# SEA++: Multi-Graph-based High-Order Sensor Alignment for Multivariate Time-
Series Unsupervised Domain Adaptation

Yucheng Wang, Yuecong Xu, Jianfei Yang, Min Wu, Xiaoli Li, Lihua Xie, Fellow,
IEEE, Zhenghua Chen111Corresponding Author Yucheng Wang and Yuecong Xu are
with Institute for Infocomm Research, A∗STAR, Singapore and the School of
Electrical and Electronic Engineering, Nanyang Technological University,
Singapore (Email<EMAIL_ADDRESS>xuyu0014@e.ntu.edu.sg).Zhenghua
Chen and Min Wu are with Institute for Infocomm Research, A∗STAR, Singapore
(Email<EMAIL_ADDRESS>wumin@i2r.a-star.edu.sg).Xiaoli Li is with
Institute for Infocomm Research, A∗STAR, Singapore and the School of Computer
Science and Engineering, Nanyang Technological University, Singapore (Email:
xlli@i2r.a-star.edu.sg).Jianfei Yang and Lihua Xie are with the School of
Electrical and Electronic Engineering, Nanyang Technological University,
Singapore (Email<EMAIL_ADDRESS>elhxie@ntu.edu.sg).

###### Abstract

Unsupervised Domain Adaptation (UDA) methods have been successful in reducing
label dependency by minimizing the domain discrepancy between a labeled source
domain and an unlabeled target domain. However, these methods face challenges
when dealing with Multivariate Time-Series (MTS) data. MTS data typically
consist of multiple sensors, each with its own unique distribution. This
characteristic makes it hard to adapt existing UDA methods, which mainly focus
on aligning global features while overlooking the distribution discrepancies
at the sensor level, to reduce domain discrepancies for MTS data. To address
this issue, a practical domain adaptation scenario is formulated as
Multivariate Time-Series Unsupervised Domain Adaptation (MTS-UDA). In this
paper, we propose SEnsor Alignment (SEA) for MTS-UDA, aiming to reduce domain
discrepancy at both the local and global sensor levels. At the local sensor
level, we design endo-feature alignment, which aligns sensor features and
their correlations across domains. To reduce domain discrepancy at the global
sensor level, we design exo-feature alignment that enforces restrictions on
global sensor features. We further extend SEA to SEA++ by enhancing the endo-
feature alignment. Particularly, we incorporate multi-graph-based high-order
alignment for both sensor features and their correlations. High-order
statistics are employed to achieve comprehensive alignment by capturing
complex data distributions across domains. Meanwhile, a multi-graph alignment
technique is introduced to effectively align the evolving distributions of MTS
data. Extensive empirical results have demonstrated the state-of-the-art
performance of our SEA and SEA++ on public MTS datasets for MTS-UDA.

Unsupervised Domain Adaptation (UDA) methods have been successful in reducing
label dependency by minimizing the domain discrepancy between a labeled source
domain and an unlabeled target domain. However, these methods face challenges
when dealing with Multivariate Time-Series (MTS) data. MTS data typically
consist of multiple sensors, each with its own unique distribution. This
characteristic makes it hard to adapt existing UDA methods, which mainly focus
on aligning global features while overlooking the distribution discrepancies
at the sensor level, to reduce domain discrepancies for MTS data. To address
this issue, a practical domain adaptation scenario is formulated as
Multivariate Time-Series Unsupervised Domain Adaptation (MTS-UDA). In this
paper, we propose SEnsor Alignment (SEA) for MTS-UDA, aiming to reduce domain
discrepancy at both the local and global sensor levels. At the local sensor
level, we design endo-feature alignment, which aligns sensor features and
their correlations across domains. To reduce domain discrepancy at the global
sensor level, we design exo-feature alignment that enforces restrictions on
global sensor features. We further extend SEA to SEA++ by enhancing the endo-
feature alignment. Particularly, we incorporate multi-graph-based high-order
alignment for both sensor features and their correlations. Extensive empirical
results have demonstrated the state-of-the-art performance of our SEA and
SEA++ on public MTS datasets for MTS-UDA.

###### Index Terms:

Multivariate Time-Series Data; Unsupervised Domain Adaptation; Graph Neural
Network

##  I Introduction

Time-Series (TS) data have gained significant attention in various fields due
to their wide range of applications. Deep Learning (DL) methods have played a
pivotal role in addressing TS-related problems by leveraging their ability to
effectively capture latent dependencies within the data [1, 2, 3, 4, 5, 6, 7].
However, the effectiveness of DL models heavily relies on the availability of
large-scale labeled TS data, which is often a challenge in real-world
applications due to the high cost associated with data labeling.

To reduce the cost of labeling, Unsupervised Domain Adaptation (UDA) methods
have been proposed to transfer the knowledge from a labeled source domain to
an unlabeled target domain, enabling effective learning in the absence of
target domain labels [8]. Due to domain shifts, existing UDA methods focused
on reducing discrepancy across domains by learning domain-invariant features
[9, 10, 11, 12, 13, 14]. These methods have achieved decent performance,
showing the effectiveness of UDA in reducing label dependency. Subsequently,
researchers applied UDA to TS data, aiming to reduce domain discrepancy by
learning domain-invariant temporal features extracted by Recurrent Neural
Network (RNN) [15], Long Short-Term Memory (LSTM) [16], and Convolutional
Neural Network (CNN) [17].

Figure 1: Comparisons between existing UDA methods and ours, where before and
after represent before and after alignment respectively. Left: Only global
sensor features are aligned, resulting in the poor alignment of the red
sensor. Right: SEA aligns the sensor information between domains, so the
feature discrepancy of each sensor can be reduced.

While substantial progress has been made, existing TS UDA methods may be
inapplicable for real-world applications where multiple sensors are deployed
simultaneously. For instance, in Remaining Useful Life (RUL) prediction,
various types of sensors are employed to measure different physical
parameters. On the other hand, in Human Activity Recognition (HAR), multiple
sensors are positioned at various locations on the human body to capture
activity data. To address these application scenarios, we formulate a more
challenging yet practical problem as Multivariate Time-Series Unsupervised
Domain Adaptation (MTS-UDA). Applying current TS UDA methods directly to MTS-
UDA poses two key limitations. First, MTS data typically consist of signals
from multiple sensors, where each sensor collects data with distinct
distributions due to variations in sensor types or deployment locations [18].
When adapting existing TS UDA methods to MTS-UDA, they mainly focus on
learning global features to facilitate alignment across domains by treating
the signals from various sensors as a whole [16, 19]. However, these methods
ignore the diverse distributions across sensors, resulting in misalignment at
the sensor level and thus leading to suboptimal solutions for MTS-UDA. As
presented in Fig. 1 (Left), existing UDA methods focused on aligning the
global features between domains, resulting in the poor alignment of the
features from the red sensor and limiting the transferability of the model.
Second, MTS data contain essential spatial-temporal dependency information due
to its multi-source nature. Specifically, the spatial dependency refers to the
correlations between sensors, which capture the critical interactive
relationships among them. For instance, the spatial dependency in the context
of machine RUL prediction corresponds to the correlation between a temperature
sensor and a fan speed sensor, as they exhibit strong correlations. Besides,
the temporal dependency refers to the temporal relationships between
consecutive timestamps. Existing UDA methods are limited in effectively
modelling and transferring both dependencies across domains, thus leading to
suboptimal solutions for MTS-UDA. These characteristics within MTS-UDA render
it a more complex and challenging task compared to standard TS UDA.

To address the above challenges, we propose SEnsor Alignment (SEA) for MTS-
UDA. We first aim to reduce domain discrepancy at both the local and global
sensor levels. At the local sensor level, we propose endo-feature alignment to
align sensor-level information between domains. This alignment process
includes aligning both the sensor features and sensor correlations, which
capture the characteristics of individual sensors and their interactions,
respectively. To achieve this alignment, we introduce contrastive sensor
alignment and sensor correlation alignment. Further, at the global sensor
level, we design exo-feature alignment to reduce global feature discrepancy.
This is achieved by enforcing restrictions on the global features derived from
stacked sensor features. Moreover, to capture the spatial-temporal
dependencies inherent in MTS data and enable their effective transfer across
domains, we introduce a graph-based encoder that is able to capture the
dependency information. Given that MTS data may not come with pre-defined
graphs that represent the relationships between sensors, we propose a multi-
branch self-attention mechanism to model these dependencies.

Although SEA presents a feasible solution for sensor alignment in MTS-UDA, it
employs a low-order statistics-based scheme for achieving endo-feature
alignment. This scheme may be insufficient to achieve comprehensive alignment
when dealing with complex data distributions in certain scenarios.
Furthermore, SEA can benefit from leveraging the dynamically changing
distributions within MTS data to enhance its performance. For instance, as a
machine’s health deteriorates, the distribution of a fan speed sensor
undergoes dynamic changes due to increased friction. Taking these issues into
consideration, we develop SEA++ as an extension of SEA by introducing multi-
graph-based high-order alignment for enhancing our endo-feature alignment. It
incorporates improved sensor correlation alignment and enhanced contrastive
sensor alignment, utilizing high-order statistics to achieve comprehensive
sensor-level alignment by capturing complex data distributions across domains.
Additionally, SEA++ introduces a multi-graph alignment technique specifically
tailored for aligning the evolving distributions within MTS data. This
involves the construction of sequential graphs that represent the
distributions within local temporal intervals, followed by the alignment of
corresponding sequential graphs between domains. To optimize the alignment
process, weights for balancing sequential graphs are learned based on
distribution discrepancies, allowing us to prioritize alignment for
challenging graphs. This module effectively aligns the evolving distributions,
enabling adaptation to the changing nature of the data. With the two types of
enhancements in the improved version, we represent the name with two plus
signs as SEA++.

Our contributions can be summarized as following.

  * •

We formulate a challenging scenario of Multivariate Time-Series Unsupervised
Domain Adaptation (MTS-UDA) in accordance with the characteristics of MTS
data. To our best knowledge, this is the first work to design a UDA method
specifically for MTS data.

  * •

We analyze the problems underlying MTS-UDA and propose SEnsor Alignment (SEA)
that reduces domain discrepancy at both the local and global sensor levels.
Meanwhile, SEA captures the spatial-temporal dependencies within MTS data,
enabling effective transfer of the dependency information across domains.

  * •

To cope with scenarios with complex data distributions, we introduce SEA++ by
enhancing our endo-feature alignment. High-order statistics alignment methods
are employed to achieve comprehensive sensor correlation and sensor feature
alignment.

  * •

Considering the dynamically changing distributions within MTS data, SEA++
further incorporates a multi-graph alignment technique. It aligns sequential
graphs representing local distributions, effectively aligning evolving
distributions between domains.

  * •

We evaluate the effectiveness of SEA and SEA++ with various real-world MTS
datasets through extensive experiments, demonstrating that the both methods
achieve state-of-the-art performance for MTS-UDA.

This journal paper presents an extended version of our previous work [20] by
introducing multi-graph-based high-order alignment techniques. First, we
introduce high-order statistic alignment to improve endo-feature alignment,
aiming to achieve more comprehensive sensor-level alignment. Second, we
analyze the properties of MTS data and emphasize the significance of
dynamically changing distributions inherent within MTS data. Thus, we
introduce a multi-graph alignment technique to align the evolving
distributions across domains. Third, we have conducted additional experiments,
including more compared methods, comprehensive ablation studies, and extensive
sensitivity analysis, to fully assess SEA and SEA++.

##  II Related Work

###  II-A Unsupervised domain adaptation

UDA aims to mitigate the need for labeled data by transferring knowledge from
a labeled source domain to an unlabeled target domain. To achieve good
performance on the target domain, existing UDA methods have mainly focused on
minimizing domain discrepancies by utilizing two categories of approaches. The
first category comprises adversarial-based methods, which utilize domain
discriminator networks to encourage a feature extractor to learn domain-
invariant representations. Prominent models in this category include Domain
Adversarial Neural Network (DANN) [10] and Adversarial Discriminative Domain
Adaptation (ADDA) [11]. These models have served as foundations for various
improved variants that address specific challenges [12, 21, 22, 23, 24, 25].
For instance, CDAN [12] and MADA [21] were both extensions of DANN that
incorporate multi-mode structure information for classification tasks. DAAN
[22] introduced class-aware adversarial loss to align the conditional
distribution using DANN. The second category encompasses metric-based methods,
which enforce metric restrictions to enable the learning of invariant
features. Commonly employed metrics include Maximum Mean Discrepancy (MMD) [9]
and deep CORrelation ALignment (CORAL) [13]. Within this category, several
variants have been proposed to tackle specific challenges [14, 26, 27, 28].
For example, JAN [14] was developed based on MMD to align the joint
distribution of feature and label distributions. MK-MMD [26] employed MMD to
align features in the final layers, enhancing their transferability.
Furthermore, high-order moment matching called HoMM [27] was developed as a
high-order moment matching technique to align feature distributions in hidden
layers, addressing the limitations of low-order alignment methods. In
addition, central moment discrepancy [29] and maximum density divergence [30]
are another two criterias to align feature distributions in hidden layers.
Overall, these methods have contributed to the advancement of UDA by
effectively reducing the domain discrepancy and enabling knowledge transfer
between different domains.

###  II-B Unsupervised domain adaptation for time-series data

In recent years, there has been a growing trend of exploring existing UDA
methods in various domains, showing their effectiveness in different
applications [31, 32]. In the context of TS data, researchers have primarily
focused on reducing domain discrepancy by aligning temporal features across
different domains [16, 17, 15, 19, 33]. Several notable works have made
significant contributions to this area. For instance, VRADA [15] leveraged a
variational RNN to learn temporal features and employs adversarial-based
methods to reduce the domain discrepancy. Similarly, Zhao et al. [34]
introduced a UDA method combining DANN and CNN for EEG classification, which
incorporated a center loss to minimize intra-class variation and maximize
inter-class distance. CoDATS [35] and CLADA [36], as multi-source UDA methods
for TS data, applied DANN and 1D-CNN to handle the scenarios where multiple
source domains are available. Moreover, some researchers have specifically
designed UDA methods tailored to TS data. ADATIME [19] considered the temporal
properties of TS data and evaluated various temporal encoders for TS UDA.
Besides, AdvSKM [37] mapped features into a high-dimensional space by
proposing adversarial Spectral Kernels and then aligned them using MMD. To
consider the invariant associative structure between variables exhibited by
MTS data, SASA [16] aligned this structural information for TS UDA.

These works are great pioneers for TS UDA, but they may not perform well for
the scenarios requiring multiple sensors, i.e., MTS data. In MTS data, signals
from different sensors often follow various distributions due to their varied
physical locations and measurements. Existing TS UDA methods can be adapted to
MTS data by treating all sensors as a whole [38, 19, 15, 34], facilitate the
learning of global features for alignment, but they are limited in accounting
for the distinct distributions of sensors within MTS data. Meanwhile, they
ignored the multi-source nature of MTS data, restricting their ability to
model and transfer spatial-temporal dependency information across domains. To
address these limitations, we propose our SEA and SEA++ that accounts for
aligning sensor-level information and modelling dependency information within
MTS data to achieve better MTS-UDA.

Figure 2: The overall structure. (1) To transfer spatial-temporal dependencies
across domains, source and target domains share the same graph-based encoder,
including MSGC, GNN, and LSTM. Each sample is segmented as multiple patches,
which are constructed as sequential graphs. With the combination of GNN and
LSTM, decent sensor information is learned, including sensor features and
correlations. (2) In addition to supervised learning on the source domain,
endo-feature alignment and exo-feature alignment are designed to reduce domain
discrepancy at the local and global sensor levels. The endo-feature alignment
includes aligning sensor features and their correlations between domains by
SCA and SFA, both of which are enhanced by incorporating multi-graph alignment
and high-order statistic alignment. Exo-feature alignment aligns global
features mapped by sensor features.

##  III SEA Model

###  III-A Problem Definition

For MTS-UDA, we are given a source domain with $N_{s}$ labeled samples
$\mathcal{D}_{S}=\\{(x_{i}^{s},y_{i}^{s})\\}_{i=1}^{N_{s}}$ and a target
domain with $N_{t}$ unlabeled samples
$\mathcal{D}_{T}=\\{x_{i}^{t}\\}_{i=1}^{N_{t}}$. Each MTS sample $x_{i}$
(either $x_{i}^{s}$ or $x_{i}^{t}$) originates from $N$ sensors with different
distributions, i.e., $x_{i}=\\{x_{im}\\}_{m=1}^{N}\in\mathbb{R}^{N\times{L}}$,
where $L$ represents the time length. The goal of SEA is to train an encoder
by transferring the knowledge from the source domain to the target domain,
enabling the effective learning of features $h_{i}\in\mathbb{R}^{F}$ from the
MTS data $x_{i}^{t}$ in the target domain. The features can then be used for
downstream tasks such as RUL prediction and HAR. Notably, the SEA framework
consists of two stages. In the first stage, we focus on feature extraction
from MTS data. To simplify the notation and enhance clarity, we omit the index
$i$ in this stage. Thus, each sample is represented as $x^{s}$ and $x^{t}$,
and the data from the $m$-th sensor is denoted as $x_{m}^{s}$ and $x_{m}^{t}$.
While in the second stage that involves domain alignment, we retain the use of
the index to facilitate a clear description of the alignment process.

Furthermore, to model the evolving distributions within MTS data, we construct
sequential graphs
$\\{\mathcal{G}_{T}\\}_{T=1}^{\hat{L}},\mathcal{G}_{T}=(Z_{T},E_{T})$ from the
sample $x$, where $Z_{T}=\\{z_{m,T}\\}_{m=1}^{N}$ and
$E_{T}=\\{e_{mn,T}\\}_{m,n=1}^{N}$ represent the sensor features and
correlations, respectively, in the $T$-th graph.
$\\{\mathcal{G}^{s}_{T}\\}_{T=1}^{\hat{L}}$ and
$\\{\mathcal{G}^{t}_{T}\\}_{T=1}^{\hat{L}}$ represent the sequential graphs in
the source and target domains respectively.

###  III-B Preliminary

####  III-B1 Graph Neural Network

Several previous studies have demonstrated the effectiveness of Graph Neural
Network (GNN) in capturing spatial dependencies [39, 40]. Motivated by this
capability, we employ GNN to capture the spatial dependencies, i.e., sensor
correlations, within our constructed sequential graphs. Specifically, we
utilize a variant of GNN known as Message Passing Neural Network (MPNN) [41]
to process each graph. The MPNN framework consists of two stages: propagation
and updating. During the propagation stage, the features of neighboring nodes
are propagated to the central node. Subsequently, the aggregated features are
updated using a non-linear function, such as a Multi-Layer Perceptron (MLP)
network. With MPNN, we can capture the spatial dependencies between sensors,
enabling the effective transfer of the dependency information across domains
in the subsequent alignment processes. The details of MPNN are shown in Eq.
(1), where sensor $m$ is the central node, and $\mathcal{N}(m)$ is the set of
its neighboring nodes.

$\begin{split}{h_{m,T}}&=\sum_{j\in\mathcal{N}(m)}{e_{mj,T}}{z_{j,T}},\\\
{z_{m,T}}&=ReLU({h_{m,T}}W_{G}).\end{split}$ (1)

####  III-B2 Deep Coral

Deep Coral [13] is one of the most typical metric-based methods for UDA.
Different from $\mathcal{L}_{2}$ or cosine distances which mainly employ
first-order statistics, Deep Coral focuses on minimizing the discrepancy in
second-order statistics between domains [42], making it possible to capture
complex data distributions for alignment. Given training batches with $n^{s}$
and $n^{t}$ samples in source and target domains respectively, we denote the
extracted features from these domains as
$\mathcal{H}^{s}\in\mathbb{R}^{n^{s}\times{f}}$ and
$\mathcal{H}^{t}\in\mathbb{R}^{n^{t}\times{f}}$, where $f$ represents the
feature dimension. With the features, we can compute the discrepancy of
second-order statistics with Deep Coral
$M_{c}(\mathcal{H}^{s},\mathcal{H}^{t})$ as Eq. (2), where $||\cdot||_{F}^{2}$
is the squared matrix Frobenius norm. By minimizing the Deep Coral discrepancy
$\mathcal{L}_{Coral}$, the second-order statistics of the features are aligned
between the source and target domains, leading to comprehensive alignment
across domains.

$\begin{split}&C^{s}=\frac{(\mathcal{H}^{s})^{T}\mathcal{H}^{s}-\frac{1}{n^{s}}(\textbf{1}^{T}\mathcal{H}^{s})(\textbf{1}^{T}\mathcal{H}^{s})}{n^{s}-1},\\\
&C^{t}=\frac{(\mathcal{H}^{t})^{T}\mathcal{H}^{t}-\frac{1}{n^{t}}(\textbf{1}^{T}\mathcal{H}^{t})(\textbf{1}^{T}\mathcal{H}^{t})}{n^{t}-1},\\\
&M_{c}(\mathcal{H}^{s},\mathcal{H}^{t})=\frac{1}{4f^{2}}||C^{s}-C^{t}||_{F}^{2}.\end{split}$
(2)

###  III-C Overall Structure

The overall structure of SEA is shown in Fig. 2. The superscripts $s$ and $t$
are omitted in Section III-C and III-D, as the forward processes in source and
target domains share the same graph-based encoder, including Multi-branch
Self-attention based Graph Construction (MSGC), GNN, and LSTM. First, we model
the spatial-temporal dependencies within MTS data for effective transfer
across domains. Given the sample $x$, we segment it into multiple patches,
which are constructed as sequential graphs by MSGC. Then, with the sequential
graphs, GNN and LSTM are leveraged to capture the spatial-temporal information
for learning sensor-level information, i.e., sensor features and correlations.
Second, in addition to supervised learning on the source domain, we
incorporate endo-feature alignment and exo-feature alignment techniques to
reduce domain discrepancy at both the local and global sensor levels. At the
local sensor level, we design endo-feature alignment, which involves aligning
sensor features and sensor correlations by proposed Sensor Feature Alignment
(SFA) and Sensor Correlation Alignment (SCA) respectively. The two components
are further improved by both the high-order statistic alignment and multi-
graph alignment to form iSFA and iSCA. The high-order statistic alignment aims
to achieve comprehensive alignment by capturing complex data distributions
while the multi-graph alignment aims to align evolving distributions at the
sensor level across domains. At the global sensor level, we design exo-feature
alignment to align global features by enforcing restrictions. We introduce the
details of each module in subsequent sections.

###  III-D Graph-based Encoder

####  III-D1 Segmentation

To capture the evolving distributions within MTS data, we employ a
segmentation approach. In this approach, a sample
$x\in\mathbb{R}^{N\times{L}}$ is segmented into multiple patches, each with a
size of $d$, yielding sequential features denoted as
$\\{Z_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{N\times{\hat{L}\times{d}}}$. Here,
$\hat{L}=\lfloor{\frac{L}{d}}\rfloor$ represents the number of patches, and
$d$ represents the feature dimension. The features within the $T$-th patch
$Z_{T}\in\mathbb{R}^{N\times{d}}$ capture the distributions in local temporal
intervals. Within each patch, the features of the $m$-th sensor are denoted as
$z_{m,T}\in\mathbb{R}^{d}$. Additionally, we incorporate a nonlinear function
$f_{L}(\cdot)$ to enhance the nonlinear expressiveness of the model.

####  III-D2 MSGC

We construct sequential graphs with the sequential features. Traditionally,
various methods have been employed to construct graphs for MTS data, such as
dot-product[43], Euclidean distance[40], and concatenation [44]. However, the
traditional methods are insufficient to model comprehensive relationships of
sensors due to the complex topological structures presented within MTS data.
To solve the limitation, we propose MSGC, which constructs a graph by
combining multiple graphs generated from various weight initializations across
multiple branches. Each branch employs differently weighted matrices to model
sensor correlations from distinct perspectives, allowing us to capture
comprehensive sensor correlations. Specifically, we adopt query and key to
learn the edge weights between sensors. Eq. (3) describes the edge between
sensors $m$ and $n$ in the $i$-th branch.

$\begin{split}q_{m,T}^{i}&=z_{m,T}W_{Q}^{i},k_{n,T}^{i}=z_{n,T}W_{K}^{i},\\\
e_{mn,T}^{i}&=\frac{\frac{q_{m,T}^{i}(k_{n,T}^{i})^{T}}{\sqrt{d}}}{\sum_{j=1}^{N}\frac{q_{m,T}^{i}(k_{j,T}^{i})^{T}}{\sqrt{d}}},\\\
\end{split}$ (3)

where we treat the $m$-th sensor $z_{m,T}$ as the query sensor and the $n$-th
sensor $z_{n,T}$ as the key sensor, and use this information to learn the edge
weight between them for the $i$-th branch, i.e., $e_{mn,T}^{i}$. The edge
weights are then normalized using the softmax function to ensure they fall
within the range of [0, 1].

To combine the graphs from multiple branches, we calculate the average of the
corresponding elements, i.e.,
$e_{mn,T}=\sum_{i=1}^{n_{b}}\frac{e_{mn,T}^{i}}{n_{b}}$, where $n_{b}$
represents the number of branches. This operation allows us to obtain the
sensor correlations
$E_{T}=\\{e_{mn,T}\\}_{m,n=1}^{N}\in\mathbb{R}^{N\times{N}}$ for the $T$-th
graph. By conducting similar operations in other patches, we can finally
obtain the sequential correlations
$\\{E_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{\hat{L}\times{N}\times{N}}$.

Figure 3: Left: Sensor correlation alignment is to make the correlations
between domains identical. Middle: Sensor feature alignment is to make the
features from the corresponding sensors in two domains similar and the data
from the different sensors in two domains different. Right: Exo-feature
alignment reduces the discrepancy between the global features in different
domains.

####  III-D3 Capturing Spatial-Temporal Information

With the constructed sequential graphs, we employ two models, MPNN and LSTM,
to capture the spatial-temporal dependencies within MTS data. First, we
utilize MPNN (Eq. (1)) to capture the sensor correlations within each
sequential graph. In this process, the graphs are processed with shared
weights to save training costs and trainable parameters. Once the features of
each sensor in the sequential graphs have been updated using MPNN, our focus
shifts to capture temporal dependencies. It is noted that features from
corresponding sensors across the sequential graphs show temporal dependencies.
To capture these dependencies, we employ LSTM, which captures the dependency
information for each sensor by taking its features
$z_{m}\in\mathbb{R}^{{\hat{L}}\times{d}}$ across sequential graphs. After GNN
and LSTM, we finally obtain updated sequential features
$\\{Z_{T}\\}_{T=1}^{\hat{L}}\in\mathbb{R}^{\hat{L}\times{N}\times{d}}$. By
combining the capabilities of the two networks, we can effectively capture the
spatial-temporal dependencies within MTS data, allowing us to transfer the
dependency information across domains through the alignment at the local and
global sensor levels in the subsequent stages.

###  III-E SEnsor Alignment

To reduce the discrepancy between domains for MTS-UDA, we design endo-feature
alignment and exo-feature alignment at the local and global sensor levels
respectively.

####  III-E1 Endo-feature alignment

Existing UDA methods reduce domain discrepancy by aligning global features
across domains, which is applicable for the data coming from a single source.
However, MTS data originate from multiple sensors, with each sensor following
various distributions. When adapting existing UDA methods to MTS-UDA, they can
consider global distributions by treating all sensors as a whole yet ignore
the distinct distributions of each sensor. This results in the misalignment at
the sensor level, restricting the performance of the learned model
transferring from the source domain to the target domain. To address this
limitation, we propose an endo-feature alignment method designed to align
sensor-level information and avoid misalignment for each sensor when
transferring knowledge across domains.

To make better endo-feature alignment, it is necessary to note that the
sensor-level information consists of both the sensor features and sensor
correlations. Sensor features represent the properties of each sensor, while
sensor correlations represent the interactive information between sensors. For
illustration, we present a scenario with temperature and fan speed sensors
deployed together to monitor the status of a machine. It is desirable for the
data from temperature sensors in two domains to follow similar distributions.
Additionally, the correlation between the sensors should also demonstrate
consistent patterns across domains, such as the fan speed consistently
increasing with temperature rises. Based on these considerations, aligning
sensor features and sensor correlations across domains simultaneously is
important to ensure comprehensive alignment at the sensor level.

##### Sensor Correlation Alignment

To explore consistent patterns within the interactions of sensors across
domains, we propose to align the sensor correlations. Specifically, our aim is
to align the distribution of correlations between any two sensors across
domains. To achieve this, we propose sensor correlation alignment (SCA), which
involves individually aligning each edge of the graphs as shown in Fig. 3
Left.

Given training batches with $n^{s}$ and $n^{t}$ samples in source and target
domains, we represent sensor correlations across sequnetial graphs as
$\mathcal{E}^{s}\in\mathbb{R}^{n^{s}\times{\hat{L}}\times{N}\times{N}}$ and
$\mathcal{E}^{t}\in\mathbb{R}^{n^{t}\times{\hat{L}}\times{N}\times{N}}$
respectively, with
$\mathcal{E}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{N}}$ and
$\mathcal{E}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{N}}$ signifying sensor
correlations for the $T$-th graph. Here,
$\mathcal{E}_{T}^{s}=\\{\mathcal{E}_{mn,T}^{s}\\}_{m,n=1}^{N}$, wherein
$\mathcal{E}_{mn,T}^{s}=\\{e^{s}_{i,mn,T}\\}_{i}^{n^{s}}$. For SCA, alignment
of sensor correlations across domains is achieved based on averaged sensor
correlations. Specifically, for the edge connecting sensors $m$ and $n$, its
mean value across the training batch is computed as
$e^{s}_{mn,T}=\sum_{i}^{n^{s}}e^{s}_{i,mn,T}/n^{s}$. Subsequently, the
averaged sensor correlations over sequential graphs are calculated as
$e^{s}_{mn}=\sum_{T}^{\hat{L}}e^{s}_{mn,T}/{\hat{L}}$. Next, we minimize the
averaged sensor correlations across domains as shown in Eq. (4), where
$|\cdot|$ is absolute value. This involves minimizing the expected discrepancy
of sensor correlations between the source and target domains. By doing so, we
can reduce distribution discrepancies for sensor correlations in source and
target domains.

$\begin{split}e_{mn}^{st}&=e_{mn}^{s}-e_{mn}^{t},\\\
\mathcal{L}_{SCA}&=\mathbb{E}(|e_{mn}^{st}|).\end{split}$ (4)

##### Sensor Feature Alignment

Next, we focus on aligning sensor features between domains. Our assumption is
that features originating from corresponding sensors in two domains should
exhibit similar distributions. Meanwhile, compared with features from
corresponding sensors, features from different sensors in two domains should
have distinct properties. For example, as shown in Fig. 3 Middle, the
distribution of sensor 1 in the source domain should be more similar to the
distribution of its corresponding sensor in the target domain than to the
distributions of other sensors (e.g., sensors 2, 3, and 4) in the target
domain. To achieve this goal, we propose a sensor contrasting mechanism that
facilitates the alignment of sensor features between domains.

Given training batches with $n^{s}$ and $n^{t}$ samples, we define sensor
features as
$\mathcal{Z}^{s}\in\mathbb{R}^{n^{s}\times{\hat{L}}\times{N}\times{d}}$ and
$\mathcal{Z}^{t}\in\mathbb{R}^{n^{t}\times{\hat{L}}\times{N}\times{d}}$, with
$\mathcal{Z}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{d}}$ and
$\mathcal{Z}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{d}}$ denoting sensor
features for the $T$-th graph. Here,
$\mathcal{Z}_{T}^{s}=\\{\mathcal{Z}_{n,T}^{s}\\}_{n=1}^{N}$, wherein
$\mathcal{Z}_{n,T}^{s}=\\{z^{s}_{i,n,T}\\}_{i}^{n^{s}}$. In SFA, domain
contrasting is realized based on averaged sensor features. In Eq. (5),
$p^{s}_{m}$ represents the averaged sensor features over multiple graphs
within a training batch, calulated as
$p^{s}_{m}=\sum_{T}^{\hat{L}}\sum_{i}^{n^{s}}z^{s}_{i,n,T}/n^{s}\hat{L}$.

$\begin{split}\mathcal{L}_{SFA}&=-\frac{1}{N}\sum_{n}^{N}log\frac{e^{\varphi(p_{m}^{s},p_{m}^{t})}}{\sum_{p_{j}^{t}\in{P^{t}}}e^{\varphi(p_{m}^{s},p_{j}^{t})}},\\\
\varphi(p_{m}^{s},p_{m}^{t})&=p_{m}^{s}(p_{m}^{t})^{T}.\end{split}$ (5)

##### Exo-feature alignment.

To reduce the domain discrepancy at the global level, we enforce restrictions
on the global features mapped by sensor features between domains.
Specifically, we first need to learn the global features of each sample. This
involves stacking all sensor features across the sequential graphs and
subsequently employing an MLP network to map the stacked features as global
features $\mathcal{P}^{s}$ and $\mathcal{P}^{t}$ in source and target domains,
as shown in Fig. 3 Right. With the global features, we can then align them
with Deep Coral, expressed as
$\mathcal{L}=\mathcal{M}_{c}(\mathcal{P}^{s},\mathcal{P}^{t})$. Finally, the
overall loss function (6) is minimized:

$\begin{split}\min\mathcal{L}=\mathcal{L}_{C}+\mathcal{L}_{EXO}+\mathcal{L}_{Endo},\\\
\mathcal{L}_{Endo}=\lambda_{SCA}\mathcal{L}_{SCA}+\lambda_{SFA}\mathcal{L}_{SFA}.\end{split}$
(6)

where $\mathcal{L}_{C}$ is the cost function (such as mean square error loss
or cross-entropy loss, depending on specific tasks) computed by the source
data and labels, and $\lambda_{SCA}$ and $\lambda_{SFA}$ are hyperparameters
to balance the effect of SCA and SFA respectively.

##  IV SEA++ Model

While the method described above offers a viable approach for achieving sensor
correlation and sensor feature alignment, there is still room for improvement
to enhance endo-feature alignment. Two key directions warrant consideration.
First, SEA employs first-order statistics for alignment, such as averaged
sensor features over a training batch used in the sensor contrastive
mechanism. However, the first-order statistical alignment method is limited in
dealing with scenarios with complex data distributions. This limitation may
restrict the ability to achieve comprehensive alignment across domains,
leading to suboptimal performance in certain scenarios. To address that, we
propose introducing second-order statistical metrics for alignment to achieve
more comprehensive endo-feature alignment.

Second, the constructed sequential graphs, comprising sequential features and
correlations, capture evolving distributions within local temporal intervals
and should be aligned across domains. We consider the example of monitoring
machine health status for illustration. As the machine’s health deteriorates,
the distribution of a fan speed sensor may change due to increased friction or
other factors. Similarly, the interaction between the fan speed sensor and a
temperature sensor will also be affected and undergo changes. Specifically,
the same fan speeds might lead to higher temperature increases due to a
machine in poor health, e.g., friction increasing. Thus, we emphasize the
importance of aligning the evolving distributions at the sensor level between
domains. However, SEA employs averaged sensor correlations and sensor features
over the sequential graphs, limiting its ability to capture the evolving
distributions for alignment. To improve this, we introduce a multi-graph
alignment technique, aiming to effectively align evolving distributions
between domains.

Next, we begin by discussing SCA and SFA enhanced by high-order statistical
alignment within each sequential graph. Subsequently, we focus on combining
the alignment across all sequential graphs by employing our multi-graph
alignment.

###  IV-A Sensor Correlation Alignment

Given training batches with $n^{s}$ and $n^{t}$ samples in source and target
domains, we have sensor correlations for the $T$-th graph, denoted as
$\mathcal{E}_{T}^{s}\in\mathbb{R}^{n^{s}\times{N}\times{N}}$ and
$\mathcal{E}_{T}^{t}\in\mathbb{R}^{n^{t}\times{N}\times{N}}$ respectively.
Instead of leveraging averaged sensor correlations within batches, we
introduce second-order statistics with Deep Coral, i.e., $M_{c}$ in Eq. (2),
to align the sensor correlations between the source and target domains.
Building upon this idea, we enhance SCA and introduce improved SCA (iSCA) as
shown in Eq. (7).

$\begin{split}d_{mn,T}^{st}&=M_{c}(\mathcal{E}_{mn,T}^{s},\mathcal{E}_{mn,T}^{t}),\\\
\mathcal{L}_{iSCA,T}&=\mathbb{E}(d_{mn,T}^{st})=\frac{\sum_{m}^{N}\sum_{n}^{N}d_{mn,T}^{st}}{mn}.\end{split}$
(7)

By aligning each edge between domains with the second-order statistical metric
and minimizing the distribution discrepancy of sensor correlations, we can
achieve more comprehensive alignment for sensor correlations.

###  IV-B Sensor Feature Alignment

Next, we focus on improving sensor feature alignment. Similar to SCA, the
sensor contrasting mechanism in SFA is realized with the averaged sensor
features over the training batches in source and target domains. Thus, SFA
encounters a challenge similar to SCA. Specifically, the contrasting-based
alignment procedure in Eq. (5) relies on first-order statistics, which might
not achieve comprehensive alignment when dealing with complex distributions.
To overcome this limitation, we enhance SFA and introduce improved SFA (iSFA)
by incorporating Deep Coral into the contrasting process.

In Eq. (8), iSFA begins by calculating sensor-level distribution discrepancies
using Deep Coral, which takes into account second-order statistics.
Subsequently, iSFA leverages the sensor-level discrepancies to perform
contrasting between domains. This involves minimizing the distribution
discrepancies between corresponding sensors across domains while maximizing
the distribution discrepancies between different sensors across domains. This
approach allows for more comprehensive alignment of sensor features.

$\begin{split}\mathcal{L}_{iSFA,T}=-\frac{1}{N}\sum_{n}^{N}log\frac{e^{M_{c}(\mathcal{Z}_{n,T}^{s},\mathcal{Z}_{n,T}^{t}})}{\sum^{N}_{j}e^{M_{c}(\mathcal{Z}_{n,T}^{s},\mathcal{Z}_{j,T}^{t})}}.\end{split}$
(8)

###  IV-C Multi-Graph Alignment

To combine the alignment for sensor correlations and sensor features across
all sequential graph, a straightforward approach would involve computing the
average alignment for these graphs. However, the distributions between domains
for different graphs might exhibit various discrepancies. To account for this
and enhance domain adaptation, we propose a method to learn adaptive weights
for adjustment, as shown in Eq. (9).

$\begin{split}\mathcal{L}_{iEndo}&=\sum_{T}\mathcal{W}_{T}(\lambda_{SCA}\mathcal{L}_{iSCA,T}+\lambda_{SFA}\mathcal{L}_{iSFA,T}),\\\
\mathcal{W}_{T}&=M_{c}([\mathcal{Z}_{1,T}^{s},...,\mathcal{Z}_{\hat{L},T}^{s}],[\mathcal{Z}_{1,T}^{t},...,\mathcal{Z}_{\hat{L},T}^{t}]),\end{split}$
(9)

where $[a,...,b]$ represents concatenating the elements from $a$ to $b$. In
Eq. (9), the learnable weights $\mathcal{W}_{T}$ are determined by evaluating
the distribution discrepancy of the $T$-th graph between domains.
Specifically, $\mathcal{W}_{T}$ will be large when there is a substantial
discrepancy in the distributions of the $T$-th graph between the source and
target domains. By assigning larger weights to the graphs with larger
distribution discrepancies between domains, we prioritize alignment in those
graphs. This adaptive weighting scheme allows us to better adapt the model to
the evolving distributions between domains. Furthermore, we also introduce
$\lambda_{SCA}$ and $\lambda_{SFA}$ to balance the effect of iSCA and iSFA. By
combining the exo-feature alignment and the improved endo-feature alignment,
the improved overall loss function (10) can be obtained as:

$\begin{split}\min\mathcal{L}&=\mathcal{L}_{C}+\mathcal{L}_{EXO}+\mathcal{L}_{iEndo}.\end{split}$
(10)

##  V Experiment

###  V-A Datasets and Setup

To evaluate the performance of SEA and SEA++ under various MTS scenarios, we
conducted experiments on two public datasets: C-MAPSS for remaining useful
life prediction [45], which collects MTS data from different sensor types, and
Opportunity HAR for human activity recognition [46], which collects MTS data
from different locations.

C-MAPSS describes the degradation process of aircraft engines. The MTS data in
C-MAPSS were originated from 14 sensors to measure various physical
parameters, such as temperature, fan speed, and pressure. The dataset includes
four sub-datasets collected under different working conditions and fault
modes, where each sub-dataset represents one domain. We processed the sub-
datasets following the data preparation in the previous work [47], and data
annotations represent the remaining useful life circle of engines.

Opportunity HAR describes human activities. The MTS data in Opportunity HAR
were originated from 113 sensors deployed to various locations of the human
body, such as hand and leg. After removing the sensors with constant values,
110 sensors were used for evaluation. The dataset includes the data collected
from four subjects, each representing one domain. The data annotations include
two levels: 1). locomotion representing the low-level tasks including four
categories, sitting, standing, walking, and lying down; 2). gestures
representing the high-level tasks including 17 various actions. Following the
experimental settings in the previous work [48], we adopted the low-level
tasks. As some values in the data were missing, we adopted the linear
interpolation approach to fill in the missing positions. To construct the
training dataset, we adopted a sliding window with a size of 128 and an
overlapping of 50% following the previous work [48].

The experiments include three parts, comparisons with state-of-the-art
methods, the ablation study, and sensitivity analysis. All experiments were
run ten times with average results reported to remove the effect of random
initialization. Besides, we set the batch size as 50 and employed the
optimizer as Adam with a learning rate of 0.001. Meanwhile, we adopted 20
training epochs for training our model. Furthermore, we built and trained our
model based on Pytorch 1.9 and NVIDIA GeForce RTX 3080Ti GPU.

We employed different evaluation indicators to assess the performance of our
method on the two distinct datasets. For C-MAPSS experiments, which involve
predicting the RUL of an engine, we utilized Root Mean Square Error (RMSE) and
the Score function [47, 49, 50, 51] as evaluation metrics. Lower values for
both indicators indicate better model performance. For Opportunity HAR
experiments, which entail a classification task, we employed accuracy as the
evaluation metric. High accuracy values signify superior model performance.

###  V-B Comparisons with State-of-the-Art

TABLE I: The Comparisons with SOTAs in C-MAPSS (R: RMSE; S: Score) Models (R) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Target | 12.93 | 13.33 | 14.66 | 12.36 | 13.33 | 14.66 | 12.36 | 12.93 | 14.66 | 12.36 | 12.93 | 13.33 | 13.32
Source | 28.76 | 35.28 | 27.21 | 21.86 | 39.80 | 33.50 | 37.90 | 30.33 | 24.10 | 33.76 | 25.31 | 21.75 | 29.96
DDC | 43.25 | 39.48 | 42.99 | 40.07 | 39.46 | 43.01 | 40.93 | 43.43 | 43.72 | 41.58 | 43.38 | 39.61 | 41.74
Coral | 16.72 | 26.49 | 25.03 | 14.47 | 37.41 | 31.13 | 32.82 | 23.40 | 19.84 | 28.37 | 32.00 | 22.47 | 25.85
DANN | 17.02 | 23.33 | 24.20 | 14.16 | 30.00 | 25.91 | 19.89 | 20.31 | 19.95 | 31.74 | 24.68 | 17.94 | 22.43
BNM | 48.18 | 41.61 | 46.45 | 56.66 | 48.40 | 62.02 | 49.39 | 48.89 | 42.98 | 40.76 | 43.71 | 41.89 | 47.58
SDAT | 16.50 | 24.11 | 23.07 | 13.57 | 29.76 | 26.38 | 25.34 | 20.29 | 18.63 | 20.13 | 23.23 | 17.65 | 21.55
AdvSKM | 31.09 | 34.16 | 40.74 | 23.93 | 32.41 | 35.73 | 28.47 | 22.17 | 26.45 | 28.56 | 22.75 | 23.51 | 29.16
CoDATs | 15.91 | 26.67 | 25.24 | 15.59 | 24.52 | 25.64 | 21.25 | 21.04 | 19.77 | 19.41 | 18.99 | 17.85 | 20.99
CtsADA | 18.31 | 29.23 | 29.49 | 23.20 | 54.69 | 41.60 | 38.14 | 24.67 | 19.86 | 31.63 | 25.71 | 23.97 | 30.04
CLUDA | 25.31 | 36.80 | 33.24 | 22.51 | 33.82 | 34.63 | 26.48 | 24.09 | 26.29 | 24.33 | 25.14 | 22.18 | 27.90
SEA | 15.63 | 21.09 | 23.21 | 13.29 | 24.62 | 25.32 | 19.69 | 19.44 | 19.37 | 18.92 | 17.70 | 16.45 | 19.56
SEA++ | 15.41 | 19.38 | 22.12 | 13.00 | 24.60 | 24.90 | 19.19 | 19.25 | 17.75 | 18.43 | 18.16 | 16.79 | 19.08
Models (S) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
Target | 678 | 330 | 1027 | 241 | 330 | 1027 | 241 | 678 | 1027 | 241 | 678 | 330 | 569
Source | 9341 | 4751 | 4699 | 2340 | 8335 | 11018 | 24892 | 26878 | 11006 | 16318 | 8771 | 2047 | 10866
DDC | 23323 | 6164 | 34891 | 11581 | 6578 | 35122 | 22236 | 59400 | 80774 | 29592 | 58029 | 15239 | 31911
Coral | 1158 | 1468 | 3613 | 406 | 6241 | 8492 | 15298 | 12661 | 3048 | 6742 | 25763 | 4758 | 7471
DANN | 1073 | 1612 | 3340 | 392 | 3160 | 4667 | 1666 | 3896 | 2381 | 22225 | 4620 | 762 | 4150
BNM | 24265 | 44650 | 19017 | 526203 | 56769 | 29689 | 46262 | 29188 | 39559 | 11618 | 20566 | 65807 | 76133
SDAT | 974 | 1403 | 2729 | 300 | 4360 | 3859 | 3756 | 3134 | 2289 | 966 | 2369 | 678 | 2235
AdvSKM | 13313 | 5012 | 18444 | 2344 | 4406 | 14624 | 6601 | 4516 | 11857 | 6332 | 4474 | 5218 | 8095
CoDATs | 1043 | 2854 | 5074 | 836 | 1533 | 5094 | 1300 | 3993 | 2352 | 1065 | 2205 | 697 | 2337
CtsADA | 1467 | 1965 | 6832 | 4136 | 29153 | 22515 | 47682 | 17418 | 2138 | 13019 | 15436 | 1593 | 13613
CLUDA | 2686 | 4754 | 6337 | 873 | 3501 | 7159 | 2389 | 3232 | 3078 | 1658 | 5566 | 1093 | 3527
SEA | 922 | 1302 | 3786 | 287 | 2090 | 3782 | 1163 | 2990 | 2278 | 939 | 1930 | 635 | 1842
SEA++ | 881 | 803 | 2669 | 295 | 1510 | 3220 | 1117 | 3045 | 1558 | 859 | 2072 | 572 | 1550
TABLE II: The Comparisons with SOTAs in Opportunity HAR Models | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Target | 91.90 | 97.95 | 95.60 | 98.40 | 97.95 | 95.60 | 98.40 | 91.90 | 95.60 | 98.40 | 91.90 | 97.95 | 95.96
Source | 47.50 | 64.50 | 60.83 | 73.00 | 71.00 | 65.67 | 64.83 | 56.67 | 72.17 | 66.67 | 65.50 | 74.67 | 65.25
DDC | 40.50 | 58.50 | 55.50 | 43.35 | 58.30 | 55.50 | 45.00 | 40.50 | 55.50 | 45.00 | 40.50 | 58.50 | 49.72
Coral | 78.55 | 82.71 | 78.17 | 81.58 | 79.93 | 79.07 | 78.50 | 75.40 | 84.88 | 78.56 | 80.38 | 87.88 | 80.47
DANN | 69.62 | 85.12 | 79.00 | 85.62 | 81.62 | 74.88 | 76.50 | 75.75 | 86.12 | 84.25 | 74.12 | 86.62 | 79.94
BNM | 44.00 | 61.00 | 52.00 | 45.00 | 61.00 | 52.00 | 45.00 | 44.00 | 52.00 | 45.50 | 44.00 | 61.00 | 50.54
LMMD | 56.25 | 61.17 | 59.08 | 45.33 | 58.50 | 55.50 | 49.42 | 46.83 | 57.33 | 59.00 | 44.75 | 61.75 | 54.58
SDAT | 84.92 | 72.42 | 77.00 | 87.25 | 83.00 | 76.17 | 87.75 | 84.25 | 87.25 | 74.58 | 66.75 | 91.50 | 81.07
AdvSKM | 58.10 | 58.25 | 51.75 | 45.40 | 40.95 | 48.40 | 69.60 | 72.20 | 70.60 | 68.75 | 55.50 | 58.20 | 58.14
CoDATs | 80.17 | 83.33 | 82.33 | 88.67 | 80.50 | 81.00 | 89.25 | 79.12 | 85.75 | 84.67 | 79.17 | 86.17 | 83.34
CtsADA | 80.00 | 73.67 | 76.08 | 69.75 | 78.17 | 79.08 | 67.75 | 77.33 | 64.75 | 74.58 | 52.17 | 75.00 | 72.36
SLARDA | 80.38 | 80.50 | 79.50 | 82.75 | 78.88 | 79.25 | 75.88 | 78.62 | 70.12 | 80.62 | 84.00 | 77.25 | 78.98
CLUDA | 70.45 | 73.75 | 74.40 | 73.20 | 74.25 | 73.95 | 61.65 | 55.10 | 64.30 | 65.75 | 61.05 | 71.55 | 68.28
SEA | 86.35 | 86.15 | 86.00 | 86.35 | 84.35 | 81.75 | 87.25 | 83.05 | 88.70 | 84.90 | 84.25 | 92.25 | 85.42
SEA++ | 86.85 | 90.30 | 90.85 | 90.25 | 83.85 | 85.40 | 89.80 | 83.10 | 88.10 | 91.70 | 86.95 | 92.55 | 88.31

We compared SEA and SEA++ with State-Of-The-Art methods (SOTAs), including
general UDA methods and TS UDA methods. General UDA methods were designed to
be applicable to various tasks and domains, regardless of the specific data
type, including DDC [9], Deep Coral [52], DANN [53], BNM [54], LMMD [55], and
SDAT [56]. TS UDA methods were specifically designed for TS data, including
AdvSKM [37], CoDATs [35], CtsADA [47], SLARDA [48], and CLUDA [57]. For fair
comparisons, all methods were run ten times based on the same feature
extractor (i.e., GNN-LSTM). LMMD and SLARDA were designed specifically for
multi-class classification UDA, so they only had results on Opportunity HAR.
Notably, in the previous SEA paper, different cross-domain scenarios were
based on various sample lengths. For example, the cross-domain scenario
1$\to$2 was based on a time length of 72 while 3$\to$4 was based on a time
length of 45. However, this training setting may be inapplicable in real-world
systems where obtaining samples with a fixed time length is desired. Thus, all
results in this paper, particularly in the context of the C-MAPSS dataset,
have been re-conducted based on a consistent time length of 60 timestamps.
Additionally, we have included the results of Target-only (Target) and Source-
only (Source) baselines for comparison. The target-only baseline represents
the upper bound of UDA, where training and testing were performed on the
target domain only. The source-only baseline represents the lower bound of
UDA, where only the source data were used for training without any UDA methods
applied.

TABLE I shows the RMSE and Score results for 12 cross-domain scenarios in
C-MAPSS. From the results, we observe that both SEA and SEA++ achieve better
performance than SOTAs in most cross-domain scenarios. Specifically, compared
to the methods excluding SEA++, SEA performs the best performance in 9 out of
12 scenarios and the second-best performance in the remaining scenarios with
very marginal gaps compared to the best methods. We utilize the RMSE results
in C-MAPSS for illustration. In the cross-domain scenario 1$\to$3, SEA
outperforms the second-best method by a significant 9.6%. In scenario 1$\to$4,
SEA is the second-best method and is only 0.6% weaker than the top-performing
method. These results demonstrate the effectiveness of SEA compared to SOTAs.
However, SEA exhibits minor limitations that make it slightly less effective
in specific cases, motivating the development of SEA++.

Regarding the results of SEA++, it not only achieves better performance in
scenarios where SEA performs the best but also achieves the best performance
in 2 out of 3 scenarios where SEA is the second-best. For example, in the
cross-domain scenario 1$\to$4 where SEA is not the top-performing, SEA++
improves by 4.1% compared to the best conventional method. Although SEA++ is
slightly weak in 2$\to$3, the gap is further reduced, being only 0.3% weaker
than the best method. In terms of average RMSE results, SEA outperforms the
best conventional methods by 9.2%. With improved endo-feature alignment and
multi-graph alignment, SEA++ further enhances the performance, surpassing SEA
by 2.45% and outperforming the best conventional method by 11.4%. Similar
improvements can be observed in the Score results of C-MAPSS and Opportunity
HAR in TABLE II. Taking the average accuracy in Opportunity HAR for
illustration, SEA improves by 2.08% compared to the best conventional method.
Additionally, SEA++ outperforms SEA by 2.89% and outperforms the best
conventional method by 4.97%.

These results indicate the effectiveness of SEA, showing the necessity of
aligning sensor-level information. Furthermore, SEA++ addresses the minor
limitations of SEA, resulting in improved performance that establishes its
state-of-the-art status compared to traditional methods.

###  V-C Ablation Study

TABLE III: The Ablation Study in C-MAPSS (R: RMSE; S: Score)

Variants (R) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
w/ Exo (w/o endo) | 16.96 | 27.74 | 28.28 | 14.47 | 33.70 | 28.18 | 32.88 | 25.45 | 19.98 | 30.75 | 23.19 | 20.03 | 25.13
w/ Exo + SCA | 15.96 | 23.83 | 25.40 | 14.12 | 27.32 | 27.13 | 26.89 | 23.97 | 19.87 | 25.94 | 20.62 | 17.97 | 22.42
w/ Exo + SCA + MGA | 15.96 | 21.95 | 22.68 | 13.65 | 28.93 | 26.31 | 27.98 | 21.64 | 18.70 | 26.15 | 20.31 | 17.33 | 21.80
w/ Exo + SCA + 2-order | 15.94 | 21.61 | 23.05 | 13.32 | 28.76 | 26.94 | 23.60 | 20.73 | 18.41 | 22.40 | 19.85 | 17.25 | 20.99
w/ Exo + iSCA | 15.90 | 21.39 | 22.62 | 13.28 | 27.91 | 26.08 | 22.01 | 19.86 | 18.30 | 22.37 | 19.55 | 17.18 | 20.54
w/ Exo + SFA | 16.71 | 22.03 | 24.96 | 14.03 | 29.01 | 26.15 | 26.71 | 21.28 | 19.71 | 25.97 | 19.63 | 17.30 | 21.96
w/ Exo + SFA + MGA | 16.11 | 21.61 | 22.47 | 13.75 | 28.97 | 26.05 | 27.35 | 21.20 | 18.97 | 23.67 | 19.58 | 17.08 | 21.40
w/ Exo + SFA + 2-order | 16.15 | 22.08 | 22.57 | 13.77 | 28.51 | 26.45 | 23.99 | 20.67 | 18.19 | 22.57 | 19.87 | 17.85 | 21.06
w/ Exo + iSFA | 15.90 | 20.37 | 22.47 | 13.59 | 27.80 | 25.34 | 21.44 | 19.95 | 18.10 | 20.54 | 19.30 | 17.28 | 20.17
SEA++ | 15.41 | 19.38 | 22.12 | 13.00 | 24.60 | 24.90 | 19.19 | 19.25 | 17.75 | 18.43 | 18.16 | 16.79 | 19.08
Variants (S) | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
w/ Exo (w/o endo) | 1156 | 1863 | 7246 | 406 | 3705 | 5160 | 17374 | 17854 | 2240 | 10247 | 6635 | 1224 | 6259
w/ Exo + SCA | 1101 | 1537 | 6392 | 360 | 2358 | 4657 | 11756 | 7351 | 2458 | 4082 | 3925 | 735 | 3893
w/ Exo + SCA + MGA | 980 | 979 | 2936 | 356 | 2255 | 3893 | 9340 | 7284 | 1974 | 4052 | 3652 | 664 | 3197
w/ Exo + SCA + 2-order | 995 | 1070 | 2838 | 328 | 2517 | 4287 | 3489 | 5954 | 1820 | 1976 | 3207 | 696 | 2431
w/ Exo + iSCA | 978 | 1045 | 2674 | 306 | 2076 | 3704 | 2270 | 3906 | 1571 | 1815 | 3084 | 679 | 2009
w/ Exo + SFA | 1052 | 1445 | 4347 | 368 | 2152 | 3855 | 6638 | 7266 | 2111 | 4406 | 3082 | 654 | 3115
w/ Exo + SFA + MGA | 949 | 1194 | 2634 | 364 | 1873 | 3697 | 6193 | 6023 | 1915 | 2443 | 3048 | 627 | 2580
w/ Exo + SFA + 2-order | 987 | 1099 | 2674 | 367 | 2233 | 4214 | 3654 | 7640 | 1621 | 2212 | 3208 | 707 | 2551
w/ Exo + iSFA | 946 | 1083 | 2634 | 342 | 1873 | 3598 | 1902 | 4515 | 1694 | 1395 | 2814 | 630 | 1952
SEA++ | 881 | 803 | 2669 | 295 | 1510 | 3220 | 1117 | 3045 | 1558 | 859 | 2072 | 572 | 1550

  * •

**MGA represents Multi-Graph Alignment; 2-order represents high-order
statistical alignment.

TABLE IV: The Ablation Study in Opportunity HAR Variants | 1$\to$2 | 1$\to$3 | 1$\to$4 | 2$\to$1 | 2$\to$3 | 2$\to$4 | 3$\to$1 | 3$\to$2 | 3$\to$4 | 4$\to$1 | 4$\to$2 | 4$\to$3 | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---
w/ Exo (w/o endo) | 78.55 | 82.71 | 78.17 | 81.58 | 79.93 | 79.07 | 78.50 | 75.40 | 84.88 | 78.56 | 80.38 | 87.88 | 80.47
w/ Exo + SCA | 84.45 | 84.90 | 82.20 | 84.00 | 81.95 | 81.00 | 81.50 | 75.81 | 86.95 | 81.80 | 81.60 | 87.35 | 82.79
w/ Exo + SCA + MGA | 85.00 | 86.75 | 84.05 | 88.25 | 82.60 | 82.20 | 83.25 | 79.75 | 85.80 | 83.90 | 81.85 | 88.35 | 84.31
w/ Exo + SCA + 2-order | 85.20 | 87.40 | 84.90 | 88.35 | 82.30 | 82.00 | 81.10 | 76.75 | 87.15 | 81.35 | 84.40 | 87.30 | 84.02
w/ Exo + iSCA | 85.70 | 87.20 | 88.20 | 89.10 | 81.30 | 84.10 | 86.75 | 79.15 | 88.05 | 89.80 | 85.50 | 87.50 | 86.03
w/ Exo + SFA | 82.94 | 83.62 | 80.38 | 84.25 | 83.45 | 80.55 | 78.80 | 75.38 | 86.20 | 79.05 | 81.90 | 87.50 | 82.00
w/ Exo + SFA + MGA | 85.95 | 87.60 | 85.75 | 87.05 | 82.75 | 82.90 | 79.60 | 79.05 | 87.35 | 83.50 | 79.95 | 88.90 | 84.20
w/ Exo + SFA + 2-order | 84.50 | 87.40 | 84.25 | 87.00 | 84.90 | 82.45 | 82.15 | 80.95 | 87.75 | 85.45 | 83.60 | 88.45 | 84.90
w/ Exo + iSFA | 85.70 | 87.90 | 85.85 | 87.90 | 83.70 | 82.90 | 86.45 | 82.05 | 86.90 | 87.10 | 84.75 | 90.15 | 85.95
SEA++ | 86.85 | 90.30 | 90.85 | 90.25 | 83.85 | 85.40 | 89.80 | 81.50 | 88.10 | 91.70 | 86.95 | 92.55 | 88.18

To comprehensively evaluate the effectiveness of our endo-feature alignment
and the associated improvements, we have conducted extensive ablation studies
consisting of nine different variants. These variants can be categorized into
three aspects. The first aspect compares our method with a baseline that only
performs exo-feature alignment without utilizing any sensor-level alignment.
In the second and third aspects, we consider variants that combine exo-feature
alignment with one of the endo-feature alignment techniques: SCA or SFA. Each
aspect further includes four variants based on different improvements. We
utilize the variants with SCA for illustration. The “w/ Exo + SCA” variant
represents the use of the vanilla SCA for sensor correlation alignment. By
introducing Multi-Graph Alignment (MGA), we have the “w/ Exo + SCA + MGA”
variant, while still using alignment with first-order statistics. By
introducing alignment with second-order statistics without MGA, we have the
variant “w/ Exo + SCA + 2-order”. Finally, by incorporating both improvements,
i.e., MGA and second-order statistic alignment, we obtain the “w/ Exo + iSCA”
variant. The same variations were applied for SFA, resulting in four
additional variants.

TABLE III and TABLE IV show the ablation study on C-MAPSS and Opportunity HAR
respectively. Focusing on the average RMSE results on C-MAPSS for
illustration, we consider the variant “w/ Exo” as the baseline. We can first
observe that introducing SCA and SFA leads to improvements of 10.8% and 12.6%,
respectively, which indicates the effectiveness of endo-feature alignment. As
the endo-feature alignment still has minor limitations, we further introduce
MGA and second-order statistic alignment to improve the sensor-level
alignment. Introducing MGA leads to performance improvements of 13.3% (SCA)
and 14.9% (SFA), while introducing second-order statistic alignment improves
it by 16.5% (SCA) and 16.2% (SFA). The improvements indicate that MGA and
second-order statistic alignment both show effectiveness for better sensor-
level alignment. Thus, it is useful to combine them to further improve the
performance of our method. Combining both parts, the variants “w/ iSCA” and
“w/ iSFA” improve the performance by 18.3% and 19.7% respectively compared to
the baseline. Finally, both the sensor features and sensor correlations can be
aligned simultaneously. Thus, by combining iSCA and iSFA for endo-feature
alignment, we achieve a remarkable improvement of 24.1% compared to the
baseline. Similar trends are also observed in the experiments conducted on
Opportunity HAR, as shown in TABLE IV. For example, the average accuracy of
SEA++ increases by 7.61% compared to the model with exo-feature alignment
only. The results in the cross-domain scenarios of Opportunity HAR can further
verify the effectiveness of our proposed modules.

These experiments highlight the significance of endo-feature alignment in
achieving effective alignment at the sensor level for MTS-UDA. Furthermore,
the additional modules for endo-feature alignment, namely MGA and high-order
statistics alignment, prove to be highly effective in further improving our
model’s performance.

###  V-D Sensitivity Analysis

Our method includes hyperparameters that require sensitivity analysis to
evaluate their impact on the method’s performance. We focus on four
hyperparameters: $\lambda_{SCA}$, $\lambda_{SFA}$, number of heads, and patch
sizes. The first two hyperparameters control the balance between sensor
correlation alignment and sensor feature alignment. The number of heads
determines the number of graph branches used to model spatial dependencies
between sensors. The patch sizes determine the number of sequential graphs
constructed. Conducting sensitivity analysis on these hyperparameters helps
determine their sensitivity and find optimal values. Notably, the sensitivity
analysis presented here pertains specifically to SEA++. Additional details
regarding the analysis related to SEA can be found in our appendix.

####  V-D1 Analysis for $\lambda_{SCA}$ and $\lambda_{SFA}$

Figure 4: The sensitivity analysis for $\lambda_{SCA}$ on C-MAPSS. Figure 5:
The sensitivity analysis for $\lambda_{SFA}$ on C-MAPSS. Figure 6: The
sensitivity analysis for $\lambda_{SCA}$ and $\lambda_{SFA}$ on Opportunity
HAR.

We have conducted experiments with various values of $\lambda_{SCA}$ and
$\lambda_{SFA}$ varying from 0.001 to 10 with an interval of $10\times$. Fig.
4 and 5 present the analysis of $\lambda_{SCA}$ and $\lambda_{SFA}$ on C-MAPSS
respectively, and Fig. 6 shows the analysis for $\lambda_{SCA}$ and
$\lambda_{SFA}$ on Opportunity HAR. From the results, we can find similar
trends for the two hyperparameters, which include two points. Firstly, our
method exhibits insensitivity to changes in the hyperparameters in most cases.
For example, our method is insensitive to $\lambda_{SCA}$ in cross-domain
scenarios such as 1$\to$4, 2$\to$1, and 2$\to$4 of C-MAPSS, and 1$\to$3,
2$\to$3, and 4$\to$2 of Opportunity HAR. Similarly, regarding $\lambda_{SFA}$,
we can also find the insensitivity in the scenarios 1$\to$2, 2$\to$1, and
3$\to$4 of C-MAPSS, and 1$\to$4, 3$\to$2, and 4$\to$3 of Opportunity HAR.
These results indicate that our method can achieve stable performance without
requiring extensive hyperparameter tuning, making it applicable to real-world
scenarios. Second, in the cases where our method is sensitive to the
hyperparameters, our method tends to achieve good performance when their
values are within the range of [0.001,1]. Taking the results on C-MAPSS as
examples, we observe that our method achieves poor performance when
$\lambda_{SCA}$ is set to 10 in the case of 3$\to$2, and when $\lambda_{SCA}$
is set to 0.001 in the case of 4$\to$1\. Similar trends can also be observed
for $\lambda_{SFA}$, e.g., $\lambda_{SFA}$=10 in 2$\to$3\. Based on these
observations, we recommend setting the values of these hyperparameters within
the range of 0.01 to 1, where our method consistently achieves good
performance.

####  V-D2 Analysis for the number of heads

Figure 7: The sensitivity analysis for different heads on C-MAPSS. Figure 8:
The sensitivity analysis for different heads on Opportunity HAR.

For the construction of each graph with MSGC, we have multiple branches with
differently initialized weights to model various aspects of sensor
correlations. In this part, we aim to evaluate the effect of the number of
branches on the model performance. Fig. 7 and 8 show the results on C-MAPSS
and Opportunity HAR respectively. In our analysis, we analyze the effect using
a maximum of 4 branches, as we observe that further increasing the number of
branches does not yield significant improvements. The results consistently
show that SEA++ obtains performance improvements with increasing the number of
branches. Specifically, SEA++ with just one branch tends to exhibit poorer
performance in most cases, such as in the 2$\to$3 cross-domain scenario of
C-MAPSS and the 2$\to$4 scenario of Opportunity HAR. With increasing the
number of branches, the performance improves. It is worth noting that the
improvements become less pronounced when the number of branches is large
enough, particularly when transitioning from 3 to 4 branches. This suggests
that using 3 branches is generally sufficient for effectively capturing the
spatial dependencies between sensors. Overall, the analysis demonstrates the
importance of leveraging multiple branches in MSGC for modeling the complex
relationships among sensors. Increasing the number of branches allows our
method to capture a more comprehensive understanding of sensor correlations,
leading to improved performance for MTS-UDA.

####  V-D3 Analysis for patch size

Figure 9: The sensitivity analysis for patch sizes on C-MAPSS. Figure 10: The
sensitivity analysis for patch sizes on Opportunity HAR.

To align the evolving distributions within MTS data, we segment an MTS sample
into multiple patches with a size of $d$, which are then utilized to construct
sequential graphs and conduct multi-graph alignment. Evaluating the effect of
patch size $d$ is important, as the size affects the construction of
sequential graphs. Notably, we conducted the analysis with different patch
sizes for C-MAPSS and Opportunity HAR as their samples have different lengths.
For C-MAPSS, the time length of each sample is fixed as 60, so we use the
patch sizes within [4,5,6,7,8,9,10] for patch size analysis. For Opportunity
HAR, the time length of each sample is fixed as 128, so we use the patch sizes
within [2,4,8,16,32].

Based on the analysis of C-MAPSS in Fig. 9 and Opportunity in Fig. 10, it is
evident that relatively large patch sizes tend to result in better performance
for our method. This observation holds true for several cross-scenarios, such
as 3$\to$1, 3$\to$2, and 3$\to$4 of C-MAPSS, where the best performance is
achieved with patch sizes of 8, 9, or 10. Similar observations can be found
for Opportunity HAR, such as 1$\to$3, 2$\to$1, and 3$\to$1, where the best
performance is achieved with patch sizes of 16 or 32. Conversely, setting too
small patch sizes leads to poor performance in most scenarios, such as 1$\to$3
and 1$\to$4 of C-MAPSS, and 1$\to$2 and 1$\to$3 of Opportunity HAR. The reason
behind this trend lies in the construction of sequential graphs and the
capturing of spatial dependencies for transferring across domains. By
segmenting MTS data into patches, we construct sequential graphs whose spatial
dependencies are captured for transferring across domains. In this process,
too small patch sizes may not contain enough information to fully capture the
complex spatial dependencies between sensors for each patch, resulting in
suboptimal performance in transferring the dependency information. Therefore,
it is recommended to use relatively large patch sizes to obtain better
performance using SEA++ for MTS-UDA.

##  VI Conclusion

In this paper, we formulate Multivariate Time-Series Unsupervised Domain
Adaptation (MTS-UDA). We analyze the problems underlying this task and propose
SEnsor Alignment (SEA) to address these issues. To reduce the domain
discrepancy at both the local and global sensor levels, we design endo-feature
alignment and exo-feature alignment. At the local sensor level, endo-feature
alignment aligns sensor features and sensor correlations across domains,
preventing misalignment at the sensor level. To enhance the endo-feature
alignment, we further design SEA++, incorporating high-order statistic
alignment and a multi-graph alignment technique. These enhancements are
specifically designed to facilitate comprehensive alignment across domains and
effectively capture the evolving data distributions within MTS data. At the
global sensor level, we enforce restrictions on global sensor features to
reduce domain discrepancy. Our extensive experiments demonstrate the
effectiveness of SEA and SEA++ for MTS-UDA.

## References

  * [1] E. Eldele, M. Ragab, Z. Chen, M. Wu, C.-K. Kwoh, X. Li, and C. Guan, “Self-supervised contrastive representation learning for semi-supervised time-series classification,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2023.
  * [2] Q. Ma, S. Li, and G. W. Cottrell, “Adversarial joint-learning recurrent neural network for incomplete time series classification,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 44, no. 4, pp. 1765–1776, 2020.
  * [3] R. Zhao, R. Yan, Z. Chen, K. Mao, P. Wang, and R. X. Gao, “Deep learning and its applications to machine health monitoring,” _Mechanical Systems and Signal Processing_ , vol. 115, pp. 213–237, 2019.
  * [4] K. Chen, D. Zhang, L. Yao, B. Guo, Z. Yu, and Y. Liu, “Deep learning for sensor-based human activity recognition: Overview, challenges, and opportunities,” _ACM Computing Surveys (CSUR)_ , vol. 54, no. 4, pp. 1–40, 2021.
  * [5] H. He, Q. Zhang, S. Bai, K. Yi, and Z. Niu, “Catn: Cross attentive tree-aware network for multivariate time series forecasting,” _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 36, no. 4, pp. 4030–4038, Jun. 2022. [Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/20320
  * [6] W. Zheng and J. Hu, “Multivariate time series prediction based on temporal change information learning method,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2022.
  * [7] A. Garg, W. Zhang, J. Samaran, R. Savitha, and C.-S. Foo, “An evaluation of anomaly detection and diagnosis in multivariate time series,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 33, no. 6, pp. 2508–2517, 2021.
  * [8] S. Zhao, X. Yue, S. Zhang, B. Li, H. Zhao, B. Wu, R. Krishna, J. E. Gonzalez, A. L. Sangiovanni-Vincentelli, S. A. Seshia _et al._ , “A review of single-source deep unsupervised visual domain adaptation,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 33, no. 2, pp. 473–493, 2020.
  * [9] E. Tzeng, J. Hoffman, N. Zhang, K. Saenko, and T. Darrell, “Deep domain confusion: Maximizing for domain invariance,” _arXiv preprint arXiv:1412.3474_ , 2014. [Online]. Available: https://arxiv.org/abs/1412.3474
  * [10] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, “Domain-adversarial training of neural networks,” _J. Mach. Learn. Res._ , vol. 17, no. 1, p. 2096–2030, jan 2016\.
  * [11] E. Tzeng, J. Hoffman, K. Saenko, and T. Darrell, “Adversarial discriminative domain adaptation,” in _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 2962–2971.
  * [12] M. Long, Z. Cao, J. Wang, and M. I. Jordan, “Conditional adversarial domain adaptation,” _Advances in neural information processing systems_ , vol. 31, 2018.
  * [13] B. Sun and K. Saenko, “Deep coral: Correlation alignment for deep domain adaptation,” in _Computer Vision – ECCV 2016 Workshops_ , G. Hua and H. Jégou, Eds. Cham: Springer International Publishing, 2016, pp. 443–450.
  * [14] M. Long, H. Zhu, J. Wang, and M. I. Jordan, “Deep transfer learning with joint adaptation networks,” in _International conference on machine learning_. PMLR, 2017, pp. 2208–2217.
  * [15] S. Purushotham, W. Carvalho, T. Nilanon, and Y. Liu, “Variational recurrent adversarial deep domain adaptation,” in _ICLR_ , 2017.
  * [16] R. Cai, J. Chen, Z. Li, W. Chen, K. Zhang, J. Ye, Z. Li, X. Yang, and Z. Zhang, “Time series domain adaptation via sparse associative structure alignment,” _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 35, no. 8, pp. 6859–6867, May 2021. [Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/16846
  * [17] Q. Liu and H. Xue, “Adversarial spectral kernel matching for unsupervised time series domain adaptation,” in _Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, IJCAI-21_ , Z.-H. Zhou, Ed. International Joint Conferences on Artificial Intelligence Organization, 8 2021, pp. 2744–2750, main Track. [Online]. Available: https://doi.org/10.24963/ijcai.2021/378
  * [18] Y. Shi, X. Ying, and J. Yang, “Deep unsupervised domain adaptation with time series sensor data: A survey,” _Sensors_ , vol. 22, no. 15, p. 5507, 2022\.
  * [19] M. Ragab, E. Eldele, W. L. Tan, C.-S. Foo, Z. Chen, M. Wu, C.-K. Kwoh, and X. Li, “Adatime: A benchmarking suite for domain adaptation on time series data,” _arXiv preprint arXiv:2203.08321_ , 2022. [Online]. Available: https://arxiv.org/abs/2203.08321
  * [20] Y. Wang, Y. Xu, J. Yang, Z. Chen, M. Wu, X. Li, and L. Xie, “Sensor alignment for multivariate time-series unsupervised domain adaptation,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 37, no. 8, 2023, pp. 10 253–10 261.
  * [21] Z. Pei, Z. Cao, M. Long, and J. Wang, “Multi-adversarial domain adaptation,” _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 32, no. 1, Apr. 2018. [Online]. Available: https://ojs.aaai.org/index.php/AAAI/article/view/11767
  * [22] C. Yu, J. Wang, Y. Chen, and M. Huang, “Transfer learning with dynamic adversarial adaptation network,” in _2019 IEEE International Conference on Data Mining (ICDM)_. IEEE, 2019, pp. 778–786.
  * [23] Z. Du, J. Li, H. Su, L. Zhu, and K. Lu, “Cross-domain gradient discrepancy minimization for unsupervised domain adaptation,” in _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 2021, pp. 3937–3946.
  * [24] Z. Gao, S. Zhang, K. Huang, Q. Wang, and C. Zhong, “Gradient distribution alignment certificates better adversarial domain adaptation,” in _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , 2021, pp. 8937–8946.
  * [25] L. Chen, H. Chen, Z. Wei, X. Jin, X. Tan, Y. Jin, and E. Chen, “Reusing the task-specific classifier as a discriminator: Discriminator-free adversarial domain adaptation,” in _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 2022, pp. 7181–7190.
  * [26] M. Long, Y. Cao, Z. Cao, J. Wang, and M. I. Jordan, “Transferable representation learning with deep adaptation networks,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 41, no. 12, pp. 3071–3085, 2018.
  * [27] C. Chen, Z. Fu, Z. Chen, S. Jin, Z. Cheng, X. Jin, and X.-S. Hua, “Homm: Higher-order moment matching for unsupervised domain adaptation,” in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 34, no. 04, 2020, pp. 3422–3429.
  * [28] S. Li, C. Liu, Q. Lin, B. Xie, Z. Ding, G. Huang, and J. Tang, “Domain conditioned adaptation network,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, no. 07, 2020, pp. 11 386–11 393.
  * [29] W. Zellinger, T. Grubinger, E. Lughofer, T. Natschläger, and S. Saminger-Platz, “Central moment discrepancy (cmd) for domain-invariant representation learning,” _arXiv preprint arXiv:1702.08811_ , 2017.
  * [30] J. Li, E. Chen, Z. Ding, L. Zhu, K. Lu, and H. T. Shen, “Maximum density divergence for domain adaptation,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 43, no. 11, pp. 3918–3930, 2020.
  * [31] Y. Xu, J. Yang, H. Cao, K. Wu, M. Wu, Z. Li, and Z. Chen, “Multi-source video domain adaptation with temporal attentive moment alignment network,” _IEEE Transactions on Circuits and Systems for Video Technology_ , 2023.
  * [32] Y. Xu, H. Cao, K. Mao, Z. Chen, L. Xie, and J. Yang, “Aligning correlation information for domain adaptation in action recognition,” _IEEE Transactions on Neural Networks and Learning Systems_ , 2022.
  * [33] J. Li, S. Qiu, C. Du, Y. Wang, and H. He, “Domain adaptation for eeg emotion recognition based on latent representation similarity,” _IEEE Transactions on Cognitive and Developmental Systems_ , vol. 12, no. 2, pp. 344–353, 2019.
  * [34] H. Zhao, Q. Zheng, K. Ma, H. Li, and Y. Zheng, “Deep representation-based domain adaptation for nonstationary eeg classification,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 32, no. 2, pp. 535–545, 2020.
  * [35] G. Wilson, J. R. Doppa, and D. J. Cook, “Multi-source deep domain adaptation with weak supervision for time-series sensor data,” in _Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining_, 2020, pp. 1768–1778.
  * [36] ——, “Calda: Improving multi-source time series domain adaptation with contrastive adversarial learning,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , 2023.
  * [37] Q. Liu and H. Xue, “Adversarial spectral kernel matching for unsupervised time series domain adaptation.” in _IJCAI_ , 2021, pp. 2744–2750.
  * [38] T. Yang, X. Yu, N. Ma, Y. Zhao, and H. Li, “A novel domain adaptive deep recurrent network for multivariate time series prediction,” _Engineering Applications of Artificial Intelligence_ , vol. 106, p. 104498, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0952197621003468
  * [39] A. Deng and B. Hooi, “Graph neural network-based anomaly detection in multivariate time series,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 35, 2021, pp. 4027–4035.
  * [40] Z. Jia, Y. Lin, J. Wang, R. Zhou, X. Ning, Y. He, and Y. Zhao, “Graphsleepnet: Adaptive spatial-temporal graph convolutional networks for sleep stage classification,” in _Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI-20_ , C. Bessiere, Ed. International Joint Conferences on Artificial Intelligence Organization, 7 2020, pp. 1324–1330, main track.
  * [41] J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl, “Neural message passing for quantum chemistry,” _arXiv preprint arXiv:1704.01212_ , 2017.
  * [42] H. Guo, R. Pasunuru, and M. Bansal, “Multi-source domain adaptation for text classification via distancenet-bandits,” in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 34, no. 05, 2020, pp. 7830–7838.
  * [43] Y. Wang, M. Wu, X. Li, L. Xie, and Z. Chen, “Multivariate time series representation learning via hierarchical correlation pooling boosted graph neural network,” _IEEE Transactions on Artificial Intelligence_ , 2023.
  * [44] R.-G. Cirstea, B. Yang, and C. Guo, “Graph attention recurrent neural networks for correlated time series forecasting,” _MileTS19@ KDD_ , pp. 1–6, 2019\.
  * [45] A. Saxena, K. Goebel, D. Simon, and N. Eklund, “Damage propagation modeling for aircraft engine run-to-failure simulation,” in _2008 International Conference on Prognostics and Health Management_ , 2008, pp. 1–9.
  * [46] D. Roggen, A. Calatroni, M. Rossi, T. Holleczek, K. Förster, G. Tröster, P. Lukowicz, D. Bannach, G. Pirkl, A. Ferscha, J. Doppler, C. Holzmann, M. Kurz, G. Holl, R. Chavarriaga, H. Sagha, H. Bayati, M. Creatura, and J. d. R. Millàn, “Collecting complex activity datasets in highly rich networked sensor environments,” in _2010 Seventh International Conference on Networked Sensing Systems (INSS)_ , 2010, pp. 233–240.
  * [47] M. Ragab, Z. Chen, M. Wu, C. S. Foo, C. K. Kwoh, R. Yan, and X. Li, “Contrastive adversarial domain adaptation for machine remaining useful life prediction,” _IEEE Transactions on Industrial Informatics_ , vol. 17, no. 8, pp. 5239–5249, 2021.
  * [48] M. Ragab, E. Eldele, Z. Chen, M. Wu, C.-K. Kwoh, and X. Li, “Self-supervised autoregressive domain adaptation for time series data,” _IEEE Transactions on Neural Networks and Learning Systems_ , pp. 1–11, 2022.
  * [49] Z. Chen, M. Wu, R. Zhao, F. Guretno, R. Yan, and X. Li, “Machine remaining useful life prediction via an attention-based deep learning approach,” _IEEE Transactions on Industrial Electronics_ , vol. 68, no. 3, pp. 2521–2531, 2021.
  * [50] Q. Xu, Z. Chen, K. Wu, C. Wang, M. Wu, and X. Li, “Kdnet-rul: A knowledge distillation framework to compress deep neural networks for machine remaining useful life prediction,” _IEEE Transactions on Industrial Electronics_ , vol. 69, no. 2, pp. 2022–2032, 2022.
  * [51] S. Fu, S. Zhong, L. Lin, and M. Zhao, “A novel time-series memory auto-encoder with sequentially updated reconstructions for remaining useful life prediction,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 33, no. 12, pp. 7114–7125, 2021.
  * [52] B. Sun, J. Feng, and K. Saenko, _Correlation Alignment for Unsupervised Domain Adaptation_. Cham: Springer International Publishing, 2017, pp. 153–171. [Online]. Available: https://doi.org/10.1007/978-3-319-58347-1_8
  * [53] Y. Ganin, E. Ustinova, H. Ajakan, P. Germain, H. Larochelle, F. Laviolette, M. Marchand, and V. Lempitsky, _Domain-Adversarial Training of Neural Networks_. Cham: Springer International Publishing, 2017, pp. 189–209. [Online]. Available: https://doi.org/10.1007/978-3-319-58347-1_10
  * [54] S. Cui, S. Wang, J. Zhuo, L. Li, Q. Huang, and Q. Tian, “Towards discriminability and diversity: Batch nuclear-norm maximization under label insufficient situations,” in _IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , June 2020.
  * [55] Y. Zhu, F. Zhuang, J. Wang, G. Ke, J. Chen, J. Bian, H. Xiong, and Q. He, “Deep subdomain adaptation network for image classification,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 32, no. 4, pp. 1713–1722, 2021.
  * [56] H. Rangwani, S. K. Aithal, M. Mishra, A. Jain, and R. V. Babu, “A closer look at smoothness in domain adversarial training,” in _Proceedings of the 39th International Conference on Machine Learning_ , 2022.
  * [57] M. Vayyat, J. Kasi, A. Bhattacharya, S. Ahmed, and R. Tallamraju, “Cluda: Contrastive learning in unsupervised domain adaptation for semantic segmentation,” _arXiv preprint arXiv:2208.14227_ , 2022.