Dynamic Spatio-Temporal Adaptive Graph Convolutional Recurrent Networks for Vacant Parking Space Prediction

Abstract

The prediction of vacant parking spaces (VPSs) can reduce the time drivers spend searching for parking, thus alleviating traffic congestion. However, previous studies have mostly focused on modeling the temporal features of VPSs using historical data, neglecting the complex and extensive spatial characteristics of different parking lots within the transportation network. This is mainly due to the lack of direct physical connections between parking lots, making it challenging to quantify the spatio-temporal features among them. To address this issue, we propose a dynamic spatio-temporal adaptive graph convolutional recursive network (DSTAGCRN) for VPS prediction. Specifically, DSTAGCRN divides VPS data into seasonal and periodic trend components and combines daily and weekly information with node embeddings using the dynamic parameter-learning module (DPLM) to generate dynamic graphs. Then, by integrating gated recurrent units (GRUs) with the parameter-learning graph convolutional recursive module (PLGCRM) of DPLM, we infer the spatio-temporal dependencies for each time step. Furthermore, we introduce a multihead attention mechanism to effectively capture and fuse the spatio-temporal dependencies and dynamic changes in the VPS data, thereby enhancing the prediction performance. Finally, we evaluate the proposed DSTAGCRN on three real parking datasets. Extensive experiments and analyses demonstrate that the DSTAGCRN model proposed in this study not only improves the prediction accuracy but can also better extract the dynamic spatio-temporal characteristics of available parking space data in multiple parking lots.

1. Introduction

Parking problems have become one of the main problems encountered in the development of smart cities. It has been reported that more than one-third of urban traffic congestion is caused by the search for vacant parking spaces (VPSs) [1,2]. With the development of communication technology, smart cities can collect massive amounts of parking data through cameras, sensors, etc. [3]. Based on historical parking data, parking prediction [4] combined with existing parking management and guidance systems can help drivers find VPSs in a shorter time, thereby improving overall traffic efficiency. Therefore, the development of smart cities requires accurate and efficient parking prediction methods.

Parking prediction aims to predict the number of future VPSs based on historical data, which is a spatio-temporal problem with complex time and space dependencies [5]. First, in terms of time dependency, the number of VPSs in a single parking lot exhibits intricate periodicity, such as variations in parking demand between weekdays and weekends. Second, as illustrated in Figure 1, parking lots are distributed across the road network, and different parking lots possess spatial heterogeneity (e.g., distinct road types, POIs, population distribution) depending on their location, such as residential or commercial areas. Furthermore, when many vehicles are parked in a parking lot around a destination, people often choose to park their vehicles in another parking lot within a tolerable distance from the destination. However, previous VPS prediction methods mainly focused on the time dependency of parking data, neglecting the spatial dependencies among parking lots.

In recent years, graph neural networks have been widely used in traffic prediction due to their ability to capture spatial correlations among network nodes [6]. The distribution of parking lots in road networks is typically non-Euclidean, and graph neural networks have advantages in handling non-Euclidean structured data. To model the complex spatio-temporal correlations between multiple parking lots, we propose a dynamic spatio-temporal adaptive graph convolutional recursive network (DSTAGCRN) for predicting available parking spaces in parking lots. In contrast to previous methods, DSTAGCRN considers hidden dependencies between different parking lots from both periodic and spatial perspectives. Specifically, we decompose the parking history data into trend and seasonal components as inputs using data decomposition. We combine daily and weekly information with node embeddings using a dynamic parameter-learning module (DPLM) to generate dynamic adaptive graphs. We design a parameter-learning graph convolutional recursive module (PLGCRM) that integrates gated recurrent units (GRUs) and the DPLM to infer the spatio-temporal dependencies at each time step. Meanwhile, we use a multihead attention mechanism module to automatically allocate relatively more attention to valuable information, enabling more effective predictions of parking availability.

Under our improvement, the proposed dynamic spatio-temporal adaptive graph convolutional recursive network (DSTAGCRN) method has the following main advantages compared with existing methods:

(1)

Dynamic Graph Generation

Existing methods mostly rely on static predefined graphs to model spatial dependencies between parking lots, such as ASTGCN and AGCRN. These static images cannot capture the dynamic relationship between parking lots. Our DSTAGCRN method utilizes a dynamic parameter-learning module (DPLM) to combine daily and weekly information to generate dynamic graphs, thereby better capturing the implicit dependency relationships between parking lots over time.

(2)

Spatio-temporal Feature Fusion

Traditional methods, such as LSTM and DCRNN, mainly focus on temporal dependencies while ignoring spatial correlations between parking lots. Our method utilizes a parameter-learning graph convolutional recursive module (PLGCRM), combined with a multihead attention mechanism, to not only capture temporal dependencies but also effectively integrate temporal and spatial dependencies, making our model more robust and accurate when facing complex spatio-temporal data and improving the prediction accuracy of the model.

(3)

Dynamic Adaptive Ability

Due to the functional role of dynamic graphs, our model can automatically adjust weights based on real-time changes in parking lot data, thereby better reflecting the dynamic spatial correlation between different parking lots. This dynamic adaptive ability enables our model to maintain high predictive performance when facing different time periods and spatial layouts. In addition, we have validated the proposed DSTAGCRN model on various parking lot datasets of different scales and characteristics, such as Zurich, Guangzhou, and Singapore, to demonstrate its strong applicability and scalability and its ability to adapt to different parking lot network scenarios.

The rest of this paper is organized as follows: Section 2 provides a brief review of related work. Section 3 presents the definition of the research problem, while Section 4 describes the proposed method in detail. The experimental results of our approach are analyzed in Section 5. Finally, Section 6 concludes the study and outlines plans.

2. Literature Review

In recent years, with the development of Internet of Things (IoT) and communication technologies, many cities have implemented monitoring systems to record the number of vacant parking spaces [7,8]. VPS prediction can be seen as a spatio-temporal forecasting task that utilizes historical data collected from devices to predict the future availability of parking spaces. Research on vacant parking space prediction can generally be divided into two categories. One type is based on statistics, and the other type is based on machine learning (ML) and deep learning (DL).

2.1. Mathematical Statistics

For statistical-based methods, Guo et al. [9] proposed a data-driven autoregressive integrated moving average (ARIMA) approach for short-term prediction. Building upon this, Jose et al. [10] introduced a seasonal autoregressive integrated moving average (SARIMA) model for vacant parking space prediction. Although ARIMA and SARIMA are simple and do not rely on external variables, they are not suitable for nonlinear time series data, and parameter estimation can be challenging [11]. Additionally, Wu et al. [12] proposed a successful parking probability prediction algorithm, which can improve the accuracy of parking space prediction after short-term or long-term driving. In another study, Zheng et al. [13] created a parking demand prediction model based on the distribution of parking arrival and departure patterns. They proposed a parking trend-based method to determine the prediction interval and used curve fitting and the method of undetermined coefficients to calibrate parameters and predict parking demand. Xiao et al. [14] presented a framework based on continuous-time Markov M/M/C/C queues for vacant parking space prediction.

Subsequently, Rong et al. [15] used linear interpolation as a benchmark and employed a distance-weighted interpolation algorithm for parking prediction. Bock et al. [1] also used a Kalman filter in a study to predict on-street parking availability, though the use of the Kalman filter did not result in significant statistical improvement. Additionally, Peng et al. [16] employed time-varying analysis of historical data and modeled the discrete occupancy of parking lots as a nonstationary Poisson process, proposing an economically efficient parking spot search method. However, statistical-based methods primarily focus on theoretical analysis and are not suitable for nonlinear time series data, making them ill suited for complex real-world parking scenarios [17].

2.2. Machine Learning Methods

With the advancement in computational power, machine learning methods have increasingly been applied in parking prediction, effectively addressing the shortcomings of statistical methods by handling complex nonlinear time series data. machine learning algorithms such as decision tree (DT) [18], least absolute shrinkage and selection operator (LASSO) [19], and K-nearest neighbors (KNN) [20] have been employed for parking prediction. However, most machine learning studies fail to consider the impact of contextual features in parking data [2]. In a study by Rajabioun et al. [21] a combination of long short-term memory (LSTM) and autoregressive integrated moving average (ARIMA) methods was used to predict the occupancy status of parking lots.

Similarly, Sampathkumar et al. [22] proposed a majority voting ensemble model that combines the outputs of seven different forecasting methods for parking prediction. On the other hand, Tekouabou et al. [23] conducted a further analysis of parking space availability using a combination of four regression methods (bagging, random forest, adaptive boosting, and gradient boosting). Zhao et al. [24] designed a model based on the LASSO regression analysis machine learning method, improving prediction accuracy by performing variable selection and regularization to capture the nonlinear characteristics of parking events. Bilotta et al. [25] highlighted the differences between models using LSTM and bidirectional LSTM, feature correlation (baseline, weather, traffic, etc.), and the impact of seasonality on predictions through feature correlation analysis using interpretable artificial intelligence techniques based on gradients and integrated gradients. Zhang et al. [26] proposed the PewLSTM model, which leverages a periodic weather-aware LSTM module to forecast parking behavior. Zeng et al. [27] developed a parking availability prediction model by incorporating wavelet transform (WT) and bidirectional long short-term memory (Bi-LSTM). The model utilizes WT for denoising historical parking data and then employs a Bi-LSTM network to learn the time correlation of parking data.

However, the aforementioned studies mostly focus on the temporal correlation of parking data, neglecting the spatial correlation between parking lots. Feng et al. [28] proposed a bi-convolutional long short-term network with dense convolutional layers that uses a CNN to extract spatial features of parking lots by partitioning grids. However, parking lots often exhibit non-Euclidean structures (as shown in Figure 1), making them less suitable for modeling spatial dependencies using a CNN.

2.3. Spatio-Temporal Graph Neural Networks

In recent years, Graph Neural Networks (GNNs) have been widely used in traffic prediction, utilizing graph theory to filter signals on local subgraphs. The graph convolution in GNN models combines the central node representation with neighboring node representations to create a new representation for the node, exploring spatio-temporal correlations in traffic data. GNNs can handle non-Euclidean structured data, using graphs to model parking networks and capture spatial dependencies, which is particularly suitable for irregularly distributed parking lots.

Guo et al. [29] enhanced the modeling capability of dynamic spatio-temporal correlations in traffic data by proposing an attention-based spatio-temporal graph convolutional network (ASTGCN) model for traffic flow forecasting. Leveraging the robustness of neural controlled differential equations (NCDEs) to irregular time series, Choi et al. [30] combined NCDEs with graph neural networks for spatio-temporal processing of traffic data. Furthermore, Guo et al. [31] introduced a latent network for extracting spatio-temporal features and proposed a dynamic graph convolutional network that adaptively constructs the dynamic road network graph matrix for traffic prediction.

In the field of parking prediction, researchers have gradually started using graph neural networks. For example, Chen et al. [32] used an Elman neural network to predict short-term available parking spaces. Vlahogianni et al. [33] employed a genetic algorithm optimized multilayer perceptron (MLP) to predict the occupancy rates of regional parking lots for the next 30 min. Liu et al. [34] also proposed an APS prediction algorithm based on a CNN-LSTM model, which considered both spatial and temporal correlations, although it only accounted for roadside parking directly affected by traffic flow. With technological advancements, Chen et al. [35] proposed a residual spatio-temporal graph convolutional neural network prediction model, which uses GNNs and temporal convolutional networks to capture spatial and temporal features, respectively, improving the accuracy and efficiency of the prediction process. Gao et al. [36] introduced a parking prediction model named DWT-ConvGRU-BRC, which uses discrete wavelet transform (DWT) to denoise historical parking data and then employs a convolutional gated recurrent unit network (ConvGRU) to extract temporal correlations of the parking lot itself and spatial correlations between different parking lots. Yang et al. [4] utilized graph convolutional neural networks (GCNNs) to extract spatial relationships in parking data from a large-scale network and used long short-term memory (LSTM) networks to capture temporal features. Zhao et al. [25] extracted discriminative features from multisource data and combined multigraph convolutional neural networks (MGCNs) with LSTM networks to capture complex spatio-temporal correlations. Zhang et al. [37] proposed a hierarchical graph convolution module to capture local and global spatial dependencies and then used a hierarchical attentional recurrent network module to capture dynamic short-term and long-term temporal dependencies in each parking lot.

However, the aforementioned GCN-based approaches require predefining connection graphs through similarity or distance measures [38] to capture spatial correlations. The graphs generated in this way are often intuitive, incomplete, and not directly tailored to the prediction task [39]. To fill this gap, this paper proposes a novel method called the dynamic spatio-temporal adaptive graph convolutional recurrent network (DSTAGCRN), which integrates dynamic graphs, GRUs, and attention mechanisms to create an accurate and efficient VPS prediction model.

3. Preliminaries

Definition 1. 

Parking lot network: We define a parking lot network as an undirected graph $G=(V,E,A)$, where $V$ is a set of nodes $N$, with each node corresponding to a parking lot. $E$ represents a set of $M$ edges, and $A\in R^{N\times N}$ denotes the adjacency matrix (constructed based on distance or semantic similarity, for example).

Definition 2. 

Parking signal matrix: The observed values of parking lot network $G$ at time slot $t$ are represented by the parking signal matrix $X_t={(x_{t,1},x_{t,2},\dots ,x_{t,N})}^T\in R^{N\times C}$. Here, $x_{t,v}\in R^C$ represents the feature vector of node $V$ at time $t$ (such as the number of available parking spaces), and $C$ denotes the number of features.

Definition 3. 

Graph convolution: According to [40], the graph convolution operation can be well approximated by a first-order Chebyshev polynomial. This can be expressed as:

$$Z=(I_N+D^{-\frac{1}{2}}A{D}^{-\frac{1}{2}})X\Theta +b$$

(1)

In the equation, $A\in R^{N\times N}$ represents the graph structure as a positive semidefinite matrix, and $D$ is the degree matrix of $A$. $X\in R^{N\times C}$ and $Z\in R^{N\times F}$ refer to the input and output of the GCN, respectively. $\Theta \in R^{C\times F}$ and $b\in R^F$ represent the weights and biases, respectively. The vacant parking space prediction problem can be described as follows: Given a parking lot network $G=(V,E,A)$ and a Pth-order parking signal matrix $X_{(t-P):t}=(X_{t-p},X_{t-p+1},\dots ,X_{t-1})\in R^{P\times N\times C}$, the mapping function $f(·)$ maps $X_{(t-P):t}$ to the next Qth-order parking signal matrix $X_{t:(t+Q)}=(X_t,X_{t+1},\dots ,X_{t+Q-1})\in R^{Q\times N\times C}$. The representation is as follows:

$$X_{t:(t+Q)}=f(X_{(t-P):t};G)$$

(2)

4. Model Architecture

This section introduces the details of DSTAGCRN. DSTAGCRN includes: (1) a sequence decomposition module, which decomposes the input parking history data into periodic trends and seasonal trends and extracts long-term stationary trends from hidden variables; (2) a dynamic parameter-learning module (DPLM), which combines daily information and weekly information with node embeddings to generate dynamic graphs to infer hidden spatial correlations in the parking lot network; (3) a parameter-learning graph convolutional recursive module (PLGCRM) to infer temporal dependencies at each time step; and (4) a multihead attention mechanism module. The spatio-temporal information is adjusted by the attention mechanism and is then fed into the PLGCRM block, allowing the network to automatically capture and fuse the spatio-temporal dependencies and dynamic changes in parking data to improve prediction performance. The architecture of DSTAGCRN is shown in Figure 2.

4.1. Sequence Decomposition

Inspired by Autoformer [41] and MICN [42], this paper also uses the moving average sequence decomposition method. This method divides parking history data into periodic trend parts and seasonal parts and can extract long-term stationary trends from hidden variables. Therefore, in this article, for the parking data $X\in R^{T\times N}$ with input time step $t$, the formula is:

$$X_c=\mathrm{AvgPool}(\mathrm{Padding}(X))$$

(3)

$$X_s=X-X_c$$

(4)

where $X_s,X_c\in R^{T\times N}$ represent the seasonal component and periodic trend component extracted from the original parking data, respectively. The use of $\mathrm{AvgPool}(•)$ and padding operations helps maintain the length of the decomposed sequences. After the sequence decomposition operation, we substitute the decomposed results into the mapping function $f(•)$ and perform an addition operation to obtain the predicted VPS data.

4.2. Dynamic Parameter-Learning Module (DPLM)

Existing approaches usually pre-calculate the graph based on geographical distances between different nodes or the similarity of the flow sequences [29,43]. However, predefined graphs may not capture the complete information about spatial dependencies [44]. To better capture the hidden dependencies between each node, some scholars have proposed graph learning methods. For example, Graph WaveNet [45], AGCRN [39], and MTGNN [46] use learnable node embeddings to construct static graph structures. Building on these works, our dynamic parameter-learning module (DPLM) employs a method that combines daily information, weekly information, and node embeddings to construct a dynamic graph, thereby learning the hidden interdependencies between different parking lots.

First, a node embedding $E_N\in R^{N\times d}$ is randomly initialized to represent node-specific features common to all time steps, where each row of $E_N$ represents the embedding of a node and $d$ denotes the dimension of node embedding. Then, we perform an elementwise product operation on node embedding $E_N$, daily information $X_T^D\in R^{T\times N}$, and weekly information $X_T^W\in R^{T\times N}$ to obtain a new space-time embedding $E_N^T\in R^{T\times N\times d}$, whose expression is:

$$E_N^T=X_T^D\odot X_T^W\odot E_N$$

(5)

where $X_T^D$ and $X_T^W$ represent the position of the parking data $X_T$ of the current time step $t$ every day and the position of each week, respectively. $N$ represents the number of parking lots, $d$ represents the embedding dimension, and $\odot$ represents the elementwise product operation. The specific process is shown in Figure 3.

In time step $t$, the input $X_T$ of the current time step and the hidden state $H_{T-1}$ of the previous step are passed through the $\mathrm{MLP}$ layer, and the dynamic signal is extracted and multiplied in an elementwise manner with the spatio-temporal embedding $E_N^T$ to generate a dynamic graph embedding $E_N^D\in R^{T\times N\times d}$, whose expression is:

$$E_N^D=\mathrm{MLP}(\mathrm{Concat}[X_T,H_{T-1}])\odot E_N^T$$

(6)

Similar to the node similarity definition graph, the spatial dependence can be inferred by multiplying $E_N^D$ by ${(E_N^D)}^T$. Therefore, the dynamic matrix $A_T^D\in R^{T\times N\times N}$ we generated is substituted into Equation (1) to obtain the dynamic convolution formula, which can be expressed as:

$$A_T^D=E_N^D{(E_N^D)}^T$$

(7)

$$\begin{array}{ll}{Z}_t& =(I_N+D^{-\frac{1}{2}}{A}_T^D{D}^{-\frac{1}{2}})X\Theta +b\\ & =(I_N+D^{-\frac{1}{2}}(\mathrm{ReLU}(E_N^D{(E_N^D)}^{\mathrm{T}}))D^{-\frac{1}{2}})X\Theta +b\end{array}$$

(8)

4.3. Parameter-Learning Graph Convolution Recursive Module (PLGCRM)

In addition to spatial dependency, parking data also exhibit complex temporal patterns such as periodicity and temporal correlations. Previous works [39,47] have addressed this by replacing the linear layers in GRUs with static graph convolutions to capture both spatial and temporal features simultaneously. As shown in Figure 4, we propose the parametric learning graph convolutional recursive module (PLGCRM), which integrates DPLM and GRU, as the basic unit for spatio-temporal modeling. We further utilize dynamic graph convolutions to learn the dynamic correlations between nodes.

$$\begin{array}{l}{r}_t=\sigma (\stackrel{\sim }{A}[X_t,H_{t-1}]{\Theta }_r+b_r)\\ u_t=\sigma (\stackrel{\sim }{A}[X_t,H_{t-1}]{\Theta }_u+b_u)\\ {\stackrel{\wedge }{h}}_t=\tan \mathrm{h}(\stackrel{\sim }{A}[X_t,r_t\odot H_{t-1}]{\Theta }_u+b_u)\\ h_t=u_t\odot H_{t-1}+(1-u_t){\stackrel{\wedge }{h}}_t\end{array}$$

(9)

where $X_t$ and $h_t$ are the input and output at time $t$, respectively. $[•]$ represents the concat operation; $r_t$ and $u_t$ are the reset gate and update gate, respectively; ${\Theta }_r$, ${\Theta }_u$, $b_r$, and $b_u$ are learnable parameters; and $\sigma$ is the sigmoid activation function.

4.4. Attention Module

To address the limitation of GRU in modeling long-term temporal dependencies, we combine the output of the hidden layers in the PLGCRM module with a multihead attention mechanism [48]. The multihead attention mechanism allows for the observation of long-term temporal trends by linearly transforming and concatenating the self-attention results of each group. This parallel computation enhances the dynamic spatio-temporal dependencies. The mathematical formula for this mechanism is as follows:

$$\begin{array}{l}MultiHead(Q,K,V)=\mathrm{Concat}(hea{d}_1,hea{d}_2,\dots ,hea{d}_h)W_o\\ hea{d}_i=\mathrm{Attention}(Q_i,K_i,V_i)=\mathrm{softmax}(\frac{Q_i{K_i}^{\mathrm{T}}}{\sqrt{d_k}})V_i\end{array}$$

(10)

Among them, $Q_i=H_o{W}_q$, $K_i=H_o{W}_k$, and $V_i=H_o{W}_v$ are all obtained by linear transformation of matrix $H_o$. $H_o$ is the result output by the PLGCRM module, $\sqrt{d_k}$ is the weight-scaling factor, and $W_q$, $W_k$, and $W_v$ are the learnable parameters corresponding to the $h$ head. The specific process of the multihead attention mechanism is shown in Figure 5. We adjust the output of the previous PLGCRM module through the multihead attention mechanism and then feed it as input to the next PLGCRM module. The DSTAGCRN model can be formed by stacking multiple layers of PLGCRM blocks and can be trained end-to-end via backpropagation.

5. Experiment

5.1. Selected Cities and Their Data Description

In this study, we selected cases from Singapore, Zurich, and Guangzhou to test the effectiveness of the constructed model. Singapore, Zurich, and Guangzhou are all representative cities in the world, facing their own challenges and opportunities in parking management and smart city construction. And the three cities have different geographical locations, economic development levels, and urban planning models. Singapore, as one of the financial centers in Asia, has highly intelligent urban management; Zurich is a typical city in Europe, emphasizing the coordinated development of transportation and environment; Guangzhou is a mega city in China, facing transportation and parking pressures brought about by rapid urbanization. Studying the data of these three cities can effectively help us verify the applicability and robustness of the model in different urban environments.

From the perspective of parking, these cities are facing the problem of an imbalanced supply and demand of parking spaces. The high demand for parking in the urban centers of Singapore and Zurich often leads to limited supply of parking spaces, resulting in difficulties in parking and traffic congestion. As a densely populated metropolis, Guangzhou has a more prominent problem of parking difficulties, especially in commercial and residential areas. By comprehensively reviewing the previous literature, we downloaded available parking space data from public data platforms: Zurich’s [49] available parking space data, Singapore’s [50] available parking space data, and Guangdong’s [51] available parking space data. VPS data are summarized at 5 min intervals, that is, there are 288 time steps for a day’s VPS data, which includes street parking and off-street parking. During the data processing, this study encountered some outliers and missing values, which we solved for by applying the linear interpolation method.

Table 1. Description and statistics of the dataset.

Dataset	Zurich	Guangzhou	Singapore
Start Time	1 January 2023	1 June 2018	1 June 2022
End Time	13 April 2023	30 June 2018	11 August 2022
Time Interval	5 min	5 min	5 min
Timesteps	29,664	8640	20,736
Parking Lots Number	33	18	122

summarizes the data spatio-temporal information of three cities.

5.2. Experiment Preparation

In our experiments, we select 60% of the data as the training set, 20% as the validation set, and the remaining as the test set. This proportion was chosen to construct the relevant dataset because we made multiple attempts from multiple proportion components and ultimately found that the dataset with this proportion performed the best, including the dataset allocation ratios of 70%:20%:10% and 80%:10%:10% that we tried. Additionally, we normalize the dataset using standard normalization methods to stabilize the training process. For these three available parking space datasets, the observation step size and prediction step size are both set to 12, and 1 h of historical data is used to predict the status of vacant parking spaces in the next 60 min. All experiments in this section were conducted on an Intel Core i5-12600 CPU@3.6-GHz computer with a GeForce RTX 3060Ti GPU card. Our experiments are based on the PyTorch framework, with the batch size set to 64 and the number of training rounds set to 50. We use the ADAM optimizer for training with a learning rate of 0.001. To prevent overfitting, we also set up an early stopping strategy with a patience value of 10. In the experiments, the mean absolute error (MAE), root mean square error (RMSE), and mean absolute percent error (MAPE) are used to evaluate the prediction performance of our proposed model.

$$\mathrm{MAE}(x,y)=\frac{1}{|\Omega |}{\displaystyle \sum _{i\in \Omega }\left|x_i-y_i\right|}$$

(11)

$$\mathrm{RMSE}(x,y)=\sqrt{\frac{1}{|\Omega |}{\displaystyle \sum _{i\in \Omega }{\left(x_i-y_i\right)}^2}}$$

(12)

$$\mathrm{MAPE}(x,y)=\frac{1}{|\Omega |}{\displaystyle \sum _{i\in \Omega }\left|\frac{x_i-y_i}{x_i}\right|}$$

(13)

where $x_1,\dots ,x_n$ represents the true value and $y_1,\dots y_n$ represents the predicted value. $\Omega$ represents the number of observation samples. In our experiment, the value of $|\Omega |$ is 12.

5.3. Comparing Models

To validate and evaluate the performance of the proposed DSTAGCRN method, we employed nine baseline models, including HA, LSTM, ASTGCN, DGCN, MegaCRN, STGNCDE, DCRNN, AGCRN, and DGCRN. Below is a brief explanation of each baseline model:

HA [52]: Historical average. It uses past VPS data to calculate the average and predict the next value.

LSTM [53]: Long short-term memory recurrent neural network model. Due to its memory function, LSTM can build learning models using long sequences of VPS data.

ASTGCN [29]: Attention-based spatio-temporal graph convolutional network. It utilizes spatial and temporal attention mechanisms to extract temporal and spatial features from the data.

DGCN [31]: Dynamic graph convolutional network. It proposes a latent network of graph Laplacian matrices to adaptively represent the spatio-temporal connections in traffic data.

MegCRN [54]: Meta graph convolution recursive network (MegaCRN). It relies solely on observed data to maintain robustness and adapt to any traffic situation, from normal to nonstationary.

STGNCDE [30]: Spatio-temporal graph neural control differential equation (STGNCDE). The neural control differential equation (NCDE) is a breakthrough concept for processing sequence data, with both time and space NCDEs designed.

DCRNN [47]: Diffusion convolutional recurrent neural network. It combines the bidirectional random walk of the graph and the recurrent neural network in an encoder–decoder structure.

AGCRN [39]: Adaptive graph convolutional recursive network. It is a deep learning model based on graph convolution, an adaptive adjacency matrix, and a recurrent neural network. An adaptive learning of node-specific parameters for graph convolution used to explore spatio-temporal correlations in data.

DGCRN [55]: Dynamic graph convolutional recurrent network. It extracts dynamic features from node attributes to generate a dynamic graph, capturing dynamic features of correlation between positions on the road network.

5.4. Experimental Results

Table 2. Performance comparison between the model proposed in this article and other models.

Models	Zurich			Guangzhou			Singapore
	MAE	RMSE	MAPE (%)	MAE	RMSE	MAPE (%)	MAE	RMSE	MAPE (%)
HA	9.29	17.78	29.59	18.19	38.23	20.36	31.1	53.35	18.89
LSTM	8.66	17.04	44.78	17.01	23.75	26.03	24.26	73.53	8.36
ASTGCN	4.57	10.13	18.47	8.28	17.03	3.65	4.9	8.74	3.16
DGCN	7.41	12.18	28.2	21.48	36.19	12.49	5.04	7.86	3.22
MegaCRN	4.37	9.26	17.46	7.78	15.17	3.48	5.06	9.83	3.14
STGNCD	4.62	9.34	17.54	9.12	17.69	4.51	4.78	8.07	3.17
DCRNN	4.65	10.69	16.39	10.16	17.72	5.11	5.97	11.71	3.79
AGCRN	4.36	10.09	16.09	9.6	17.41	4.71	4.89	9.45	3.13
DGCRN	4.59	9.84	16.78	9.21	16.32	5.58	5.58	10.51	3.59
Our	3.61	8.66	13.54	6.87	14.21	2.88	4.16	8.06	2.71

shows the evaluation results of our proposed DSTAGCRN model and the main baseline methods on three VPS datasets. Our method (DSTAGCRN) consistently achieves the best performance and is highlighted in bold. HA and LSTM only consider the temporal correlation of parking data, resulting in large prediction errors. These models ignore the spatial correlation among parking lots, and hence, their accuracy is lower than that of the graph neural network-based models, which also demonstrates the importance of considering spatial correlation for VPS prediction. Among the graph neural network models, the proposed DSTAGCRN model achieves the best performance. Predefined graph-based methods such as ASTGCN exhibit good performance as they can capture the spatial dependencies present in the VPS data. Additionally, methods such as AGCRN that utilize adaptive adjacency matrices can better extract complex dependency relationships present in VPS data and therefore exhibit good performance. Compared with dynamic graph networks such as DGCN and DGCRN, our proposed STDGCRN model explores the complex spatio-temporal dependency relationship in parking data more robustly by considering factors such as daily and weekly cycles. The three parking datasets used in the experiment have large differences in node numbers and temporal span, indicating that our proposed STDGCRN is capable of handling prediction tasks for different parking network scenarios. Note that according to the different roles of the training set, validation set, and test set in the model development process, all evaluation metrics (MAE, RMSE, MAPE) presented in this article are calculated based on the test set. This is because the training set is used for learning model parameters, the validation set is used for hyperparameter adjustment and model selection, and the test set is used to evaluate the model’s performance on unseen data, thereby verifying its generalization ability. Therefore, displaying the results of the test set can better reflect the actual application effect of the model.

To provide a comprehensive evaluation of the models, we present the 15 min, 30 min, and 60 min prediction accuracies for all models on the three parking datasets in Table 2. It can be observed that, except for the Singapore parking dataset, the DSTAGCRN model achieves the best prediction performance regardless of the time step variations in the prediction task. Overall, our proposed STDGCRN model effectively captures the complex spatio-temporal features present in the parking data. Across all time ranges, the error rate of DSTAGCRN is significantly lower than that of the other baselines, which is crucial for real-world VPS prediction. Figure 6 illustrates the results of ASTGCN, MegaCRN, AGCRN, DCRNN, DGCRN, and our proposed DSTAGCRN model at different prediction horizons on the three VPS datasets. It can be observed that the prediction results of all models decrease as the prediction range expands, but the DSTAGCRN model demonstrates higher prediction accuracy and smaller prediction error fluctuations compared with these models.

To clearly demonstrate the spatio-temporal correlations explored by our proposed approach and the performance differences with other baseline models, we plot the predicted values and true values of our method (DSTAGCRN) and STGNCDE for two parking lots within a given period in Figure 7. Both Parking Lot 1 and Parking Lot 2 are commercial parking lots with similar trends in VPS counts, which illustrates the spatio-temporal correlation of the VPS. DSTAGCRN and STGNCDE can fit the true situation of VPS count changes in many periods well, but our model not only accurately adapts to the rapidly changing trends but also captures similar trends such as traffic peak hours, demonstrating a more accurate prediction performance.

This paper takes Parking Lot 2 and Parking Lot 7 in the Guangzhou parking dataset as examples to verify the effectiveness of the dynamic graph inferred by the DSTAGCRN model. Figure 8 shows the available parking space data of the two parking lots for one day, and Figure 9 visualizes the weights of the dynamic spatial correlation of the two parking lots. As shown in these two figures, in cycle 1, the trends of the number of available parking spaces in the two parking lots are different, resulting in relatively low weights being learned by the dynamic graph. In cycle 2, the trends of the two parking lots are similar, so the weights learned by the dynamic graph are higher. Combining Figure 8 and Figure 9, it can be seen that the dynamic graph can effectively adapt to these changes and automatically adjust the weights between multiple parking lots, which reflects that the dynamic graph in the DSTAGCRN model can effectively capture the dynamic spatial associations between multiple parking lots.

5.5. Other Experimental Studies

Due to space limitations in this paper, we chose to present data from Zurich as a representative example to complete the model ablation experiment. This is because the data from Zurich has good representativeness, which can effectively demonstrate the performance of our proposed method in dealing with different parking lot networks. In addition, Zurich’s high data quality and wide coverage provide a sufficient testing basis for our model. We let DSTAGCRN denote the complete network; W/o Trend represents the removal of the trend term, W/o Attention represents the removal of the multihead attention mechanism, W/o Embeddings represents the removal of weekly and daily embeddings, and W/o Dynamic represents the use of only static graphs without dynamics. The evaluation results for the Zurich parking data using these modules are shown in

Table 3. Ablation experiment based on the Zurich parking dataset.

Algorithms	MAE	RMSE	MAPE
DSTAGCRN	3.61	8.66	13.54%
W/o Attention	3.64	8.8	13.80%
W/o Trend	3.63	8.67	14.33%
W/o Embeddings	3.69	8.73	15.42%
W/o Dynamic	4.36	10.09	16.09%

.

Throughout the entire ablation experiment, we applied the same dataset as Zurich in Table 2, and the specific implementation and result calculation were completed in our research environment, which used Python and related machine learning libraries as well as the PyTorch framework. Overall, the performance using only static graphs was the worst, demonstrating the limitations of considering spatial correlations between different parking lots on the road network solely from the perspective of available parking bay data. The performance of using weekly and daily information embeddings and trend terms was better than that of using only dynamic graphs, indicating that VPS data have periodic features. In addition, the DSTAGCRN model achieved the best performance, indicating that the weight design changed by adding the multihead attention mechanism would greatly affect the graph convolution and that this method could better learn the spatio-temporal correlation between different parking lots.

Figure 10 gives the results of our model with different embedding dimensions on the Zurich parking dataset. DSTAGCRN achieved a relatively good performance on all tested embedding dimensions, which demonstrates the robustness of our method. Furthermore, DSTAGCRN achieved the best performance when the embedding dimension was set to 10. Node embedding dimensions that are too small or too large will lead to poor performance. Appropriate embedding dimensions can derive more accurate spatial correlations and prevent overfitting. Figure 11 shows the impact of dimensions and layers in the PLGCRM module on the model. When the hyperparameters are different, the results of the model are different. This article selects the best parameters.

6. Conclusions

Aiming at the lack of ability of static graphs to model the dynamic spatial association of multiple parking lots, this paper proposed a novel dynamic graph model, DSTAGCRN, for predicting available parking spaces. The proposed model combines node embedding with dynamic graphs to explore the spatial correlation and dynamic information between parking lots in the road network, providing initial evidence of the effectiveness of dynamic graphs in improving prediction performance. We decomposed the original parking data into cyclic trends and seasonal trends, extracting long-term stationary trends from hidden variables to enhance predictive accuracy. Additionally, we introduced a multihead attention mechanism module that self-adjusts based on the dynamic information at each time step to explore more effective hidden information. Ablation experiments validated the effectiveness of these components, indicating the high potential of the model in exploring the spatio-temporal structure of available parking spaces.

Compared with existing prediction models, we conducted extensive experiments on three actual datasets, and the results showed that the proposed DSTAGCRN model could more accurately and efficiently model the number of available parking spaces. In these experiments, we used datasets from Singapore, Zurich, and Guangzhou, which covered different urban environments and parking lot types, validating the widespread applicability and robustness of our model. Specifically, the mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) of DSTAGCRN on the test set were significantly lower than those of other benchmark models. For example, compared with the ASTGCN model proposed by Guo et al. [30], our DSTAGCRN model reduced MAE by 0.89, RMSE by 1.47, and MAPE by 4.93% on the Zurich dataset. This advantage is mainly focused on handling dynamic spatial correlations and temporal dependencies. Meanwhile, compared with other benchmark models such as LSTM, MegaCRN, DGCN, etc., it also showed a stronger generalization ability and robustness.

In future research, we plan to integrate more external factors such as temperature, weather, points of interest (POIs), and traffic accidents into the model to further improve the performance and reliability of parking space prediction. In addition, we also would like to explore the applicability of the model in other cities and more application scenarios to promote the development of smart cities. We believe that through continuous optimization and expansion, the graph neural network model would provide more effective technical support for parking management and traffic optimization.