## 4 Results
### Rfsp and Quantile Prediction Intervals Results  
For the data across the entire area, we have applied the RFsp method [27]. It has been trained on the data described in Section 3.1 on the ozone and aerosol data. This produces a spatiotemporal model (see Section 3.3). We present the output of the model when test data are passed into it and examine the statistical performance of the model using some common tests that are widely applied in analyses where RF models are used; specifically, we have applied k-fold cross-validation (CV), and out-of-bag (OOB), in each case yielding an R\({}^{2}\) and an MSE.  
The RFsp methodology applied to all (\(r\times n\)) data rows in the design matrix produces a model predicting ozone level given all \(r\) predictor variables. Predicted values from the RFsp algorithm are presented for gridded ozone in Figures 11-13, for the 20 consecutive days starting on 3 November 2019, 1 December 2019 and 20 January 2020, respectively. These predictions should be compared to the actual ozone level data shown in Figure 7 in Section 3.1. For ease of comparison, note that the figures such as Figure 7 should be compared with Figure 11. The other two regions and their appropriate dates produce similar comparisons; we refrain from showing these.  
As well as predicted values, corresponding data relevant to quantile prediction intervals covering the same set of days are presented in Figures 14-16. The choice of each starting date in the above-mentioned figures, together with the 20 consecutive days thereafter, allows for effective coverage across most of the change-points in the data, as discussed below. We investigate these change-points for both total sum ozone and total sum aerosol time-series separately for each of the three regions.  
Figure 11: Spatial-temporal predictions of daily gridded ozone values over 20 days from 3 November 2019.  
Figure 12: Spatial-temporal predictions of daily gridded ozone values over 20 days from 1 December 2019.  
Figure 14: Quantile prediction intervals of daily gridded ozone values over 20 days from 3 November 2019.  
Figure 13: Spatial-temporal predictions of daily gridded ozone values over 20 days from 20 January 2020.  
Figure 16: Quantile prediction intervals of daily gridded ozone values over 20 days from 20 January 2020.  
Figure 15: Quantile prediction intervals of daily gridded ozone values over 20 days from 1 December 2019.  
We anticipate that these quantile prediction intervals will be wider around the change-points. The change-points in the total summed data, which are not represented in the above-mentioned figures, serve to focus our subsequent investigation into the data, specifically, in Region 1 (Ocean), Region 2 (Vic), and Region 3 (NSW).  
It can be seen from Figures 11-13, that the parts of the map where the gradient of the ozone value is large in absolute terms, typically correspond to a distinctive ridge-like pattern in the quantile prediction interval plots. These ridge lines roughly follow the contours of the three-dimensional surfaces of the predicted ozone levels. These predicted ozone levels are represented by the colors in the ozone color bar (e.g., 'Ozone (DU)' legend; RHS of Figure 11). The value of the quantile prediction interval also appears to be, at least visually, closely correlated with the magnitude of these gradients of the predicted ozone value maps. In the same vein, areas where the predicted ozone values have sharp high-value peaks are associated with a higher concentration of wider quantile prediction intervals. The darker blue areas, with values ranging between 380 and 420 DU of the ozone prediction plots roughly match up with the purple areas within the range 12 to 15 on the color bar such as in Figure 14.