Noise Pollution Prediction in a Densely Populated City Using a Spatio-Temporal Deep Learning Approach

Abstract

Noise pollution in densely populated urban areas is a major issue that affects both quality of life and public health. This study explores and evaluates the application of deep learning techniques to predict urban noise levels, using the city of Madrid, Spain, as a case study. Several complementary approaches are compared: Convolutional Neural Networks (CNNs), Long Short-Term Memory (LSTM) networks, and Graph Convolutional Networks (GCNs). Each technique contributes specific strengths to the modeling of spatiotemporal series: CNNs are effective at capturing local spatial patterns, while LSTM networks excel at modeling long-term temporal dependencies. In turn, GCNs integrate spatial structure and temporal dynamics through graph representations, achieving superior performance compared to traditional approaches or models based solely on CNN or LSTM architectures. This study provides empirical evidence of the potential of GCNs to effectively address the spatiotemporal complexity of urban noise and highlights new possibilities for their application in urban planning and environmental management. Our hybrid model, CNN1D+LSTM+TransformerConv, achieves a root mean squared error (RMSE) of 0.0169, reducing the error by 5.1% compared to the second-best model (Transformer, RMSE = 0.0178), and reaches a correlation coefficient of 0.9601. The results demonstrate that explicitly integrating the spatial component through graphs, alongside temporal sequence modeling, leads to improved prediction accuracy over alternative methods.

1. Introduction

Noise pollution in urban areas has become a growing concern due to rising population density, urban expansion, and increased vehicular traffic. Numerous studies have confirmed that prolonged exposure to high noise levels can have significant adverse effects on human health, impacting multiple bodily systems. For example, constant exposure to noise—especially at night—can lead to sleep disorders, increasing the risks of cardiovascular disease, hypertension, and stroke [1,2].

Moreover, it has been demonstrated that long-term noise exposure can cause chronic stress, negatively affecting both physical and mental health. Recent studies indicate that continuous exposure to urban noise is associated with psychological disorders such as anxiety and depression, due to the sustained activation of biological mechanisms related to stress. Additionally, disruptions to circadian rhythms caused by nighttime noise exacerbate these negative effects [3].

Against this backdrop, mitigating urban noise becomes a key priority for public health policies and urban planning. The accurate prediction of acoustic levels would allow for the implementation of more effective strategies, such as sound barriers, dynamic traffic regulation, or improvements in urban design, thereby reducing population exposure and improving well-being [4].

Despite the recognized importance of this issue, there is a clear lack of research leveraging recent deep learning techniques for the spatiotemporal prediction of urban noise. In particular, the use of Graph Convolutional Networks (GCNs) to simultaneously model the spatial and temporal relationships present in urban noise remains limited.

In this context, the present study addresses this gap through the development and comparative evaluation of deep-learning-based models—CNN, LSTM, and particularly GCN—to predict noise pollution in a densely populated city such as Madrid, Spain. The specific choice of this analysis period allows for the evaluation of exceptional events such as the COVID-19 pandemic, as well as recurring patterns associated with holidays and weekends, and their impact on urban noise levels. The goal is not only to anticipate acoustic patterns but also to provide a deeper understanding of how these levels vary under different temporal and social conditions.

This study contributes to current knowledge by explicitly evaluating the effectiveness of different deep learning models and proposing an integrated methodology that can be adapted to other cities facing similar noise-related challenges.

2. Related Work

Urban noise pollution prediction has been addressed using a wide range of methods, from classical statistical techniques to deep-learning-based approaches. Traditional time series models, such as ARIMA and SARIMAX, have been widely used to model acoustic variables by identifying trends and seasonal patterns in historical data. These approaches have proven effective in scenarios where time series are linear and stationary. However, their performance is limited when dealing with non-linear dynamics and the large data volumes characteristic of modern urban environments [5].

With the rise of deep learning, new methodologies have emerged to overcome the limitations of classical models. Recurrent Neural Networks (RNNs), particularly Long Short-Term Memory (LSTM) networks, have demonstrated strong capabilities in capturing long-term temporal dependencies, significantly improving the accuracy of acoustic predictions [6,7]. Similarly, Convolutional Neural Networks (CNNs) have shown effectiveness in extracting spatial and temporal patterns in complex urban soundscapes [8].

Recently, hybrid architectures that combine LSTMs with Internet of Things (IoT) technologies have been proposed, enabling short-interval predictions and more dynamic urban noise management. These approaches have outperformed traditional methods such as Random Walk (RW), Stacked Autoencoders (SAEs), and Support Vector Machines (SVMs) in dynamic urban scenarios [7,9].

Other recent studies have explored the integration of Artificial Neural Networks with Geographic Information Systems (GISs) to explicitly incorporate spatial factors into noise prediction, thereby enhancing responsiveness to local phenomena [10]. Complementary efforts have combined fuzzy logic with autoregressive techniques to manage noise in specific environments, such as construction sites [11].

An emerging trend in the prediction of dynamic urban phenomena is the use of Graph Neural Networks (GNNs), particularly Graph Convolutional Networks (GCNs). These models enable the representation of spatial relationships through adaptive graphs and attention mechanisms, achieving notable improvements in related tasks such as traffic prediction and air quality forecasting [12,13,14,15,16].

Although the direct applications of GCNs to urban noise prediction are still limited, their success in analogous domains suggests substantial potential. For instance, recent approaches using CNNs on traffic flow data have enabled the indirect estimation of noise levels, although these methods face challenges in spatial generalization [17].

Explicitly modeling urban spatial structure through graphs could help overcome these limitations, allowing for a more robust representation of the spatiotemporal interactions that characterize noise in urban environments.

In conclusion, despite recent progress, the explicit application of Graph Convolutional Networks (GCNs) to urban noise prediction remains an emerging research direction with considerable promise. Exploring new graph-based spatiotemporal architectures offers the opportunity to improve predictive accuracy, generalization capacity, and dynamic noise management in smart cities. In this study, we propose a hybrid architecture composed of CNN1D, LSTM, and TransformerConv (GCN) layers, which integrates spatiotemporal relationships to forecast future noise levels from real multivariate records, demonstrating significant improvements over previous models.

3. Methodology

This section outlines the methodological approach used for predicting urban noise pollution. Historical noise data are first collected and preprocessed, along with the geographic coordinates of the monitoring stations. A spatial graph is then constructed by connecting each node to its 15 nearest neighbors based on the Haversine distance.

The dataset is partitioned into training (70%), validation (15%), and test (15%) subsets and organized into three-dimensional tensors representing spatial, temporal, and feature dimensions.

The proposed model (see Figure 1) integrates three core components: one-dimensional convolutional (CNN1D) layers to extract local temporal patterns, a Long Short-Term Memory (LSTM) layer to model more complex and long-range temporal dependencies, and a TransformerConv (Graph Convolutional Network) layer, which employs attention mechanisms to emphasize relevant spatial relationships between stations. A final linear layer produces the output prediction by integrating the extracted spatiotemporal features.

Model performance is assessed using the Huber loss function and compared against several baseline models for validation.

3.1. Formal Problem Definition

To apply Graph Convolutional Networks (GCNs) in noise pollution prediction, we first need to represent the data as spatiotemporal graphs. Let the graph $G_t$ at a specific time t be defined as

$$G_t=(V,E_t,X_t)$$

where

• V is the set of nodes, defined as

  $$V=\{v_1,v_2,\dots ,v_{\left|V\right|}\},$$

  where each node represents a noise measurement station in the city.
• $E_t$ is the set of edges at time t, which connects the nodes based on their spatial proximity. Specifically, each node is connected by edges to its k nearest neighbors according to the geodesic distance calculated using the Haversine formula. Thus, it is formally defined as

  $$E_t=\left\{(v_i,v_j)\right|v_i,v_j\in V,v_j\in {\mathcal{N}}_k\left(v_i\right)\},$$

  where ${\mathcal{N}}_k\left(v_i\right)$ is the set of the k nearest neighbors of node $v_i$.
  The distance associated with each edge $(v_i,v_j)$ is defined by the Haversine distance as

  $$d(v_i,v_j)=2R\arcsin \left(\sqrt{{\sin }^2\left({\displaystyle \frac{{\varphi }_j-{\varphi }_i}{2}}\right)+\cos \left({\varphi }_i\right)\cos \left({\varphi }_j\right){\sin }^2\left({\displaystyle \frac{{\lambda }_j-{\lambda }_i}{2}}\right)}\right),$$

  (1)

  where ${\varphi }_i,{\lambda }_i$ and ${\varphi }_j,{\lambda }_j$ are the geographic latitudes and longitudes of stations $v_i$ and $v_j$, respectively, and R is the average radius of the Earth.
• $X_t\in {\mathbb{R}}^{\left|V\right|\times F}$ is the feature matrix of the graph at time t, where each row represents a node (station) and each column an acoustic or temporal feature associated with that station. Therefore,

  $$X_t={[x_{v_1}^t,x_{v_2}^t,\dots ,x_{v_{\left|V\right|}}^t]}^{\top },x_{v_i}^t\in {\mathbb{R}}^F.$$

Since our goal is to make future predictions based on historical sequences of graphs, we mathematically formulate the problem as follows:

Given a historical sequence of graphs:

$$\{G_{T-n},G_{T-(n-1)},\dots ,G_{T-1},G_T\},$$

the goal is to learn a predictive function $\mathcal{F}$ such that

$$\mathcal{F}:\{G_{T-n},G_{T-(n-1)},\dots ,G_T\}\to \{G_{T+1},G_{T+2},\dots ,G_{T+N}\},$$

where N is the number of future steps to predict. The prediction involves estimating the feature matrix $X_{T+i}$ for each future graph $G_{T+i}$, with $i=1,2,\dots ,N$, while keeping the node structure V and edges $E_t$ previously defined fixed.

3.2. Data Modeling

To impute missing values, temporal interpolation is applied to intermediate time values $T-1$, $T_{\mathrm{imputed}}$, and $T+1$ in combination with Bayesian regression. Before and after imputation, the average time series are examined to ensure that key properties—such as seasonality and trend—are preserved, using visual inspection techniques. Additionally, it is verified that statistical metrics such as the mean, median, maximum, and minimum values remain consistent with those observed prior to imputation.

We normalize the data to the range [0, 1]. To prevent data leakage, only the training set is used to compute the minimum and maximum values, ensuring that the model has no access to information from the validation or test sets. Normalization is performed using the standard min–max scaling formula:

$$x^{\prime }={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}},$$

where x is the original value; $x_{\min }$ and $x_{\max }$ are the minimum and maximum values from the training set, respectively; and $x^{\prime }$ is the normalized value.

3.3. Models Used

As previously mentioned, the following deep learning models were used to address the problem. Below, the main technologies used in each predictive model are listed for comparison with our proposed model:

• LSTM [18]: Long Short-Term Memory (LSTM) networks are a type of Recurrent Neural Network (RNN) designed to capture long-term temporal dependencies in sequential data.
• CNN+LSTM [19]: This hybrid architecture combines Convolutional Neural Networks (CNNs) for extracting local spatial features with LSTM networks for modeling temporal dependencies, making it suitable for multivariate time series prediction.
• GAT [20]: Graph Attention Networks (GATs) apply attention mechanisms to assign adaptive weights to neighboring nodes in a graph, enhancing predictive performance in structured data scenarios.
• GraphSAGE [21]: GraphSAGE is a graph learning method that generates node embeddings by aggregating features from neighboring nodes, suitable for inductive learning.
• GraphSAGE+LSTM: This combined model integrates GraphSAGE for capturing spatial relationships within a graph and LSTM networks for modeling temporal dynamics, offering a spatiotemporal solution for sequential graph-structured data.
• Transformer [22]: The Transformer architecture leverages self-attention mechanisms to model long-range dependencies without relying on recurrence. It allows for the parallel processing of sequences and has demonstrated strong performance in tasks such as natural language processing, machine translation, and time series forecasting.

3.4. Our Model, Spatio-Temporal Graph Convolution—Transformer Conv (STGC-TC)

This section details the methodological approach used for predicting noise pollution in an urban setting. As previously noted, the proposed framework integrates three deep learning techniques: Convolutional Neural Networks (CNNs), Long Short-Term Memory networks (LSTMs), and Graph Convolutional Networks (GCNs).

This combination addresses the need to model the spatiotemporal dynamics that characterize urban noise pollution. While each method offers specific advantages, none alone is sufficient to capture the complex spatial and temporal dependencies involved. CNNs are particularly effective at extracting local patterns and relevant features within fixed temporal windows, facilitating the identification of recurring structures in high-dimensional time series. However, their ability to model long-range dependencies is limited. This shortcoming is addressed by LSTM networks, which are well suited for capturing long-term temporal relationships and significant variations over extended periods—essential for modeling daily and weekly noise patterns. Nevertheless, both CNNs and LSTMs neglect the spatial dimension and the interactions between geographically distributed stations. GCNs overcome this limitation by explicitly modeling the spatial structure through graphs that connect measurement stations based on geographic proximity. This integration enables the model to learn not only how noise evolves over time but also how it propagates spatially across locations.

By combining CNN, LSTM, and GCN components, the proposed model overcomes the limitations of each individual technique, providing a robust and effective solution for urban noise prediction.

To implement this integration, the model (shown in Figure 2) takes as input three-dimensional tensors of shape $[num\text{\_}nodes,num\text{\_}timesteps,num\text{\_}features]$, where each node represents a measurement station and each feature corresponds to an acoustic variable recorded over time.

First, the input data are rearranged to apply two 1D convolutional layers along the temporal axis of each node, transforming the original input into a tensor of shape $[num\text{\_}nodes,num\text{\_}features,num\text{\_}timesteps]$. These convolutions allow local patterns in the data to be extracted, highlighting variations over short time periods. Subsequently, the data are rearranged again into shape $[num\text{\_}nodes,num\text{\_}timesteps,num\text{\_}features]$, and processed through an LSTM layer responsible for capturing more complex and long-term temporal dependencies. The output of this LSTM layer, shaped $[num\text{\_}nodes,hidden\text{\_}channels]$, is then passed to a TransformerConv layer [23], which uses attention mechanisms to evaluate and weigh the importance of spatial connections between nodes, enabling the model to emphasize those spatial relationships most relevant to prediction. Finally, a linear layer produces the final predictions for each node, integrating all temporal and spatial information extracted in the previous stages.

3.5. Evaluation Metrics

To evaluate model performance on the test set (15% of the data), several statistical metrics are employed: Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), accuracy, Pearson correlation coefficient, and the coefficient of determination ($R^2$). These metrics are computed by comparing predicted values against ground truth observations and are widely used in the evaluation of time series forecasting models.

• RMSE (Root Mean Squared Error): Square root of the average of the squared errors:

  $$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2},$$

  where $y_i$ are the actual values and ${\widehat{y}}_i$ are the predicted values.
• MAE (Mean Absolute Error): Average of the absolute errors:

  $$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-{\widehat{y}}_i|.$$
• Accuracy: A measure of model precision; in this case, a prediction is considered correct if the difference between the predicted and actual value is less than 10% of the actual value:

  $$\mathrm{Accuracy}={\displaystyle \frac{\mathrm{Number}\mathrm{of}\mathrm{predictions}\mathrm{within}\mathrm{the}10\%\mathrm{threshold}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{predictions}}}.$$
  The 10% threshold is justified considering that the analyzed ambient noise levels have average values around 60 dB. Thus, a relative error of less than 10% corresponds approximately to an absolute error of less than 6 dB, which is within the acceptable range for urban noise prediction and operational monitoring tasks.
• Pearson Correlation: Measures the strength and direction of the linear relationship between two variables:

  $$r={\displaystyle \frac{{\sum }_{i=1}^n(y_i-\overline{y})({\widehat{y}}_i-\overline{\widehat{y}})}{\sqrt{{\sum }_{i=1}^n{(y_i-\overline{y})}^2{\sum }_{i=1}^n{({\widehat{y}}_i-\overline{\widehat{y}})}^2}}},$$

  where $y_i$ represents the actual values, ${\widehat{y}}_i$ represents the predicted values, $\overline{y}$ is the mean of the actual values, and $\overline{\widehat{y}}$ is the mean of the predicted values.

4. Experiments

To evaluate our model, we conducted a comparative study using a case study: the city of Madrid (Spain).

This section describes the experiment carried out. The code is available for replication (Code: https://github.com/Semllo/AcusticPollution (accessed on 10 May 2025)).

4.1. Dataset

The data used in this study come from the Madrid City Council (Spain), are open access, and are available on its website [24]. The dataset used in this study contains the following information:

• NMT (Measurement Station Number): A unique numerical identifier for each station where noise measurements are taken. This code allows for the specific identification of each station’s location.
• Year: The year in which the measurements were taken, represented as a number.
• Month: The month when the noise data were recorded, indicated by a number from 1 to 12.
• Day: The day of the month when the measurement was taken, represented by a numerical value between 1 and 31.
• Type: Time of day corresponding to the measurement. Expressed using the following options:
  • “D”: Daytime.
  • “N”: Nighttime.
• LAeq (Equivalent Continuous Sound Level): The average sound pressure level during the measurement period, expressed in decibels (dB). This value reflects overall exposure to noise over the observed period.
• L1: Sound pressure level exceeded during 1% of the observation time. Used to identify noise peaks that occur rarely.
• L10: Sound pressure level exceeded during 10% of the observation time. Typically associated with frequent loud noises.
• L50: Sound pressure level exceeded during 50% of the observation time. Often used as a reference for “typical” or “median” noise levels in an area.
• L90: Sound pressure level exceeded during 90% of the observation time. Usually represents background or constant noise levels.
• L99: Sound pressure level exceeded during 99% of the observation time. Indicates the most persistent noise levels, reflecting minimum values present most of the time.

The target variable in this study is LAeq (Equivalent Continuous A-weighted Sound Pressure Level), which represents the continuous equivalent A-weighted sound pressure level in decibels (dB) over a specified time period. This variable is calculated to reflect the total energy of noise exposure, providing a representative measure of the average noise level in a given environment.

Using LAeq as the target variable is appropriate because it captures overall environmental noise exposure, integrating both fluctuating noise levels and their cumulative effects over time. Since LAeq is widely recognized in noise studies and public health regulations, it provides a solid framework for assessing and predicting noise pollution across various urban contexts.

According to WHO guidelines, the acceptable noise levels for urban areas are as follows [25,26]:

• Environmental noise during the day (outdoor): 55 dB(A)—maximum recommended to avoid annoyance.
• Environmental noise at night (outdoor): 40 dB(A)—recommended limit to avoid adverse health effects related to sleep.
• Indoor noise during sleep: 30 dB(A)—ideal limit for indoor environments such as bedrooms at night.
• Maximum noise in short events (peaks) during the night: 45 dB(A)—limit for isolated noise events like passing cars or sirens.

The dataset consists of approximately 240,932 measurements collected by 31 stations over a period of 11 years. Measurements are taken during two periods—day and night—due to the different recommended thresholds for health protection. Since the goal is to predict 14 future time intervals, our model focuses on estimating noise levels for the next 7 days.

The average levels recorded by the stations are presented in Figure 3.

As shown, the COVID-19 pandemic period exhibits anomalous measurement points, which aligns with expectations given the reduced urban activity and traffic during that time. To mitigate the impact of these anomalies on model performance, data from 2020 and 2021 were excluded from the training process.

Additional features were incorporated to enhance the models. For instance, weekends may exhibit distinct patterns compared to weekdays, and holidays may influence measurements at specific noise monitoring stations, leading to unique patterns.

The plots in Figure 4 illustrate the distribution of the equivalent sound pressure level (LAeq) under different conditions: holidays versus non-holidays (left) and weekdays versus weekends (right).

In the left plot, which compares the LAeq distributions on holidays versus non-holidays, both curves exhibit similar shapes, although a slight difference in peak density is evident. Non-holiday days generally show marginally higher noise levels, as indicated by the rightward shift of the peak. This suggests that human activities—such as traffic and work—contribute to increased noise levels on regular days.

In the right plot, comparing weekdays and weekends, a similar pattern is observed. Weekdays display a slightly higher density of high LAeq values compared to weekends. This supports the idea that routine activities during the workweek generate more noise than those typical of weekends and holidays. While these differences are subtle, they may be significant for a predictive model that incorporates such temporal factors.

It is also plausible that noise patterns shifted after the COVID-19 pandemic. The Figure 5 presents the LAeq distribution before and after the pandemic. It is evident that, overall, post-pandemic noise levels (orange curve) tend to be lower than pre-pandemic levels (blue curve). The peak of the post-pandemic distribution is shifted to the left, indicating a higher density of values in lower sound intensity ranges.

Although this is not the primary objective of the study, we provide evidence suggesting that mobility restrictions and changes in social and work habits during and after the pandemic have contributed to a reduction in environmental noise levels in the city of Madrid, as will be discussed in subsequent sections. The differences in the shape and position of the curves emphasize the importance of incorporating social and economic changes into predictive models of noise pollution.

To train Graph Convolutional Networks (GCNs), a spatiotemporal graph must be constructed, considering the nearest distances to each station. Various connectivity criteria were tested, including linking each node to its 10 and 20 nearest neighbors. Performance was evaluated using 10 and 20 neighbors, with results showing a deterioration of approximately $3.33\%$ and $0.33\%$ (RMSE), respectively. Connections to 15 neighbors yielded slightly better results for some GCN models. Therefore, this criterion was selected for graph construction. The resulting graph is shown in Figure 6.

The choice of geodesic distance (Haversine) is based on several practical and methodological considerations. First, since the measurement stations are relatively close to one another (with typical distances under 20 km), the spherical Earth distance offers a good approximation of spatial connectivity relevant to acoustic propagation without introducing significant distortion. Second, the Haversine metric is computationally efficient, which simplifies preprocessing and facilitates scaling to large datasets. Finally, several studies in spatiotemporal prediction have demonstrated that graphs constructed with geodesic distances deliver comparable performance to those based on road-network distances, but with lower complexity and fewer external data dependencies [12,14].

Each graph in time encapsulates information about the spatial relationships between nodes (measurement stations) and their temporal features. Six past graphs are constructed to represent the historical evolution of noise, while fourteen future graphs are what the model aims to predict. The connections between nodes reflect the distances between nearby measurement stations, thereby capturing spatial relationships within the urban environment. The dataset is organized into temporal snapshots, enabling the models to predict node features in future graphs.

• Six past graphs: $G_{T-6},G_{T-5},G_{T-4},G_{T-3},G_{T-2},G_{T-1},G_T$.
• Fourteen future graphs: $G_{T+1},G_{T+2},G_{T+3},G_{T+4},G_{T+5},\dots ,G_{T+14}$.
• Vertex connections to the 15 closest nodes based on Haversine distance.
• Edge attributes, including distance.
• Geographic coordinates used for future model evaluation.

4.2. Model Configuration

The configuration of the models used in this article are in

Table 1. Model structure and main parameters.

Model	Structure
LSTM	LSTM: 12 inputs, 14 units, 10% dropout
GAT	GATConv: 12 to 128, 8 attention heads GATConv: 1024 to 128, 1 attention head Linear layer: 128 to 14
CNN3D+LSTM	Conv3D: 1 to 32 channels, 3 × 3 × 3 kernel, padding 1 LSTM: 32 to 64 units Dropout: 10% Linear layer: 64 to 14
GraphSAGE+LSTM	GraphSAGE: 12 to 12, mean aggregation LSTM: 12 to 32 units Dropout: 10% Linear layer: 32 to 14
TransformerConv	TransformerConv: 12 to 128, 1 attention head Linear layer: 128 to 14
GraphSAGE	GraphSAGE: 12 to 128, mean aggregation Linear layer: 128 to 14
CNN1D+LSTM+ TransformerConv	Conv1D: 12 to 16, kernel 3, padding 1 Conv1D: 16 to 32, kernel 3, padding 1 LSTM: 32 to 64 units TransformerConv: 64 to 64, 2 attention heads, no concatenation Dropout: 40% Linear layer: 64 to 14
Transformer	Transformer encoder: dimension 12, 2 attention heads Dropout: 40% Linear layer: 12 to 14

. These models were selected to evaluate different levels of complexity, ranging from relatively simple approaches to more advanced hybrid architectures that simultaneously capture spatial and temporal patterns. The specific structure of each model and its hyperparameters were determined through an exploratory process based on trial and error, empirically optimizing performance on the validation set.

Huber Loss is used as the loss function due to its robustness to outliers and predictions that deviate significantly from actual values. The Huber loss combines the advantages of Mean Squared Error (MSE), which heavily penalizes large deviations, with those of Mean Absolute Error (MAE), which is less sensitive to extreme values. The mathematical expression of the Huber loss is:

$$L_{\delta }(y,\widehat{y})=\left\{\begin{array}{cc}\frac{1}{2}{(y-\widehat{y})}^2\hfill & \mathrm{if}|y-\widehat{y}| \le \delta ,\hfill \\ \delta |y-\widehat{y}|-\frac{1}{2}{\delta }^2\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where y is the true value, $\widehat{y}$ is the model prediction, and $\delta$ is a hyperparameter that sets the threshold between quadratic (for small errors) and linear behavior (for large errors). In our case, $\delta$ is determined empirically through trial and error to maximize predictive performance on the validation set.

The Huber loss is integrated into the training process using the standard gradient-based optimization procedure. In each training epoch, predictions are made on the training set, and the loss is computed using the Huber function. The error is then backpropagated, and the model’s weights are adjusted based on the gradients. This process is repeated iteratively until a convergence criterion is met, based on the model’s performance on the validation set, ensuring greater stability and robustness in the presence of outliers or anomalies in the data.

To optimize training, a learning rate scheduler is used, which dynamically adjusts the rate according to the following expression:

$$\mathrm{lr}=\mathrm{lr}\times {\gamma }^{\frac{\mathrm{epoch}}{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\text{\_}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}}},$$

where lr is the current learning rate, $\gamma$ is the reduction factor, epoch is the current epoch, and $\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}\text{\_}\mathrm{s}\mathrm{i}\mathrm{z}\mathrm{e}$ is the epoch interval at which the rate is adjusted.

The dataset consists of a total of 4403 time steps (day and night), with 70% used for training, 943 for validation, and 944 for testing. The models are trained using sequences of six past time steps to predict the next six time steps.

During training, the model iteratively updates its parameters using the training data. After each epoch, inference is performed on the validation data to monitor performance. Once training is complete, the model is evaluated on the test set. All results reported in this study correspond to the average obtained from 10 independent runs, each performed with a different random seed, ensuring the stability and reproducibility of the results.

4.3. Results

This section presents the results obtained by the different models for predicting noise levels (LAeq).

Table 2. Model results for LAeq.

Model	RMSE	MAE	R2	Correlation	Accuracy
GraphSage+LSTM	0.0297	0.0274	0.6985	0.8982	0.8013
CNN3D+LSTM	0.0340	0.0340	0.6827	0.8803	0.6763
GAT	0.0361	0.0365	0.6382	0.8586	0.6485
LSTM	0.0305	0.0290	0.7188	0.8939	0.7725
GraphSage	0.0294	0.0268	0.7234	0.8997	0.8188
Transformer	0.0178	0.0171	0.6684	0.8807	0.9090
TransformerConv	0.0238	0.0205	0.6333	0.9330	0.7157
CNN1D+LSTM+TransformerConv	0.0169	0.0174	0.8927	0.9601	0.9158

summarizes the evaluation metrics used: Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Coefficient of Determination (R²), Correlation, and Accuracy. These metrics allow us to evaluate and compare the performance of each model in terms of accuracy and its ability to capture temporal and spatial relationships in the data. The metrics are computed as the average values from the predictions on the test data, i.e., the last 15%, which corresponds to predictions for 492 days.

Our proposed model, CNN1D+LSTM+TransformerConv, demonstrates superior performance compared to the other evaluated models. With an accuracy of 0.9158 and a correlation of 0.9601, it significantly outperforms the other approaches. Specifically, it reduces the MAE by 35% and the RMSE by 37% compared to the next best model. These results highlight the effectiveness of our approach in accurately predicting noise levels.

Therefore, Figure 7 shows the evaluation metrics per time step for the best model, CNN1D+LSTM+TransformerConv, illustrating how model performance varies across different forecast horizons. This figure reinforces the robustness of the proposed model, emphasizing its accuracy in predicting noise levels over time.

In Figure 7, we observe how the evaluation metrics change across different prediction horizons. Both RMSE and MAE increase slightly as longer-term intervals are predicted, indicating a slight reduction in accuracy for more distant time steps. Correlation and R² remain high for most steps but also exhibit a slight decline in the longer forecast horizons.

5. Discussion

When analyzing the results, it is observed that the model CNN1D+LSTM+ TransformerConv outperforms the others in all evaluation metrics, obtaining the lowest RMSE (0.0169) and the highest R² (0.8927). In addition, this model achieves the highest correlation (0.9601) and accuracy (0.9158), indicating a superior ability to capture the complex spatio-temporal relationships in noise data. Our model, like GraphSage+LSTM or CNN3D+LSTM, is designed to capture spatio-temporal relationships, but the main difference is that CNN1D+LSTM+TransformerConv captures temporal relationships first and then spatial ones, which could explain its superior performance. In comparison, models like GraphSage+LSTM and GraphSage show relatively good performance, with an RMSE of 0.0294 and correlations close to 0.9. These results indicate that the explicit combination of spatial information via graphs and temporal dependencies via LSTM provides significant predictive value. However, the precision and predictive capacity of these simpler approaches are limited compared to the proposed hybrid model. On the other hand, the CNN3D+LSTM and GAT models show lower performance, with higher errors (RMSE of 0.0340 and 0.0361, respectively) and lower correlation. This may be due to the fact that these architectures fail to effectively capture spatial and temporal relationships simultaneously, possibly due to their structural complexity or limitations in spatial information processing. A noteworthy aspect is the performance of the simple LSTM model, with an RMSE of 0.0305 and a correlation of 0.8939. This suggests that temporal dependencies are key in predicting noise levels and that purely sequential models can capture much of this complexity without explicitly incorporating spatial components. Regarding model performance over multiple time steps, the RMSE fluctuates during the first 10 steps and then shows a steady increase, suggesting a decline in predictive capacity for more distant steps. MAE behaves more erratically, with noticeable peaks around steps 4 and 13, indicating that the model struggles to capture specific patterns. R² remains high (0.89) for most steps but shows a slight decreasing trend after step 10, confirming a decline in explanatory power over longer horizons. Correlation initially decreases until step 4, then fluctuates with a peak around step 10, and decreases again in later steps, suggesting difficulty maintaining a linear relationship between real and predicted variables as the forecast horizon is extended. Nevertheless, the metrics and predictions remain accurate, with relatively low error margins. The following figure shows a scatter plot of actual versus predicted values for all time steps. Most points cluster around the red diagonal line, indicating good overall model performance. Although the model captures the general trend, it struggles to precisely predict some extreme values. It is likely that specific and anomalous events, such as protests or other activities that generate noise peaks, challenge the model’s predictive ability. Unfortunately, adding variables to help the model anticipate these social factors is difficult due to their variability and unpredictability. When evaluating the model’s predictive ability for the L1 variable (representing noise peaks), we found that R² drops to 0.7046 and correlation decreases to 0.8892, compared to the 0.8927 and 0.9601 obtained for LAeq. This indicates that although the model can adequately predict average noise levels, its accuracy decreases for extreme events. Interestingly, the RMSE and MAE for L1 remain relatively similar, suggesting that the model’s errors relate more to its ability to capture extreme fluctuations than to general prediction errors. These results highlight the need to explore additional methods that help the model better anticipate anomalous peaks, such as using external data or models specialized in anomaly detection. Nonetheless, as shown in the Figure 8, the results fit the real values quite well.

As an example of a spatio-temporal prediction, we present the Figure 9 (real noise) and Figure 10 (predicted noise) that illustrates a forecast for a specific time step.

As observed, predictions are quite accurate. If we inspect a specific node at a specific moment (see Figure 11), we can observe variations between day and night, as well as the fact that the model slightly underestimates real values, possibly because the training data are mostly from the pre-COVID-19 period, while the test set is post-pandemic, where noise pollution is slightly lower.

This study presents a hybrid methodology integrating Convolutional Neural Networks (CNN), Long Short-Term Memory Networks (LSTM), and Graph Convolutional Networks (GCN) for urban noise pollution prediction. The experimental results show improvements compared to simpler baseline models. However, several aspects require detailed analysis to better understand the scope of the proposed model. First, the complexity of the hybrid architecture must be considered. While the model yields better predictive performance, the combined use of CNN, LSTM, and TransformerConv must be clearly justified. Experimental results show that simpler methods like LSTM or GraphSAGE alone do not effectively capture the complex spatio-temporal dynamics of urban noise. Thus, the proposed combination provides an advantage by simultaneously modeling local temporal patterns, long-term dependencies, and spatial relationships between stations, justifying the approach’s greater complexity. Additionally, it is necessary to conduct more extensive comparisons with current state-of-the-art models, especially recent architectures using attention mechanisms or purely Transformer-based time series models. While the comparisons made in this study are satisfactory, future work could include further comparisons with emerging techniques, reinforcing the validity and relevance of the proposed approach. In practical terms, the presented model has potential real-world applications, contributing to improved urban noise pollution management. For instance, municipalities could use the predictions to prioritize corrective or preventive measures. Furthermore, since the model uses data typically available in modern cities (historical noise levels and station locations), it is highly replicable in other cities with different acoustic dynamics by simply adapting the graph’s spatial configuration. Finally, while the proposed model shows solid predictive performance, its effectiveness depends significantly on the quality and representativeness of the data used. Exceptional acoustic situations or extraordinary events may affect its accuracy. Therefore, future research should incorporate additional sources of contextual information, such as one-off urban events (holidays, public works, and mass events) and weather variables influencing noise propagation. Additionally, exploring adaptive approaches capable of dynamically updating the model could help improve its predictive performance in continuously changing real-world scenarios.

5.1. Analysis of Individual Components (Ablation Study)

To analyze the individual contribution of the main components of the proposed model (CNN, LSTM, and TransformerConv), additional experiments were conducted using only specific modules. In particular, we evaluated a model focused solely on temporal dependencies (LSTM) and one focused solely on spatial relationships (GraphSAGE). The results show that the LSTM model achieved an RMSE of 0.0305 and a correlation of 0.8939, while the GraphSAGE model obtained an RMSE of 0.0294 and a correlation of 0.8997. These results indicate that both temporal and spatial relationships individually contribute significant predictive value. However, the ordered integration of both dimensions—first capturing temporal dependencies via CNN1D+LSTM and then spatial relationships via TransformerConv—improves predictive capability, as reflected in the superior performance of the CNN1D+LSTM+TransformerConv hybrid model (RMSE of 0.0169 and correlation of 0.9601). This provides functional evidence of the relevance of each component in urban noise prediction.

5.2. Limitations

According to Directive 2002/49/EC, the day–evening–night level ($L_{den}$) is defined over three time intervals—day (12 h), evening (4 h), and night (8 h)—with penalties of +5 dB in the evening and +10 dB at night. Data limitation: our records include only daytime (12 h) and nighttime (12 h) noise levels, without an independent measurement for the evening period. Consequence: it is not possible to compute the standard $L_{den}$, which prevents direct comparison with studies using this full metric. Future work: it is recommended to include a third measurement for the evening period (4 h) to implement the official formula:

$$L_{den}=10{\log }_{10}\left({\displaystyle \frac{12·{10}^{L_{\mathrm{day}}/10}+4·{10}^{(L_{\mathrm{evening}}+5)/10}+8·{10}^{(L_{\mathrm{night}}+10)/10}}{24}}\right)$$

and enhance the analysis of acoustic exposure.

6. Conclusions

This study has demonstrated the ability of various deep learning models to predict noise levels in densely populated urban areas, such as Madrid, in a spatio-temporal manner. Among the evaluated models, the CNN1D+LSTM+TransformerConv model has shown significantly superior performance across all key metrics compared to the others, suggesting that this approach better captures the complex spatio-temporal relationships inherent to noise pollution.

By combining 1D convolutions to capture temporal patterns, LSTM networks for long-term dependencies, and the TransformerConv layer to analyze spatial relationships among nodes, our approach offers a robust solution for noise pollution prediction. While models like GraphSAGE+LSTM and CNN3D+LSTM have also proven effective, their performance is lower, reinforcing the importance of integrating multiple processing mechanisms into a single model.

Despite these achievements, the model faces challenges in predicting noise peaks and anomalous events, paving the way for future research. Improving the accuracy of predicting extreme events, such as noise peaks, will require additional approaches, such as incorporating external data or using models specialized in anomaly detection.

In summary, the CNN1D+LSTM+TransformerConv model stands out as an efficient and accurate solution for urban noise prediction, with potential for further improvements in capturing extreme events. This work contributes to the literature by demonstrating the relevance of complex spatio-temporal models in the study of noise pollution.