

VAR 4.42 7.89 5.41 9.13 6.13 9.76 6.52 10.11

LSTM 3.44 6.30 3.77 7.23 4.08 7.97 4.37 8.69

Graph WaveNet 2.67 5.51 3.07 6.22 3.32 6.77 3.53 7.37

STGCN 2.88 5.74 3.47 7.24 4.12 8.76 4.59 9.43

DCRNN 2.77 5.38 3.15 6.45 3.43 7.14 3.61 7.62

AGCRN 2.86 6.27 3.23 6.58 3.34 7.05 3.62 7.51

ASTGCN 3.31 7.62 4.13 9.77 4.54 10.08 5.06 11.66

AGLG-GRU 2.74 4.87 3.1265 5.710 3.36 6.27 3.706 6.784

PEMS04 ARIMA 24.36 43.47 30.58 61.30 37.62 66.43 44.31 71.02

VAR 32.47 49.84 34.07 52.06 35.15 53.61 36.18 55.05

LSTM 22.90 37.10 29.11 44.98 34.38 51.48 39.41 60.84

Graph WaveNet 24.45 35.45 26.81 37.29 27.81 39.82 29.89 42.63

STGCN 23.36 36.04 24.58 38.01 25.64 40.15 28.55 42.53

DCRNN 20.62 32.11 21.22 34.33 23.16 36.11 24.87 37.12

AGCRN 21.08 33.23 22.16 34.89 23.54 35.73 25.65 36.79

ASTGCN 20.46 31.83 21.75 33.76 22.82 35.34 24.15 36.93

AGLG-GRU 19.60 29.85 20.53 30.13 21.41 31.32 22.26 32.54

PEMS08 ARIMA 18.25 29.58 22.98 39.61 28.16 50.03 33.87 60.15

VAR 18.42 26.96 21.47 31.37 23.95 34.83 25.93 37.41

LSTM 17.46 28.14 23.24 37.05 27.05 43.37 33.67 50.31

Graph WaveNet 19.13 31.05 20.52 33.94 21.87 34.19 23.03 34.23

STGCN 17.16 26.04 18.04 27.48 18.95 28.86 19.55 29.93

DCRNN 15.98 23.67 16.82 26.36 17.54 26.49 19.02 28.67

AGCRN 15.54 23.34 17.52 27.09 17.89 26.77 18.87 27.74

ASTGCN 15.67 23.76 16.59 25.29 17.35 26.43 18.28 28.05

AGLG-GRU 14.99 22.24 15.94 23.81 16.77 25.13 17.51 26.28
11



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Table 4

Summary of models for ablation studies.

Model Name Description of the model

AGLG-GRU The full model of Adaptive Global-Local Graph

structure learning with Gated Recurrent Units (AGLG-GRU)

AGLG-GRU-ndi The graph diffusion convolution operation is replaced by

graph convolution.

AGLG-GRU-ndg The dynamic local graph structure learning module is removed.

The local adjacency matrices are replaced by an identity matrix.

AGLG-GRU-nsg The static global graph structure learning module is removed.

The global adjacency matrix is replaced by an identity matrix

Fig. 5. Ablation study on METR-LA.

2) Graph-based methods allow the application of deep learning models to non-Euclidean geometric structures by considering both 
the correlation and topology of graph structures. Graph-based models (such as Graph WaveNet, STGCN, AGCRN, ASTGCN, and our 
model) outperform classic deep learning approaches (Long Short-term Memory and DCRNN). Consequently, the experimental results 
indicate that non-euclidean data should be taken into account when estimating traffic flow.

3) Our AGLG-GRU achieves the best results for the four forecasting intervals. According to Fig. 4, the forecasting accuracy varies 
with the length of the interval for each model. As only the temporal correlation of the time series is modeled, the VAR and ARIMA 
models are advantageous for short-term time series forecasting. With the forecasting interval increases, the performances deteriorate 
dramatically. With the addition of spatial correlation to DCRNN and Long Short-term Memory, the forecasting accuracy gradually 
decreases as time intervals increase. The non-Euclidean geometry of the road network plays a significant role in the accuracy of some 
road network models, such as Graph WaveNet, STGCN, AGCRN, ASTGCN, and our own. Experiments demonstrate that non-Euclidean 
data is essential for predicting traffic flows, and the AGLG-GRU model outperforms all other models on all three datasets.

By combining the global spatial characteristics of the network with the dynamic temporal characteristics of the data, our AGLG-

GRU model achieves a higher forecasting accuracy than other baseline techniques during the experiment. When the forecasting 
interval is longer than 30 minutes, the advantages of our model are even more evident. The model produces better results for three 
main reasons.

• The global structure learning module uses a more efficient mapping of static spatial-temporal features of the time series as input 
to the forecasting model by learning the probabilistic properties of the graph.

• The local structure learning modules modify the weights of the graph feature vectors across batches of data, capturing the 
dynamic relevance of the data.

• Hierarchically using gated recurrent unit improved by bivariate diffusion of graph random walks, the model captures global and 
local spatial features of multivariate time series in a targeted manner.

A longer-term forecasting with more precision will be advantageous for practical applications since this will provide road man-

agers and travelers sufficient time to plan and prepare for travel. Through dynamically capturing the hierarchical spatial-temporal 
features of traffic flow data, we have enhanced the performance of long-term forecasting of traffic networks with complex traffic 
topologies.

4.6. Ablation study

We use the METR-LA data to conduct ablation research in order to evaluate the effectiveness of key components of our proposed 
model. As shown in Table 4, we test the full model of AGLG-GRU with three other models that have certain component removed 
or replaced. Following [38], we repeat each experiment ten times with 50 epochs per repetition and calculate the MAE, RMSE, and 
12

MAPE on the validation set. The average values of the three metrics are given in Fig. 5.



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 6. Parameter Study on METR-LA and PEMS04.

The introduction of a global graph structure learning module significantly improves the results as it enables the improvement 
of input quality by learning the probabilistic properties from space features. The effect of the diffusion convolution mechanism 
is evident as well: it confirms that bidirectional graph random walks are effective in capturing global information in diffusion 
convolution layers. Lastly, our local graph structure learning modules prove to be effective in terms of improving long-term traffic 
flow forecasting (horizon 6 and 12). It enables our model to hierarchically capture dynamic parameters step by step as the level of 
learning difficulty increases.

4.7. Hyper-parameter study

This section investigates the influence of hyper-parameters using a parameter analysis of the AGLG-GRU model. Each experiment 
is conducted ten times, with the MAE on the test set acting as a comparison indicator. During each experiment, only one parameter 
is examined, while all the other variables are held constant. Fig. 6 displays the experimental results of hyper-parameter study on the 
datasets METR-LA and PEMS04 for the 30-minute traffic flow forecasting. The hyper-parameters 𝛽 and 𝐾 range are set between 1 and 
30. The dynamic local graph convolution module can be simplified by setting 𝛽 to 0, where the dynamic local graph convolution is 
eliminated. Fig. 6-(a) and 6-(b) depicts the experimental findings for the MAE indicator. It is evident from the curves that decreasing 
𝐾 or raising 𝛽 increases forecasting accuracy.

Node embedding dimensions range from 20 to 50, and hidden state dimensions range from 32 to 128. Fig. 6-(c) and 6-(d) 
demonstrates that by increasing the dimension of the embedded node in the dynamic graph and the hidden state of the gated 
recurrent unit, the representational capability of the AGLG-GRU will be strengthened, while reducing the MAE loss. We find from the 
experimental results that the forecasting accuracy initially increases as the hidden unit dimension increases, but declines with time 
in METR-LA. According to the results shown in Fig. 6-(e), it appears that the AGLG-GRU model is insensitive to the concealed state 
of the large size nodes in PEMS04. Due to the significantly more complex traffic conditions of the PEMS04 dataset, embeddings of 
nodes must have a greater dimension to hold sufficient information. Fig. 6-(f) illustrates the excellent performance of the AGLG-GRU 
on the PEMS04 dataset with 40 embedded nodes. The increase in the hidden cell dimension or the number of hidden cells makes 
13

the model more complicated and hence captures a wider range of data attributes. However, overfitting arises when these variables 



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 7. Data Visualization on PEMS04 for node 45.

Fig. 8. Data Visualization on PEMS04 for node 125.

take on more than a certain value, and the ideal values for the hidden state dimension and node embedding for the METR-LA dataset 
are around 64 and 30 respectively. Therefore, it is vital for this study to determine the correct hyper-parameter values based on 
individual data sets, as the selection of each hyper-parameter value will have a significant impact on the final performance of the 
model.

4.8. Model interpretation

For a deeper understanding of the model, we compare the predicted speed values with the ground-truth data of some nodes in 
the PEMS04 traffic network. As depicted in the illustration of comparison of data of node 45 in Fig. 7, our AGLG-GRU model can 
predict traffic peaks and trends with a high degree of accuracy and works well when simulating rapid changes in traffic speeds. 
Fig. 8 illustrates the comparison between predicted data and ground-truth data for node 125. This demonstrates that by utilizing our 
AGLG-GRU model, we can obtain accurate and smooth forecasting at nodes with moderate fluctuations in traffic speed.

We design a hierarchy experiment to demonstrate the effectiveness of the model in acquiring dynamic features by training several 
batches of test data and examining the weight matrix hierarchically. Three separate sets of data spanning different observation scales 
were fed into the trained model from the METR-LA test set. Fig. 9 depicts the heat map of the dynamic adjacency matrix obtained 
through the encoder and predictor structures at the scale (0,1). The weights of the 𝑖𝑡ℎ rows and 𝑗𝑡ℎ columns in the heatmap represent 
the influence of 𝑛𝑜𝑑𝑒𝑖 on 𝑛𝑜𝑑𝑒𝑗 . As shown in Fig. 9, some rows and columns have significantly higher weights than others, indicating 
that the nodes corresponding to the high-weight rows and columns have more influence than others during that period of time.

To test the effectiveness of our proposed dynamic local graphs, we compare the differences in the attributes of the same node in 
the dynamic adjacency matrix at different observation scales. In our experiment, we select the node 75 in METR-LA dataset for case 
study. Some subsets of METR-LA data collected from 8:00 am to 10:00 am and 4:00 pm to 6:00 pm were fed into a trained dynamic 
local graph learning module to generate dynamic local graphs.

Fig. 10 distinguishes the ten most influential nodes for node 75 at two different observation scales, where the red dots represent 
node 75, the green dots represent the ten most influential nodes between 8:00 and 10:00 am, and the yellow dots represent the ten 
most influential nodes between 4:00 and 6:00 pm. Since the relationships between neighboring nodes in our model are dynamically 
acquired at different observation scales, our model allows for the dynamic discovery of spatial-temporal dependencies in the time 
series and goes beyond the limitations of a fixed adjacency matrix to perform more efficiently.

We calculate the importance of METR-LA data nodes at different observation scales and plot the top ten nodes in terms of 
14

importance on Fig. 11. The following Equation (19) is used to calculate node importance:



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 9. Heatmaps of the dynamic adjacency matrix for each encoder layer.

Fig. 10. Top 10 nodes affecting node 75 on METR-LA at various times.

𝑰𝑴𝑞 =
∑

𝑨𝐺
𝑖𝑞
+
∑

𝑨𝐺
𝑞𝑗

(19)

where 𝑰𝑴𝑞 represents the importance of the node 𝑞, and 𝑨𝐺 represents the dynamic adjacency matrix. A sampling of data from the 
METR-LA dataset was taken between 8:00 AM and 10:00 AM. Fig. 11 displays the distribution of the more influential nodes over the 
several network layers of the encoder. Most of the prominent nodes are located near junctions, which is a characteristic feature of 
traffic networks.

4.9. Model computational complexity analysis

Table 5 presents a comparison of computational complexity, measured in terms of FLOP (Floating Point Operations) counts. It is 
evident that our method exhibits higher FLOP counts compared to other methods. This is primarily attributed to the graph structure 
learning process. The inclusion of graph structure learning introduces a considerable number of FLOPs since it entails learning the 
correlation features of the data’s graph structure. It should be noted that although our method may introduce additional complexity, 
they possess the potential to offer superior accuracy and capture intricate dependencies, global trends and local randomness within 
15

time series data.



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

Fig. 11. The top 10 nodes as evaluated at different encoder levels in terms of influence.

Table 5

A comparison of compu-

tational complexity on 
METR-LA.

Method Flops

DCRNN 1.0e07

AGCRN 1.0e07

LSTM 8.1e08

ASTGCN 2.3e09

AGLG-GRU 4.5e09

5. Conclusion

In this paper, we propose AGLG-GRU, a novel multivariate time series forecasting method that combines the advantages of 
graph-based methods and global-local methods. Our AGLG-GRU model consists of four modules: a global graph structure learning 
module, a local graph structure learning module, a GRU module for hierarchical weighted feature integration, and a GRU-based 
encoder-predictor module for multi-step forecasting. The learned global graph captures the global trends and shared information of 
variables. The learned local graphs capture the local randomness of variables and their dynamic correlations. The GRU module fuses 
the information captured by the global graph and local graphs to update the gates for encoding input features and forecasting multi-

step future values. Experiments demonstrate that our approach improves multivariate time series forecasting accuracy, particularly 
for long-term forecasting.

In the future work, we will investigate to combine our method with deep stochastic configuration networks (SCN) [35,7]. Robust 
SCN [40] has shown advantages in processing data with noises and outliers, and federated SCN [6] is capable of analyzing distributed 
large-scale data. We will explore using robust SCN to learn adaptive graphs from data with noises and outliers. We expect to be able 
to process distributed multivariate time series data by using federated SCN. We will also try the latest methods of graph stream 
summarization [17], preference modeling in next location prediction [23], and graph structure learning-based knowledge tracing 
16

[32].



Information Sciences 648 (2023) 119560T. Guo, F. Hou, Y. Pang et al.

CRediT authorship contribution statement

Ting Guo: Conceptualization, Methodology, Software, Writing – original draft. Feng Hou: Conceptualization, Writing – original 
draft. Yan Pang: Funding acquisition, Investigation. Xiaoyun Jia: Writing – review & editing. Zhongwei Wang: Project administra-

tion, Validation. Ruili Wang: Supervision, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 
influence the work reported in this paper.

Data availability

All the datasets are publicly available

Acknowledgement

This study was supported by the Key Research and Development Project of Hunan Province (2022GK2025); Hunan Key Laboratory 
of Intelligent Logistics Technology, China (No. 2019TP1015).

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	Learning and integration of adaptive hybrid graph structures for multivariate time series forecasting
	1 Introduction
	2 Related work
	2.1 Graph-based methods for multivariate time series forecasting
	2.2 Global-local methods for multivariate time series forecasting

	3 Methodology
	3.1 Preliminaries and problem formulation
	3.2 The architecture of AGLG-GRU
	3.3 Adaptive global-local graph structure learning
	3.3.1 Global graph structure learning
	3.3.2 Local graph structure learning

	3.4 Hierarchical feature integration with GRUs
	3.4.1 Hierarchical weighted feature integration
	3.4.2 Updating GRUs for the GRU-based encoder-predictor model

	3.5 Training strategy

	4 Experiments
	4.1 Datasets
	4.2 Experimental settings
	4.3 Evaluation metrics
	4.4 Baselines
	4.4.1 Single-step forecasting baselines
	4.4.2 Multi-step forecasting baselines

	4.5 Main results
	4.5.1 Single-step forecasting performance
	4.5.2 Multi-step forecasting performance

	4.6 Ablation study
	4.7 Hyper-parameter study
	4.8 Model interpretation
	4.9 Model computational complexity analysis

	5 Conclusion
	CRediT authorship contribution statement
	Declaration of competing interest
	Data availability
	Acknowledgement
	References