## 3 Results
So as an intermediate conclusion, we can note that the balance equation provided valuable guidance for choice of predictors, and also that due to understanding of underlying physics we could inspect and verify the sign of predictors. On the other hand, despite all these advantages, the explained variance seems low, which makes physical interpretation of model parameters rather approximate. Furthermore, the relatively low explained variance would negatively influence predictive ability of the model. Let us now examine possible predictive power of the statistical models developed around balance equation. In the further text we will focus on School 1, since during this particular campaign we have collected more data, and furthermore data gaps in particulate predictors were smaller compared to School 2. Having sufficient data for training and testing is paramount for developing models and also for testing them in a meaningful way. Under optimal conditions, stages of model validation, selection, and predictive errors should be calculated using independent i.e., previously unseen data, however this is often not possible. For smaller datasets it can easily happen that data is incomplete, and there could be different ratios of incompleteness for the test and training sets. Best way for doing cross validation is when it resembles and mimics the way in which model is to be practically used. This brings issues of optimal training test split, both in terms of amount of data and also temporal position of the data. Similar issues are encountered in low cost sensor calibration [19; 27]. Additionally, since we are dealing with time series analysis test data will always need to be temporally separated and have timestamp later in time compared to training data. Thus, to satisfy these requirements of complete data sets and optimal timestamps of training and test data we opted to use 50-50 training test split. One additional requirement for the predictive models of small ion concentration is that they must produce positive output. A way to ensure positive prediction of the statistical model, is to take logarithm of the concentration when training the model and then to exponentiate the prediction of the model [25], and this was applied here. Figure 4 shows comparison of a linear model based on radon and particle aggregates for School 1 and a measurement of small ions. The score metric (R\({}^{2}\)) on the training set is 0.44 and on test set, it is 0.49. While the model shows similar trend as the measurements it is also evident that it significantly under predicts small ions concentration. Gaps in the figure correspond to periods where data for particles was missing. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameters of The Linear Model** & **Intercept** & **Radon Term** & **Aggr1** & **Aggr2** & **Aggr3** & **Aggr4** \\ \hline School 1 parameters & 291.6 & 1.61 & \(-5.08\times 10^{-3}\) & \(-8.24\times 10^{-3}\) & \(-5.08\times 10^{-1}\) & \(-4.88\times 10^{1}\) \\ \hline School 1 attachment & & & & & & \\ \(\beta_{\text{}}[\text{cm}^{3}/\text{s}]\) & & & & & & \\ \hline
**Parameters of The Linear Model** & **Intercept** & **Radon Term** & **Aggr1** & **Aggr2** & **Aggr3** & **Aggr4** & **Aggr5** \\ \hline School 2 parameters & 111.8 & 1.52 & \(-2.00\times 10^{-3}\) & \(-6.88\times 10^{-3}\) & \(-2.08\times 10^{-2}\) & \(-3.75\times 10^{-1}\) & \(-2.29\times 10^{9}\) \\ \hline School 2 attachment & & & & & & \\ \(\beta_{\text{}}[\text{cm}^{3}/\text{s}]\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Linear model based on Equation (6), with radon as a proxy for volumetric rate and aggregated particle channels. Let us now examine if ANNs can improve the situation compared to the linear model. If we move away from a requirement that parameters of the statistical models have simple and precisely defined physical interpretation, we can bring several improvements to our modeling methodology.