# Using Graph Neural Networks and Frequency Domain Data for Automated
Operational Modal Analysis of Populations of Structures

Xudong Jian  Yutong Xia  Gregory Duthé  Kiran Bacsa  Wei Liu  Eleni Chatzi
Future Resilient Systems, Singapore-ETH Centre, Singapore, 138602, Singapore
Institute of Data Science, National University of Singapore, Singapore,
117602, Singapore<EMAIL_ADDRESS>Department of Civil, Environmental
and Geomatic Engineering, ETH Zurich, Zurich, 8049, Switzerland.  Department
of Industrial Systems Engineering and Management, National University of
Singapore, Singapore, 117576, Singapore

###### Abstract

The Population-Based Structural Health Monitoring (PBSHM) paradigm has
recently emerged as a promising approach to enhance data-driven assessment of
engineering structures by facilitating transfer learning between structures
with some degree of similarity. In this work, we apply this concept to the
automated modal identification of structural systems. We introduce a Graph
Neural Network (GNN)-based deep learning scheme to identify modal properties,
including natural frequencies, damping ratios, and mode shapes of engineering
structures based on the Power Spectral Density (PSD) of spatially-sparse
vibration measurements. Systematic numerical experiments are conducted to
evaluate the proposed model, employing two distinct truss populations that
possess similar topological characteristics but varying geometric (size,
shape) and material (stiffness) properties. The results demonstrate that, once
trained, the proposed GNN-based model can identify modal properties of unseen
structures within the same structural population with good efficiency and
acceptable accuracy, even in the presence of measurement noise and sparse
measurement locations. The GNN-based model exhibits advantages over the
classic Frequency Domain Decomposition (FDD) method in terms of identification
speed, as well as against an alternate Multi-Layer Perceptron (MLP)
architecture in terms of identification accuracy, rendering this a promising
tool for PBSHM purposes.

###### keywords:

Population-based structural health monitoring, Operational modal
identification, Deep learning, Graph neural network, Feature Propagation

###### keywords:

[MSC Codes][Primary]CODE1; [Secondary]CODE2, CODE3

††journal: Data-Centric Engineering††articletype: Research article

2024

Xudong Jian et al.

Xudong Jian et al.

## Impact Statement

We have developed a Graph Neural Network (GNN)-based scheme for automated
operational modal analysis. Numerical experiments demonstrate that the
proposed model can efficiently and effectively identify natural frequencies,
damping ratios, and mode shapes of different engineering structures that
belong to a population. Notably, the GNN-based model outperforms the
traditional Frequency Domain Decomposition method in terms of efficiency and
surpasses a Multi-layer Perceptron in accuracy, positioning it as a promising
tool for population-based structural health monitoring.

##  1 Introduction

Structural Health Monitoring (SHM) has evolved into a useful tool for
engineering practice, showcasing its importance through various applications
in the management and maintenance of engineering structures ([58, 47, 72]).
Although further steps are needed for standardization of SHM schemes, which
are often bespoke to the structure at hand, the overall practice has matured,
with the primary methods for extracting valuable information from SHM data
categorized into physics-based and data-driven approaches [48, 73]. The
notable rise of data-driven approaches in recent years can be attributed not
only to advancements in powerful Deep Learning (DL) tools, which favors
computation, but also to the apparent robustness of schemes that capitalize on
data against discrepancies between physical models and real-world structures
[53, 67]. However, these approaches face challenges in interpretability and
generalization, especially in civil engineering, where structures such as
bridges are often uniquely designed to meet site-specific requirements [76].
Training data-driven models with large datasets that represent a variety of
structures can enhance their generalization capabilities. However, data from
real structures often lack, since SHM tends to so far mostly be applied onto
specific case studies, of higher importance, due to budget and labor
constraints. Consequently, the rich and diverse datasets necessary for
training learning (data-driven) models are generally unavailable ([85]). To
address this challenge, new sensing technologies, such as mobile sensing
schemes ([71, 84, 70]), are developed to gather more measurement data from a
greater number of structures. Additionally, advanced analysis methods can be
used to more efficiently extract deeper insights from the available data.
Treating structures as a population of shared features could enable learning
and knowledge transfer among group members. For instance, although bridges are
often unique, they can be categorized into broader typologies, such as truss
or girder bridges. This approach is embodied in the recently introduced
concept of Population-Based Structural Health Monitoring (PBSHM) ([63, 87, 88,
52]).

According to structural analysis theory, a population of structures can be
represented by graphs that share certain morphological or topological
characteristics, despite each instance comprising unique geometric or material
properties. Following the logic of graph representations as suggested by [62],
Graph Neural Networks (GNNs) can work as efficient representations for PBSHM.
GNNs are a type of deep learning models that are specifically designed to
operate on graph-structured data, comprising a set of nodes and edges that
capture interrelations amongst elements of the network. Compared with
traditional deep learning models, GNNs present several unique advantages
([83]): 1) Flexibility: The flexibility of the GNN architecture allows them to
handle graphs with varying node counts and diverse connectivity patterns. This
characteristic makes them particularly suitable for PBSHM-related datasets,
where structures within a population share similarities but also possess
individual differences. 2) Accuracy: GNNs are explicitly designed to handle
data that is structured as graphs, which can process data in three different
spatial levels: node level, edge level, and graph level. The message passing
function of GNNs ensures more effective utilization of spatial information
hidden in the data, consequently enhancing accuracy in processing graph-
structured data. 3) Interpretability: We can naturally correlate data within a
given dataset with the node-, edge-, and graph-level features of a GNN. This
characteristic facilitates a more straightforward interpretation of both the
model itself and the resulting outcomes.

Due to the above-mentioned advantages of GNNs, a number of studies have
already attempted to integrate GNNs with forward analysis tasks. Examples
include the use of GNNs to predict the shear stress in wall structures ([55,
54]), airflow of wind energy systems ([79, 56, 57]), and estimating the main
natural frequencies of truss structures given specific environmental
conditions (temperature) and different member types ([86]). For SHM, however,
inverse problems play a more critical role, where the goal is to identify
structural parameters from measurements of structures and, thus, evaluate the
condition of a monitored structure ([59]). To date, a paucity of studies
involving GNNs for inverse problem solutions is noted. In this study, we take
this task on by introducing GNNs for a fundamental SHM task, namely Operation
Modal Analysis (OMA), aiming to identify modal properties of a structure based
solely on measured vibration response data ([81, 49]). As typically conducted
within an OMA setting, the identified modal properties can be used for
downstream tasks such as damage identification ([68, 64, 65]) or response
prediction ([75]).

Given the background introduced above, this study proposes a novel deep
learning architecture, with the GNN serving as the pivotal building block,
aimed at automatically performing output-only modal identification for
structures within a population. Key contributions of this study include:

  * •

We design the architecture of a GNN-based deep learning model, in which the
structural population is modelled by a GNN. This model is trained to capture
the meta behavior of the structural population, enabling it to output natural
frequencies, damping ratios, and mode shapes from the input that is the Power
Spectral Density (PSD) of the vibration acceleration on structure nodes.

  * •

We adopt the Feature Propagation (FP) algorithm to reconstruct the full-field
acceleration PSD using partial measurements of a small subset of structure
nodes, in an effort to reflect a realistic SHM context where only sparse
dynamic response measurements are available. The proposed GNN-based model can
then be used for modal identification based on the reconstructed PSD.

  * •

We conduct systematic numerical experiments to assess the performance of the
proposed GNN-based model. The accuracy and efficiency of the model are
thoroughly investigated.

##  2 Methodology

###  2.1 Problem formulation

Modal identification of structural systems plays a fundamental role in the
context of SHM. In general, the identification task aims at ascertaining the
natural frequencies, damping ratios and mode shapes of the monitored structure
by processing measurements relating to dynamic response (structural output),
such as accelerations, from a finite- and typically sparse- set of Degrees of
Freedom (DOFs). OMA schemes rely on use of output-only information under the
assumption of broadband and random unmeasured excitation. In keeping true to
this requirement, this study employs Gaussian white noise to model the
excitation sources of the simulated structures.

To further define the problem, let us denote the available acceleration
measurements as $\mathbf{X}(t)\in\mathbb{R}^{N\times P}$, where $t$ represents
the time components, $N$ denotes the amount of monitored DOFs, and $P$ denotes
the amount of available time samples per signal. In this study, we aim to
learn a function $func\left(\cdot\right)$ that can identify natural
frequencies, damping ratios, and mode shapes of the first $k$ structural modes
based on the measured signals, $\mathbf{X}(t)$. Then the identification
process can be expressed as:

$\left[\hat{\mathbf{F}},\hat{\mathbf{Z}},\hat{\mathbf{\Phi}}\right]=func(\mathbf{X}(t))$
(2.1)

where $\hat{\mathbf{F}}\in\mathbb{R}_{>0}^{1\times k}$,
$\hat{\mathbf{Z}}\in\mathbb{R}_{>0}^{1\times k}$, and
$\hat{\mathbf{\Phi}}\in\mathbb{R}^{N\times k}$ denote identified natural
frequencies (in Hz), damping ratios, and mode shapes, respectively.

Within an OMA context, $P$ is usually chosen to reflect a large number of time
samples, since typically long time series are required for construction of the
appropriate spectra. As a consequence, the input of the deep learning model,
$\mathbf{X}(t)$, will comprise a high dimensionality, which is often
undesirable as it compromises efficiency ([89]). Therefore, in this study, we
reduce the dimension of $\mathbf{X}(t)$ by employing the frequency domain
representation of the measured signals, i.e., the Power Spectral Density
(PSD). The converted PSD is denoted as ${\mathbf{S}}(f)\in\mathbb{R}_{\geq
0}^{N\times M}$, where $f$ denotes the frequency components and $M$ is the
dimension of the PSD representation. Since the PSD contains only amplitude and
not phase information of the time-history signals, only absolute mode shapes
$\left|\hat{\mathbf{\Phi}}\right|\in\mathbb{R}_{\geq 0}^{N\times k}$ can be
identified from the proposed approach, which are nonetheless still meaningful
for SHM tasks. Thus, the problem in this study can be reduced to learn a
function $func\left(\cdot\right)$ that can be expressed as:

$\left[\hat{\mathbf{F}},\hat{\mathbf{Z}},\left|\hat{\mathbf{\Phi}}\right|\right]=func({\mathbf{S}}(f))$
(2.2)

###  2.2 Model architecture

To obtain the modal identification function $func\left(\cdot\right)$ shown in
Equation (2.2), we design a deep learning model to learn the mapping between
${\mathbf{S}}(f)$ and
$\left[\hat{\mathbf{F}},\hat{\mathbf{Z}},\left|\hat{\mathbf{\Phi}}\right|\right]$.
The architecture of the proposed model is visualized in Figure 1.

Figure 1: Architecture of the proposed model, in which the model input and
output are marked in red color, hidden features are marked in yellow color,
and deep learning blocks are marked in blue color.

As indicated in Figure 1, five deep learning blocks (marked in blue color) are
employed in the proposed model. Their description is as follows:

  * •

MLP 1: The multi-layer perceptron (MLP) 1 block serves as an encoder that
compresses the input data $\mathbf{S}$ ($M$-dimensional) into more compact
hidden features $\mathbf{H}1$ ($D$-dimensional). This encoding process is
widely adopted in the design of DL models ([61]). The ablation study (section
3.4) demonstrates that the usage of this block helps to improve the accuracy
of the proposed model.

  * •

GNN: The main building block in the proposed architecture is the GNN, which is
constructed on the basis of engineering intuition. More specifically, as
illustrated in Figure 2, we can reasonably designate the joint locations of a
structure as the nodes of the GNN and the structural elements connecting those
nodes (e.g. beams, truss bars) to correspond to the GNN edges. Unlike a common
MLP architecture, where the number of neurons is fixed, GNNs can process
graphs of various configurations, corresponding to varying numbers of nodes
and edges ([92]), which is particularly beneficial for PBSHM tasks, since each
structure usually comprises a different number of nodes and elements. The
flexibility of GNNs comes from their message passing mechanism, which
aggregates information from neighboring nodes rather than a fixed set of
neurons, with the model parameters being shared across all nodes and edges.
Once the skeleton of the GNN is built, the input quantities need to be
determined on the basis of available measurements from the monitored systems.
For the framework we propose, as shown in Figure 2, this involves computation
of the PSDs of vibration acceleration signals $\mathbf{S}$ as well as the
absolute mode shapes $\left|\hat{\mathbf{\Phi}}\right|$ as the node-level
features of the GNN, while the natural frequencies $\hat{\mathbf{F}}$ and
damping ratios $\hat{\mathbf{Z}}$ serve as the graph-level features of the
GNN. This study adopts an encoder-decoder architecture, which implies that the
GNN does not directly process $\mathbf{S}$ as the input to produce
$\left|\hat{\mathbf{\Phi}}\right|$, $\hat{\mathbf{F}}$, and $\hat{\mathbf{Z}}$
as the output. Instead, the GNN takes the encoded features $\mathbf{H}1$ as
the nodal input and executes two individual tasks, which in Figure 1 are
correspondingly indicated as the graph-level task and the node-level task. The
graph-level task corresponds to the ‘readout’ operation, which is essential
for graph-level downstream tasks ([60]), and serves for generating a single
hidden feature $\mathbf{H}2$ by aggregating node features. In parallel, the
node-level task performs the message passing operation, which is the GNN’s
fundamental mechanism that allows nodes in a graph to exchange information
with their neighbors in order to exploit the interrelations that lay latent in
the graph-structured data ([91]). The product of the node-level task is the
hidden feature $\mathbf{H}3$ on each node.

Figure 2: An example of the graph dataset used in this study. A truss
structure can be naturally modeled as a graph. See text about the GNN block
for more details.

  * •

MLP 2&3&4: Finally, three MLPs are employed as the decoder architecture in
order to project the $D$-dimensional hidden features back to the physical
domain, yielding the modal properties of the first $k$ modes. The MLP 2 and
MLP 3 blocks decode the graph-level hidden feature $\mathbf{H}2$, and thus
output the graph-level modal properties, which are natural frequencies
$\hat{\mathbf{F}}$ and damping ratios $\hat{\mathbf{Z}}$, respectively. The
MLP 4 block decodes the node-level hidden feature $\mathbf{H}3$ and outputs
the mode shapes $\left|\hat{\mathbf{\Phi}}\right|$ of every node.

###  2.3 Loss function

To train our model, we employ an objective function based on the Mean Squared
Error (MSE):

$\mathcal{L}=\frac{\lambda_{1}}{kN}\sum_{i=1}^{N}\sum_{j=1}^{k}\left(\left|\mathbf{\Phi}\right|_{i,j}-\hat{\left|\mathbf{\Phi}\right|}_{i,j}\right)^{2}+\frac{\lambda_{2}}{k}\sum_{j=1}^{k}\left(\frac{\hat{\mathbf{F}}_{j}}{\mathbf{F}_{j}}-1\right)^{2}+\frac{\lambda_{3}}{k}\sum_{j=1}^{k}\left(\frac{\hat{\mathbf{Z}}_{j}}{\mathbf{Z}_{j}}-1\right)^{2}$
(2.3)

where ${\mathbf{F}}$, ${\mathbf{Z}}$, and $\left|{\mathbf{\Phi}}\right|$ are
the target natural frequencies, damping ratios, and absolute mode shapes,
respectively. These quantities are known during training, and the methods for
deriving them will be elaborated in Section 2.4. $\lambda_{1}$, $\lambda_{2}$,
and $\lambda_{3}$ are coefficients that are adopted to balance the
contribution of the individual components related to ${\mathbf{F}}$,
${\mathbf{Z}}$, and $\left|{\mathbf{\Phi}}\right|$ to the loss function. In
this study, we determine $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ by
trial and error.

It is also noteworthy that, in this study the PSDs (model input) and absolute
mode shapes (model output) are normalized by scaling their amplitudes to a
maximum of 1 unit before training to boost the generalization ability of the
deep learning model. However, natural frequencies and damping ratios cannot be
normalized by max normalization because their original values matter, and this
can cause imbalance during model training. For instance, natural frequencies
are generally much larger than damping ratios, so the loss term regarding
natural frequencies will outweigh the damping ratio loss if they are not
normalized. To diminish this possible imbalance, as shown in the loss
function, we normalize natural frequencies and damping ratios by dividing the
estimated values (model output, which are ${\mathbf{\hat{F}}}$,
${\mathbf{\hat{Z}}}$) by their corresponding target values (${\mathbf{F}}$,
${\mathbf{Z}}$).

###  2.4 Framework

On the basis of above-mentioned problem formulation, model architecture, and
loss function, the proposed GNN-based PBSHM framework for modal identification
is summarized in Figure 3.

Figure 3: Framework of using the GNN-based model for population-based
structural modal identification

As shown in Figure 3, the framework is divided into two stages, namely model
training and model implementation. In the model training stage, a subset of
the employed structural population is used as the training set. To establish
such a structural population, numerical simulation is necessary because for
real-world datasets it is impossible to monitor the dynamic response of all
DOFs of structures, while the full-field measurements and mode shapes are
required during the training stage. For the simulated population, structures
within it are randomly generated and modelled by the finite element method.
Then the modal properties of each structure are calculated by means of
eigenvalue analysis. To make the simulated population more realistic, some of
the members could be modelled using the information from available monitored
structures in the real world, and model updating could be performed based on
the measurements to enhance the fidelity of the simulation. Nonetheless, in
this study we only verify the proposed model with pure numerical experiments,
so the model updating part is not involved. More details about dataset
generation can be seen in Section 3.1 below.

After the training is completed, the trained model can be readily used to
automatically identify the modal properties of other unseen structures within
the structural population, even though their configuration may be different to
what is met in the training set, i.e., correspondent to a different number of
nodes and edges and connectivity. This identification process is end-to-end,
automatic, and fast, delivering a main advantage over existing modal
identification schemes. A more detailed comparison can be found in Section
3.8.

##  3 Numerical experiments

In this section, we present the conducted numerical experiments, which were
designed to answer the following research questions:

  * •

Question 1: Which type of GNN is more suitable for the population-based OMA
task?

  * •

Question 2: How do the deep learning modules in the model architecture,
especially the GNN module, contribute to the overall model performance?

  * •

Question 3: Can the proposed model effectively handle real-world challenges
such as incomplete measurements, measurement noise, and limited number of
structures in the dataset?

  * •

Question 4: What are the advantages and disadvantages of the proposed model
versus existing modal identification methods?

  * •

Question 5: Does the proposed model demonstrate sufficient generalization
ability across diverse structural populations?

###  3.1 Dataset description

The extraction of datasets from real-world structural populations is non-
trivial, although recent efforts in Asia [78] and Europe [77] do involve
denser instrumentation of structural populations (such as bridges). These
endeavors are expected to deliver datasets that can support PBSHM tasks.
However, these datasets have not been made public yet, so in this work, we
initiate from simulated scenarios for two datasets that correspond to two
different truss populations. The first truss population consists of 500
trusses assumed to be arranged within a trapezoidal boundary that are meant to
approximate the geometrical configuration of simply-supported beam structures.
Each truss is generated by randomly meshing the trapezoidal area with Delaunay
triangles ([80]). The first 400 trusses are used to train the model, and the
remaining 100 trusses are used to test the model. The geometric boundary and
some generated truss examples are shown in Figure 4(a) and Figure 4(c),
respectively. Based on the generated geometric configurations, corresponding
finite element models are straightforwardly created using truss elements. The
density and area of truss elements are constantly set as 8015 kg/m3 and 0.5
m2, respectively. In order to reflect varying material properties, the Young’s
modulus of the employed truss elements is set as a random number ranging from
100 GPa to 300 GPa. Figure 4 (a) further indicates the boundary conditions
(simply-supported type) and the external excitation (Gaussian white noise)
that are imposed on the bottom boundary. Linear time history analyses
(Newmark-$\beta$ method) are then performed to obtain the in-plane vertical
nodal acceleration (response) time series, of 60 seconds duration, sampled at
a time step of 0.005 second (200 Hz). Welch’s method is next applied to
convert the time series into PSDs, which serves as the input of the suggested
DL architecture. An eigenvalue analysis is further conducted on all 500 finite
element models, generating the reference modal properties of the primary
(first 4 in our study) modes for model training and testing. It should be
noted that, for model training, the complete full-field nodal acceleration
measurements are used, whereas the trained model is tested with both the
complete and incomplete measurements. More details in this regard are offered
in Section 3.5.

In order to further test the generalization ability of the proposed model, we
simulate a separate structural population comprised of 100 samples that are
meant to approximate the geometry of cantilevered trusses. This population is
unseen for the learning PBSHM model, which is trained on the simply-supported
structural population. As illustrated in Figure 4(b), the second population
only differs from the first population in the structural supports. Except for
the structural supports, all the other settings, including geometric
boundaries, material properties, and excitation, are the as same as those of
the first structural population. More details in this regard can be found in
the following Section 3.9.

Figure 4: Visualization of the two generated datasets: (a) Geometric
configuration meant to approximate a simply-supported truss population; (b)
Geometric configuration serving to approximate a cantilevered truss
population; (c) Some representative truss samples from the two datasets (only
nodes and elements are displayed)

###  3.2 Implementation details

The environment we implement our model is set up in PyTorch 2.0.1, CUDA 11.7,
and DGL 1.11. The Adam optimizer is adopted to train our model with the
default setting (learning rate: 0.001, first momentum decaying parameter: 0.9,
second momentum decaying parameter: 0.999), and we use a NVIDIA GeForce RTX
3060 to accelerate the batch training process. The training batch size is 64,
with 5,000 epochs deployed. The MLPs and GNNs used in our study all have three
layers, and the adopted dimension is 64 for every layer. In the loss function,
we set the coefficients $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ as 2,
1, and 1, respectively. More implementation details can be found in our GitHub
repository ([69]), which will be made public upon publication of this work.

###  3.3 Effects of different GNN models

Depending on the type of message passing function that is adopted, different
GNN modeling instances can be created. In this study, we compare three
commonly-used GNN models, namely Graph Convolutional Networks (GCNs, proposed
by [74]), Graph Attention Networks (GATs, proposed by [90]), and GraphSAGE
(proposed by [66]). To control the comparison, we only replace the GNN block
in the model architecture shown in Figure 1 with different GNN models. Then
different models are trained using the K-fold cross validation, which means we
evenly divide the training set that consists of 400 trusses into 5 folds. In
each training process, 4 folds of trusses (320 in number) are used to train
the model, and the remaining 1 fold of trusses (80 in number) are used to
validate the model. We train the model for 5 times, so that all the 5 folds
can serve as the validation set in turn. The mean values and Standard
Deviation (SD) of the final validation loss and training time of the 5
training processes are shown in Table 1.

Table 1: Comparison between the different GNN models | GCN | GAT | GraphSAGE
---|---|---|---
| Mean | SD | Mean | SD | Mean | SD
Final validation loss | 0.076 | 0.019 | 0.066 | 0.012 | 0.059 | 0.010
Training time (s) | 519.696 | 10.293 | 699.911 | 14.639 | 443.838 | 11.881
*Indicators showing better performance are shown in bold.

Based on Table 1, we observe that the GraphSAGE model demonstrates superior
performance, since both its final validation loss and required training time
are statistically smaller than the GCN and GAT model. This is because,
according to [66], the GraphSAGE exhibits inductive learning capabilities,
scalability via neighborhood sampling, and flexibility in aggregation methods.
These features render it particularly suitable for large-scale and dynamic
graphs such as those employed herein within the PBSHM context, where
efficiency and the ability to handle new nodes without re-training are
crucial. GCNs are less scalable and inductive, while GATs offer sophisticated
attention mechanisms at the cost of increased computational complexity. As a
result, hereafter we choose to adopt the GraphSAGE model as the GNN block,
unless mentioned otherwise.

Now that the particular type of GNN models is determined, it is necessary to
elaborate on the details of the training and testing process in order to
comprehensively report on the proposed model. Figure 5 shows the loss curves
corresponding to one training process among the K-Fold Cross-Validation. The
total loss (Equation 2.3), the loss terms regarding mode shapes
($\mathbf{\hat{\Phi}}$), damping ratios ($\mathbf{\hat{Z}}$), and natural
frequencies ($\mathbf{\hat{F}}$) are all displayed.

|  |
---|---|---
(a) | (b) | (c)
Figure 5: Loss curves of one training process among the K-Fold Cross-
Validation: (a) Total training and validation loss; (b) Different loss terms
in the training loss; (c) Different loss terms in the validation loss

Figure 5 allows for the following observations:

  * •

The training loss results smaller than the validation loss in Figure 5(a),
which is reasonable as the validation loss is not used in backpropagation for
updating the model weights.

  * •

Nonetheless, the validation loss is introduced for deciding when to terminate
the training process and for facilitating the comparison of different GNN
models in this subsection as well as different model architecture in the
ablation study below. Although the training loss appears to continually
decrease in Figure 5(b), the validation loss appears to have converged in
Figure 5(c), which promoted termination of the training process after 5,000
epochs.

After the training is completed, we firstly use the trained model to identify
the modal properties of truss configurations lying within the simply-supported
structural population. Identification results of training set (320 trusses)
and testing set (100 trusses) are visualized in Figure 6 and Figure 7,
respectively, in which each dot represents the identification result per
individual truss per mode and different order of modes are distinguished by
different colors. Figure 6(a) and Figure 7(a) show the Modal Assurance
Criterion (MAC) values of identified mode shapes, of which is defined by
Equation (3.1), along with their box plots. The closer the MAC value is to 1,
the higher the accuracy of the mode shape identification.

$\text{MAC}(|\hat{\phi}|,|\phi|)=\frac{\left(|\hat{\phi}|^{T}|\phi|\right)^{2}}{(|\hat{\phi}|^{T}|\hat{\phi}|)(|\phi|^{T}|\phi|)}$
(3.1)

where $|\hat{\phi}|$ and $|\phi|$ are vectors of the identified and target
absolute mode shapes, respectively.

The identified damping ratios and natural frequencies are shown in Figure
6(b), Figure 7(b), Figure 6(c), and Figure 7(c), respectively. In these
figures, the further a point lies from the $\pm 0\%$-error baseline, the
larger the weighing error. Two extra reference lines representing the $\pm
10\%$ relative error are also plotted in the figure to further evaluate the
identification accuracy and reliability.

|  |
---|---|---
(a) | (b) | (c)
Figure 6: Performance of the trained model on the training set: (a) Box plot of Modal Assurance Criterion (MAC) values of identified mode shapes; (b) Scatter plot of identified damping ratios; (c) Scatter plot of identified natural frequencies |  |
---|---|---
(a) | (b) | (c)
Figure 7: Performance of the trained model on the testing set: (a) Box plot of
MAC values of identified mode shapes; (b) Scatter plot of identified damping
ratios; (c) Scatter plot of identified natural frequencies

Figure 6 and Figure 7 allow for the following findings:

  * •

The training process is deemed as successful, since the majority of evaluation
points, as reported in Figure 6(a), lie close to 1, and the points plotted in
Figure 6(b) and Figure 6(c) are scattered near the $\pm 0\%$-error baseline.

  * •

The identification results on the testing set are noticeably worse than those
on the training set. Despite a few instances of very low identification
accuracy, the errors generally lie below 10%, which is considered to be
acceptable, particularly considering that the identification process is fast
and completely end-to-end. To quantitatively understand the performance of the
model on the testing set, Table 2 presents the statistics of the MAC values of
identified mode shapes and the relative errors (%) for the identified damping
ratios and natural frequencies. As shown, the average MAC values for the
identified mode shapes for all four modes are above 0.9, and the average
errors corresponding to the damping ratios and natural frequencies are merely
1% - 2%. This indicates that the trained model statistically performs well in
the modal identification of unseen structures among the same structural
population.

Table 2: Performance indicators on the testing set from the simply-supported structural population | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.995 | 0.006 | 0.975 | -0.069 | 0.033 | 0.132 | -0.445 | 6.740 | 24.915
Mode 2 | 0.952 | 0.072 | 0.603 | -1.853 | 4.479 | 16.905 | -1.351 | 7.201 | 32.007
Mode 3 | 0.902 | 0.124 | 0.336 | -0.608 | 5.974 | 16.259 | -1.804 | 9.180 | 30.053
Mode 4 | 0.905 | 0.148 | 0.302 | 0.304 | 5.867 | 15.076 | -0.765 | 10.911 | 34.681
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

###  3.4 Ablation study

To justify the model architecture shown in Figure 1, we perform an ablation
study by comparing the following model variants:

  * •

No encoder: remove the encoder (MLP1) and retain the remaining blocks in the
model architecture.

  * •

No GNN for mode shape identification: replace the GNN block with a MLP when
converting the hidden features $\mathbf{H1}_{N\times D}$ to
$\mathbf{H3}_{N\times D}$ in the model architecture. It should be noted that
the GNN cannot be completely removed from the architecture because the
identification of damping ratios and natural frequencies require the graph
readout operation. Without the GNN, the MLP that has fixed input dimensions
will not be able to convert the hidden features $\mathbf{H1}_{N\times D}$ to
$\mathbf{H2}_{1\times D}$, making the architecture inapplicable to structures
that posse different number of nodes among a structural population.

To achieve this comparison, we train the originally proposed model and the two
variants for 5 times with the K-Fold Cross-Validation method. The mean values
and standard deviation (SD) of the final validation loss and training time of
the 5 training processes are shown in Table 3.

Table 3: Comparison for ablation study | No encoder | No GNN | Original model
---|---|---|---
| Mean | SD | Mean | SD | Mean | SD
Final validation loss | 0.075 | 0.024 | 0.085 | 0.021 | 0.059 | 0.010
Training time (s) | 396.511 | 5.434 | 287.386 | 9.391 | 443.838 | 11.881
*Indicators showing better performance are shown in bold.

Based on Table 3, we can draw the following conclusions:

  * •

The final validation loss indicates that the original model, whose
architecture is illustrated in Figure 1, is the optimal choice as this
exhibits the lowest validation loss, which justifies the design of the
proposed model. The validation loss for the case of ’No GNN’ model is not only
higher than the other two models in this ablation study, but also higher than
GCN and GAT models, confirming that the GNN does have an advantage of
processing graph-structured data over the MLP.

  * •

Removing the encoder and the GNN from the architecture can reduce the
complexity of the model and thus shorten the training time, but this comes at
the expense of accuracy. In the context of the modal identification problem,
the identification accuracy far outweighs the training time. Hence, the
original model is preferable.

###  3.5 Effects of incomplete measurements

In practice, the DOFs of civil engineering structures usually significantly
outnumber the measurement points, so it is essential to investigate whether
the proposed model can identify complete mode shapes from incomplete
measurements. As stated previously, we construct the GNN corresponding to the
analytical model of a structure. For those structural nodes that have no
measurements, their node features are unknown, and the GNN model used in this
study cannot work with unknown node features. To make the proposed model work
in the case of unknown nodes features, we introduce the Feature Propagation
(FP) algorithm ([82]), to fill the unknown features, which are nodal
acceleration PSD, on the basis of known features and the graph structure.
Applying FP transforms the incomplete measurements into complete ones,
enabling our GNN-based model to conduct modal identification.

Figure 8 illustrates the mode shape identification results of a truss example
in the testing set with different assumed ratios of unknown node features,
meaning under availability of a lower amount of sensors. We can see that even
when as much as 82% of node features are unknown, which corresponds to a
realistic instrumentation setting, the proposed model can still identify mode
shapes with an acceptable accuracy.

---
(a)
(b)
(c)
Figure 8: Mode shape identification results of one truss example with
incomplete measurements: (a) 0% node features are unknown; (b) 66% nodes node
features are unknown; (c) 82% node features are unknown

For further evaluation, Table 4 and Table 5 offer statistics of the MAC values
of identified mode shapes and the relative errors (%) of identified damping
ratios and natural frequencies, under 66% and 82% unknown node features,
respectively.

Table 4: Performance indicators on the testing set from the simply-supported structural population - 66% node features are unknown | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.988 | 0.010 | 0.940 | -0.063 | 0.038 | 0.143 | -2.125 | 6.965 | 25.400
Mode 2 | 0.920 | 0.097 | 0.589 | -1.465 | 4.838 | 17.971 | -2.305 | 7.551 | 33.343
Mode 3 | 0.874 | 0.143 | 0.046 | 0.011 | 6.346 | 18.002 | -2.524 | 9.078 | 29.748
Mode 4 | 0.860 | 0.161 | 0.245 | 1.488 | 6.283 | 17.766 | -1.073 | 11.150 | 35.750
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.
Table 5: Performance indicators on the testing set from the simply-supported structural population - 82% node features are unknown | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.971 | 0.018 | 0.906 | -0.064 | 0.040 | 0.150 | -2.945 | 7.885 | 36.507
Mode 2 | 0.847 | 0.122 | 0.422 | -1.520 | 5.435 | 19.572 | -2.196 | 7.370 | 30.076
Mode 3 | 0.793 | 0.160 | 0.070 | 0.099 | 7.337 | 22.738 | -2.357 | 8.905 | 27.269
Mode 4 | 0.744 | 0.192 | 0.136 | 1.757 | 6.942 | 19.019 | -0.566 | 12.228 | 35.295
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

By comparing Table 4, Table 5, and Table 2 (no unknown node features), we can
conclude that:

  * •

As the ratio of unknown node features increases, the identification accuracy
drops, which is naturally expected. Moreover, the accuracy of identifying
high-order mode shapes drops more than low-order mode shapes due to the
greater complexity of high-order mode shapes.

  * •

Compared with the identification of mode shapes, the identification of natural
frequencies and damping ratios are much less influenced by the increase of
unknown node measurements. This is reasonable because, in theory, natural
frequencies and damping ratios can be successfully identified with only a few
node measurements, whereas accurate identification of mode shapes requires
measurements on more structural nodes. This observation implies that the
trained model successfully learns the mechanism of modal identification.

  * •

The modal identification accuracy is statistically acceptable despite being as
high as 82% unknown node features, which manifests the robustness of the
proposed model.

###  3.6 Effects of measurement noise

In real-world applications, noise corruption is inevitable when acquiring
signals that measure the vibration acceleration of structures. To test our
model’s sensitivity to measurement noise, we artificially pollute the
acceleration time series generated by the testing set by injecting white
Gaussian noise of which the Root Mean Square (RMS) is 10% of the signal power.
Then we use the Welch’s method to obtain the PSD of the polluted time series
and input these to the model trained with noise-free data for modal
identification. Table 6 reports the statistics of the MAC values of identified
mode shapes and the relative errors (%) of identified damping ratios and
natural frequencies on the noise-polluted testing set.

Table 6: Performance indicators on the testing set from the simply-supported structural population - adding 10% white Gaussian noise | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.994 | 0.005 | 0.973 | -0.066 | 0.040 | 0.141 | -11.717 | 8.775 | 37.612
Mode 2 | 0.944 | 0.076 | 0.610 | -2.657 | 4.607 | 17.549 | -8.856 | 7.669 | 38.872
Mode 3 | 0.872 | 0.131 | 0.278 | -0.249 | 6.113 | 15.956 | -8.752 | 9.126 | 37.960
Mode 4 | 0.876 | 0.149 | 0.330 | 1.515 | 6.260 | 15.042 | -7.107 | 11.747 | 47.254
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

The difference between Table 6 and Table 2 (no noise) demonstrates that the
introduction of measurement noise slightly reduces the accuracy of identifying
mode shapes and damping ratios. However, the average and maximum errors for
the identified natural frequencies increase more heavily. This discrepancy may
arise because mode shapes and damping ratios are relative measures, while
natural frequencies are absolute values. The addition of noise alters the
absolute amplitude of the resulting PSDs without perceptibly changing the
relative amplitude between the PSD of different nodes.

###  3.7 Effect of training set size

As mentioned above, the proposed population-based modal identification method
necessitates data from a group of structures to train the GNN-based model.
Intuitively, a larger training set provides more diversified data, which in
turn enhance the performance of the trained model. While this is easier to
configure when employing a simulated dataset, where populations are
artificially generated, the situation is different when aiming to use real-
world datasets for training. In real-world applications, the number of
monitored structures that can serve for model training is typically scarce,
while the number of available sensors is also limited. The latter challenge is
often tackled via using of mobile or wide-coverage (such as computer vision
measurements) sensor setups. In accounting for the first challenge, we here
investigate the impact of the training set size on the model’s performance. In
this section, we use the first 40 and 200 trusses (10% and 50% of the original
training set size) among the simply-supported population to train the model.
Accordingly, the training batch size is changed to 8 and 32 to accommodate the
shrinkage of the training set. The trained models are then used to identify
the modal properties of 100 trusses from the testing set, with the results
shown in Table 7 and Table 8.

Table 7: Performance indicators on the testing set from the simply-supported structural population - using 40 trusses for training | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.980 | 0.028 | 0.851 | -0.409 | 0.061 | 0.603 | 1.200 | 17.916 | 84.798
Mode 2 | 0.907 | 0.114 | 0.373 | 3.255 | 7.734 | 24.472 | -2.144 | 15.192 | 72.254
Mode 3 | 0.796 | 0.129 | 0.280 | 2.990 | 10.015 | 27.516 | -1.869 | 12.753 | 34.363
Mode 4 | 0.871 | 0.147 | 0.148 | 6.033 | 8.407 | 33.502 | -1.114 | 18.301 | 67.220
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.
Table 8: Performance indicators on the testing set from the simply-supported structural population - using 200 trusses for training | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.995 | 0.005 | 0.969 | -0.419 | 0.057 | 0.564 | 0.776 | 8.953 | 31.528
Mode 2 | 0.952 | 0.074 | 0.616 | 1.073 | 5.478 | 17.137 | 0.049 | 7.560 | 31.599
Mode 3 | 0.894 | 0.119 | 0.392 | 3.538 | 8.338 | 26.735 | -1.032 | 9.446 | 26.195
Mode 4 | 0.897 | 0.137 | 0.283 | 0.139 | 6.725 | 22.277 | -0.795 | 11.570 | 42.272
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

By comparing Table 7 (40 trusses for training), Table 8 (200 trusses for
training), and Table 2 (400 trusses for training), we find that:

  * •

As anticipated, reducing the size of the training set will decrease the
accuracy of modal identification. However, even with a training set that
amounts to only 40 trusses (10% of the original size), the decrease in
accuracy can be deemed acceptable. A training set that amounts to 50% of the
original size can produce results similar to those obtained with the full set.
Thus, while a larger training set improves modal identification, a smaller one
can still be effective when more data is unavailable.

  * •

Compared against mode shapes, the decrease in the size of the training set
induces higher errors in the identification of damping ratios and natural
frequencies, especially in terms of maximum relative errors. A possible
explanation to this is postulated in that mode shapes form spatial quantities,
whose nature is more graph-structured than natural frequencies and damping
ratios (which are global variables). Thus, the GNN can likely more robustly
tackle mode shape identification, even at smaller training sets.

###  3.8 Comparison against existing modal identification methods

To thoroughly evaluate our model with respect to modal identification, we
compare it against one of most widely adopted modal identification methods,
namely the Frequency Domain Decomposition (FDD) method ([51, 50]), which also
operates in the frequency domain. The original FDD method requires a manual
peak-picking process to determine the modes that will be identified. For the
population-based modal identification, however, manually picking peaks would
be too time-consuming given the high number of trusses in our testing set. To
tackle this problem, we couple the FDD with an automated function (in this
case the ‘findpeaks’ function in MATLAB) to automatically pick the peaks that
are corresponding to the first four structural modes. The FDD is subsequently
implemented to identify mode shapes, damping ratios, and natural frequencies
of the picked four modes. The MATLAB codes of our automated FDD method will be
made public in a dedicated GitHub repository ([69]) once this work is
published. Table 9 outlines the statistical indicators of FDD’s modal
identification performance on the testing set of the simply-supported
structural population.

Table 9: Performance indicators of FDD on the testing set from the simply-supported structural population | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 1.000 | 0.000 | 1.000 | 168.034 | 61.472 | 374.388 | -0.021 | 1.004 | 2.998
Mode 2 | 0.990 | 0.100 | 0.002 | 72.789 | 52.787 | 310.786 | -0.806 | 5.469 | 0.980
Mode 3 | 0.987 | 0.086 | 0.174 | 0.042 | 27.217 | 96.304 | -0.824 | 2.827 | 6.636
Mode 4 | 0.949 | 0.215 | 0.000 | -10.741 | 21.554 | 41.097 | -0.175 | 10.040 | 69.684
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

The comparison between Table 9 and Table 2 (the GNN-based model) shows that
our GNN-based model significantly outperforms the FDD in identifying damping
ratios. However, when it comes to identifying mode shapes and natural
frequencies, the FDD method shows substantial advantages over the GNN-based
model according to the average accuracy indicators. Nevertheless, due to
outliers caused by incorrectly picked peaks, the extreme errors of FDD result
significantly larger when compared against those of the GNN-based model. This
outcome can be attributed to the general effectiveness of FDD in identifying
natural frequencies and mode shapes, coupled with its typical limitations in
accurately identifying damping ratios.

Although the FDD is more accurate in general, our GNN-based model is much more
efficient when performing automated modal identification for a structural
population, since the FDD consumes substantial computation time to construct
the PSD matrix and conduct the singular value decomposition. For example, in
this numerical experiment we run both the FDD and our GNN-based model on the
same CPU. The FDD spends 1096.392 seconds identifying the modal properties of
100 trusses, whereas our GNN-based model completes the same task in just 9.402
seconds. Even if we count in the training time of the GNN-based model, which
is 443.838 seconds as shown in Table 1, our model remains faster than FDD.
Considering the training time would, however, be an unfair comparison, since
the intention (and main benefit) is to deploy the GNN-based modal
identification approach in testing mode. This advantage becomes even more
significant when dealing with larger structural populations, rendering the
proposed GNN-based model an efficient tool for population-based modal
identification.

###  3.9 Effect of different structural populations

Last but not least, we investigate the generalization capability of the
proposed model on different structural populations. We apply the model trained
on the simply-supported truss population, as shown in Figure 4(a), to the
cantilevered truss population, as shown in Figure 4(b). As an example, Figure
9 illustrates the identified mode shapes of a truss from the cantilevered
population. The modal identification results of all (100) trusses are
visualized in Figure 10, where each dot represents the identification result
for a certain mode of a truss. Table 10 statistically summarizes the
identification results illustrated in Figure 10.

Figure 9: Mode shape identification results of a truss from the cantilevered population |  |
---|---|---
(a) | (b) | (c)
Figure 10: Performance of the trained model on the cantilevered truss dataset: (a) Box plot of MAC values of identified mode shapes; (b) Scatter plot of identified damping ratios; (c) Scatter plot of identified natural frequencies Table 10: Performance indicators on the testing set from the cantilevered structural population | MAC of $\hat{\mathbf{\Phi}}$ | Errors (%) of $\hat{\mathbf{Z}}$ | Errors (%) of $\hat{\mathbf{F}}$
---|---|---|---
| Mean | SD | Min | Mean | SD | Max | Mean | SD | Max
Mode 1 | 0.732 | 0.062 | 0.581 | -0.180 | 0.938 | 9.496 | 151.939 | 106.071 | 559.304
Mode 2 | 0.874 | 0.074 | 0.625 | 48.659 | 16.558 | 97.241 | 30.832 | 44.656 | 213.919
Mode 3 | 0.730 | 0.128 | 0.446 | 10.032 | 9.172 | 33.428 | -12.744 | 25.746 | 108.983
Mode 4 | 0.826 | 0.141 | 0.302 | 11.352 | 9.787 | 45.934 | -5.295 | 20.704 | 83.603
*$\hat{\mathbf{\Phi}}$/$\hat{\mathbf{Z}}$/$\hat{\mathbf{F}}$: identified mode shapes/damping ratios/natural frequencies; SD: Standard Deviation.

The identification results on the cantilevered structural population are
clearly worse than those on the simply-supported structural population. This
disparity is expected, given the differences between the cantilevered and
simply-supported datasets, with the former not being utilized during the
training of the GNN-based model. Nonetheless, the identification results are
not entirely unsatisfactory. For instance, although the relative errors of
identified natural frequencies and damping ratios are high, as shown in Table
10, the identified values do not deviate significantly from their target
values as Figure 10 (b) and (c) shows. This indicates that the GNN-based model
exhibits a degree of generalization capability, enabling it to leverage
commonalities between two distinct yet similar structural populations.

##  4 Conclusions

To exploit the information contained within a structural population for
population-based structural health monitoring, this study develops an
automated operational modal analysis method capable of autonomously
identifying modal properties of structures among a population by using a GNN-
based deep learning model. A series of numerical experiments are carried out
to validate the proposed method. Main findings emerging from this study are
summarized as follows:

1) Model architecture. The model comparison suggests that the GraphSAGE model
outperforms the GCN and GAT model. The ablation study reveals that using an
MLP to encode the PSD input of the model is beneficial. More importantly, the
non-GNN model is inferior to all the three GNN-based models, confirming the
superiority of GNNs in processing graph-structured data.

2) Modal identification performance. Upon training on data from a structural
population built using the finite element method and model updating, the
proposed GNN-based model can efficiently and effectively identify natural
frequencies, damping ratios, and mode shapes of various structures in the same
structural population. The performance is robust even in the presence of
measurement noise and spatially sparse measurements. To tackle the issue of
measurement sparsity, the feature propagation algorithm is needed to
reconstruct the full-field node input features, specifically the PSD of node
acceleration, before using the proposed model to process incomplete
measurements.

3) Limitations. While the proposed model is significantly faster than the
classic frequency domain decomposition method and provides more accurate
damping ratio identification, it is less accurate in identifying natural
frequencies and mode shapes. Furthermore, when training data is insufficient,
or the testing structure belongs to a different population, the model accuracy
decreases, albeit still considered acceptable. This is expected due to the
purely data-driven nature of the proposed approach. Future research could
incorporate some physical information to enhance the model’s generalization
ability. A further limitation is that the model can only identify absolute
mode shapes, instead of signed mode shapes, owing to the use of the auto-PSD
that lacks phase information. While this is not prohibitive for SHM purposes,
additional node features besides PSD could be utilized to address this issue.
Lastly, it would be valuable to validate the proposed method with real-world
data.

#### Funding Statement

The research was conducted at the Singapore-ETH Centre, which was established
collaboratively between ETH Zurich and the National Research Foundation
Singapore. This research is supported by the National Research Foundation,
Prime Minister’s Office, Singapore under its Campus for Research Excellence
and Technological Enterprise (CREATE) program.

#### Competing Interests

The authors declare no competing interests exist.

#### Data Availability Statement

Demonstrative MATLAB and Python codes that implement the proposed method are
openly available at https://github.com/JxdEngineer.

#### Ethical Standards

The research meets all ethical guidelines, including adherence to the legal
requirements of the study country.

#### Author Contributions

Conceptualization: Eleni Chatzi; Xudong Jian. Methodology: Xudong Jian; Yutong
Xia; Gregory Duthé; Kiran Bacsa; Wei Liu. Data curation: Xudong Jian. Formal
analysis: Xudong Jian. Software: Yutong Xia. Data visualisation: Xudong Jian.
Writing - original draft: Xudong Jian. Writing - Review & Editing: Eleni
Chatzi; Xudong Jian; Kiran Bacsa; Yutong Xia; Gregory Duthé. Supervision:
Eleni Chatzi. Funding Acquisition: Eleni Chatzi. Project administration: Eleni
Chatzi. All authors approved the final submitted draft.

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