# A Survey on Graph Neural Networks for Time Series: Forecasting,
Classification, Imputation, and Anomaly Detection

Ming Jin, Huan Yee Koh, Qingsong Wen, Daniele Zambon, Cesare Alippi, ,
Geoffrey I. Webb, , Irwin King, , Shirui Pan  Ming Jin, Huan Yee Koh, and
Geoff Webb are with the Department of Data Science and AI, Monash University,
Melbourne, Australia. E-mail: {ming.jin, huan.koh<EMAIL_ADDRESS>Qingsong Wen is with Alibaba DAMO Seattle, WA, USA. E-mail:
<EMAIL_ADDRESS>Daniele Zambon and Cesare Alippi are with Swiss AI Lab
IDSIA, Università della Svizzera italiana, Lugano, Switzerland. E-mail:
{daniele.zambon<EMAIL_ADDRESS>Irwin King is with the Department of
Computer Science & Engineering, The Chinese University of Hong Kong. E-mail:
<EMAIL_ADDRESS>Shirui Pan is with the School of Information and
Communication Technology and Institute for Integrated and Intelligent Systems
(IIIS), Griffith University, Queensland, Australia. E-mail:
<EMAIL_ADDRESS>M. Jin and H. Y. Koh contributed equally to this work.
Corresponding Author: Shirui Pan.GitHub Page:
https://github.com/KimMeen/Awesome-GNN4TSVersion date:

###### Abstract

Time series are the primary data type used to record dynamic system
measurements and generated in great volume by both physical sensors and online
processes (virtual sensors). Time series analytics is therefore crucial to
unlocking the wealth of information implicit in available data. With the
recent advancements in graph neural networks (GNNs), there has been a surge in
GNN-based approaches for time series analysis. These approaches can explicitly
model inter-temporal and inter-variable relationships, which traditional and
other deep neural network-based methods struggle to do. In this survey, we
provide a comprehensive review of graph neural networks for time series
analysis (GNN4TS), encompassing four fundamental dimensions: forecasting,
classification, anomaly detection, and imputation. Our aim is to guide
designers and practitioners to understand, build applications, and advance
research of GNN4TS. At first, we provide a comprehensive task-oriented
taxonomy of GNN4TS. Then, we present and discuss representative research works
and introduce mainstream applications of GNN4TS. A comprehensive discussion of
potential future research directions completes the survey. This survey, for
the first time, brings together a vast array of knowledge on GNN-based time
series research, highlighting foundations, practical applications, and
opportunities of graph neural networks for time series analysis.

###### Index Terms:

Time series, graph neural networks, deep learning, forecasting,
classification, imputation, anomaly detection.

###### Contents

  1. 1 Introduction
    1. 2 Definition and Notation
      1. 3 Framework and Categorization
        1. 3.1 Task-Oriented Taxonomy
        2. 3.2 Unified Methodological Framework
          1. 4 GNNs for Time Series Forecasting
            1. 4.1 Modeling Inter-Variable Dependencies
            2. 4.2 Modeling Inter-Temporal Dependencies
            3. 4.3 Forecasting Architectural Fusion
              1. 5 GNNs for Time Series Anomaly Detection
                1. 5.1 General Framework for Anomaly Detection
                2. 5.2 Discrepancy Frameworks for Anomaly Detection
                  1. 6 GNNs for Time Series Classification
                    1. 6.1 Univariate Time Classification
                    2. 6.2 Multivariate Time Series Classification
                      1. 7 GNNs for Time Series Imputation
                        1. 7.1 In-Sample Imputation
                        2. 7.2 Out-of-Sample Imputation
                          1. 8 Practical Applications
                            1. 9 Future Directions
                              1. 10 Conclusions

##  1 Introduction

The advent of advanced sensing and data stream processing technologies has led
to an explosion of time series data [1, 2, 3]. The analysis of time series not
only provides insights into past trends but also facilitates a multitude of
tasks such as forecasting [4], classification [5], anomaly detection [6], and
data imputation [7]. This lays the groundwork for time series modeling
paradigms that leverage on historical data to understand current and future
possibilities. Time series analytics have become increasingly crucial in
various fields, including but not limited to cloud computing, transportation,
energy, finance, social networks, and the Internet-of-Things [8, 9, 10, 11].

Figure 1: Graph neural networks for time series analysis (GNN4TS). In this
example of wind farm, different analytical tasks can be categorized into time
series forecasting, classification, anomaly detection, and imputation.

Many time series involve complex interactions across time (such as lags in
propagation of effects) and variables (such as the relationship among the
variables representing neighboring traffic sensors). By treating time points
or variables as nodes and their relationships as edges, a model structured in
the manner of a network or graph can effectively solve the task at hand by
exploiting both data and relational information. Indeed, much time series data
is spatial-temporal in nature, with different variables in the series
capturing information about different locations – space – too, meaning it
encapsulates not only time information but also spatial relationships [12].
This is particularly evident in scenarios such as urban traffic networks,
population migration, and global weather forecasting. In these instances, a
localized change, such as a traffic accident at an intersection, an epidemic
outbreak in a suburb, or extreme weather in a specific area, can propagate and
influence neighboring regions. This spatial-temporal characteristic is a
common feature of many dynamic systems, including the wind farm in Figure 1,
where the underlying time series displays a range of correlations and
heterogeneities [13, 14]. Traditional analytic tools, such as support vector
regression (SVR) [15, 16], gradient boosting decision tree (GBDT) [17, 18],
vector autoregressive (VAR) [19, 20], and autoregressive integrated moving
average (ARIMA) [21, 22], struggle to handle complex time series relations
(e.g., nonlinearities and inter-variable relationships), resulting in less
accurate prediction results [23]. The advent of deep learning technologies has
led to the development of different neural networks based on convolutional
neural networks (CNN) [24, 25], recurrent neural networks (RNN) [26, 27], and
Transformers [28], which have shown significant advantages in modeling real-
world time series data. However, one of the biggest limitations of the above
methods is that they do not explicitly model the spatial relations existing
between time series in non-Euclidean space [13, 29], which limits their
expressiveness [29].

In recent years, graph neural networks (GNNs) have emerged as a powerful tool
for learning non-Euclidean data representations [30, 31, 32], paving the way
for modeling real-world time series data. This enables the capture of diverse
and intricate relationships, both inter-variable (connections between
different variables within a multivariate series) and inter-temporal
(dependencies between different points in time). Considering the complex
spatial-temporal dependencies inherent in real-world scenarios, a line of
studies has integrated GNNs with various temporal modeling frameworks to
capture both spatial and temporal dynamics and demonstrate promising results
[13, 33, 34, 35, 36]. While early research efforts were primarily concentrated
on various forecasting scenarios [13, 34, 35], recent advancements in time
series analysis utilizing GNNs have demonstrated promising outcomes in other
mainstream tasks. These include classification [37, 38], anomaly detection
[39, 40], and imputation [41, 42]. In Figure 1, we provide an overview of
graph neural networks for time series analysis (GNN4TS).

Related Surveys. Despite the growing body of research performing various time
series analytic tasks with GNNs, existing surveys tend to focus on specific
perspectives within a restricted scope. For instance, the survey by Wang et
al. [12] offers a review of deep learning techniques for spatial-temporal data
mining, but it does not specifically concentrate on GNN-based methods. The
survey by Ye et al. [36] zeroes in on graph-based deep learning architectures
in the traffic domain, primarily considering forecasting scenarios. A recent
survey by Jin et al.[13] offers an overview of GNNs for predictive learning in
urban computing, but neither extends its coverage to other application domains
nor thoroughly discusses other tasks related to time series analysis. Finally,
we mention the work by Rahmani et al. [43], which expands the survey of GNNs
to many intelligent transportation systems, but tasks other than forecasting
remain overlooked. A detailed comparison between our survey and others is
presented in section 1.

TABLE I: Comparison between our survey and other related surveys.

Survey | Domain | Scope
---|---|---
Specific |  General |  Forecasting |  Classification |  Anomaly Detection |  Imputation
Wang et al. [12] |  |  2714 |  |  |  |
Ye et al. [36] |  2714 |  |  |  |  |
Jiang and Luo [34] |  2714 |  |  |  |  |
Bui et al. [35] |  2714 |  |  |  |  |
Jin et al. [13] |  2714 |  |  |  |  |
Al Sahili and Awad [33] |  |  2714 |  |  |  |
Rahmani et al. [43] |  |  2714 |  |  |  |
Our Survey |  |  2714 |  |  |  |

  * *

Specifically,  represents “Not Covered”,  signifies “Partially Covered”, and
corresponds to “Fully Covered”.

To fill the gap, this survey offers a comprehensive and up-to-date review of
graph neural networks for time series analysis, encompassing the majority of
tasks ranging from time series forecasting, classification, anomaly detection,
and imputation. Specifically, we first provide two broad views to classify and
discuss existing works from the task- and methodology-oriented perspectives.
Then, we delve into six popular application sectors within the existing
research of GNN4TS, and propose several potential future research directions.
The key contributions of our survey are summarized as follows:

  * •

First Comprehensive Survey. To the best of our knowledge, this is the first
comprehensive survey that reviews the recent advances in mainstream time
series analysis tasks with graph neural networks. It covers a wide range of
recent research and provides a broad view of the development of GNN4TS without
restricting to specific tasks or domains.

  * •

Unified and Structured Taxonomy. We present a unified framework to
structurally categorize existing works from task- and methodology-oriented
perspectives. In the first classification, we offer an overview of tasks in
time series analysis, covering different problem settings prevalent in GNN-
based research; and in the second classification, we dissect GNN4TS in terms
of spatial and temporal dependencies modeling and the overall model
architecture.

  * •

Detailed and Current Overview. We conduct a comprehensive review that not only
covers the breadth of the field but also delves into the depth of individual
studies with fine-grained classification and detailed discussion, providing
readers with an up-to-date understanding of the state-of-the-art in GNN4TS.

  * •

Broadening Applications. We discuss the expanding applications of GNN4TS
across various sectors, highlighting its versatility and potential for future
growth in diverse fields.

  * •

Future Research Directions. We shed light on potential future research
directions, offering insights and suggestions that could guide and inspire
future research in the field of GNN4TS.

The remainder of this survey is organized as follows: section 2 provides
notations used throughout the paper. section 3 presents the taxonomy of GNN4TS
from different perspectives. section 4, section 5, section 6, and section 7
review the four major tasks in the GNN4TS literature. section 8 surveys
popular applications of GNN4TS across various fields, while section 9 examines
open questions and potential future directions.

##  2 Definition and Notation

Time series data comprises a sequence of observations gathered or recorded
over a period of time. This data can be either regularly or irregularly
sampled, with the latter also referred to as time series data with missing
values. Within each of these cases, the data can be further classified into
two primary types: univariate and multivariate time series. In the sequel, we
employ bold uppercase letters (e.g., $\mathbf{X}$), bold lowercase letters
(e.g., $\mathbf{x}$), and calligraphic letters (e.g., $\mathcal{V}$) to denote
matrices, vectors, and sets, respectively.

######  Definition 1 (Univariate Time Series)

A univariate time series is a sequence of scalar observations collected over
time, which can be regularly or irregularly sampled. A regularly sampled
univariate time series is defined as
$\mathbf{X}=\\{x_{1},x_{2},...,x_{T}\\}\in\mathbb{R}^{T}$, where
$x_{t}\in\mathbb{R}$. For an irregularly sampled univariate time series,
observations are collected at non-uniform time intervals, such as
$\mathbf{X}=\\{(t_{1},x_{1}),(t_{2},x_{2}),...,(t_{T},x_{T})\\}\in\mathbb{R}^{T}$,
where time points are non-uniformly spaced.

######  Definition 2 (Multivariate Time Series)

A multivariate time series is a sequence of $N$-dimensional vector
observations collected over time, i.e., $\mathbf{X}\in\mathbb{R}^{N\times T}$.
A regularly sampled multivariate time series has vector observations collected
at uniform time intervals, i.e., $\mathbf{x}_{t}\in\mathbb{R}^{N}$. In an
irregularly sampled multivariate time series, there are possibly $N$ unaligned
time series with respect to time steps, which implies only $0\leq n\leq N$
observations available at each time step.

The majority of research based on GNNs focuses on modeling multivariate time
series, as they can be naturally abstracted into spatial-temporal graphs. This
abstraction allows for an accurate characterization of dynamic inter-temporal
and inter-variable dependencies. The former describes the relations between
different time steps within each time series (e.g., the temporal dynamics of
red nodes between $t_{1}$ and $t_{3}$ in Figure 2), while the latter captures
dependencies between time series (e.g., the spatial relations between four
nodes at each time step in Figure 2), such as the geographical information of
the sensors generating the data for each variable. To illustrate this, we
first define attributed graphs.

######  Definition 3 (Attributed Graph)

An attributed graph is a static graph that associates each node with a set of
attributes, representing node features. Formally, an attributed graph is
defined as $\mathcal{G}=(\mathbf{A},\mathbf{X})$, which consists of a
(weighted) adjacency matrix $\mathbf{A}\in\mathbb{R}^{N\times N}$ and a node-
feature matrix $\mathbf{X}\in\mathbb{R}^{N\times D}$. The adjacency matrix
represents the graph topology, which can be characterized by
$\mathcal{V}=\\{v_{1},v_{2},\dots,v_{N}\\}$, the set of $N$ nodes, and
$\mathcal{E}=\\{e_{ij}:=(v_{i},v_{j})\in\mathcal{V}\times\mathcal{V}\mid\mathbf{A}_{ij}\neq
0\\}$, the set of edges; $\mathbf{A}_{ij}$ is the $(i,j)$-th entry in the
adjacency matrix $\mathbf{A}$. The feature matrix $\mathbf{X}$ contains the
node attributes, where the $i$-th row $\mathbf{x}_{i}\in\mathbb{R}^{D}$
represents the $D$-dimensional feature vector of node $v_{i}$.

In attributed graphs, multi-dimensional edge features can be considered too,
however, this paper assumes only scalar weights encoded in the adjacency
matrix to avoid overwhelming notations.

In light of this, a spatial-temporal graph can be described as a series of
attributed graphs, which effectively represent (multivariate) time series data
in conjunction with either evolving or fixed structural information over time.

######  Definition 4 (Spatial-Temporal Graph)

A spatial-temporal graph can be interpreted as a discrete-time dynamic graph
[44], i.e.,
$\mathcal{G}=\\{\mathcal{G}_{1},\mathcal{G}_{2},\cdots,\mathcal{G}_{T}\\}$,
where $\mathcal{G}_{t}=(\mathbf{A}_{t},\mathbf{X}_{t})$ denotes an attributed
graph at time $t$. $\mathbf{A}_{t}\in\mathbb{R}^{N\times N}$ and
$\mathbf{X}_{t}\in\mathbb{R}^{N\times D}$ are corresponding adjacency and
feature matrices. $\mathbf{A}_{t}$ may either evolve over time or remain
fixed, depending on specific settings. When abstracting time series data, we
let $\mathbf{X}_{t}:=\mathbf{x}_{t}\in\mathbb{R}^{N}$.

Figure 2: Examples of spatial-temporal graphs.

We introduce graph neural networks as modern deep learning models to process
graph-structured data. The core operation in typical GNNs, often referred to
as graph convolution, involves exchanging information across neighboring
nodes. In the context of time series analysis, this operation enables us to
explicitly rely on the inter-variable dependencies represented by the graph
edges. Aware of the different nuances, we define GNNs in the spatial domain,
which involves transforming the input signal with learnable functions along
the dimension of $N$.

######  Definition 5 (Graph Neural Network)

Given an attributed graph $\mathcal{G}=(\mathbf{A},\mathbf{X})$, we define
$\mathbf{x}_{i}=\mathbf{X}[i,:]\in\mathbb{R}^{D}$ as the $D$-dimensional
feature vector of node $v_{i}$. A GNN learns node representations through two
primary functions: $\textsc{Aggregate}(\cdot)$ and $\textsc{Combine}(\cdot)$
[31]. The $\textsc{Aggregate}(\cdot)$ function computes and aggregates
messages from neighboring nodes, while the $\textsc{Combine}(\cdot)$ function
merges the aggregated and previous states to transform node embeddings.
Formally, the $k$-th layer in a GNN is defined by the extended

$\displaystyle\mathbf{a_{i}^{(k)}}$
$\displaystyle=\textsc{Aggregate}^{(k)}\left(\left\\{\mathbf{h_{j}^{(k-1)}}:v_{j}\in\mathcal{N}(v_{i})\right\\}\right),$
(1) $\displaystyle\mathbf{h_{i}^{(k)}}$
$\displaystyle=\textsc{Combine}^{(k)}\left(\mathbf{h_{i}^{(k-1)}},\mathbf{a_{i}^{(k)}}\right),$

or, more generally, aggregating messages computed from both sending and
receiving nodes $v_{j}$ and $v_{i}$, respectively. Here,
$\mathbf{a_{i}^{(k)}}$ and $\mathbf{h_{i}^{(k)}}$ represent the aggregated
message from neighbors and the transformed node embedding of node $v_{i}$ in
the $k$-th layer, respectively. The input and output of a GNN are
$\mathbf{h_{i}^{(0)}}:=\mathbf{x}_{i}$ and
$\mathbf{h_{i}^{(K)}}:=\mathbf{h_{i}}$.

The above formulation in Equation 1 is referred to as spatial GNNs, as opposed
to spectral GNNs which defines convolution from the lens of spectral graph
theory. We refer the reader to recent publication [29] for a deeper analysis
of spectral versus spatial GNNs, and [30, 45] for a comprehensive review of
GNNs.

To employ GNNs for time series analysis, it is implied that a graph structure
must be provided. However, not all time series data have readily available
graph structures and, in practice, two types of strategies are utilized to
generate the missing graph structures from the data: heuristics or learned
from data.

Heuristic-Based Graphs. This group of methods extracts graph structures from
data based on heuristics, such as:

  * •

Spatial Proximity: This approach defines the graph structure by considering
the proximity between pairs of nodes based on, e.g., their geographical
location. A typical example is the construction of the adjacency matrix
$\mathbf{A}$ based on the shortest travel distance between nodes when the time
series data have geospatial properties:

$\mathbf{A}_{i,j}=\begin{cases}\frac{1}{d{ij}},&\text{if}\ d_{ij}\neq 0,\\\
0,&\text{otherwise},\end{cases}$ (2)

where $d_{ij}$ denotes the shortest travel distance between node $i$ and node
$j$. Some common kernel functions, e.g., Gaussian radial basis, can also be
applied [13].

  * •

Pairwise Connectivity: In this approach, the graph structure is determined by
the connectivity between pairs of nodes, like that determined by
transportation networks. The adjacency matrix $\mathbf{A}$ is defined as:

$\mathbf{A}_{i,j}=\begin{cases}1,&\text{if}\ v_{i}\ \text{and}\ v_{j}\
\text{are directly linked},\\\ 0,&\text{otherwise}.\end{cases}$ (3)

Typical scenarios include edges representing roads, railways, or adjacent
regions [46, 47]. In such cases, the graph can be undirected or directed,
resulting in symmetric and asymmetric adjacency matrices.

  * •

Pairwise Similarity: This method constructs the graph by connecting nodes with
similar attributes. A simple example is the construction of adjacency matrix
$\mathbf{A}$ based on the cosine similarity between time series:

$\mathbf{A}_{i,j}=\frac{\mathbf{x}_{i}^{\top}\mathbf{x}_{j}}{\|\mathbf{x}_{i}\|\|\mathbf{x}_{j}\|},$
(4)

where $\|\cdot\|$ denotes the Euclidean norm. There are also several variants
for creating similarity-based graphs, such as Pearson Correlation Coefficient
(PCC) [48] and Dynamic Time Warping (DTW) [49].

  * •

Functional Dependence: This approach defines the graph structure based on the
functional dependence between pairs of nodes. These include the construction
of adjacency matrix $\mathbf{A}$ based on Granger causality [50]:

$\mathbf{A}_{i,j}=\begin{cases}1,&\text{if node}\ j\ \text{Granger-causes}\\\
&\text{node}\ i\ \text{at a significance level}\ \alpha,\\\
0,&\text{otherwise}.\end{cases}$ (5)

Other examples involve transfer entropy (TE) [51] and directed phase lag index
(DPLI) [52].

Learning-Based Graphs. In contrast to heuristic-based methods, learning-based
approaches aim to directly learn the graph structure from the data end-to-end
with the downstream task. These techniques typically involve optimizing the
graph structure alongside model parameters during the training process, e.g.,
embedding-based [53], attention-based [54, 55], sampling-based [56, 57]. These
learning-based approaches enable the discovery of more complex and potentially
more informative graph structures compared to heuristic-based methods.

Figure 3: Task-oriented taxonomy of graph neural networks for time series
analysis in the existing literature.

##  3 Framework and Categorization

In this section, we present a comprehensive task-oriented taxonomy for GNNs
within the context of time series analysis (subsection 3.1). Subsequently, we
investigate how to encode time series across various tasks by introducing a
unified methodological framework for GNN architectures (subsection 3.2).
According to the framework, all architectures are composed of a similar graph-
based processing module $f_{\theta}$ and a second module $p_{\phi}$
specialized in downstream tasks.

###  3.1 Task-Oriented Taxonomy

(a) Graph neural networks for time series forecasting.

(b) Graph neural networks for time series anomaly detection.

(c) Graph neural networks for time series imputation.

(d) Graph neural networks for time series classification: Formulates green
series classification as a graph (top) or node (bottom) classification task.

Figure 4: Four categories of graph neural networks for time series analysis.
For the sake of simplicity and illustrative purposes, we assume the graph
structures are fixed in all subplots.

In Figure 3, we illustrate a task-oriented taxonomy of GNNs encompassing the
primary tasks and mainstream modeling for time series analysis, and showcasing
the potential of GNN4TS. This survey focuses on four categories: time series
forecasting, anomaly detection, imputation, and classification. These tasks
are performed on top of the time series representations learned by spatial-
temporal graph neural networks (STGNNs), which serve as the foundation for
encoding time series data in existing literature across various tasks. We
detail this in subsection 3.2.

Time Series Forecasting. This task is centered around predicting future values
of the time series based on historical observations, as depicted in 3a.
Depending on application needs, we categorize this task into two types:
single-step-ahead forecasting and multi-step-ahead forecasting. The former is
meant to predict single future observations of the time series once at a time,
i.e., the target at time $t$ is $\mathbf{Y}:=\mathbf{X}_{t+H}$ for some
$H\in\mathbb{N}$ steps ahead, while the latter makes predictions for a time
interval, e.g., $\mathbf{Y}:=\mathbf{X}_{t+1:t+H}$. Parameterized solutions to
both predictive cases can be derived by optimizing

$\theta^{*},\phi^{*}=\mathop{\arg\min}\limits_{{\theta,\phi}}\mathcal{L}_{\textsc{f}}\Big{(}p_{\phi}\big{(}f_{\theta}(\mathbf{X}_{t-T:t},\mathbf{A}_{t-T:t})\big{)},\mathbf{Y}\Big{)},$
(6)

where $f_{\theta}(\cdot)$ and $p_{\phi}(\cdot)$ represent a spatial-temporal
GNN and the predictor, respectively. Details regarding the $f_{\theta}(\cdot)$
architecture are given in subsection 3.2 while the predictor is, normally, a
multi-layer perceptron. In the sequel, we denote by $\mathbf{X}_{t-T:t}$ and
$\mathbf{A}_{t-T:t}$ a spatial-temporal graph
$\mathcal{G}=\\{\mathcal{G}_{t-T},\mathcal{G}_{t-T+1},\cdots,\mathcal{G}_{t}\\}$
with length $T$. If the underlying graph structure is fixed, then
$\mathbf{A}_{t}:=\mathbf{A}$. $\mathcal{L}_{\textsc{f}}(\cdot)$ denotes the
forecasting loss, which is typically a squared or absolute loss function,
e.g., STGCN [58] and MTGNN [53]. Most existing works minimize the error
between the forecasting and the ground truth $\mathbf{Y}$ through Equation 6;
this process is known as deterministic time series forecasting. Besides, we
have probabilistic time series forecasting methods, such as DiffSTG [59], that
share the same objective Equation 6 function though it is not directly
optimized. Based on the size of the forecasting horizon $H$, we end up in
either short-term or long-term forecasting.

Time Series Anomaly Detection. This task focuses on detecting irregularities
and unexpected event in time series data (3b). Detecting anomalies requires
determining when the anomalous event occurred, while diagnosing them requests
gaining insights about how and why the anomaly occurred. Due to the general
difficulty of acquiring anomaly events, current research commonly treats
anomaly detection as an unsupervised problem that involves the design of a
model describing normal, non-anomalous data. The learned model is then used to
detect anomalies by generating a high score whenever an anomaly event occurs.
This model learning process mirrors the forecasting optimization, Equation 6,
with $f_{\theta}(\cdot)$ and $p_{\phi}(\cdot)$ denote the spatial-temporal GNN
and the predictor, respectively. In general, the spatial-temporal GNN and the
predictor are trained on normal, non-anomalous data using either forecasting
[40, 60] or reconstruction [39, 61] optimization approaches, with the aim of
minimizing the discrepancy between the normal input and the forecast (or
reconstructed) series. Nonetheless, when these models are put to use for
detecting anomalies, they are expected to fail in minimizing this discrepancy
upon receiving anomalous input. This inability to conform to the expected low-
discrepancy model behavior during anomaly periods creates a detectable
difference, facilitating the detection of anomalies. The threshold separating
normal and anomalous data is a sensitive hyperparameter that should be set
considering the rarity of anomalies and aligned with a desired false alarm
rate [62]. Lastly, to diagnose the causes of anomalies, a common strategy
involves calculating discrepancies for each channel node and consolidating
these into a single anomaly score [63]. This approach allows for the
identification of the channel variables responsible for the anomaly events by
calculating their respective contributions to the final score.

Time Series Imputation. This task is centered around estimating and filling in
missing or incomplete data points within a time series (3c). Current research
in this domain can be broadly classified into two main approaches: in-sample
imputation and out-of-sample imputation. In-sample imputation involves filling
missing values in a given time series, while out-of-sample imputation pertains
to inferring missing data not present in the training dataset. We formulate
the learning objective as follows:

$\theta^{*},\phi^{*}=\mathop{\arg\min}\limits_{{\theta,\phi}}\mathcal{L}_{\textsc{i}}\Big{(}p_{\phi}\big{(}f_{\theta}(\tilde{\mathbf{X}}_{t-T:t},\mathbf{A}_{t-T:t})\big{)},\mathbf{X}_{t-T:t}\Big{)},$
(7)

where $f_{\theta}(\cdot)$ and $p_{\phi}(\cdot)$ denote the spatial-temporal
GNN and imputation module to be learned, respectively. The imputation module
can e.g., be a multi-layer perceptron. In this task,
$\tilde{\mathbf{X}}_{t-T:t}$ represents input time series data with missing
values (reference time series), while $\mathbf{X}_{t-T:t}$ denotes the same
time series without missing values. As it is impossible to access the
reference time series during training, a surrogate optimization objective is
considered, such as generating synthetic missing values [41]. In Equation 7,
$\mathcal{L}_{\textsc{i}}(\cdot)$ refers to the imputation loss, which can be,
for instance, an absolute or a squared error, similar to forecasting tasks.
For in-sample imputation, the model is trained and evaluated on
$\tilde{\mathbf{X}}_{t-T:t}$ and $\mathbf{X}_{t-T:t}$. Instead, for out-of-
sample imputation, the model is trained and evaluated on disjoint sequences,
e.g., trained on $\tilde{\mathbf{X}}_{t-T:t}$ but evaluated on
$\mathbf{X}_{t:t+H}$, where the missing values in $\tilde{\mathbf{X}}_{t:t+H}$
will be estimated. Similar to time series forecasting and anomaly detection,
the imputation process can be either deterministic or probabilistic. The
former predicts the missing values directly (e.g., GRIN [41]), while the
latter estimates the missing values from data distributions (e.g., PriSTI
[42]).

Figure 5: Methodology-oriented taxonomy of graph neural networks for time
series analysis.

Time Series Classification. This task aims to assign a categorical label to a
given time series based on its underlying patterns or characteristics. Rather
than capturing patterns within a time series data sample, the essence of time
series classification resides in discerning differentiating patterns that help
separate samples based on their class labels. The optimization problem can be
expressed as:

$\theta^{*},\phi^{*}=\mathop{\arg\min}\limits_{{\theta,\phi}}\mathcal{L}_{\textsc{c}}\Big{(}p_{\phi}\big{(}f_{\theta}(\mathbf{X},\mathbf{A})\big{)},\mathbf{Y}\Big{)},$
(8)

where $f_{\theta}(\cdot)$ and $p_{\phi}(\cdot)$ denote, e.g., a GNN and a
classifier to be learned, respectively. Using univariate time series
classification as an example 3d, the task can be formulated as either a graph
or node classification task. In the case of graph classification (Series-As-
Graph) [64], each series is transformed into a graph, and the graph will be
the input of a GNN to generate a classification output. This can be achieved
by dividing a series into multiple subsequences with a window size, $W$,
serving as graph nodes, $\mathbf{X}\in\mathbb{R}^{N\times W}$, and an
adjacency matrix, $\mathbf{A}$, describing the relationships between
subsequences. A simple GNN, $f_{\theta}(\cdot)$, then employs graph
convolution and pooling to obtain a condensed graph feature to be exploited by
a classifier $p_{\phi}(\cdot)$ which assigns a class label to the graph.
Alternatively, the node classification formulation (Series-As-Node), treats
each series as a node in a dataset graph. Series-As-Node constructs an
adjacency matrix representing the relationships between multiple distinct
series in a given dataset [65]. With several series of length $T$ stacked into
a matrix $\mathbf{X}\in\mathbb{R}^{N\times T}$ as node features and
$\mathbf{A}$ representing pairwise relationships, the GNN operation,
$f_{\theta}(\cdot)$, aims at leveraging the relationships across different
series for accurate node series classification [66]. In all cases,
$\mathbf{Y}$ is typically a one-hot encoded vector representing the
categorical label of a univariate or multivariate time series.

###  3.2 Unified Methodological Framework

Figure 6: General pipeline for time series analysis using graph neural
networks.

In Figure 5, we present a unified methodological framework of STGNNs mentioned
in subsection 3.1 for time series analysis. Specifically, our framework serves
as the basis for encoding time series data in the existing literature across
various downstream tasks (Figure 3). As an extension, STGNNs incorporate
spatial information by considering the relationships between nodes in the
graph and temporal information by taking into account the evolution of node
attributes over time. Similar to [13], we systematically categorize STGNNs
from three perspectives: spatial module, temporal module, and overall model
architecture.

  * •

Spatial Module. To model dependencies between time series over time, STGNNs
employ the design principles of GNNs on static graphs. These can be further
categorized into three types: spectral GNNs, spatial GNNs, and a combination
of both (i.e., hybrid) [30]. Spectral GNNs are based on spectral graph theory
and use the graph shift operator (like the graph Laplacian) to capture node
relationships in the graph frequency domain [67, 68, 29]. Differently, spatial
GNNs simplify spectral GNNs by directly designing filters that are localized
to each node’s neighborhood. Hybrid approaches combine both spectral and
spatial methodologies to capitalize on the strengths of each method.

  * •

Temporal Module. To account for temporal dependencies in time series, STGNNs
incorporate temporal modules that work in tandem with spatial modules to model
intricate spatial-temporal patterns. Temporal dependencies can be represented
in either the time or frequency domains. In the first category of methods,
approaches encompass recurrence-based (e.g., RNNs [26]), convolution-based
(e.g., TCNs [69]), attention-based (e.g., Transformers [28]), and a
combination of these (i.e., hybrid). For the second category, analogous
techniques are employed, followed by orthogonal space projections [29], such
as the Fourier transform.

  * •

Model Architecture. To integrate the two modules, existing STGNNs are either
discrete or continuous in terms of their overall neural architectures. Both
types can be further subdivided into two subcategories: factorized and
coupled. With typical factorized STGNN model architectures, the temporal
processing is performed either before or after the spatial processing, whether
in a discrete (e.g., STGCN [58]) or continuous manner (e.g., STGODE [70]).
Conversely, the coupled model architecture refers to instances where spatial
and temporal modules are interleaved, such as DCRNN [71] (discrete) and MTGODE
[23] (continuous). Other authors refer to very related categories as time-
then-space and time-and-space [72].

General Pipeline. In Figure 6, we showcase a general pipeline that shows how
STGNNs can be integrated into time series analysis. Given a time series
dataset, we first process it using the data processing module, which performs
essential data cleaning and normalization tasks, including the extraction of
time series topology (i.e., graph structures). Subsequently, STGNNs are
utilized to obtain time series representations, which can then be passed to
different handlers (i.e., downstream task prediction module) to execute
various analytical tasks, such as forecasting and anomaly detection.

##  4 GNNs for Time Series Forecasting

TABLE II: Summary of representative graph neural networks for time series
forecasting. Task notation: The first letter, “M” or “S”, indicates multi-step
or single-step forecasting, and the second letter, “S” or “L”, denotes short-
term or long-term forecasting. Architecture notation: “D” and “C” represent
“Discrete” and “Continuous”; “C” and “F” stand for “Coupled” and “Factorized”.
Temporal module notation: “T” and “F” signify “Time Domain” and “Frequency
Domain”; “R”, “C”, “A”, and “H” correspond to “Recurrence”, “Convolution”,
“Attention”, and “Hybrid”. Input graph notation: “R” indicates that a pre-
calculated graph structure (with a certain graph heuristic) is a required
input of the model, “NR” that such graph is not required (not a model’s
input), while “O” signifies that the model can optionally exploit given input
graphs. Notation of learned graph relations: “S” and “D” indicate “Static” and
“Dynamic”. Notation of adopted graph heuristics: “SP”, “PC”, “PS”, and “FD”
denote “Spatial Proximity”, “Pairwise Connectivity”, “Pairwise Similarity”,
and “Functional Dependency”, respectively. The “Missing Values” column
indicates whether corresponding methods can handle missing values in input
time series.

Approach Year Venue Task Architecture Spatial Module Temporal Module Missing
Values Input Graph Learned Relations Graph Heuristics DCRNN [71] 2018 ICLR M-S
D-C Spatial GNN T-R No R - SP STGCN [58] 2018 IJCAI M-S D-F Spectral GNN T-C
No R - SP ST-MetaNet [73] 2019 KDD M-S D-F Spatial GNN T-R No R - SP, PC NGAR
[74] 2019 IEEE IJCNN S-S D-F Spatial GNN T-R No R - - ASTGCN [75] 2019 AAAI
M-S D-F Spectral GNN T-H No R - SP, PC ST-MGCN [46] 2019 AAAI S-S D-F Spectral
GNN T-R No R - SP, PC, PS Graph WaveNet [76] 2019 IJCAI M-S D-F Spatial GNN
T-C No O S SP MRA-BGCN [77] 2020 AAAI M-S D-C Spatial GNN T-R No R - SP MTGNN
[53] 2020 KDD S-S, M-S D-F Spatial GNN T-C No NR S - STGNN* [78] 2020 WWW M-S
D-C Spatial GNN T-H No R - SP GMAN [79] 2020 AAAI M-S D-C Spatial GNN T-A No R
- SP SLCNN [80] 2020 AAAI M-S D-F Hybrid T-C No NR S - STSGCN [81] 2020 AAAI
M-S D-C Spatial GNN T No R - PC StemGNN [54] 2020 NeurIPS M-S D-F Spectral GNN
F-C No NR S - AGCRN [82] 2020 NeurIPS M-S D-C Spatial GNN T-R No NR S - LSGCN
[83] 2020 IJCAI M-S D-F Spectral GNN T-C No R - SP STAR [84] 2020 ECCV M-S D-F
Spatial GNN T-A No R - PC GTS [56] 2021 ICLR M-S D-C Spatial GNN T-R No NR S -
GEN [85] 2021 ICLR S-S D-F Spatial GNN T-R No R - - Z-GCNETs [86] 2021 ICML
M-S D-C Spatial GNN T-C No NR S - STGODE [70] 2021 KDD M-S C-F Spatial GNN T-C
No R - SP, PS STFGNN [49] 2021 AAAI M-S D-F Spatial GNN T-C No R - SP, PS
DSTAGNN [87] 2022 ICML M-S D-F Spectral GNN T-H No R - PC, PS TPGNN [88] 2022
NeurIPS S-S, M-S D-F Spatial GNN T-A No NR D - MTGODE [23] 2022 IEEE TKDE S-S,
M-S C-C Spatial GNN T-C No NR S - STG-NCDE [89] 2022 AAAI M-S C-C Spatial GNN
T-C Yes NR S - STEP [90] 2022 KDD M-S D-F Spatial GNN T-A No NR S - Chauhan et
al. [91] 2022 KDD M-S - - - Yes O S SP RGSL [92] 2022 IJCAI M-S D-C Spectral
GNN T-R No R S SP, PC FOGS [93] 2022 IJCAI M-S - - - No NR S - METRO [94] 2022
VLDB M-S D-C Spatial GNN T No NR D - SGP [95] 2023 AAAI M-S D-F Spatial GNN
T-R No R - SP, PS HiGP [96] 2023 arXiv M-S D-F Spatial GNN T-R No R S SP, PS
Jin et al. [29] 2023 arXiv M-S, M-L D-F Spectral GNN F-H No NR S -

Time series forecasting aims to predict future time series values based on
historical observations. In recent years, deep learning-based approaches have
demonstrated considerable success in forecasting time series by capturing
nonlinear temporal and spatial patterns more effectively than the linear
counterpart [23]. Techniques such as recurrent neural networks (RNNs),
convolutional neural networks (CNNs), and attention-based neural networks have
been employed. However, many of these approaches, such as LSTNet [97] and TPA-
LSTM [98], overlook and implicitly model the rich underlying dynamic spatial
correlations between time series. Recently, graph neural network (GNN)-based
methods have shown great potential in explicitly and effectively modeling
spatial and temporal dependencies in multivariate time series data, leading to
enhanced forecasting performance.

GNN-based forecasting models can be categorized and examined from multiple
perspectives. In terms of forecasting tasks, while many models focus on multi-
step forecasting (i.e., predicting multiple consecutive steps ahead based on
historical observations), a minority also discuss single-step forecasting
(i.e., predicting the next or one arbitrary step ahead). From a methodological
standpoint, these models can be dissected from three aspects: (1) modeling
spatial (i.e., inter-variable) dependencies, (2) modeling inter-temporal
dependencies, and (3) the fusion of spatial and temporal modules for time
series forecasting. A summary of representative works is given in Table II.

###  4.1 Modeling Inter-Variable Dependencies

Spatial dependencies, or inter-variable relationships, play a pivotal role in
affecting a model’s forecasting capability [29]. When presented with time
series data and corresponding graph structures that delineate the strength of
interconnections between time series, current studies typically employ (1)
spectral GNNs, (2) spatial GNNs, or (3) a hybrid of both to model these
spatial dependencies. At a high level, these methods all draw upon the
principles of graph signal processing (as detailed in Definition 5 and
subsequent discussion). Considering input variables $\mathbf{X}_{t}$ and
$\mathbf{A}_{t}$ at a given time $t$, the goal here is to devise an effective
GNN-based model, termed $\textsc{Spatial}(\cdot)$, to adeptly capture salient
patterns from different time series at time $t$. This can be expressed as
$\hat{\mathbf{X}}_{t}=\textsc{Spatial}(\mathbf{X}_{t},\mathbf{A}_{t})$, where
$\hat{\mathbf{X}}_{t}$ collects all time series representations at time $t$
with spatial dependencies embedded.

Spectral GNN-Based Approaches. Early GNN-based forecasting models
predominantly utilized ChebConv [99] to approximate graph convolution with
Chebyshev polynomials, thereby modeling inter-variable dependencies. For
instance, STGCN [58] intersects temporal convolution [100] and ChebConv layers
to capture both spatial and temporal patterns. StemGNN [54] further proposes
spectral-temporal graph neural networks to extract rich time series patterns
by leveraging ChebConv and frequency-domain convolution neural networks. Other
relevant research has largely followed suit, employing ChebConv to model
spatial time series dependencies, while introducing innovative modifications.
These include attention mechanisms [75, 83], multi-graph construction [46,
92], and combinations of the two [87]. Recently, building upon StemGNN, Jin et
al. [29] have theoretically demonstrated the benefits of using spectral GNNs
to model different signed time series relations, such as strongly positive and
negative correlated variables within a multivariate time series. They also
observed that any orthonormal family of polynomials could achieve comparable
expressive power for such tasks, albeit with varying convergence rates and
empirical performances.

Spatial GNN-Based Approaches. Inspired by the recent success of spatial GNNs
[30], another line of research has been modeling inter-variable dependencies
using message passing [101] or graph diffusion [102, 103]. From the graph
perspective, these methods are certain simplifications compared to those based
on spectral GNNs, where strong local homophilies are emphasised [29, 104].
Early methods such as DCRNN [71] and Graph WaveNet [76] incorporated graph
diffusion layers into GRU [105] or temporal convolution to model time series
data. Subsequent works including GTS [56] and ST-GDN [106] also applied graph
diffusion. In contrast, STGCN$(1^{st})$ (a second version of STGCN [58]) and
ST-MetaNet [73] modelled spatial dependencies with GCN [107] and GAT [108] to
aggregate information from adjacent time series. Related works, such as MRA-
BGCN [77], STGNN*[78], GMAN[79], and AGCRN [82], proposed variants to model
inter-variable relations based on message passing. To enhance learning
capabilities, STSGCN [81] proposed spatial-temporal synchronous graph
convolution, extending GCN to model spatial and temporal dependencies on
localized spatial-temporal graphs. STFGNN [49] constructed spatial-temporal
fusion graphs based on dynamic time wrapping (DTW) before applying graph and
temporal convolutions. GSS models [109] rely on state-space representations
with attributed graphs as inputs, states and outputs, and where the states’
topology is learned from data. Z-GCNETs [86] enhanced existing methods with
salient time-conditioned topological information, specifically zigzag
persistence images. METRO [94] introduced multi-scale temporal graphs to
characterize dynamic spatial and temporal interactions in time series data,
together with the single-scale graph update and cross-scale graph fusion
modules to unify the modeling of spatial-temporal dependencies. Another line
of improvements incorporates graph propagation, allowing for the mixing of
neighborhood information from different hops to learn high-order relations and
substructures in the graph. Example are MTGNN [53], SGP [95] and HiGP [96]. In
particular, SGP exploits reservoir computing and multi-hop spatial processing
to precompute spatio-temporal representations yielding effective, yet scalable
predictive models, while HiGP exploits learned cluster assignments to
regularize predictions and propagate messages hierarchically among nodes.
Follow-up works such as MTGODE [23] and TPGNN [88] proposed continuous graph
propagation and graph propagation based on temporal polynomial coefficients.
Other similar works include STGODE [70] and STG-NCDE [89]. Distinct from GAT-
based methods, approaches based on graph transformer [110] can capture long-
range spatial dependencies due to their global receptive field, making them a
separate branch of enhanced methods. Examples include STAR [84], ASTTN [111],
and ASTTGN [112]. At last, we mention NGAR [74] and GEN [85] as works that can
predict the graph topology alongside node-level signals.

Hybrid Approaches. Some hybrid methodologies also exist, integrating both
spectral and spatial GNNs. For instance, SLCNN [80] employs ChebConv [99] and
localized message passing as global and local convolutions to capture spatial
relations at multiple granularities. Conversely, Auto-STGNN [113] integrates
neural architecture search to identify high-performance GNN-based forecasting
models. In this approach, various GNN instantiations, such as ChebConv, GCN
[107], and STSGCN [81], can be simultaneously implemented in different
spatial-temporal blocks.

###  4.2 Modeling Inter-Temporal Dependencies

The modeling of temporal dependencies within time series represents another
important element in various GNN-based forecasting methods. These dependencies
(i.e., temporal patterns) are capable of being modeled in the time or/and
frequency domains. A summary of representative methods, along with their
temporal module classifications, is presented in Table II. Given a univariate
time series $\mathbf{X}_{n}$ with length $T$, the primary goal here is to
learn an effective temporal model, referred to as $\textsc{Temporal}(\cdot)$.
This model is expected to accurately capture the dependencies between data
points within $\mathbf{X}_{n}$, such that
$\hat{\mathbf{X}}_{n}=\textsc{Temporal}(\mathbf{X}_{n})$, where
$\hat{\mathbf{X}}_{n}$ symbolizes the representation of time series
$\mathbf{X}_{n}$. In the construction of $\textsc{Temporal}(\cdot)$, both the
time and frequency domains can be exploited within convolutional and attentive
mechanisms. Recurrent models can also be employed for modeling in the time
domain specifically. Additionally, hybrid models exist in both domains,
integrating different methodologies such as attention and convolution neural
networks.

Recurrent Models. Several early methodologies rely on recurrent models for
understanding inter-temporal dependencies in the time domain. For instance,
DCRNN [71] integrates graph diffusion with gated recurrent units (GRU) [105]
to model the spatial-temporal dependencies in traffic forecasting. ST-MetaNet
[73] incorporates two types of GRU to encode historical observations and
capture diverse temporal correlations that are tied to geographical
information. Inspired by [71], MRA-BGCN [77] combines the proposed multi-range
attention-based bicomponent graph convolution with GRU. This model is designed
to better capture spatial-temporal relations by modeling both node and edge
interaction patterns. On a different note, AGCRN [82] merges GRU with a
factorized variant of GCN [107] and a graph structure learning module. Some
studies, such as GTS [56] and RGSL [92], share similar designs but primarily
emphasize different graph structure learning mechanisms. Recently, echo state
networks (ESN) [114] – a type of RNN with sparse and randomized connectivity
producing rich dynamics – have been employed to design scalable models without
compromising the performance [95, 115]. Lastly, Graph Kalman Filters [116]
introduce a feedback loop in GSS models [109] to improve the accuracy of state
estimates and the predictions when actual system outputs are acquired.

Convolution Models. Convolutional Neural Networks (CNNs), on the other hand,
provide a more efficient perspective for modeling inter-temporal dependencies,
with the bulk of existing studies in the time domain. An instance of this is
STGCN [58], which introduces temporal gated convolution that integrates 1-D
convolution with gated linear units (GLU) to facilitate tractable model
training. Works that adopt a similar approach include DGCNN [117], SLCNN [80],
and LSGCN [83]. Building on these foundations, Graph WaveNet [76] incorporated
dilated causal convolution, which notably expands the receptive field with
only a minimal increase in model layers. STGODE [70] and STFGNN [49] have
produced similar designs in capturing temporal dependencies. MTGNN[53] also
uses these underpinning concepts, but it enhances temporal convolution by
utilizing multiple kernel sizes. Further expanding on this, MTGODE[23] adopts
a neural ordinary differential equation[118] to generalize this modeling
process. There are some other studies, such as Z-GCNETs[86], that directly
apply canonical convolution to capture temporal patterns within the time
domain, albeit with other focuses. An alternative strand of methodologies,
including StemGNN[54] and TGC[29], focuses on modeling temporal clues in the
frequency domain. StemGNN applies gated convolution to filter the frequency
elements generated by the discrete Fourier transform of the input time series.
In contrast, TGC convolves frequency components individually across various
dimensions to craft more expressive temporal frequency-domain models.

Attention Models. Recently, a growing number of methodologies are turning
towards attention mechanisms, such as the self-attention used in the
Transformer model[119], to embed temporal correlations. For instance, GMAN
[79] attentively aggregates historical information by considering both spatial
and temporal features. ST-GRAT [120] mirrors the Transformer’s architecture,
employing multi-head self-attention layers within its encoder to embed
historical observations in conjunction with its proposed spatial attention
mechanism. STAR [84], TPGNN [88], and STEP [90] similarly employ Transformer
layers to model the temporal dependencies within each univariate time series.
There are also variations on this approach, like the multi-scale self-
attention network proposed by ST-GDN [106], aiming to model inter-temporal
dependencies with higher precision.

Hybrid Models. Hybrid models also find application in modeling inter-temporal
dependencies. For example, ASTGCN[75], HGCN[121], and DSTAGNN[87] concurrently
employ temporal attention and convolution in learning temporal correlations.
STGNN*[78] amalgamates both GRU and Transformer to capture local and global
temporal dependencies. Auto-STGCN[113], on the other hand, potentially
facilitates more diverse combinations when searching for high-performance
neural architectures. In the frequency domain, the nonlinear variant of TGC
[29] is currently the only hybrid model proposing to capture temporal
relations through the combination of spectral attention and convolution
models.

###  4.3 Forecasting Architectural Fusion

Given the spatial and temporal modules discussed, denoted as
$\textsc{Spatial}(\cdot)$ and $\textsc{Temporal}(\cdot)$, four categories of
neural architectural fusion have been identified as effective means to capture
spatial-temporal dependencies within time series data: (1) discrete
factorized, (2) discrete coupled, (3) continuous factorized, and (4)
continuous coupled. In discrete factorized models, spatial and temporal
dependencies are usually learned and processed independently. This approach
may involve stacking and interleaving spatial and temporal modules within a
model building block[58, 76, 53]. Discrete coupled models, on the other hand,
explicitly or implicitly incorporate spatial and temporal modules into a
singular process when modeling spatial-temporal dependencies, such as in
[122], [77], and [81]. Different from discrete models, some methods abstract
the underlying modeling processes with neural differential equations, which we
categorize as continuous models. Specifically, continuous factorized models
involve distinct processes, either partially or entirely continuous (e.g.,
[70]), to model spatial and temporal dependencies. In contrast, continuous
coupled models employ a single continuous process to accomplish this task,
such as [23] and [89].

Discrete Architectures. Numerous existing GNN-based time series forecasting
methods are models processing discrete data. For instance, factorized
approaches like STGCN [58] employ a sandwich structure of graph and temporal
gated convolution layers as its fundamental building block, facilitating the
modeling of inter-variable and inter-temporal relations. Subsequent works,
such as DGCNN [117], LSGCN [83], STHGCN [123], and HGCN [121], retain this
model architecture while introducing enhancements such as dynamic graph
structure estimation [117], hypergraph convolution [121], and hierarchical
graph generation [121]. A multitude of other studies adhere to similar
principles, stacking diverse spatial and temporal modules in their core
building blocks. For example, ST-MetaNet [73] interweaves RNN cells and GAT
[108] to model evolving traffic information. Comparable works include ST-MGCN
[46], DSATNET [124], and EGL [125]. In contrast, ASTGCN [75], DSTAGNN [87],
and GraphSleepNet [126] are constructed upon spatial-temporal attention and
convolution modules, with the latter module comprising of stacking ChebConv
[99] and the convolution in the temporal dimension. Graph WaveNet [76], SLCNN
[80], StemGNN [54], MTGNN [53], STFGNN [49], and TGC [29] share a similar
model architecture without the attention mechanism. There are also alternative
designs within the realm of discrete factorized forecasting models. For
instance, STAR [84] integrates the proposed spatial and temporal Transformers,
while ST-GDN [106] initially performs attention-based temporal hierarchical
modeling before applying various graph domain transformations. TPGNN [88]
employs temporal attention and the proposed temporal polynomial graph module
to more effectively capture time-evolving patterns in time series data.
MTHetGNN [51] stacks the proposed temporal, relational, and heterogeneous
graph embedding modules to jointly capture spatial-temporal patterns in time
series data. CausalGNN [127] models multivariate time series with causal
modeling and attention-based dynamic GNN modules. Auto-STGCN [113] explores
high-performance discrete combinations of different spatial and temporal
modules.

In the realm of discrete coupled models, early works such as DCRNN [71] and
Cirstea et al.[128] straightforwardly incorporate graph diffusion or attention
models into RNN cells. This approach models spatial-temporal dependencies in
historical observations for forecasting. Subsequent works, including ST-
UNet[129], MRA-BGCN [77], STGNN*[78], AGCRN[82], RGSL [92], and MegaCRN [130],
are based on similar concepts but with varying formulations of graph
convolutional recurrent units. Some studies integrate spatial and temporal
convolution or attention operations into a single module. For instance, GMAN
[79] proposes a spatial-temporal attention block that integrates the spatial
and temporal attention mechanisms in a gated manner. Z-GCNETs [86] initially
learns time-aware topological features that persist over time (i.e., zigzag
persistence representations), then applies spatial and temporal graph
convolutions to capture salient patterns in time series data. TAMP-S2GCNETS
[131] is slightly more complex, modeling the spatial-temporal dependencies in
time series by coupling two types of GCN layers, Dynamic Euler-Poincare
Surface Representation Learning (DEPSRL) modules, and CNNs. Another line of
research direction models spatial-temporal dependencies by convolving on
specially crafted graph structures (e.g., STSGCN [81]), performing graph
convolutions in a sliding window manner (e.g., STSGCN [81] and STG2Seq [132]),
or utilizing the concept of temporal message passing (e.g., METRO [94] and
ASTTN [111]).

Continuous Architectures. To date, only a handful of existing methods fall
into the category of continuous models. For factorized methods, STGODE [70]
proposes to depict the graph propagation as a continuous process with a neural
ordinary differential equation (NODE) [118]. This approach allows for the
effective characterization of long-range spatial-temporal dependencies in
conjunction with dilated convolutions along the time axis. For coupled
methods, MTGODE [23] generalizes both spatial and temporal modeling processes
found in most related works into a single unified process that integrates two
NODEs. STG-NCDE [89] shares a similar idea but operates under the framework of
neural controlled differential equations (NCDEs) [133]. Similarly, a recent
work, TGNN4I [134], integrates GRU [105] and MPNN [101] as the ODE function to
model continuous-time latent dynamics.

##  5 GNNs for Time Series Anomaly Detection

Time series anomaly detection aims to identify data observations that do not
conform with the nominal regime of the data-generating process [135]. We
define anomaly as any such data point, and use the term normal data otherwise;
we note however that different terminologies, like novelty and outlier, are
used almost interchangeably to anomaly in the literature [136]. Deviations
from the nominal conditions could take the form of a single observation
(point) or a series of observations (subsequence) [137]. However, unlike
normal time series data, anomalies are difficult to characterize for two main
reasons. First, they are typically associated with rare events, so collecting
and labeling them is often a daunting task. Secondly, establishing the full
range of potential anomalous events is generally impossible, spoiling the
effectiveness of supervised learning techniques. Consequently, unsupervised
detection techniques have been widely explored as a practical solution to
real-world problems.

Traditionally, methods [138] such as distance-based [139, 140, 141], and
distributional techniques [142] have been widely used for detecting
irregularities in time series data. The former family uses distance measures
to quantify the discrepancy of observations from representative data points,
while the latter looks at points of low likelihood to identify anomalies.

The advent of deep learning has sparked significant advancements, drawing
lessons from earlier methods. Early research in this area proposed recurrent
models with reconstruction [143] and forecasting[144] strategies respectively
to improve anomaly detection in multivariate time series data. The forecasting
and reconstruction strategies rely on forecast and reconstruction errors as
discrepancy measures between anticipated and real signals. These strategies
rely on the fact that, if a model trained on normal data fails to forecast or
reconstruct some data, then it is more likely that such data is associated
with an anomaly. However, recurrent models [145] are found to lack explicit
modeling of pairwise interdependence among variable pairs, limiting their
effectiveness in detecting complex anomalies [39, 146]. Recently, GNNs have
shown promising potential to address this gap by effectively capturing
temporal and spatial dependencies among variable pairs [40, 60, 147].

TABLE III: Summary of representative graph neural networks for time series
anomaly detection. Strategy notation: “CL”, “FC”, “RC”, and “RL” indicate
“Class”, “Forecast”, “Reconstruction”, and “Relational Discrepancies”,
respectively. The remaining notations are shared with Table II.

Approach Year Venue Strategy Spatial Module Temporal Module Missing Values
Input Graph Learned Relations Graph Heuristics CCM-CDT [52] 2019 IEEE TNNLS RC
Spatial GNN T-R No R - PC, FD MTAD-GAT [39] 2020 IEEE ICDM FC+RC Spatial GNN
T-A No NR - - GDN[40] 2021 AAAI FC Spatial GNN - No NR S - GTA[147] 2021 IEEE
IoT FC Spatial GNN T-H No NR S - EvoNet[148] 2021 WSDM CL Spatial GNN T-R No R
- PS Event2Graph[149] 2021 arXiv RL Spatial GNN T-A No R - PS GANF[61] 2022
ICLR RC+RL Spatial GNN T-R No NR S - Grelen[150] 2022 IJCAI RC+RL Spatial GNN
T-H No NR D - VGCRN[151] 2022 ICML FC+RC Spatial GNN T-R No NR S -
FuSAGNet[60] 2022 KDD FC+RC Spatial GNN T-R No NR S - GTAD[152] 2022 Entropy
FC+RC Spatial GNN T-C No NR - - HgAD[153] 2022 IEEE BigData FC Spatial GNN -
No NR S HAD-MDGAT[154] 2022 IEEE Access FC+RC Spatial GNN T-A No NR - -
STGAN[154] 2022 IEEE TNNLS RC Spatial GNN T-R No R - SP GIF [155] 2022 IEEE
IJCNN RC Spatial GNN - No R - SP, PC, FD DyGraphAD[156] 2023 arXiv FC+RL
Spatial GNN T-C No R - PS GraphSAD[157] 2023 arXiv CL Spatial GNN T-C No R -
PS, PC CST-GL[158] 2023 arXiv FC Spatial GNN T-C No NR S -

###  5.1 General Framework for Anomaly Detection

Treating anomaly detection as an unsupervised task relies on models to learn a
general concept of what normality is for a given dataset [159, 160]. To
achieve this, deep learning architectures deploy a bifurcated modular
framework, constituted by a backbone module and a scoring module [161].
Firstly, a backbone model, $\textsc{Backbone}(\cdot)$, is trained to fit given
training data, assumed to be nominal, or to contain very few anomalies.
Secondly, a scoring module, $\textsc{Scorer}(\mathbf{X},\mathbf{\hat{X}})$,
produces a score used to identify the presence of anomalies by comparing the
output $\mathbf{\hat{X}}=\textsc{Backbone}(\mathbf{X})$ of the backbone module
with the observed time series data $\mathbf{X}$. The score is intended as a
measure of the discrepancy between the expected signals under normal and
anomalous circumstances. When there is a high discrepancy score, it is more
likely that an anomaly event has occurred. Furthermore, it is also important
for a model to diagnose anomaly events by pinpointing the responsible
variables. Consequently, a scoring function typically computes the discrepancy
for each individual channel first, before consolidating these discrepancies
across all channels into a single anomaly value.

To provide a simple illustration of the entire process, the backbone can be a
GNN forecaster that makes a one-step-ahead forecast for the scorer. The scorer
then computes the anomaly score as the sum of the absolute forecast error for
each channel variable, represented as
$\sum_{i}^{N}|x_{t}^{i}-\hat{x}_{t}^{i}|$ across $N$ channel variables. Since
the final score is computed based on the summation of channel errors, an
operator can determine the root cause variables by computing the contribution
of each variable to the summed error.

Advancements in the anomaly detection and diagnosis field have led to the
proposal of more comprehensive backbone and scoring modules [161, 137],
primarily driven by the adoption of GNN methodologies [39, 40, 60, 162].

###  5.2 Discrepancy Frameworks for Anomaly Detection

All the proposed anomaly detection methods follow the same backbone-scorer
architecture. However, the way the backbone module is trained to learn data
structure from nominal data and the implementation of the scoring module
differentiate these methods into three categories: reconstruction, forecast,
and relational discrepancy frameworks.

Reconstruction Discrepancy. Reconstruction discrepancy frameworks rely on the
assumption that reconstructed error should be low during normal periods, but
high during anomalous periods. From a high-level perspective, they are
fundamentally designed to replicate their inputs as outputs like autoencoders
[163]. However, it is assumed the backbone is sufficiently expressive to model
and reconstruct well the training data distribution, but not out-of-sample
data. Therefore, a reconstruction learning framework often incorporates
certain constraints and regularization terms, e.g., to enforce a low-
dimensional embedded code [164] or applying variational objectives [165].

Once the data structure has been effectively learned, the backbone model
should be able to approximate the input during non-anomalous periods, as this
input would closely resemble the normal training data. In contrast, during an
anomalous event, the backbone model is expected to struggle with
reconstructing the input, given that the input patterns deviate from the norm
and is situated outside the manifold. Using the reconstructed outputs from the
backbone module, a discrepancy score is computed by the
$\textsc{Scorer}(\cdot)$ to determine whether an anomalous event has occurred.
Although deep reconstruction models generally follow these principles for
detecting anomalies, a key distinction between GNNs and other architectural
types rests in the backbone reconstructor, $\textsc{Backbone}(\cdot)$, which
is characterized by its spatiotemporal GNN implementation.

MTAD-GAT [39] utilizes a variational objective [165] to train the backbone
reconstructor module. During inference, the reconstructor module will provide
the likelihood of observing each input channel signal to the scorer. The
scorer then summarizes the likelihoods into a single reconstruction
discrepancy as an anomaly score. With the availability of reconstruction
probability for each channel variable, MTAD-GAT can diagnose the anomaly
scorer by computing the contribution of each variable to the discrepancy
score. While MTAD-GAT shares the same variational objective as LSTM-VAE [143],
MTAD-GAT differs by employing graph attention network as a spatial-temporal
encoder to learn inter-variable and inter-temporal dependencies. Empirically,
it is shown to outperform LSTM on the same VAE objective111While MTAD-GAT also
optimizes their network using forecasting objective, its ablation shows that
using reconstruction alone on GNN can outperform its LSTM counterpart..
Interestingly, MTAD-GAT also shows that attention scores in the graph
attention network reflect substantial differences between normal and anomalous
periods.

GNNs require knowledge of graph structures that is often not readily available
for time series anomaly detection data [145, 166]. To solve this issue, MTAD-
GAT plainly assumes a fully-connected graph between the spatial variables in a
multivariate time series. This assumption may not necessarily hold true in
real-world scenarios and can potentially create unnecessary noise that weakens
the ability to learn the underlying structure of normal data.

In response, VGCRN [151] first assigns learnable embeddings to the channel
variates of a time series data. VGCRN then generates the channel-wise
similarity matrix as the learned graph structure by computing the dot product
between the embeddings. Under the same graph structural learning framework,
FuSAGNet [60] proposes to learn a static and sparse directed graph by taking
only the top-$k$ neighbors for each node in the similarity matrix. FuSAGNet
however differs in the reconstruction framework by learning a sparse embedded
code [164] rather than optimizing for variational objectives.

In the category of reconstruction-based methods, a research direction focused
on graph-level embeddings to represent the input graph data as vectors to
enable the application of well-established and sophisticated detection methods
designed for multivariate time series. The works by Zambon et al. [62, 167]
laid down the general framework which has been instantiated in different
nuances. Some papers design low-dimensional embedding methods trained so that
the distance between any two vector representations best preserves the graph
distance between their respective input graphs [168, 169]. Conversely, GIF
[155] employs a high-dimension graph-level embedding method based on the idea
of random Fourier feature to discover anomalous observations. CCM-CDT [52] is
a graph autoencoder with Riemannian manifolds as latent spaces. The
autoencoder, trained adversarially to reconstruct the input, produces a
sequence of manifold points where statistical tests are performed to detect
anomalies and changes in the data distribution.

Forecast Discrepancy. Forecast discrepancy frameworks rely on the assumption
that forecast error should be low during normal periods, but high during
anomalous periods. Here, the backbone module is substituted with a GNN
forecaster that is trained to predict a one-step-ahead forecast. During model
deployment, the forecaster makes a one-step-ahead prediction, and the forecast
values are given to the scorer. The scorer compares the forecast against the
real observed signals to compute forecast discrepancies such as absolute error
[40] or mean-squared error [147]. Importantly, it is generally assumed that a
forecasting-based model will exhibit erratic behavior during anomaly periods
when the input data deviates from the normal patterns, resulting in a
significant forecasting discrepancy.

A seminal work in applying forecasting-based GNN to detect anomalies in time
series data is GDN [40]. The forecaster of GDN consists of two main parts:
first, a graph structure module that learns an underlying graph structure,
and, second, a graph attention network that encodes the input series
representation. The graph structure module computes the graph adjacency matrix
for the graph attention network on the learned graph to obtain an expressive
representation for making a one-step-ahead forecast. Finally, the scorer
computes the forecast discrepancy as the maximum absolute forecast error among
the channel variables to indicate whether an anomaly event has occurred.

Interestingly, GDN illustrates that an anomaly event may be manifested in a
single variable, which acts as a symptom, while the underlying cause may be
traced to a separate, root-cause variable. Hence, GDN proposed to utilize the
learned relationships between variables for diagnosing the root causes of such
events rather than relying solely on the individual contributions of each
variable to the discrepancy score for diagnosing the root cause of anomaly
events. This is accomplished by identifying the symptomatic variable that
results in the maximum absolute error, followed by pinpointing its neighbour
variables. The ability of GDN to discern these associations underscores the
potential of GNNs in offering a more holistic solution to anomaly detection
and diagnosis through the automated learning of inter-variable relationships.

Within the context of statistical methods, the AZ-whiteness test [170]
operates on the prediction residuals obtained by forecasting models. Assuming
that a forecasting model is sufficiently good in modeling the nominal data-
generating process, the statistical test is able to identify unexpected
correlations in the data that indicate shifts in the data distribution. The
AZ-test is also able to distinguish between serial correlation, i.e., along
the temporal dimension, and spatial correlation observed between the different
graph nodes. In a similar fashion, the AZ-analysis of residuals [63] expands
the set of analytical tools to identify anomalous nodes and time steps, thus
providing a finer inspection and diagnosis.

Relational Discrepancy. Relational discrepancy frameworks rely on the
assumption that the relationship between variables should exhibit significant
shifts from normal to anomalous periods. This direction has been alluded to in
the MTAD-GAT work, where it was observed that attention weights in node
neighborhoods tend to deviate substantially from normal patterns during
anomaly periods. Consequently, the logical evolution of using spatiotemporal
GNN involves leveraging its capability to learn graph structures for both
anomaly detection and diagnosis. In this context, the backbone module serves
as a graph learning module that constructs the hidden evolving relationship
between variables. The scorer, on the other hand, is a function that evaluates
changes in these relationships and assigns an anomaly or discrepancy score
accordingly.

GReLeN [150] was the first method to leverage on learned dynamic graphs to
detect anomalies from the perspective of relational discrepancy. To achieve
this, the reconstruction module of GReLeN learns to dynamically construct
graph structures that adapt at every time point based on the input time series
data. The constructed graph structures serve as the inputs for a scorer that
computes the total changes in the in-degree and out-degree values for channel
nodes. GReLeN discovered that by focusing on sudden changes in structural
relationships, or relational discrepancy, at each time point, they could
construct a robust metric for the detection of anomalous events.

On the other hand, DyGraphAD adopts a forecasting approach to compute
relational discrepancy [156]. The method begins by dividing a multivariate
series into subsequences and converting these subsequences into a series of
dynamically evolving graphs. To construct a graph for each subsequence,
DyGraphAD employs the DTW distance between channel variables based on their
respective values in the subsequence. Following this preprocessing step, the
DTW distance graphs are treated as ground truth or target and the network is
trained to predict one-step-ahead graph structures. The scorer of DyGraphAD
computes the forecast error in the graph structure as the relational
discrepancy for anomaly detection.

Hybrid and Other Discrepancies. Each type of discrepancy-based framework often
possesses unique advantages for detecting and diagnosing various kinds of
anomalous events. As demonstrated in GDN[40], the relational discrepancy
framework can uncover spatial anomalies that are concealed within the
relational patterns between different channels. In contrast, the forecast
discrepancy framework may be particularly adept at identifying temporal
anomalies such as sudden spikes or seasonal inconsistencies. Therefore, a
comprehensive solution would involve harnessing the full potential of
spatiotemporal GNNs by computing a hybridized measure that combines multiple
discrepancies as indicators for anomaly detection. For instance, MTAD-GAT[39]
and FuSAGNet[60] employ both reconstruction and forecast discrepancy
frameworks, while DyGraphAD[156] utilizes a combination of forecast and
relational discrepancy frameworks to enhance the detection of anomalous
events. In these cases, the scoring function should be designed to encapsulate
the combination of either reconstruction, forecast or relational discrepancy.
In general, the anomaly score can be represented as
$S_{t}=\|\mathbf{X}_{t}-\mathbf{\hat{X}}_{t}\|_{2}^{2}+\|\mathbf{A}_{t}-\mathbf{\hat{A}}_{t}\|_{F}^{2}$,
which captures the reconstruction and relational discrepancies, respectively.
Here, $\mathbf{X}_{t}$ and $\mathbf{A}_{t}$ represent the target signals and
inter-variable relations, while $\mathbf{\hat{X}}_{t}$ and
$\mathbf{\hat{A}}_{t}$ denote the predicted signal and inter-variable
relations.

Apart from learning the underlying structures of normal training data, another
approach to detecting time series anomalies involves incorporating prior
knowledge of how a time series might behave during anomalous events. With this
aim in mind, GraphSAD [157] considers six distinct types of anomalies,
including spike and dip, resizing, warping, noise injection, left-side right,
and upside down, to create pseudo labels on the training data. By doing so,
unsupervised anomaly detection tasks can be transformed into a standard
classification task, with the class discrepancy serving as an anomaly
indicator.

##  6 GNNs for Time Series Classification

TABLE IV: Summary of graph neural networks for time series classification.
Task notation: “U” and “M” refer to univariate and multivariate time series
classification tasks. Conversion represents the transformation of a time
series classification task into a graph-level task as either graph or node
classification task, represented as “Series-As-Graph” and “Series-As-Node”,
respectively. The remaining notations are shared with Table II.

Approach Year Venue Task Conversion Spatial Module Temporal Module Missing
Values Input Graph Learned Relations Graph Heuristics MTPool[171] 2021 NN M -
Spatial GNN T-C No NR S - Time2Graph+[64] 2021 IEEE TKDE U Series-As-Graph
Spatial GNN - No R - PS RainDrop[37] 2022 ICLR M - Spatial GNN T-A Yes NR S -
SimTSC[65] 2022 SDM U+M Series-As-Node Spatial GNN T-C No R - PS LB-SimTSC[66]
2023 arXiv U+M Series-As-Node Spatial GNN T-C No R - PS TodyNet[172] 2023
arXiv M - Spatial GNN T-C No NR D -

Time series classification task seeks to assign a categorical label to a given
time series based on its underlying patterns or characteristics. As outlined
in a recent survey [173], early literature in time series classification
primarily focused on distance-based approaches for assigning class labels to
time series [174, 175, 176], and ensembling methods such as Hierarchical Vote
Collective of Transformation-based Ensembles (HIVE-COTE) [177, 178]. However,
despite their state-of-the-art performances, the scalability of both
approaches remains limited for high-dimensional or large datasets [179, 180].

To address these limitations, researchers have begun to explore the potential
of deep learning techniques for enhancing the performance and scalability of
time series classification methods. Deep learning, with its ability to learn
complex patterns and hierarchies of features, has shown promise in its
applicability to time series classification problems, especially for datasets
with substantial training labels [181, 182]. For a comprehensive discussion on
deep learning-based time series classification, we direct readers to the
latest survey by Foumani et al. [173].

One particularly intriguing development in this area not covered by the
aforementioned survey [173] is the application of GNN to time series
classification tasks. By transforming time series data into graph
representations, one can leverage the powerful capabilities of GNNs to capture
both local and global patterns. Furthermore, GNNs are capable of mapping the
intricate relationships among different time series data samples within a
particular dataset.

In the following sections, we provide a fresh GNN perspective on the
univariate and multivariate time series classification problem.

###  6.1 Univariate Time Classification

Inherent in the study of time series classification lies a distinct
differentiation from other time series analyses; rather than capturing
patterns within time series data, the essence of time series classification
resides in discerning differentiating patterns that help separate time series
data samples based on their class labels.

For example, in the healthcare sector, time series data in the form of heart
rate readings can be utilized for health status classification. A healthy
individual may present a steady and rhythmic heart rate pattern, whereas a
patient with cardiovascular disease may exhibit patterns indicating irregular
rhythms or elevated average heart rates. Unlike forecasting future points or
detecting real-time anomalies, the classification task aims to distinguish
these divergent patterns across series, thereby enabling health status
classification based on these discerned patterns.

In the following, we delve into two novel graph-based approaches to univariate
time series classification, namely Series-As-Graph and Series-As-Node.

Series-As-Graph. The Series-As-Graph approach transforms a univariate time
series into a graph to identify unique patterns that enable accurate
classification using a GNN. In this manner, each series is treated as a graph,
and the graph will be the input for a GNN to make classification outputs.

Firstly, each series is broken down into subsequences as nodes, and the nodes
are connected with edges illustrating their relationships. Following this
transformation, a GNN is applied to make graph classification. This procedure
is represented in the upper block of 3d. Fundamentally, it seeks to model
inter-temporal dependencies under a GNN framework to identify differentiating
patterns across series samples into their respective classes.

The graph classification perspective of Series-As-Graph was first proposed by
the Time2Graph technique [183]. This approach was later developed further with
the incorporation of GNN, as Time2Graph+ [64]. The Time2Graph+ modeling
process can be described as a two-step process: first, a time series is
transformed into a shapelet graph, and second, a GNN is utilized to model the
relations between shapelets. To construct a shapelet graph, the Time2Graph
algorithm partitions each time series into successive segments. It then
employs data mining techniques to assign representative shapelets to the
subsequences. These shapelets serve as graph nodes. Edges between nodes are
formed based on the conditional probability of a shapelet occurring after
another within a time series. Consequently, each time series is transformed
into a graph where shapelets form nodes and transition probabilities create
edges. After graph construction, Time2Graph+ utilizes Graph Attention Network
along with a graph pooling operation to derive the global representation of
the time series. This representation is then fed into a classifier to assign
class labels to the time series.

While we use Series-As-Graph to describe the formulation of a time series
classification task as a graph classification task, this should not be
confused with the Series2Graph [184] approach, which is utilized for the
anomaly detection tasks. The strategy of recasting a series as a graph is not
unique to time-series classification, as demonstrated by Series2Graph. The
usage of this strategy across various tasks and domains highlights its
inherent adaptability and underscores its potential worth exploring further by
researchers.

Series-As-Node. As capturing differentiating class patterns across different
series data samples is important, leveraging relationships across the
different series data samples in a given dataset can be beneficial for
classifying a time series. To achieve this, one can take the Series-As-Node
approach, where each series sample is seen as a separate node. These series
nodes are connected with edges that represent the relationships between them,
creating a large graph that provides a complete view of the entire dataset.

The Series-As-Node was originally proposed by SimTSC [65] approach. With
SimTSC, series nodes are connected using edges, which are defined by their
pairwise DTW distance, to construct a graph. During the modeling process, a
primary network is initially employed to encode each time series into a
feature vector, thus creating a node representation. Subsequently, a standard
GNN operation is implemented to derive expressive node representations,
capturing the similarities between the series. These node representations are
then inputted into a classifier, which assigns class labels to each time
series node in the dataset. LB-SimTSC [66] extends on SimTSC to improve the
DTW preprocessing efficiency by employing the widely-used lower bound for DTW,
known as LB_Keogh [185]. This allows for a time complexity of $O(L)$ rather
than $O(L^{2})$, dramatically reducing computation time.

The Series-As-Node process essentially formulates a time series classification
task as a node classification task. As illustrated in the lower block of 3d,
the Series-As-Node perspective aims to leverage the relationships across
different series samples for accurate time series node classification [66]. It
is also an attempt to marry classical distance-based approaches with advanced
GNN techniques. While not explicitly depicted in the figure, it is important
to note that the same concept can be applied to classify multivariate time
series by modifying the backbone network that transforms each series into a
node [65].

###  6.2 Multivariate Time Series Classification

In essence, multivariate time series classification maintains fundamental
similarities with its univariate counterpart, however, it introduces an
additional layer of complexity: the necessity to capture intricate inter-
variable dependencies.

For example, instead of solely considering heart rate, patient data often
incorporate time series from a multitude of health sensors, including blood
pressure sensors, blood glucose monitors, pulse oximeters, and many others.
Each of these sensors provides a unique time series that reflects a particular
aspect of the health of a patient. By considering these time series together
in a multivariate analysis, we can capture more complex and interrelated
health patterns that would not be apparent from any single time series alone.

Analogously, each node in an electroencephalogram (EEG) represents electrical
activity from a distinct brain region. Given the interconnectedness of brain
regions, analyzing a single node in isolation may not fully capture the
comprehensive neural dynamics [186]. By employing multivariate time series
analysis, we can understand the relationships between different nodes, thereby
offering a more holistic view of brain activity. This approach facilitates the
differentiation of intricate patterns that can classify patients with and
without specific neurological conditions.

In both examples, the relationships between the variables, or inter-variable
dependencies, can be naturally thought of as a network graph. Hence, they
ideally suit the capabilities of GNNs as illustrated in forecasting section 4.
As such, spatiotemporal GNNs, exemplified by those utilized in forecasting
tasks [53], are conveniently adaptable for multivariate time series
classification tasks. This adaptation can be achieved through the replacement
of the final layer with a classification component. The unique design of these
STGNN architectures enables the capture of both inter-temporal and inter-
variable dependencies. The primary aim here is to effectively distill the
complexity of high-dimensional series data into a more comprehensible, yet
equally expressive, representation that enables differentiation of time series
into their representative classes [171, 172].

The proficiency of spatiotemporal GNNs in decoding the complexities of
multivariate time series is demonstrably showcased in the Raindrop
architecture [37]. To classify irregularly sampled data where subsets of
variables have missing values at certain timestamps, Raindrop adaptively
learns a graph structure. It then dynamically interpolates missing
observations within the embedding space, based on any available recorded data.
This flexible approach ensures that the data representation remains both
comprehensive and accurate, despite any irregularities in the sampling.
Empirical studies provide evidence that Raindrop can maintain robust, high-
performance classification, even in the face of such irregularities [37].
These findings further reinforce the versatility of spatiotemporal GNNs in
time series classification, highlighting their effectiveness even in scenarios
characterized by missing data and irregular sampling patterns.

##  7 GNNs for Time Series Imputation

TABLE V: Summary of graph neural networks for time series imputation. Task
notation: “Out-of-sample”, “In-sample”, and “Both” refer to the types of
imputation problems addressed by the approach. Type represents the imputation
method as either deterministic or probabilistic. Inductiveness indicates if
the method can generalize to unseen nodes. The remaining notations are shared
with Table II.

Approach Year Venue Task Type Spatial Module Temporal Module Inductiveness
Input Graph Learned Relations Graph Heuristics IGNNK [187] 2021 AAAI Out-of-
sample Deterministic Spectral GNN - Yes R - SP, PC GACN [188] 2021 ICANN In-
sample Deterministic Spatial GNN T-C No R - PC SATCN [189] 2021 arXiv Out-of-
sample Deterministic Spatial GNN T-C Yes R - SP GRIN [41] 2022 ICLR Both
Deterministic Spatial GNN T-R Yes R - SP SPIN [190] 2022 NIPS In-sample
Deterministic Spatial GNN T-A No R - SP FUNS [191] 2022 ICDMW Out-of-sample
Deterministic Spatial GNN T-R Yes R - - AGRN [192] 2022 ICONIP In-sample
Deterministic Spatial GNN T-R No NR S - MATCN [193] 2022 IEEE IoT-J In-sample
Deterministic Spatial GNN T-A No R - - PriSTI [42] 2023 arXiv In-sample
Probabilistic Spatial GNN T-A No R - SP DGCRIN [194] 2023 KBS In-sample
Deterministic Spatial GNN T-R No NR D - GARNN [195] 2023 Neurocomputing In-
sample Deterministic Spatial GNN T-R No R - PC MDGCN [196] 2023 Transp. Res.
Part C In-sample Deterministic Spatial GNN T-R No R - SP, PS

Time series imputation, a crucial task in numerous real-world applications,
involves estimating missing values within one or more data point sequences.
Traditional time series imputation approaches have relied on statistical
methodologies, such as mean imputation, spline interpolation [197], and
regression models [198]. However, these methods often struggle to capture
complex temporal dependencies and non-linear relationships within the data.
While some deep neural network-based works, such as [199, 200, 201], have
mitigated these limitations, they have not explicitly considered inter-
variable dependencies. The recent emergence of graph neural networks has
introduced new possibilities for time series imputation. GNN-based methods
better characterize intricate spatial and temporal dependencies in time series
data, making them particularly suitable for real-world scenarios arising from
the increasing complexity of data. From a task perspective, GNN-based time
series imputation can be broadly categorized into two types: in-sample
imputation and out-of-sample imputation. The former involves filling in
missing values within the given time series data, while the latter predicts
missing values in disjoint sequences [41]. From a methodological perspective,
GNN for time series imputation can be further divided into deterministic and
probabilistic imputation. Deterministic imputation provides a single best
estimate for the missing values, while probabilistic imputation accounts for
the uncertainty in the imputation process and provides a distribution of
possible values. In Table V, we summarize most of the related works on GNN for
time series imputation to date, offering a comprehensive overview of the field
and its current state of development.

###  7.1 In-Sample Imputation

The majority of existing GNN-based methods primarily focus on in-sample time
series data imputation. For instance, GACN [188] proposes to model spatial-
temporal dependencies in time series data by interleaving GAT [108] and
temporal convolution layers in its encoder. It then imputes the missing data
by combining GAT and temporal deconvolution layers that map latent states back
to original feature spaces. Similarly, SPIN [190] first embeds historical
observations and sensor-level covariates to obtain initial time series
representations. These are then processed by multi-layered sparse spatial-
temporal attention blocks before the final imputations are obtained with a
nonlinear transformation. GRIN [41] introduces the graph recurrent imputation
network, where each unidirectional module consists of one spatial-temporal
encoder and two different imputation executors. The spatial-temporal encoder
adopted in this work combines MPNN [101] and GRU [105]. After generating the
latent time series representations, the first-stage imputation fills missing
values with one-step-ahead predicted values, which are then refined by a final
one-layer MPNN before passing to the second-stage imputation for further
processing. Similar works using bidirectional recurrent architectures include
AGRN [192], DGCRIN [194], GARNN [195], and MDGCN [196], where the main
differences lie in intermediate processes. For example, AGRN and DGCRIN
propose different graph recurrent cells that integrate graph convolution and
GRU to capture spatial-temporal relations, while GARNN involves the use of GAT
and different LSTM [202] cells to compose a graph attention recurrent cell in
its model architecture. MDGCN models time series as dynamic graphs and
captures spatial-temporal dependencies by stacking bidirectional LSTM and
graph convolution. Recently, a few research studies have explored
probabilistic in-sample time series imputation, such as PriSTI [42], where the
imputation has been regarded as a generation task. In PriSTI, a similar
architecture of denoising diffusion probabilistic models [203] has been
adopted to effectively sample the missing data with a spatial-temporal
denoising network composed of attentive MPNN and temporal attention.

###  7.2 Out-of-Sample Imputation

To date, only a few GNN-based methods fall into the category of out-of-sample
imputation. Among these works, IGNNK [187] proposes an inductive GNN kriging
model to recover signals for unobserved time series, such as a new variable or
“virtual sensor” in a multivariate time series. In IGNNK, the training process
involves masked subgraph sampling and signal reconstruction with the diffusion
graph convolution network presented in [122]. Another similar work is SATCN
[189], which also focuses on performing real-time time series kriging. The
primary difference between these two works lies in the underlying GNN
architectures, where SATCN proposes a spatial aggregation network combined
with temporal convolutions to model the underlying spatial-temporal
dependencies. It is worth noting that GRIN [41] can handle both in-sample and
out-of-sample imputations, as well as a similar follow-up work [191].

##  8 Practical Applications

Graph neural networks have been applied to a broad range of disciplines
related to time series analysis. We categorize the mainstream applications of
GNN4TS into six areas: smart transportation, on-demand services, environment &
sustainable energy, internet-of-things, and healthcare.

Smart Transportation. The domain of transportation has been significantly
transformed with the advent of GNNs, with typical applications spanning from
traffic prediction to flight delay prediction. Traffic prediction,
specifically in terms of traffic speed and volume prediction, is a critical
component of smart transportation systems. By leveraging advanced algorithms
and data analytics related to spatial-temporal GNNs, traffic conditions can be
accurately predicted [71, 58, 73, 77, 87, 124, 204], thereby facilitating
efficient route planning and congestion management. Another important
application is traffic data imputation, which involves the estimation of
missing or incomplete traffic data. This is crucial for maintaining the
integrity of traffic databases and ensuring the accuracy of traffic analysis
and prediction models [205, 193, 194, 195, 196]. There is also existing
research related to autonomous driving with 3D object detection and motion
planners based on GNNs [206, 207, 208], which has the potential to drastically
improve road safety and traffic efficiency. Lastly, flight delay prediction is
another significant application that can greatly enhance passenger experience
and optimize airline operations. This is achieved through the analysis of
various factors such as weather conditions, air traffic, and aircraft
maintenance schedules [209, 210]. In summary, smart transportation, through
its diverse applications, is paving the way for a more efficient, safe, and
convenient transportation system. The integration of advanced technologies,
such as GNNs, in these applications underscores the transformative potential
of smart transportation, highlighting its pivotal role in shaping the future
of transportation.

On-Demand Services. For systems providing goods or services upon request, GNNs
have emerged as powerful tools for modeling time series data to accurately
predict personalized real-time demands, including transportation, energy,
tourism, and more. For instance, in ride-hailing services, GNNs capture the
complex, temporal dynamics of ride demand across different regions, enabling
accurate prediction of ride-hailing needs and thereby facilitating efficient
fleet management [211, 46, 212, 213, 132, 214, 215]. Similarly, in bike-
sharing services, GNNs leverage the spatial-temporal patterns of bike usage to
accurately predict demand, contributing to the optimization of bike
distribution and maintenance schedules [216, 47, 217, 218, 219, 220]. In the
energy sector, GNNs model the intricate relationships between various factors
influencing energy demand, providing accurate predictions that aid in the
efficient management of energy resources [221]. In the tourism industry, GNNs
capture the temporal trends and spatial dependencies in tourism data,
providing accurate predictions of tourism demand and contributing to the
optimization of tourism services and infrastructure [222, 223, 224, 225].
There are also GNN-based works that model the complex spatial-temporal
dynamics of delivery demand, accurately predicting delivery needs and
facilitating efficient logistics planning and operations [226]. The advent of
GNN4TS has significantly improved the accuracy of demand prediction in on-
demand services, enhancing their efficiency and personalization. The
integration of GNNs in these applications underscores their transformative
potential, highlighting their pivotal role in shaping the future of on-demand
services.

Environment & Sustainable Energy. In the sector related to environment and
sustainable energy, GNNs have been instrumental in wind speed and power
prediction, capturing the complex spatial-temporal dynamics of wind patterns
to provide accurate predictions that aid in the efficient management of wind
energy resources [227, 228, 229, 230, 231, 232, 233]. Similarly, in solar
energy, GNNs have been used for solar irradiance and photovoltaic (PV) power
prediction, modeling the intricate relationships between various factors
influencing solar energy generation to provide accurate predictions [234, 235,
236, 237, 238]. In terms of system monitoring, GNNs have been applied to wind
turbines and PV systems. For wind turbines, GNNs can effectively capture the
temporal dynamics of turbine performance data, enabling efficient monitoring
and maintenance of wind turbines [239]. For PV systems, GNNs have been used
for fault detection, leveraging the spatial dependencies in PV system data to
accurately identify faults and ensure the efficient operation of PV systems
[240]. Furthermore, GNNs have been employed for air pollution prediction and
weather forecasting. By modeling the spatial-temporal patterns of air
pollution data, GNNs can accurately predict air pollution levels, contributing
to the formulation of effective air quality management strategies [241, 242,
243]. In weather forecasting, GNNs capture the complex, temporal dynamics of
weather patterns, providing accurate forecasts that are crucial for various
sectors, including agriculture, energy, and transportation [244, 245].

Internet-of-Things (IoTs). IoT refers to intricately linked devices that
establish a network via wireless or wired communication protocols, operating
in concert to fulfill shared objectives on behalf of their users [246]. These
IoT networks generate substantial quantities of high-dimensional, time series
data that are increasingly complex and challenging for manual interpretation
and understanding. Recent advancements have seen the application of GNNs as a
powerful tool for encoding the intricate spatiotemporal relationships and
dependencies inherent in IoT networks [166, 147, 247]. GNNs leverage their
ability to unravel these convoluted relationships, allowing for greater
insights into the structure and behavior of these networks. This approach has
garnered attention across various industrial sectors including robotics and
autonomous systems [248, 249, 250], utility plants [40], public services
[251], and sports analytics [252, 253, 254, 255], expanding the breadth of IoT
applications. With a consistent track record of state-of-the-art results[247],
GNNs have proven integral in numerous IoT applications, underpinning our
understanding of these increasingly complex systems.

Healthcare. Healthcare systems, spanning from individual medical diagnosis and
treatment to broader public health considerations, present diverse challenges
and opportunities that warrant the application of GNNs. In the sphere of
medical diagnosis and treatment, graph structures can effectively capture the
complex, temporal dynamics of diverse medical settings including electrical
activity such as electronic health data[256, 257, 258, 259], patient
monitoring sensors [260, 261, 37], EEG[126, 262, 186], brain functional
connectivity such as magnetic resonance imaging (MRI)[263, 264] and
neuroimaging data[265]. Simultaneously, for public health management, GNNs
have been proposed to predict health equipment useful life [266] and
forecasting ambulance demand [267]. More recently, GNNs have been proposed to
manage epidemic disease outbreaks as temporal graphs can provide invaluable
insights into disease spread, facilitating the formulation of targeted
containment strategies [268, 269, 127]. In summary, the integration of GNNs
with time series data holds substantial potential for transforming healthcare,
from refining medical diagnosis and treatment to strengthening population
health strategies, highlighting its critical role in future healthcare
research.

Fraud Detection. As elucidated in the four-element fraud diamond [270], the
perpetration of fraud necessitates not just the presence of incentive and
rationalization, but also a significant degree of capability - often
attainable only via coordinated group efforts undertaken at opportune moments.
This suggests that fraud is typically committed by entities possessing
sufficient capability, which can be primarily achieved through collective
endeavors during suitable periods. Consequently, it is rare for fraudsters to
operate in isolation [271, 272]. They also frequently demonstrate unusual
temporal patterns in their activities, further supporting the necessity for
sophisticated fraud detection measures [273, 274]. To this end, GNNs have been
proposed to capture these complex relational and temporal dynamics inherent to
fraud network activities. They have found successful applications in various
domains, such as detecting frauds and anomalies in social networks [273, 274,
275], financial networks and systems [276, 277, 278, 279], and in several
other sectors [272, 280, 281, 282, 283].

Other Applications. Beyond the aforementioned sectors, the application of GNNs
for time series analysis has also been extended to various other fields, such
as finance [284], urban planning [285, 286], epidemic control [287, 288, 289],
and particle physics [290]. As research in this area continues to evolve, it
is anticipated that the application of GNNs will continue to expand, opening
up new possibilities for data-driven decision making and system optimization.

##  9 Future Directions

Pre-training, Transfer Learning, and Large Models. Pre-training, transfer
learning, and large models are emerging as potent strategies to bolster the
performance of GNNs in time series analysis [90, 291, 292], especially when
data is sparse or diverse. These techniques hinge on utilizing learned
representations from one or more domains to enhance performance in other
related domains [3, 293]. Recent successful examples are Panagopoulos et al.’s
[294] model-agnostic meta-learning schema for COVID-19 spread prediction in
data-limited cities, and Shao et al.’s [90] pre-training enhanced framework
for spatial-temporal GNNs. The exploration of pre-training strategies and GNN
transferability for time series tasks is a burgeoning research area,
especially in the current era of generative AI and large models, which
showcase the potential for a single, multimodal model to address diverse tasks
[295]. However, several challenges remain, including the limited availability
of time series data for large-scale pre-training compared to language data for
large language models (LLMs) [296], to ensure wide coverage and
transferability of learned knowledge, and designing effective pre-training
strategies that capture complex spatial-temporal dependencies. Addressing
these challenges is pivotal for the future development and application of
GNN4TS.

Robustness. Robustness of GNNs refers to their ability to handle various forms
of data perturbations and distribution shifts, particularly those that are
deliberately engineered by adversaries [297]. This quality becomes critical
when dealing with time series data generated by rapidly evolving systems. Any
operational failures within GNNs can potentially precipitate adverse
consequences on the integrity of the entire system [298, 247]. For instance,
if a GNN fails to adequately handle noise or data corruption in a smart city
application, it might disrupt essential traffic management functions.
Similarly, in healthcare applications, the inability of a GNN to remain robust
amidst disturbances could lead to healthcare providers missing out on critical
treatment periods, potentially having serious health implications for
patients. While GNNs have demonstrated superior performance across numerous
applications, improving their robustness and creating effective failure
management strategies remains vital. This not only enhances their reliability
but also widens their potential usage across contexts.

Interpretability. The interpretability of GNNs plays an equally pivotal role
in facilitating the transparent and accountable use of these sophisticated
tools. This attribute sheds light on the opaque decision-making processes of
GNNs, allowing users to comprehend the reasoning behind a given output or
prediction. Such understanding fosters trust in the system and enables the
discovery of latent patterns within the data [299, 297]. For example, in drug
discovery [300] and financial time series analyses [301, 302],
interpretability may illuminate causal factors, facilitating more informed
decision-making. As we strive to harness the full potential of GNN4TS,
advancing their interpretability is paramount to ensuring their ethical and
judicious application in increasingly complex environments.

Uncertainty Quantification. Time series data, by its nature, is often fraught
with unpredictable noise and uncertainty associated with the data-generating
process. The ability of a model to account for and quantify uncertainty can
greatly enhance its reliability and utility[303, 304]. Uncertainty
quantification provides a probabilistic measure of the confidence to the
predictions made by the model and to the system state estimates [109, 116],
aiding in the understanding of the range and likelihood of potential
outcomes[305]. This becomes particularly important when GNNs are used for
decision-making processes in fields where high stakes are involved, such as
financial forecasting, healthcare monitoring[256, 259], or traffic prediction
in smart cities[73, 75, 46, 226]. Despite progress, a gap remains in the
current GNN models, which largely provide point estimates[73, 75, 53, 147, 40,
37, 65], inadequately addressing the potential uncertainties. This underlines
an essential research direction: developing sophisticated uncertainty
quantification methods for GNNs to better navigate the complexities of time
series data. This endeavor not only enhances the interpretability and
reliability of predictions but also fosters the development of advanced models
capable of learning from uncertainty. Thus, uncertainty quantification
represents a pivotal element in the ongoing advancement of GNN4TS.

Privacy Enhancing. GNNs have established themselves as invaluable tools in
time series analysis, playing crucial roles in diverse, interconnected systems
across various sectors [211, 256, 273, 209, 278, 207]. As these models gain
broader adoption, particularly in fields that require the powerful data
forecasting[40, 186] and reconstruction [39, 41] capabilities of GNNs, the
need for stringent privacy protection becomes increasingly apparent. Given the
ability of GNNs to learn and reconstruct the relationships between entities
within complex systems [121, 156], it is essential to safeguard not only the
privacy of individual entities (nodes), but also their relationship (edges)
within the time series data [306, 297]. Furthermore, the interpretability of
GNNs can serve as a double-edged sword. While it can help identify and
mitigate areas vulnerable to malicious attacks, it could also expose the
system to new risks by revealing sensitive information [307]. Therefore,
maintaining robust privacy defenses while capitalizing on the benefits of GNN
models for time series analysis requires a delicate balance, one that calls
for constant vigilance and continual innovation.

Scalability. GNNs have emerged as powerful tools to model and analyze
increasingly large and complex time series data, such as large social networks
consisting of billions of users and relationships [273, 275]. However, the
adaptation of GNNs to manage vast volumes of time-dependent data introduces
unique challenges. Traditional GNN models often require the computation of the
entire adjacency matrix and node embeddings for the graph, which can be
extraordinarily memory-intensive [308]. To counter these challenges,
traditional GNN methods utilize sampling strategies like node-wise [309, 310],
layer-wise [311], and graph-wise[312] sampling. Yet, incorporating these
methods while preserving consideration for temporal dependencies remains a
complex task. Additionally, improving scalability during the inference phase
to allow real-time application of GNNs for time series analysis is vital,
especially in edge devices with limited computational resources. This
intersection of scalability, time-series analysis, and real-time inference
presents a compelling research frontier, ripe with opportunities for
breakthroughs. Hence, exploring these areas can be a pivotal GNN4TS research
direction.

AutoML and Automation. Despite the notable success of GNNs in temporal
analytics [53, 39, 41, 37], their empirical implementations often necessitate
meticulous architecture engineering and hyperparameter tuning to accommodate
varying types of graph-structured data [313, 314]. A GNN architecture is
typically instantiated from its model space and evaluated in each graph
analysis task based on prior knowledge and iterative tuning processes [308].
Furthermore, with the plethora of architectures being proposed for different
use cases [41, 34, 173, 65, 162], discerning the most suitable option poses a
significant challenge for end users.

AutoML and automation in time series analysis using GNNs thus plays a pivotal
role in overcoming the complexities associated with diverse model
architectures. It can simplify the selection process, enhancing efficiency and
scalability while fostering effective model optimization [315, 316, 313].
Furthermore, it is important to note that GNNs may not always be the optimal
choice compared to other methods[317, 318, 319]. Their role within the broader
landscape of AutoML must therefore be thoughtfully evaluated. By encouraging
reproducibility and broadening accessibility, automation democratizes the
benefits of GNNs for advanced temporal analytics.

##  10 Conclusions

This comprehensive survey bridges the knowledge gap in the field of graph
neural networks for time series analysis (GNN4TS) by providing a detailed
review of recent advancements and offering a unified taxonomy to categorize
existing works from task- and methodology-oriented perspectives. As the first
of its kind, it covers a wide range of tasks including forecasting,
classification, anomaly detection, and imputation, providing a detailed
understanding of the state of the art in GNN4TS. We also delve into the
intricacies of spatial and temporal dependencies modeling and overall model
architecture, offering a fine-grained classification of individual studies.
Highlighting the expanding applications of GNN4TS across various sectors, we
demonstrate its versatility and potential for future growth. This survey
serves as a valuable resource for machine learning practitioners and domain
experts interested in the latest advancements in this field. Lastly, we
propose potential future research directions, offering insights to guide and
inspire future work in GNN4TS.

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