## 3 Results
As mentioned earlier we will test the predictive power of the two statistical modeling approaches, which will be inspired, in part, by Equation (6). Let us first consider linear regression model, with predictors being radon concentration and aggregated particle channels showed in Table 2. Since we are using radon concentration as a proxy for volumetric production rate we will also need a way of converting radon concentration to volumetric production rate. Since activity of 1 Bq/m\({}^{3}\) produces alpha particle of 5.49 MeV in cubic meter every second, and mean energy to create an ion pair in air is around 35.6 eV [26], one decay per second producing alpha particle will produce on average \(A_{0}\approx\frac{5.49\,\mathrm{MeV}}{35.6\,\mathrm{eV}\cdot 10^{10}\mathrm{cm}^{ 3}}=0.15\frac{\mathrm{i}\,\mathrm{p}}{\mathrm{cm}^{3}\mathrm{s}}\). Using this approximate conversion constant, we can notice using data from Table 1, that in School 1 volumetric production rate has a median of around 6  i.p/cm\({}^{3}\)s and maximum of 19  i.p/cm\({}^{3}\)s, while in School 2 median is similar and maximum is about 37  i.p/cm\({}^{3}\)s. Going back to linear regression model, which was implemented using software library [20], using whole data sets in Schools 1 and 2 we obtained coefficients given in Table 3. Explained variance of the model is 0.49 in School 1, and 0.52 in School 2. Note that this explained variance should not be interpreted as predictive power of the model, since it was calculated on the whole data set. We will further discuss the issue of data set splitting a bit later. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Nanoscan**}} & \multicolumn{4}{c}{**(a)**} \\
**ch. [nm]** & **11.5** & **15.4–20.5** & **27.4–64.9** & **86.6–154** & **205–273.8** \\ \hline S1 & & Aggr1 & & Aggr2 \\ \hline S2 & & Aggr1 & & Aggr2 & Aggr3 \\ \hline \multicolumn{1}{c}{\multirow{2}{*}{**(b)**}} & \multicolumn{4}{c}{**(b)**} \\
**OPS ch.** & & & & \\
**[um]** & **0.337–0.997** & & **1.244–9.015** \\ \hline S1 & Aggr3 & & Aggr4 \\ \hline S2 & Aggr4 & & Aggr5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Aggregation of particle channels into larger bins based on correlation matrix in Figure 2.  
Figure 3: Correlation matrix of quantities measured in **(a)** School 1 and **(b)** School 2. Shades of red are used for positive correlations, and shades of blue for negative correlations. (Plots were produced in Python 3.7.4 environment using libraries Seaborn 0.9.0 [23] for visualization and Pandas 0.25.2 [24] for data processing and calculations.) Looking at the results listed in Table 3, several conclusions can be made. In both schools, the linear model had a similar value of explained variance, despite School 2 having one additional predictor. The intercept (corresponding to \(n_{0}\) in Equation (6)) of the linear model is larger in School 1, which is in accordance with relative value of descriptive statistics for ions for two schools listed in Table 1. Note that \(n_{0}\) in Equation (6) is _not_ the median of \(n^{-}\), but rather a value corresponding to the medians of radon and particle concentrations. The radon term is very similar in both schools, indicating similar increase in ion concentration with radon concentration. In the linear model for both schools, the sign of attachment coefficients is physically justified (positive), despite the fact that we have used ordinary least squares, and did not enforce the sign of predictors a priori. Attachment coefficients become larger for aggregations corresponding to larger particle diameters, in accordance to theoretical expectations [26]. However, the magnitude of the attachment coefficients is larger compared to theoretical expectation, and this effect could be due to non-controlled PM size distribution which is to be expected in non-laboratory conditions, and also differences in number of particles in different size channels that were aggregated.