# Predicting Parking Availability in Singapore with Cross-Domain Data:
A New Dataset and A Data-Driven Approach

Huaiwu Zhang1* Yutong Xia2* Siru Zhong1 Kun Wang2 Zekun Tong2
Qingsong Wen3 Roger Zimmermann2 Yuxuan Liang1,4†
1The Hong Kong University of Science and Technology (Guangzhou), China
2National University of Singapore, Singapore
3Squirrel AI, USA
4State Key Laboratory of Resources and Environmental Information System, China
{huaiwuzhang1998, yutong.x, siruzhong<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The increasing number of vehicles highlights the need for efficient parking
space management. Predicting real-time Parking Availability (PA) can help
mitigate traffic congestion and the corresponding social problems, which is a
pressing issue in densely populated cities like Singapore. In this study, we
aim to collectively predict future PA across Singapore with complex factors
from various domains. The contributions in this paper are listed as follows:
(1) A New Dataset: We introduce the SINPA dataset, containing a year’s worth
of PA data from 1,687 parking lots in Singapore, enriched with various spatial
and temporal factors. (2) A Data-Driven Approach: We present DeepPA, a novel
deep-learning framework, to collectively and efficiently predict future PA
across thousands of parking lots. (3) Extensive Experiments and Deployment:
DeepPA demonstrates a 9.2% reduction in prediction error for up to 3-hour
forecasts compared to existing advanced models. Furthermore, we implement
DeepPA in a practical web-based platform to provide real-time PA predictions
to aid drivers and inform urban planning for the governors in Singapore. We
release the dataset and source code at https://github.com/yoshall/SINPA.

11footnotetext: H. Zhang and Y. Xia contributed equally to this
work.22footnotetext: Corresponding author. Email<EMAIL_ADDRESS>
##  1 Introduction

Urban traffic congestion significantly contributes to air pollution and poses
health risks Kelly and Fussell (2015); Liang et al. (2017). The uneven
distribution of parking spaces and inadequate parking decision-making by
drivers are significant factors contributing to traffic congestion Tilahun et
al. (2017). The challenge of finding parking spaces is expected to grow as the
population increases, thus the resulting social and environmental problems
will be especially more acute in densely populated countries such as Singapore
Shamsuzzoha et al. (2021). To address these problems, there is an urgent need
to optimize this issue by employing precise and efficient prediction of
Parking Availability (PA) using smart city technologies Xie et al. (2022).

Figure 1: Distribution of 1,687 parking lots throughout Singapore. Each lot
reports real-time parking availability every 15 minutes.

In this study, we focus on collectively predicting PA across thousands of
parking lots in Singapore, as shown in Figure 1. This undertaking is
confronted with several noteworthy challenges. Firstly, the availability of
parking spaces within a parking lot is intricately influenced by multiple
complex factors, including both temporal and spatial dimensions. Temporal
external factors (e.g., weather conditions) wield substantial influence in
shaping travel preferences and, consequently, parking patterns. Meanwhile,
spatial factors, such as land use and road density, exert a noteworthy impact
on the parking behavior of residents, consequently affecting PA. Therefore,
effectively capturing the effects of temporal and spatial factors on PA is a
necessary and formidable challenge.

Secondly, the _complex spatial correlations_ among parking lots exhibit a non-
linear relationship concerning geospatial distance measured in Euclidean
space. This non-linearity is further compounded by the inherent disparities in
their attributes, such as land functions and road network structures. In
essence, the PA patterns of two proximate parking lots may diverge
significantly due to distinctions in their intrinsic spatial characteristics.
Additionally, comprehending the interconnected dynamics among various parking
lots, influenced by the relatively stable total number of vehicles in the
region, is imperative for a better understanding of PA patterns.

Thirdly, an approach that captures long-range (i.e., global) spatial
dependencies Zhang et al. (2022) comes with another challenge in terms of
_computational efficiency_. This challenge is particularly pronounced at fine
spatial granularity, i.e., thousands of carparks in our task. Common methods
like Graph Convolutional NetworksKipf and Welling  capture global spatial
dependencies by stacking many local convolutional layers, which significantly
increases computational costs. More advanced methods Wu et al. (2019, 2020)
using an adaptive adjacency matrix also result in a high computational
complexity i.e., $\mathcal{O}(N^{2})$, and the computational load of methods
like Multi-head Self-Attention (MSA)Vaswani et al. (2017) also increases
quadratically with the number of nodes $N$. Therefore, efficiently modeling
the global spatial dependencies remains an unresolved issue.

Given the challenges outlined above, we aim to develop a model that not only
incorporates a variety of cross-domain factors but also effectively captures
the complex spatial correlations among parking lots to predict future PA
accurately. To achieve this, we first crawl and process the Singapore PA
dataset, i.e., SINPA, including PA data along with various external spatio-
temporal factors, such as meteorological and land use data, specific to
Singapore.

We then propose a data-driven model DeepPA, for collectively forecasting the
future PA readings across thousands of carparks in Singapore while maintaining
a manageable computational load. Specifically, we introduce the Graph Cosine
Operator (GCO), a new mechanism for dynamically capturing spatial dependencies
across extensive parking lots while mitigating computational complexity
through discrete cosine transform. In addition, considering the crucial role
of modeling temporal dependencies Liang et al. (2018); Guo et al. (2019);
Zheng et al. (2020), we innovatively treat temporal information as a distinct
“parking lot” entity. This allows us to integrate spatial data into position
encoding effectively. Then we utilize causal MSA Oord et al. (2016); Liang et
al. (2023) to adhere to temporal sequences, thereby facilitating the learning
of time-sensitive patterns in a manner that is both efficient and
interpretable.

In summary, our contributions lie in the following aspects:

  * •

A New Dataset. We crawl, process, and introduce SINPA, a large-scale parking
availability dataset incorporating cross-domain data in Singapore. To the best
of our knowledge, this dataset is the first publicly available dataset in the
field of PA values forecasting, providing diverse applications within spatio-
temporal domain research. The dataset is publicly accessible at
https://github.com/yoshall/SINPA. It includes data obtained from Data.gov.sg,
Urban Redevelopment Authority, and Land Transport Authority, which are made
available under the Singapore Open Data
Licence111https://beta.data.gov.sg/open-data-license.

  * •

A Data-Driven Approach. Leveraging insights gained from the analysis of multi-
domain features, we develop a simple yet effective deep learning framework,
DeepPA, to predict PA readings. DeepPA uses Graph Cosine Operator and Casual
MSA to capture the complex spatial and temporal correlations among parking
lots while ensuring an acceptable computational load.

  * •

Extensive Experiments. We evaluate DeepPA on our dataset. The empirical
results demonstrate the model’s high accuracy, efficiency, and adaptability.
These attributes make DeepPA an ideal candidate for real-world deployment for
real-time PA prediction applications.

##  2 Preliminary

Parking Availability (PA) refers to the remaining space for parking in a
parking lot. At the time $t$, the PA of the parking lots can be denoted as
$\mathbf{X}^{t}=\left\\{x^{t}_{1},x^{t}_{2},...,x^{t}_{N}\right\\}\in\mathbb{R}^{N}$,
where $N$ refers to the total number of parking lots. Each entry $x^{t}_{n}$
indicates the count of remaining parking spaces of $n$-th parking lots at time
$t$. Given the historical PA of all carparks from the past $T$ time steps and
corresponding external features, we aim to learn a function $f(\cdot)$ that
predicts their PA readings over the next $\tau$ steps:

$f(\mathbf{X}^{1:T};\mathbf{F}_{t}^{1:T},\mathbf{F}_{s})\rightarrow\mathbf{Y}^{1:\tau},$
(1)

where $\mathbf{X}^{1:T}\in\mathbb{R}^{T\times N}$ indicates historical PA of
all carparks;
$\mathbf{F}^{1:T}_{t}=\left\\{F^{1:T}_{mete},F^{1:T}_{time}\right\\}\in\mathbb{R}^{T\times
C_{t}}$ denotes external temporal features and $C_{t}$ represent its
dimension;
$\mathbf{F}_{s}=\left\\{F_{loc},F_{pln},F_{use},F_{rd}\right\\}\in\mathbb{R}^{N\times
C_{s}}$ denotes the spatially related feature and $C_{s}$ represent its
dimension; $\mathbf{Y}^{1:\tau}\in\mathbb{R}^{\tau\times N}$ is the future
PAs.

###  2.1 Related Work

####  2.1.1 Intelligent Parking

Intelligent parking has attracted considerable attention from both academia
and industry. Regarding the prediction of PA, Sii-MOBILITYBadii et al. (2018)
integrates weather and traffic flow predicting PA based on Bayesian
regularization neural networks. Subsequently, Zhang et al. (2020b) proposes a
novel method SHARE based on Gated Recurrent UnitsChung et al. (2014) and
contextual graph convolution to capture both local and global spatial
dependencies within parking lots, enabling estimation of missing PA in both
temporal and spatial dimensions.

In addition, intelligent parking systems have other downstream tasks. Rong et
al. (2018a) employs Long Short-Term Memory (LSTM) to model the PA-related
temporal closeness, period, and current general influence to achieve parking
availability imputation. Utilizing PA prediction combined with recommendation
algorithms suggested the optimal parking lots near the user’s specified
destination Xu et al. (2023). Moreover, models such as Random Forest and LSTM
have identified the influence of weather factors on drivers’ parking behavior
Zhang et al. (2020a). Besides, unlike enclosed parking lots that are
independent of the street, it has also achieved the prediction of the
availability of curbside parking spaces Roman et al. (2018).

Figure 2: The framework of our proposed DeepPA. Initially, historical PA
readings and spatio-temporal information are encoded and mapped into hidden
states, respectively. Subsequently, PA will interact with spatio-temporal
information in two sequential stages: (a) Spatial Learning Block (SLBlock)
captures spatial dependence among parking lots, in which we introduce a novel
structure Graph Cosine Operator (GCO) to enhance interaction efficiency; (b)
TLBlock simultaneously investigates the temporal dynamics of PA and temporal
information adhering to temporal patterns through Causal Multi-Head Attention
(Causal MSA). Finally, DeepPA will integrate the feature space to forecast
future PA. Concat: Concatenate. Info:Information

####  2.1.2 Data-Driven Spatio-Temporal Forecasting

Spatio-Temporal (ST) forecasting has garnered significant attention in the
past few years Jin et al. (2023). Conventional models, e.g., ARIMA and VAR,
have enjoyed widespread adoption, while their performance tends to be
suboptimal with highly volatile datasetsLiang et al. (2018); Guo et al.
(2019); Zheng et al. (2020); Liang et al. (2021). Building upon Graph Neural
Networks (GNN) Kipf and Welling , Spatio-Temporal Graph Neural Networks
(STGNN) Wang et al. (2021) have emerged for ST data by integrating temporal
components, e.g. Temporal Convolutional Networks Bai et al. (2018) or
Recurrent Neural Networks Graves (2013). Pioneering examples include DCRNN Li
et al. (2018), STGCN Yu et al. (2018), and ST-MGCN Geng et al. (2019).
Subsequent studies GWNet Wu et al. (2019) and AGCRN Bai et al. (2020),
leverage adaptive adjacency matrices to enhance predictive performance.
Meanwhile, ASTGCN Guo et al. (2019), GMAN Zheng et al. (2020), and STTNXu et
al. (2020) employ attention mechanisms to discern dynamic ST dependencies.
STGODE Fang et al. (2021), STGNCDE Choi et al. (2022), and MixRNN Liang et al.
(2022) capture continuous ST dynamics by employing neural ordinary
differential equations. Causality-based methods such as CaST Xia et al. (2024)
resolve the distribution shift problem via causal tools. Nonetheless, these
methods are not tailored to the parking problem while concurrently addressing
the above-mentioned challenges.

##  3 Methodology

The spatial and temporal external factors (i.e., weather and geo-location)
play an important role in PA forecasting. To address the challenges of complex
spatial correlations and computational efficiency, we have developed DeepPA, a
deep learning framework for efficiently predicting future PA throughout
Singapore.

###  3.1 Overall Framework

To address the challenges of complex spatial correlations and computational
efficiency, we develop DeepPA, a deep learning framework for efficiently
predicting future PA throughout Singapore, shown in Figure 2. We first
transform the historical PA $\mathbf{X}^{1:T}$, along with spatial features
$\mathbf{F}_{s}$ and temporal features $\mathbf{F}_{t}^{1:T}$ into the latent
feature space using encoders, implemented through Multi-Layer Perceptrons
(MLPs) and embedding layers. The hidden features are then fed to $L$ DeepPA
blocks to interact with each other to capture ST dependencies within data.
Each DeepPA block consists of two sub-blocks as follows.

  * •

_Spatial Learning Block (SLBlock)_ : This module is utilized to capture
intricate spatial relationships between parking lots, adopting a unique
perspective by treating time information as a new node. To efficiently capture
the non-Euclidean spatial relationship between thousands of parking lots, we
introduce a novel Graph Cosine Operator.

  * •

_Temporal Learning Block (TLBlock)_ : This transformer-structure module is
dedicated to capturing the periodic patterns that emerge over time at each
parking lot. Specifically, it simultaneously model the hidden state of PA
$\mathbf{H}^{1:T}$ and temporal features $\mathbf{F}_{t}^{1:T}$. To preserve
the sequential order, we incorporate a mask in the MSA mechanism, ensuring
that each state focuses only on its preceding states.

###  3.2 Spatial Learning Block (SLBlock)

Considering PA across different parking lots is interdependent, we aim to
develop a module that effectively captures the specific spatial dependencies
among thousands of parking lots while ensuring lower computational costs.

####  3.2.1 Overview

As depicted in the left part of Figure 2,
$\mathbf{H}_{s}\in\mathbb{R}^{B\times T\times N\times C_{hs}}$ is concatenated
with the spatial position encoding $\mathbf{P_{s}}\in\mathbb{R}^{B\times
T\times N\times(C_{p}-C_{hs})}$ and then added to $\mathbf{H}^{1:T}_{pa}$:

$\mathbf{H}^{1:T}=\mathbf{H}^{1:T}_{pa}+\operatorname{concat}(\mathbf{H}_{s},\mathbf{P_{s}}),\mathbf{H}^{1:T}\in\mathbb{R}^{B\times
T\times N\times C_{p}}$ (2)

where $\mathbf{H}^{1:T}$ is the PA with integrated the spatial features. In
order to achieve interaction with temporal information, we then map the
temporal information to the same dimension $C_{p}$ as $\mathbf{H}^{1:T}$,
treating it as an auxiliary node, i.e., a virtual node which acts as the
$(N+1)$-th parking lot, and concatenating it with $\mathbf{H}^{1:T}$:

$\mathbf{H}^{1:T}=\operatorname{concat}(\mathbf{H}^{1:T},\mathbf{H}^{1:T}_{t}),\mathbf{H}^{1:T}\in\mathbb{R}^{BT\times(N+1)\times
C_{p}}$ (3)

where $\mathbf{H}^{1:T}$ represents the PA that integrates spatio-temporal
information. After obtaining the initial features from both spatial and
temporal domains, we jointly capture local and global spatial correlations of
$\mathbf{H}^{1:T}$ via our proposed _Graph Cosine Operator (GCO)_ by
$\mathbf{\hat{H}}^{1:T}=\operatorname{GCO}(\mathbf{H}^{1:T})$, which will be
described in the subsequent part.

####  3.2.2 Graph Cosine Operator (GCO)

Since dynamic adjacency matrix or attention mechanism in our dataset is
computationally extensive, we design GCO to mitigate this issue by simulating
how spatial dependencies induce changes $\Delta\mathbf{\hat{H}}^{1:T}$ in the
hidden state of the parking lot. For a single parking lot, the variation of PA
at each moment is known to be affected by all other parking lots, which can be
expressed by the _heat conduction formula_ Wang et al. (2007):

$\Delta\hat{H}^{t}_{i}=\sum k_{ji}(\hat{H}^{t}_{j}-\hat{H}^{t}_{i})\ \ (i,j\in
N\ ,t\in T)$ (4)

where $k_{ji}$ represents the thermodynamic coefficient, which, in our study,
signifies a learnable correlation coefficient between ${\hat{H}}^{t}_{j}$ and
${\hat{H}}^{t}_{i}$. Thus, at the same time instant $t$, the changes in the
hidden states of all parking lots, i.e., $\Delta\mathbf{\hat{H}}^{t}$, can be
represented as follows.

$\displaystyle\Delta\mathbf{\hat{H}}^{t}=\left[\begin{array}[]{c}\Delta\hat{H}^{t}_{1}\\\
\Delta\hat{H}^{t}_{2}\\\ ...\\\
\Delta\hat{H}^{t}_{N}\end{array}\right]=\left[\begin{array}[]{c}\sum
k_{j1}(\hat{H}^{t}_{j}-\hat{H}^{t}_{1})\\\ \sum
k_{j2}(\hat{H}^{t}_{j}-\hat{H}^{t}_{2})\\\ ...\\\ \sum
k_{jN}(\hat{H}^{t}_{j}-\hat{H}^{t}_{N})\end{array}\right]$ (5)

Since the number of vehicles remains essentially constant throughout the
entire range, we can assume that the correlation between two parking lots is
consistent, i.e., $k_{ji}=k_{ij}$. After applying the distributive property to
Eq. 5, we can obtain the following expression:

$\displaystyle\Delta\mathbf{\hat{H}}^{t}=\left[\begin{array}[]{ccc}k_{11}&\cdots&k_{1N}\\\
\vdots&\ddots&\vdots\\\
k_{N1}&\cdots&k_{NN}\end{array}\right]\mathbf{\hat{H}}^{t}-\left[\begin{array}[]{cccc}K_{1}&&&\\\
&K_{2}&&\\\ &&...&\\\ &&&K_{N}\end{array}\right]\mathbf{\hat{H}}^{t}$ (6)

where $K_{i}$ represents $\sum k_{ji}$, $j\in N$. It can be observed that the
two learnable matrices in the equation are respectively a symmetric matrix and
a diagonal matrix. Hence, they can be regarded as a special form of degree
matrix $\mathbf{\hat{D}}$ and adjacency matrix $\mathbf{\hat{A}}$. Therefore,
Eq. 6 can be expressed in the following form:

$\displaystyle\Delta\mathbf{\hat{H}}^{t}=(\mathbf{\hat{D}}-\mathbf{\hat{A}})\mathbf{\hat{H}}^{t}=\mathbf{\hat{L}}\mathbf{\hat{H}}^{t}=\sigma\mathbf{L}\mathbf{\hat{H}}^{t}$
(7)

where $\mathbf{L}$ represents the Laplacian matrix Merris (1994), We aim to
fit $\mathbf{\hat{L}}$ by adding a set of trainable weights $\sigma$ to
$\mathbf{L}$. Let $\lambda$ and $\mathbf{v}$ be the eigenvalues and
eigenvectors of $\mathbf{L}$, respectively. Then Eq. 7 can be expressed as
follows:

$\displaystyle\mathbf{\hat{L}}\mathbf{\hat{H}}^{t}=\sigma\mathbf{L}\mathbf{v}=\sigma\lambda\mathbf{v}$
(8)

Left-multiplying by the Laplacian matrix is equivalent to solving for the
second derivative Koren (2003). Therefore, for Fourier bases, the
transformation can be denoted as follows:

$\sigma\mathbf{L}e^{-j\omega t}=\sigma\frac{\partial^{2}e^{-j\omega
t}}{\partial t^{2}}=\sigma(-\omega^{2})e^{-j\omega t}$ (9)

where $-\omega^{2}$ represents the the eigenvalues of the Laplacian matrix.
Defining a set of eigenvalues of $\mathbf{L}$ as:

$\Lambda=(\lambda_{1},\lambda_{2},...,\lambda_{N})\in(-\omega^{2})$ (10)

The corresponding unit eigenvectors are denoted as:

$\mathbf{U}=(\vec{u_{1}},\vec{u_{2}},...,\vec{u_{N}})\in e^{-j\omega t}$ (11)

Thus Eq. 8 can be rewritten as:

$\mathbf{\hat{L}}\mathbf{\hat{H}}^{t}=\sigma\mathbf{U}\Lambda\mathbf{U}^{T}\mathbf{\hat{H}}^{t}=\sigma(\mathcal{F}^{-1}(\mathcal{F}(\Lambda)\cdot\mathcal{F}(\mathbf{\hat{H}}^{t}))$
(12)

where $\mathcal{F}(\cdot)$ represents Discrete Fourier Transform (DFT).
Inspired by Fourier Neural Operator (FNO)Li et al. (2020), we parameterize
$\sigma$ and $\mathcal{F}(\Lambda)$ via MLP layers. Previous work employed
neural operators to approximate the MSAGuibas et al. (2021). According to
Euler’s formula, the Fourier transform maps data from the time domain to the
frequency domain, converting real numbers into complex numbers:

$e^{-j\omega t}=\cos{\omega t}+j\sin{\omega t}$ (13)

After this transformation, complex numbers are decomposed into their real and
imaginary components. When multiplied by weights, it is crucial to adhere to
the distributive property of multiplication, thus MLPs exhibit a fourfold
increase in computational time when operating on complex numbers compared to
real numbers. In the case of retaining only the real part, we use the Discrete
Cosine Transform (DCT) to approximate DFT to reduce computational complexity
while preserving performance efficacy, as demonstrated in Table 2. For a
clearer comprehension, the overall operational steps of GCO for
$\mathbf{H}^{1:T}$ are as illustrated in Algorithm 1.

Algorithm 1 Graph Cosine Operator

0: $\mathbf{H}^{1:T}\in\mathbb{R}^{BT\times(N+1)\times C_{p}}$;

0: Updated hidden states $\mathbf{\hat{H}}^{1:T}$;

1: Step 1: Projection into the frequency domain.
$\mathbf{\hat{H}}^{1:T}=\text{DCT}(\mathbf{H}^{1:T})$

2: Step 2: Fitting the basis functions.
$\mathbf{\hat{H}}^{1:T}=\text{MLP}(\mathbf{\hat{H}}^{1:T})$

3: Step 3: Returning to the spatial domain.
$\mathbf{\hat{H}}^{1:T}=\text{iDCT}(\mathbf{\hat{H}}^{1:T})$

4: Step 4: Normalizing the output.
$\mathbf{\hat{H}}^{1:T}=\text{LayerNorm}(\mathbf{\hat{H}}^{1:T})$

5: Step 5: Fitting the Laplacian matrix.
$\mathbf{\hat{H}}^{1:T}=\text{MLP}(\mathbf{\hat{H}}^{1:T})$

####  3.2.3 Discussion

SLBlock is designed by jointly considering the following factors: 1) _Spatial
dependencies_. Given that the strength of mutual influence between stations is
not solely dependent on distance, we introduced a dynamic correlation
coefficient $k_{ji}$ to calculate the relationship between parking lots. 2)
_Efficiency_. Via the GCO, we have managed to reduce the computational
complexity. 3) _Temporal factor interation_ : We treat the overall temporal
information as a new node interacting with the PA to take temporal external
factors into account.

###  3.3 Temporal Learning Block (TLBlock)

Accurately capturing the periodic relationship is crucial for the prediction
of PA. Meanwhile, variations in weather conditions and the impact of holidays
can alter the travel patterns of residents, thereby impacting this
periodicity. Therefore, we designed a novel module TLBlock to capture the
periodicity while taking into account the impact of special time variables.

####  3.3.1 Capturing temporal correlation

First, to enable position-aware MSA, we infuse learnable position encoding
into the input PA of TLBlock. Then, we decouple $\mathbf{\hat{H}}^{1:T}_{t}$
and $\mathbf{\hat{H}}^{1:T}$, where $\mathbf{\hat{H}}^{1:T}$ is considered as
the $(N+1)$th parking lot in the SLBlock:
$\mathbf{\hat{H}}^{1:T},\mathbf{\hat{H}}^{1:T}_{t}=\operatorname{split}(\mathbf{\hat{H}}^{1:T})$.
Next, we merge the $B$ and $N$ dimensions of $\mathbf{\hat{H}}^{1:T}$ and
concatenate them with $\mathbf{\hat{H}}^{1:T}_{t}$:

$\mathbf{\hat{H}}^{1:T}=\operatorname{concat}(\mathbf{\hat{H}}^{1:T},\mathbf{\hat{H}}^{1:T}_{t}),\mathbf{\hat{H}}^{1:T}+\mathbf{P_{t}}\in\mathbb{R}^{(BN+B)\times
T\times C_{p}}$

where $\mathbf{P_{t}}$ represents temporal position encoding. Subsequently,
temporal factors are modeled alongside PA using the same encoder to enable the
model will take into account the influence of temporal information to a
greater extent.

####  3.3.2 Temporal causality

Given that the PA at a given step is not contingent on its future, inspired by
Oord et al. (2016); Liang et al. (2023), we introduce causality into MSA to
ensure the model adheres to the temporal sequence of the input data,
implemented by masking specific entries in the attention map:

$\displaystyle\mathbf{Q}=\mathbf{W_{q}}\mathbf{\hat{H}}^{1:T};\quad\mathbf{K}=\mathbf{W_{k}}\mathbf{\hat{H}}^{1:T};\quad\mathbf{V}=\mathbf{W_{v}}\mathbf{\hat{H}}^{1:T}$
(14) $\displaystyle\text{Causal
MSA}(\mathbf{\hat{H}}^{1:T})=\text{softmax}(\frac{\mathbf{Q}\mathbf{K}^{T}}{\sqrt{\alpha}}+\mathbf{M})\mathbf{V}$

For a single state, $\hat{H}^{i}$, applying a mask
$\mathbf{M}\in\mathbb{R}^{\alpha\times T\times T\times C_{p}/\alpha}$ involves
setting its attention weights towards temporally later states $\hat{H}^{j}$ to
$-\inf$, where $j\in(t+1,T)$.

###  3.4 Prediction & Optimization

DeepPA makes predictions based on the hidden state of the last DeepPA block
$\mathbf{\hat{H}}^{1:T}$ using a predictor, which is implemented via an MLP
layer in our implementation:

$\mathbf{\hat{Y}}^{1:\tau}=\operatorname{MLP}(\mathbf{\hat{H}}^{1:T}).$ (15)

To train our model, we minimize the loss function $\mathcal{L}_{MAE}$. Here,
$\mathcal{L}_{MAE}$ is the Mean Absolute Error for evaluating the errors
between our prediction $\mathbf{\hat{Y}}^{1:\tau}$ and the corresponding
ground truth $\mathbf{{Y}^{1:\tau}}$.

##  4 Experiments

We then conduct a series of experiments on SINPA to validate the efficacy of
DeepPA. These experiments are designed to address the following Research
Questions (RQ):

  * •

RQ1: How does DeepPA compare in performance to existing PA forecasting
approaches?

  * •

RQ2: How does each module within DeepPA contribute to improving overall model
performance?

  * •

RQ3: What efficiency gains are achieved through the implementation of the GCO
module?

  * •

RQ4: Can DeepPA be effectively applied in practical scenarios for real-time
online prediction?

Model | #Param | Time / | Average (1$\sim$12 steps) | 0-1h (1$\sim$4 steps) | 1-2h (5$\sim$8 steps) | 2-3h (9$\sim$12 steps)
---|---|---|---|---|---|---
epo (s) $\downarrow$ | MAE $\downarrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | RMSE $\downarrow$ | MAE $\downarrow$ | RMSE $\downarrow$
HA | - | - | 45.28 | 69.68 | 44.41 | 68.11 | 43.68 | 67.89 | 45.62 | 71.44
VAR | - | - | 54.12 | 105.64 | 53.10 | 93.98 | 56.86 | 94.46 | 55.71 | 130.00
DCRNN | 392 | 896 |  12.68$\pm$0.02 |  30.33$\pm$0.05 |  9.53$\pm$0.04 |  27.16$\pm$0.06 |  12.80$\pm$0.04 |  30.04$\pm$0.09 |  15.71$\pm$0.03 |  33.83$\pm$0.06
STGCN | 583 | 221 |  12.70$\pm$0.11 |  31.17$\pm$0.20 |  9.64$\pm$0.06 |  27.71$\pm$0.09 |  12.94$\pm$0.18 |  31.03$\pm$0.23 |  15.65$\pm$0.23 |  34.84$\pm$0.31
GWNET | 808 | 303 |  12.76$\pm$0.21 |  31.14$\pm$0.18 |  9.54$\pm$0.14 |  27.73$\pm$0.15 |  12.91$\pm$0.27 |  30.88$\pm$0.26 |  15.84$\pm$0.28 |  34.86$\pm$0.16
MTGNN | 207 | 361 |  12.41$\pm$0.24 |  30.62$\pm$0.18 |  9.31$\pm$0.16 |  27.37$\pm$0.10 |  12.50$\pm$0.25 |  30.37$\pm$0.25 |  15.42$\pm$0.34 |  34.16$\pm$0.33
ASTGCN | 11,587 | 540 |  12.06$\pm$0.23 |  30.22$\pm$0.31 |  9.88$\pm$0.12 |  28.37$\pm$0.19 |  12.27$\pm$0.24 |  30.21$\pm$0.32 |  14.02$\pm$0.34 |  32.10$\pm$0.41
Du-Parking | 392 | 683 |  12.55$\pm$0.04 |  30.41$\pm$0.05 |  9.22$\pm$0.02 |  27.09$\pm$0.07 |  12.62$\pm$0.04 |  30.12$\pm$0.08 |  15.80$\pm$0.07 |  34.05$\pm$0.08
SHARE | 476 | 402 |  12.57$\pm$0.05 |  31.02$\pm$0.21 |  9.33$\pm$0.04 |  27.46$\pm$0.03 |  12.70$\pm$0.05 |  30.74$\pm$0.17 |  15.69$\pm$0.07 |  34.90$\pm$0.43
DeepPA (ours) | 197 | 219 | 10.95*$\pm$0.10 | 29.54*$\pm$0.06 | 9.11*$\pm$0.09 | 28.00$\pm$0.07 | 11.18*$\pm$0.09 | 29.57*$\pm$0.08 | 12.56*$\pm$0.15 | 31.06*$\pm$0.11
Table 1: 5-run results. The bold/underlined font means the best/the second
best result. #Param: the number of parameters, expressed in thousands (Kilo).
* denotes the improvement over the second best model is statistically
significant at level 0.05 .

###  4.1 Dataset Description

We crawled over three-year real-time PA data (2018/03/26 to 2021/08/11) every
5 minutes from 1,921 parking lots throughout Singapore from
Data.gov.sg222https://data.gov.sg/. For our experiments, we re-sampled the raw
dataset into the 15-minute interval and chose lots with a missing rate of PA
of less than 30%. In addition, due to the temporal distribution shift, we only
use one-year data (2020/07/01 to 2021/06/30), and the ratio of training:
validation: testing sets is set as 10:1:1. We then remove parking lots with
obvious distribution shift (i.e., high KL divergence). After sample filtering,
it remains 1,687 parking lots with stationary data distributions. We also
crawl external attributes for these lots, including meteorological data (i.e.,
temperature, humidity, and wind speed), panning areas, utilization type, and
road networks data from Data.gov.sg, the Urban Redevelopment Authority
(URA)333https://www.ura.gov.sg/ and the Land Transport Authority (LTA)
website444https://datamall.lta.gov.sg/content/datamall/en.html.

###  4.2 Experimental Settings

We implement our model by PyTorch 1.10 using a Quadro RTX A6000 GPU. The Adam
optimizer is utilized to train our model, and the batch size is 8. The
learning rate starts from $1\times 10^{-3}$, halved every three epochs. For
the hidden dimension $C$ in SLBlock and SLBlock, we conduct a grid search over
$\left\\{8,16,32,64,128\right\\}$, and $C$ = 64 obtains the best result. The
block number of SLBlock and TLBlock is 2.

We compare our DeepPA with the following baselines that belong to the
following four categories: (1) _Classical methods_ : HA Zhang et al. (2017)
and VAR Toda (1991). (2) _STGNN variants_ : DCRNN Li et al. (2018), STGCN Yu
et al. (2018), GWNET Wu et al. (2019), and MTGNN Wu et al. (2020). (3)
_Attention-based models_ : ASTGCN Guo et al. (2019). (4) _Deep learning for PA
prediction_ : SHARE Zhang et al. (2020b) and Du-parking Rong et al. (2018b).

Note that some strong baselines STTNXu et al. (2020) GMANZheng et al. (2020),
are omitted due to Out-Of-GPU-Memory (OOM) caused by a large number of
carparks within SINPA.

###  4.3 Model Comparison (RQ1)

To address RQ1, following previous studies, we perform a model comparison in
terms of Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE)Jean et
al. (2016). We run each method five times and report the average metric of
each model. As shown in Table 1, DeepPA significantly outperforms all
competing baselines on both metrics according to the Student’s T-test at level
0.05.

In terms of performance, we can observe that: 1) Deep-learning-based
approaches show a substantial improvement over classical methods like HA and
VAR due to their enhanced learning capacity. 2) Our proposed model exhibits a
pronounced advantage over the SHARE and Du-parking models in the domain of
parking space prediction, thereby providing additional validation for the
influential role of the collected spatio-temporal information on PA
prediction. 3) In comparison to the second-ranked approach (i.e., ASTGCN),
DeepPA demonstrates a remarkable 9.2% decrease in MAE for long-term future
prediction within the time interval of 3h. 4) Shifting the focus to individual
hour predictions, the performance improvements for a single hour are observed
to be 0.1%, 8.8%, and 10.4%, respectively.

Moreover, we have evaluated the efficiency of each method by reporting both
the number of parameters and the training time per epoch, as detailed in Table
1. Our proposed model stands out due to its minimal parameter count and the
fastest training duration among all baselines. This underscores our model’s
suitability for practical deployment: the reduced parameter requirement
minimizes memory demands, making it ideal for resource-constrained
environments; the rapid training capability highlights its ability to quickly
adapt to user needs, significantly enhancing user experience.

These attributes collectively make our model an optimal choice for real-world
applications where accuracy, efficiency, and adaptability are of paramount
importance.

###  4.4 Ablation Study (RQ2)

####  4.4.1 Effects of SLBlock

To investigate the effects of SLBlock, we consider the following variables for
comparison: a) w/o SLBlock: we replace SLBlock with regular convolution solely
operating on the PA. b) w/o spatial: we do not treat spatial information as a
distinct entity for interaction with the PA. c) w/o temporal: we do not
incorporate time information as a separate node for interaction with PA. The
results (see Figure 3) reveal that removing either SLBlock or the ST
information within the module leads to a significant decrease in MAE,
underscoring the critical role of modeling spatial dependencies.

Figure 3: Effects of SLBlock on MAE.

####  4.4.2 Effects of TLBlock

We then compare our model with its variants integrated with various temporal
modules to assess the effectiveness of our TLBlock as well as the components
in it: a) w/o TLBlock: we replace TLBlock with regular convolution solely
operating on the PA. b) MSA: we replace Causal MSA with vanilla MSA. c) w/o
PE: we remove the temporal position encoding from TLBlock. The results (see
Figure 4) show that all the variants incorporating temporal modules
demonstrate superior performance compared to the variants without TLBlock,
confirming the importance of temporal information.

Figure 4: Effects of TLBlock on MAE.

####  4.4.3 Hyperparameter Study

We explore the effects of the number of DeepPA blocks $L$ and the hidden
dimension $C$. First, we fix the block size at 2 and increase the hidden
dimension starting from 8. Results in Figure 5a show that once the hidden
dimension reached 64, the performance did not improve further. We thus set $C$
as 64 and increase $L$ starting from 2 (see Figure 5b) and found that when $L$
increases from 2 to 4, there is no significant improvement in the performance.
However, the training time per epoch increases by nearly double. Furthermore,
when $L$ exceeds 4, the model’s performance deteriorates sharply, possibly due
to overfitting and other factors.

Figure 5: Effects of (a) hidden dimension $C$ and (b) the number of blocks
$L$.

###  4.5 Efficiency Study (RQ3)

To address RQ3, we evaluate the efficiency gains achieved by the GCO compared
with other methods, then examine how the performance varies with the retention
of different low-frequency signals.

####  4.5.1 Efficiency of GCO

To investigate the efficiency improvements introduced by the GCO, we replace
this module with the following modules: a) MSA: while effective in capturing
relationships across all parking lots, MSA is known for its high theoretical
time complexity. b) AFNO: conceptually similar to GCO, AFNO Guibas et al.
(2021) uses neural operators to approximate MSA. Previous research has shown
its effectiveness in addressing the high time complexity issues in Vision
Transformers with increased image resolution. Table 2 shows the training time
per epoch for each variant. Compared to MSA and AFNO, GCO not only maintains
performance but also significantly improves efficiency. Notably, after
removing the SLBlock, the model’s training time per epoch is around 115s,
indicating that GCO reduces the training time by about 3.20 times and 2.37
times compared to MSA and AFNO, respectively, demonstrating GCO’s efficient
capability.

Table 2: Comparative efficiency of GCO against other methods, measured in MAE. #Param: the number of parameters, expressed in thousands (Kilo). Variant | 1-3h | 0-1h | 1-2h | 2-3h | #Param | Time(s)
---|---|---|---|---|---|---
MSA | 10.87 | 9.07 | 11.10 | 12.44 | 227 | 448
AFNO | 10.90 | 9.11 | 11.12 | 12.48 | 199 | 361
GCO | 10.95 | 9.11 | 11.18 | 12.56 | 197 | 219

###  4.6 Practicality (RQ4)

Regarding RQ4, leveraging the demonstrated efficiency and accuracy of our
model, we deploy it for real-time parking availability forecasting. An
illustrative example of our system in action is shown in Figure 6. The web is
built on the Mapbox platform Rzeszewski (2023). After selecting the target
parking space, the system will predict future PA and present it in the form of
a line graph. The shown graph displays a notable correlation between the
predicted (represented by the blue line) and actual (represented by the red
line) PA readings, demonstrating our model’s high accuracy in predicting PA
trends and variations effectively. For a direct experience of this interactive
forecasting tool, please visit our demo web-based platform at
https://sinpa.netlify.app.

Figure 6: Our web-based PA monitoring and prediction system.

##  5 Conclusion and Future Work

In this paper, we crawled, processed, and released the SINPA dataset, a
comprehensive collection of one-year PA readings from over 1,600 parking lots
across Singapore, along with various external factors. To the best of our
knowledge, SINPA is the first dataset of its kind to be made publicly
available. Considering the complex relationships and the important role of
external features, we presented DeepPA, a deep-learning model tailored for PA
prediction. Our evaluations demonstrate the accuracy, efficiency, and
adaptability of our model. In the future, we plan to explore reinforcement
learning to improve parking recommendation services.

## Acknowledgement

This work is mainly supported by a grant from State Key Laboratory of
Resources and Environmental Information System and the Guangzhou-HKUST(GZ)
Joint Funding Program (No. 2024A03J0620). This work is also funded by the
Advanced Research and Technology Innovation Centre (ARTIC), the National
University of Singapore under Grant (project number: A-8000969-00-00).

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