The global and local datasets include the annual mean NO2 concentrations (measured in µgm-3) for the year 2017 . Figure  presents the distribution of NO2 concentrations at the global and local measurement stations. The terms “global” and “local” were chosen to reflect the relative scale of the datasets, with “global” representing a broader, cross-national dataset and “local” focusing specifically on Amsterdam. While the “global” dataset includes only two neighboring countries, this terminology emphasizes its wider scope compared to the local dataset. The global dataset comprises ground station measurements from Germany and the Netherlands, while the local dataset includes the Palmes data in the Amsterdam region.

The spatial distribution of NO2 measurement stations is provided in the Supplement (Figs. S1 and S2). Urban areas generally have a higher density of measurement stations. This study focuses on the differences between global and local models, particularly in Amsterdam, while also considering the city's less densely populated areas to examine the urban impact on predicted NO2 concentrations in the local models.

To evaluate whether prediction quality varies across areas with different characteristics of spatial patterns (e.g., high vs. low road density), the global and local datasets are divided into three spatial groups based on population density and traffic-oriented variables. Population data for 2015 from the Global Human Settlement Layer are used , and road length information is sourced from . Descriptive statistics for the variables used to define the spatial groups are presented in Table .

The three spatial groups are defined as follows:

Urban. Areas within 100 m of road class 1 (highways) or 2 (primary roads) and with a population density in the highest 25 % or areas where both road class 3 (local roads) values and population density are in the highest 25 %.

This classification resulted in 85 observations being labeled as “urban”, 138 as “suburban”, and 259 as “rural”, totaling 482 observations in the global dataset. Given the smaller sample size of the local dataset, the threshold for defining “urban” was adjusted from the 75th percentile to the 50th percentile, which had a converging effect on the relative group sizes. Moreover, the increase in samples classified as “urban” is desirable, as this group exhibits relatively high heterogeneity. The local dataset consists of 56 observations classified as “urban”, 46 as “suburban”, and 30 as “rural.”

Although this adjustment introduces some inconsistency between the global and local definitions of “urban”, it addresses the challenge of unequal distributions of instances across groups in the local dataset, which could introduce bias into the statistical learning models. The threshold adjustment represents an initial step toward mitigating such effects by ensuring a more balanced representation of spatial characteristics within the local model. Figures S3 and S4 show the spatial distribution of observations among these groups for both datasets, while Figs. S5 and S6 show the measured NO2 concentrations per station.

We utilized a set of variables derived from , including data on industrial areas, road lengths, population density, Earth night lights, wind speed, temperature, elevation, TROPOMI level 3 NO2, and global radiation. A complete list of these variables is available in the Supplement (Table S1). Precipitation data were sourced from weather stations and interpolated using ordinary kriging to cover the NO2 measurement stations. Kriging parameters are detailed in the Supplement (Sect. S2.4, “Parameters”).

Building density was obtained from the World Settlement Layer 2015 dataset available on Figshare . In line with previous studies , we considered various buffer sizes (100, 500, 1000 m) around measurement stations to account for spatial proximity effects, especially in densely populated urban areas. Normalized difference vegetation index (NDVI) values were obtained from NASA .

Traffic volume data were sourced from the Nationaal Dataportaal Wegverkeer (NDW) in the Netherlands and from the Bundesanstalt für Strassenwesen (BAST) in Germany . These data, generated by automatic counting stations, are expressed as average hourly traffic in 2017, with buffer sizes of 25, 50, 100, 400, and 800 m. The formula for calculating average hourly traffic is provided in the Supplement (Eq. S1).

Feature selection for global models is initially based on Shapley values . While the variance inflation factor (VIF) is effective at detecting multicollinearity, it does not consider feature importance or interactions. The VIF results are available in Tables S4 and S5. Feature selection aims to remove irrelevant or highly correlated predictors that could generate unstable estimates and affect model interpretation .

Shapley values are calculated for each feature (i.e., predictor) based on its contribution ϕj to the prediction of NO2 concentration levels compared to the average prediction across the dataset . The contribution of a feature is determined by comparing the difference in the response variable when the feature is present vs. when it is absent (i.e., marginal contribution) . The formula for calculating Shapley values can be found in the Supplement (Eq. S6).

In this study, feature selection is guided by the out-of-sample performance in a random sampling validation repeated 10 times, where Shapley values are calculated for each iteration of the random forest models. Predictors are ranked based on the median Shapley value across all iterations. The relative positions of each predictor using the median-based approach are illustrated in Figs. S13 and S14, with the Shapley ranking of a single run shown in Fig. . The most influential predictors for the global models include nightlight intensity (450 and 3150 m buffers), population density (1000 and 3000 m buffers), road class (class 2 within 25 m and class 3 within 300 and 3000 m buffers), the annual mean NO2 column density of 2018 measured by the TROPOMI instrument on board Sentinel-5p (trop mean filter 2018), building density in the 100 m buffer, NDVI, and traffic buffers (25 and 50 m buffers). Descriptive statistics of the most influential predictors for the global models are in Table .

A random forest algorithm is applied iteratively to determine the optimal number of predictors, starting with the 2 most influential predictors and extending to the 30 most influential features. The RMSE and R2 metrics are used to evaluate the optimal number of predictors. The number of predictor variables and their corresponding evaluation scores (R2, RMSE) are shown in Fig. . In particular, prediction accuracy improves significantly when considering at least 12 predictors, but the improvement is marginal beyond this number.

Due to the random forest model's poor performance across all local station measurements (Figs. S15–S17) and per spatial group (Table S2), the random forest algorithm is deemed unsuitable for identifying the number of variables for the local models. Instead, the best-subset regression is used for variable selection in local models. This approach tests all possible combinations of predictor variables , with a maximum of 30 predictors considered. The statistical criteria include adjusted R2; Mallows criterion p – with p referring to the number of predictors in the regression model; and Bayesian information criteria (BIC) scores. As a result, nine features are identified for the local models. The most influential predictors for the local models include nightlight intensity (450 and 4950 m buffer), population density (3000 m buffer), road class (class 1 within a 5000 m buffer, class 2 within 1000 and 5000 m buffers, class 3 within 100 and 300 m buffers), and the traffic buffer (50 m buffer). Descriptive statistics of the most influential predictors for the local models are in Table .

In global modeling, comparisons are made among tree-based models, random forest and XGBoost, and linear models, with LASSO and ridge penalization (the models are also called LASSO and ridge). For local modeling, we compare linear models, mixed-effect models, and kriging models. Each model is evaluated based on the standard matrix of R2, RMSE, and MAE . For model evaluation, leave-one-out cross-validation (LOOCV) is employed for local models, while a random 90/10 training–testing split is used for global models. Additionally, the prediction patterns of the local and global models are analyzed. To benchmark the model performance, a mobile NO2 map of the study area is used for comparison. This map provides detailed spatial information collected by two Google Street View cars that continuously measured NO2 at a frequency of 1 Hz in Amsterdam from 25 May 2019 to 15 March 2020 (stopped due to the COVID-19 lockdown policy). We acknowledge that the temporal resolution of these benchmark data differs from the coarser temporal scales used in our models. The data used in are over specific, limited time periods, while our models address predictions over broader temporal spans. Despite this temporal inconsistency, the detailed spatial granularity of the annual map from provides valuable insights and remains an appropriate standard for assessing spatial prediction quality.

Table  provides an overview of the global and local models, along with selected predictors and evaluation methods. The global models are applied to areas with varying demographic characteristics, including two large cities with populations exceeding 700 000 (Amsterdam and Hamburg), a mid-sized city with around 350 000 inhabitants (Utrecht), and a small city with approximately 70 000 inhabitants (Bayreuth). The resolution of the analysis is 100 m, with raster files of the most important predictors resampled and converted into 100 m grid cells for these regions using spatial extraction methods. The influential predictor information (for global models, see Tables  and ; for local models, see Tables  and ) is recalculated at a 100 m resolution for the extent of the aforementioned regions. The 100 m by 100 m grid cells containing predictor information are used to predict NO2 values for the respective 100 m grids based on the trained local and global models . The 100 m grid resolution is consistently applied in the predictions for both local and global models. Local model predictions are applied exclusively to Amsterdam. Table  summarizes the complexity of the models and how spatial components are accounted for.

Evaluations of the different linear and nonlinear models were carried out using repeated random sampling validation (i.e., Monte Carlo cross-validation) performed 20 times. In each iteration, 90 % of the data was used for training and the remaining 10 % for testing. For testing model performance on spatial groups, 30 testing samples were used every time . This approach allowed us to evaluate the variance and median statistics for each model in terms of R2, MAE, and RMSE (Fig. ). The repeated sampling provided stable estimates.

When comparing out-of-sample performance via 20-fold repeated random sampling validation, the linear models (i.e., LASSO and ridge) exhibited performance similar to that of the nonlinear models, particularly in terms of R2. Among the models, the random forest consistently outperformed the others, with the highest median R2, lowest RMSE, and lowest MAE. The robustness of the random forest model is further emphasized by its minimal standard deviation in R2 and MAE (Fig. ).

The performance of the local models was assessed using the R2, RMSE, and MAE metrics. Table  summarizes the performance of the linear model, mixed-effects model, ordinary kriging model, and universal kriging model, all evaluated using leave-one-out cross-validation. Among these, the ordinary kriging model exhibits the poorest performance. Figure  illustrates the spatial prediction patterns for each model. Notably, the universal kriging model outperforms the ordinary kriging model significantly. However, note that with the ordinary kriging model, we could see the smoothed spatial patterns of the air pollution measurements. The simple linear model surpasses the universal kriging method in terms of prediction accuracy. Incorporating spatial groups as random effects in the mixed-effects model leads to a higher R2 and lower RMSE and MAE, indicating the importance of accounting for spatial heterogeneity.

Table  provides model performance metrics for each spatial group, again using leave-one-out cross-validation. Consistent with the global model results, local models trained on urban observations tend to perform poorly. This poor performance is likely caused by an imbalance between the relatively small number of samples and the relatively high heterogeneity. This imbalance may hinder the models' ability to capture the variability within urban areas, contributing to their poorer performance in this group. Interestingly, proximity to roads does not necessarily correlate with model performance, as the suburban group exhibits a higher R2 than the rural group. Unlike global models, which perform best in rural areas, local models perform best in suburban areas. This difference may arise because observations in rural areas within the local dataset are more similar to those in urban and suburban areas than in the global dataset due to a more uniform distribution of predictor values.

Figure  displays the predicted NO2 patterns based on the local dataset. The prediction map for the linear model (panel a) is quite similar to those for the mixed-effects (panel c) and universal kriging (panel e) models, with all identifying a high-NO2-concentration cluster in the northwestern part of Amsterdam. Further analysis suggests that this cluster is likely influenced by the predictor “road class 2 5000” (i.e., the number of primary roads within each 5000 m buffer), as this predictor exhibits a similar cluster in the same location (see Figs. S32–S41).

The two models that account for spatial groups before the modeling process (mixed-effects and universal kriging) display comparable patterns, where the influence of roads is evident through either the predictors themselves or the spatial groupings (see also Fig. S42). The relatively low NO2 values along roads in the outer Amsterdam area can be attributed to the spatial grouping. High standard deviations in predictor values within a specific spatial group can affect that group's NO2 predictions, potentially leading to overestimation or underestimation in certain areas.

The high NO2 values along roads are primarily associated with the suburban spatial group, where the observations are located within 100 m of the roads. Compared to the rural group, the data distribution for each predictor in the suburban group is substantially different, leading to distinct learning patterns that explain the relatively high prediction values along roads (see Figs. S43–S51). In some instances, negative predicted values are observed, albeit rarely. These may result from discrepancies in feature characteristics between the training and testing datasets.

Comparing local prediction patterns to global prediction patterns reveals that the local models identify a cluster of high air pollution in the northwestern part of Amsterdam that the global models do not detect. This discrepancy could be due to differences in the spatial distribution of NO2 values between the local and global datasets, leading to distinct learning patterns in the respective models (Fig. ). Moreover, Figs.  and  underscore the challenge of comparing spatial variations between global and local models, given their differing algorithms. Local models, with their focus on specific spatial groupings and detailed predictors, capture regional clusters that global models may overlook or underrepresent due to their broader scope.

For both global and local datasets, traffic and population density emerge as the most influential predictors, aligning with the findings of , which emphasize the importance of these variables for improving prediction accuracy. The strong influence of traffic on NO2 concentrations also supports the conclusions of and . However, since the pollutant sources vary , the modeling results for NO2 may not be directly applicable to other pollutants.

Without accounting for the spatial groups, the differences in terms of the accuracy assessment matrices between linear and nonlinear techniques are minimal. The random forest model generally performs with the highest R2 and lowest MAE, and the R2 of the ridge model is higher than that of the XGBoost model. When accounting for spatial groups, the differences in model performance between linear and nonlinear techniques become more pronounced, with nonlinear models generally outperforming linear models, particularly in rural areas where data are more homogeneous.

A limitation of the data is the fact that the most heterogeneous group (urban) is the least represented in terms of the number of data points, at least for the global dataset. In urban areas, the more heterogeneous nature of the data reduces the performance gap between linear and nonlinear techniques, with both performing poorly. This poor prediction accuracy in urban areas is concerning, as the impact of air pollution is often more severe in these regions due to proximity to traffic-heavy roads and industrial areas . Although spatial grouping improves predictive reliability, it can lead to counterintuitive patterns, such as lower predicted NO2 concentrations along roads compared to in the surrounding rural areas. In the local dataset, the threshold for defining “urban” areas was adjusted from the upper 75 % (0.75) to the median (0.5). This adjustment was necessary due to the limited sample size, which required a broader definition to ensure sufficient data coverage for urban areas. However, this change also resulted in a less stringent definition of “urban,” potentially including areas with lower population densities. While this adjustment expands the number of training samples available for the most heterogeneous group (urban), it introduces a limitation by diluting the urban group and affecting the comparability of results. This trade-off underscores the challenges of balancing data representation with statistical robustness in spatial analyses.

The cross-validation (CV) strategy plays a crucial role in model performance and generalizability. In this study, we opted for a random 90/10 training–testing split repeated 20 times to ensure model stability while maintaining sufficient training data. For testing on spatial groups, we further limited the testing samples to 30 to account for spatial variability and avoid data imbalance. This approach allows us to evaluate model performance across different spatial settings while mitigating the risk of overfitting.

However, the random split employed in this approach can lead to biased performance estimates, particularly with small datasets. Some points may be used multiple times, while others might not be used at all, which can skew results. To address this issue, alternative strategies like the 5-fold CV are often employed, as they provide a good balance between bias and variance. In the 5-fold CV, the dataset is divided into five partitions, ensuring that each point is used for validation exactly once. This method can be particularly useful when the dataset is small, as it ensures more comprehensive utilization of the data.

In our study, we chose to use the Monte Carlo CV because it provides a robust measure of uncertainty, particularly for relatively small datasets. The Monte Carlo CV works well when data are limited, as it generates multiple datasets through resampling, allowing us to estimate variability and model performance more effectively. The Monte Carlo CV aims to reduce bias by randomly drawing training points for each iteration, thus providing a more reliable estimate of model accuracy. Although increasing the number of iterations and reducing the testing set size could further reduce bias, leaving more data points in the testing set offers additional information about the model's performance.

Alternative techniques, such as the 5-fold CV, remain popular because they offer a straightforward balance between training and validation data. Nonetheless, the Monte Carlo CV may provide better estimates of uncertainty, especially in cases with limited data, which could make it preferable for heterogeneous spatial datasets. Although increasing the number of iterations may improve predictive reliability, the trade-off between computational efficiency and statistical robustness must be carefully considered.

criticized the imprudent use of global-scale models, in particular highlighting the issue that model performance cannot be validated in regions without observational data. They advocate for the use of spatial cross-validation to address this limitation. After careful consideration, we opted for the Monte Carlo CV to mitigate the bias introduced by spatial cross-validation . We would like to emphasize that the core issue in cross-validation for spatial modeling lies in the inclusion of spatial coordinates or distances as predictors. From this perspective, it is evident that sampling should be random in the predictor space. Therefore, we argue that cross-validation in spatially correlated data is not fundamentally different from standard cross-validation, provided that the predictor space sampling is handled appropriately.

In comparing global and local models, each approach has distinct strengths and limitations. Local models, tailored to specific spatial groupings and incorporating detailed predictors, excel at capturing regional clusters and nuances. These models can identify patterns and variations that broader, global models might miss or represent inadequately. On the other hand, global models are designed to capture overarching trends across larger areas but often overlook the finer local details crucial for accurate predictions in specific regions.

The findings of support this distinction, highlighting the fact that integrating large-scale stationary measurements with local mobile data improves modeling performance in urban areas by accounting for finer spatial variations. Their study underscores the limitations of global models, which, while providing a broad overview, may fail to capture the detailed local variations necessary for precise predictions. By combining global and local data, a more accurate and nuanced depiction of air pollution can be achieved, particularly in complex urban environments where local details are critical.

The influence of specific predictors on NO2 concentrations can vary significantly between cities. For example, building density and population are more significant contributors to air pollution in Utrecht, whereas traffic has a greater impact on high NO2 concentrations in Hamburg. Applying global models with the same predictors across different cities may conceal this finding and yield suboptimal results. It is therefore important to consider the spatial heterogeneity and at the same time ensure a consistent uncertainty assessment. However, the current number of official ground stations may not be sufficient to characterize the spatial heterogeneity and ensure a detailed and reliable prediction.

The limited number of observations in the local dataset poses challenges for fitting complex models. Transforming the original data could potentially avoid predictions falling outside the plausible range (e.g., below 0 µgm-3). However, in this study, a transformation was not applied for the following reason. Although airborne pollutant concentrations are often positively skewed , found that optimal modeling results were obtained without transforming the data and by using a Gaussian likelihood – even when other distributions, such as gamma, better reflected the data's skewed distribution. Moreover, while the LASSO and ridge models perform well with the global dataset, their predictions were less satisfactory with the local dataset. In this study, traffic volumes were a significant feature, yet no distinction was made between different types of traffic (e.g., cars, buses, trucks), vehicle types (e.g., electric, diesel), or engine types, all of which are known to influence air pollution . For example, distinguishing between vehicle types could reveal that certain roads, such as those leading to or from the port of Hamburg, have a higher proportion of trucks, which might explain localized clusters of high NO2 concentrations.