Once a clustering of our stations is obtained, we can use the clustering solution to impute missing TS (pollutants). If station j belongs to cluster Cx, (1≤x≤k, where k is the number of clusters) given the measured pollutants over time, then, to impute pollutant Pi based on the clustering results, we use three models.

We impute the average of pollutant Pi in cluster Cx, which is the hourly average of pollutant Pi in all the stations that fall in this cluster. We call this method cluster average (CA).

First, we measure the geographic distance using the Harvison metric, which calculates geographic distance on Earth based on longitude and latitude. We calculate the distance between station j and all other stations that measure pollutant Pi. Then to impute pollutant Pi for station j we use the following:

the nearest neighbour (1NN) using the Harvison-based distance to station j – this method is called 1NN; and

In this approach, for a given station j, to impute pollutant Pi, we use the median value of all the imputed values from the previous models. Those are cluster average (CA), cluster average considering the station type (CA+ENV), cluster average considering the region (CA+REG), first nearest neighbour (1NN), and the average of the two nearest neighbours (2NN). This method is called Median. This imputation approach may be computationally the most expensive as it needs for all others to be computed, but ensembles have the potential to provide very powerful solutions by combining predictions.

Table  shows the statistical analysis results. In this table N is the number of stations that measure each pollutant. The table also shows the fraction of predictions within a factor of 2 (FAC2), mean bias (MB), normalized mean bias (NMB), root mean squared error (RMSE), coefficient of correlation (R), and index of agreement (IOA).

In general, model 6 (Median), which is the model that uses the ensemble technique of other models, gives the lowest error average (RMSE), the highest Pearson correlation coefficient (R), and the highest agreement between imputed and observed concentrations (IOA) for O3, PM2.5, and PM10. However, NO2 shows different behaviour, with model 2 (CA+ENV) achieving slightly higher performance with an increase in the correlation coefficient (by 0.049) and decrease in error average (by 0.826) compared to model 6 (Median). The model bias (MB) for model 2 is 50 % higher than that of model 6. NO2 shows local patterns, as it is concentrated where it is emitted in urban areas and near the roadside. Adding to that, NO2 is shorter-lived than other pollutants and shows greater spatial variability, with concentrations being strongly influenced by the environment type (e.g. roadside, urban background, rural). This changes the NO2 concentrations from one location to another based on the environmental type .

All the selected models performed well, with 71 %–89 % of their imputations falling within a factor of 2 of the observed concentrations as shown in the FAC2 values in Table . According to , an air quality model minimum requirement is that the FAC2 value is higher than 0.50 and NMB values should be in the range between -0.2 and +0.2. Both are met by our models. NMB measures if the model underpredicts or overpredicts, as it estimates the difference between the mean observed and imputed concentrations. Negative NMB means that the model underpredicts and vice versa. All the models have very small biases.

We use a Taylor diagram to analyse three main statistics: correlation coefficient R, the standard deviation (sigma), and the root mean square error (centred). These statistics can be plotted on one (2D) graph, which can be represented through the law of cosines .

The standard deviation represents the variability between modelled and observed concentrations. The observed variability is plotted on the x axis. The magnitude of the variability is measured as the radial distance from the plot's origin. The black dashed line shows this for the observed value. The grey lines are isopleths for the correlation coefficient (R) as indicated by the arc-shaped axis; the correlation increases along the arc towards the x axis. The centred root mean square error (RMSE) is represented by the concentric brown dashed lines. The further the points or models are from the observed value, the worse performance they have . Figure  shows Taylor diagram plots for all models with all pollutants.

In almost all cases the models exhibit less variability than observed, as indicated by the points being closer to the origin than the black dashed line. In general, model 4 (1NN) followed by model 5 (2NN) show variability that is most similar to the observations, as indicated by their relative closeness to the black dashed line. However, these models tend to have the lowest correlation coefficients, as indicated by the grey lines, and the greatest RMSE, as indicated by the brown dashed lines. Models 4 and 5 use the concentrations from a single site (i.e. the nearest stations) in the imputation, whereas the other models use a cluster average (CA, CA+REG, CA+ENV) or a model ensemble average (Median), so it is reasonable for models 4 and 5 to have variability fairly similar to the observed concentrations. All the other models display less variability than the observed concentrations (as indicated by their points being further from the black dashed line); this may be consistent with their derivation methods, which may smooth out some of the variability.

Model 6 (Median), regardless of its ability to capture variability, is confirmed as having the highest correlation coefficient and the lowest centred root means squared with all the pollutants except NO2, for which it is the second-best behind model 2 (CA+ENV).

We analyse the spread of the modelled and observed pollutant concentrations using conditional quantile plots. Figures  and show the conditional quantile plots for the six imputation models (panels a to f). This visualization splits the concentrations into bins according to values of the modelled concentrations. The median line of these values as well as the 25th / 75th and the 10th / 90th quantile values are plotted together with a blue line showing a “perfect” model. Also shown are histograms of modelled concentrations (shaded grey bars) and histograms of observed concentrations (blue outline bars).

These plots show how the modelled concentrations compare with the observed concentrations and how the models capture the variability in the concentrations. The spread of the modelled concentrations around the perfect model line (blue line) is shown by the shaded portions and quantile intervals. If narrow, it indicates high agreement or precision between the modelled and observed concentrations. The quantile intervals also represent the uncertainty bands. In some cases these intervals do not extend along with the median line due to insufficient concentrations to calculate them. The model with good performance is obtained when the median (red line) coincides with the perfect model (blue line) and when the spread in the percentile is as narrow as possible.

From these plots, in general, the histograms indicate that model 4 (1NN) (panel d) has better estimation of the variability between the observed and modelled concentrations, as observed before, even though the median line does not match the perfect model. This model is positively biased at high concentrations, as shown by the departure of the median line below the blue line for all pollutants. This result supports our analysis from the Taylor diagram that model 4 (1NN) has the lowest variability between modelled and observed concentrations, but with a lower correlation coefficient, and the highest centred root means squared for all pollutants.

In Fig.  (left), the O3 models show that most modelled concentrations match the observations well for a wide range of values. The histograms indicate underestimation in general at the extreme low and high concentrations. In general, the cluster and Median imputation methods (i.e. that use averaging) will tend to struggle to reproduce the lowest and highest concentrations since they take an average approach. Moreover, the highest concentrations are typically limited to relatively few data points. The cases of high ozone concentrations typically occur during specific meteorological conditions and are episodic in nature, and there may be small differences in timings of the peak concentrations at different sites. Very low ozone concentrations are likely to occur at specific sites (near roads where emissions of nitric oxide are large) and therefore may not be reproduced in the models which take a cluster average or where a nearest-neighbour site is not a similar type of site.

Model 6 (Median) (panel f) has the best performance, as indicated by an overlapping median line with the blue line. This model has the lowest mean bias and the highest degree of agreement, as indicated by the narrow spread of the modelled concentration quantile intervals.

In the same figure (right), NO2 models show different behaviours from this analysis. Even though the statistical analysis shows that model 2 (CA+ENV) (panel b) gives the best performance, it is clear that in this model, the modelled concentrations tend to be lower than observations for most concentration levels (the medians are under the blue line), and the width of the 10th and 75th as well as the 10th and 90th percentiles is quite broad. The only advantage of using this model is its ability to capture a wide range of concentrations. Model 4 (1NN) (panel d) compared to other models can reproduce the higher concentrations (higher than 125 µg m-3) as it does not take an average approach. However, this model is positively biased (NMB = 0.095), which is shown by the departure of the median line from the blue one.

The variation between PM2.5 models in Fig.  (left) shows similar performance for the different models. The quantile intervals are wider within the area of high concentrations ≤60 µg m-3, and all models underestimate the high concentrations ≤80 µg m-3; note that these concentrations are very low-frequency events.

Model 6 (Median) (panel f) gives better performance, as indicated by the narrow spread of the modelled concentration quantile intervals and minimal bias, which is indicated by the overlaps between the red and blue lines compared to other models. Models for PM10 (right) show performance similar to PM2.5.

In this analysis, we focus on the performance of model 6 (Median) and model 2 (CA+ENV), as those performed best for the different pollutants in the previous section, but now we break down the analysis for the six environmental types (background rural, background urban, background suburban, and industrial urban, industrial suburban, and traffic urban) to which stations belong. Notice that a pollutant may or may not be measured in all stations and the number of stations of each type is different as shown in Table . We also use conditional quantiles to analyse our model's performance within each environmental type.

First, we show the monthly average concentrations for each pollutant under each environment type in our test dataset (year 2018) to understand the normal variation of the pollutant concentrations in different environment types. Figures , , , and show conditional quantile plots by the environmental types for the selected models. Table  shows the statistical measures of performance also broken down by environment type.

The most common sources of NO2 are roads; however, NO2 concentrations are influenced by traffic density, road locations, and meteorological conditions, which cause variation from one roadside location to another. Figure  shows that high NO2 concentrations are found at traffic urban followed by industrial suburban, then background urban sites, while the background rural sites have the lowest NO2 concentrations.

Figure  shows the conditional quantile plots by station type for NO2 imputation using model 2 (CA+ENV). Here, we see that modelled concentrations are higher than observed concentrations with all environmental types. This is confirmed by all the statistical model quality measures presented in Table , where we can observe a positive mean bias for NO2.

As NO2 distributions in general are skewed to the lower values and our selected model (model 2) (CA+ENV) is based on the average concentrations, the model performs better with lower concentrations.

From Table based on model RMSE, the model's best performance is associated with background rural stations, while the worst performance is shown for traffic urban stations. Contrasting this with quantile plots, Fig. a shows that for background rural stations the histogram and the median line show better performance with lower concentrations (less than 30 µg m-3). On the other hand, for traffic urban stations (panel f), the quantile intervals are wider within the area of high concentrations (higher than 25 µg m-3), and the modelled concentrations tend to be lower than observed concentrations.

For O3, Fig.  shows the monthly average observed O3 concentrations in each environment type. From that we can see that ozone in all environment types follows a similar trend. However, background rural stations have the highest concentrations and traffic urban stations have the lowest, consistent with depletion of ozone due to rapid reaction with fresh emissions of nitric oxide from vehicles. Looking at model 6 (Median) performance in Table  based on the RMSE, the best model performance is associated with industrial urban stations, and its average performance is associated with background rural stations (those with higher concentrations in Fig. ), while its worst performance is associated with traffic urban stations (those with lower concentrations in Fig. ).

Conditional quantile analysis in Fig.  shows the performance of model 6 (Median) for imputing O3 for the six environmental types (panels a to f). The model shows similar performance for industrial suburban (panel d) and background rural stations (panel a). For both types, the model is negatively biased (see also Table ), meaning that the modelled concentrations tend to be lower than observed concentrations (the median lines are above the blue lines).

The worst performance based on the RMSE is associated with traffic urban stations (panel f), which are the stations located at roadsides. With those stations, the modelled concentrations are higher than observed concentrations; i.e. the modelled histogram is shifted to the right. This is indicated by the model positive bias (0.503). The median line also extends beyond the blue line, which means that some modelled concentrations are much higher than observed measurements.

The best model performance is associated with industrial urban stations (panel e) according to the RMSE, even though background urban stations (panel c) appear to have the best performance by looking at the conditional quantile plots. The histogram in panel (c) indicates that the distributions of the observed and modelled concentrations tend to be closer to each other for higher concentrations. However, the model overestimates the average concentrations at these stations (between 25 and 70 µg m-3) and underestimates the very low concentrations.

Figure  shows that PM2.5 concentrations in rural areas are lower than those in suburban, urban background, and traffic urban areas. That is consistent with the model performance at these sites. Figure  shows corresponding conditional quantile plots by station types. Imputing PM2.5 concentrations using model 6 (Median) gives similar performance for the different station types. In general, the model underestimates the concentrations of PM2.5, especially for high concentration levels. Table  shows that the model underestimates high concentrations in suburban, urban background, and traffic urban areas, as indicated by the model negative biases, while it overestimates the concentrations at industrial urban and background rural sites. The model shows the worst performance for traffic urban (panel e), and this is also indicated by the highest RMSE (5.098) shown in Table . The model underestimates the concentrations at these stations, which is confirmed by the model bias (-0.073) in Table . On the other hand, the model's best performance is associated with background suburban sites (Fig. b), even though it underestimates PM2.5 concentrations with a mean bias of -0.013.

Finally, PM10 levels in background rural and urban areas are lower than those in industrial and traffic urban areas as shown in Fig. . For PM10, imputation performance shown in Fig.  is similar for background urban and background rural sites (panels a and b). The model overestimates the concentrations of PM10 that are ≤10 µg m-3, while it underestimates the high concentrations of PM10 at industrial urban (slightly) and traffic urban sites (panels c and d). That is confirmed by the model mean bias at these sites (-0.002, -0.106) as shown in Table .

Next, we show some examples of our imputed TS compared to the real TS for each pollutant using the selected imputation models in some stations. The following examples in Figs. , , , and show the observed and imputed hourly pollutant concentrations for the four pollutants using the selected imputation model that gave better imputation. We also apply the models for which there is a period of missing values in the observed concentrations to give some idea of how the models work for the whole TS, including when real values do not exist.

Figure compares the observed hourly concentrations of PM2.5 (red) at London Eltham station for 1 to 15 January 2018 with imputed concentrations (black) using model 6 (Median). As we can see, the variation between the imputed and the real TS is very small and the imputed TS reproduces the trend very well, even though there is a period of missing concentrations within the observed TS (red). Similarly, Fig.  represents the observed hourly concentrations of PM10 (red) at Oxford St Ebbes station for the same period of time with imputed concentrations using model 6. We can see that model 6 underestimates the high concentrations and overestimates the very low concentrations of PM10 and PM2.5, as mentioned previously in the analysis in Sect. . However, there is still a good match of the trend.

Figure shows a comparison of imputed (black) and observed (red) TS for NO2 concentrations at Birmingham Acocks Green station for the same period of time (1 to 15 January 2018), but produced by a different imputation model (model 3, CA+ENV) that gives better imputation than others for NO2. It is known that NO2 has greater spatial variability than other pollutants and it is very complex to impute; the variation between the imputed and the real TS is slightly higher when compared to the previous examples.

Figure shows a comparison of the imputed (black) and observed (red) TS for O3 concentrations at Birmingham Acocks Green station for the period 16 to 23 January 2018 produced by model 6 (Median). The imputation underestimates the concentrations but represents the trends of high and low values.