## Materials and Methods  
### Form of the Balance Equation Suitable for Statistical Modelling  
The small air ion concentration (\(n_{*}\)) is determined by the following balance equation:  
\[\frac{dn_{*}}{dt}=q-\alpha n_{*}n_{*}-n_{*}\beta Z \tag{1}\]  
where \(q\) is the volumetric production rate, \(Z\) is the aerosol number concentration, \(\alpha\) coefficient accounts for the losses of ion-to-ion recombination and \(\beta\) represents an effective ion-aerosol attachment coefficient, which is the integral over the size distribution of aerosol particles. The balance equation can include additional terms. If electrostatic deposition (occurring mainly in indoor air) is included in a model, there is an additional right-hand side term \(-\delta^{\pm}n^{\pm}\), where \(\delta^{\pm}\) is an electrostatic deposition rate coefficient of the air ions. Additional details about physics behind changes in SI concentration and relevant terms in balance equations are given in Appendix A. While Equation (1) seems very intuitive, it is worth noting that it was discovered rather painstakingly, and some terms were added to increase its scope of validity. Namely, in the first half of last century [PERSON] [17] showed that the quadratic law of recombination previously held valid (quadratic law is Equation (1) without the last (linear) term on the right-hand side. Since \(n_{+}\approx n_{-}\)equality being only approximate due to the different mobility of positive and negative small ions, the middle term in Equation (1) is a quadratic term) is not valid in ordinary air, but only for clean air (Equation (1) when \(Z\approx 0\)). As another interesting historical note ion-aerosol attachment coefficient was previously referred to as \"diminuution coefficient of small ions in the presence of nuclei and large ions\" [18]. If we neglect the quadratic term present in (1) we can obtain the following (under the assumption of a quasi-steady state):  
\[\frac{dn^{-}}{dt}\approx 0\approx q-n^{-}(\beta Z+\delta^{-}) \tag{2}\]  
(since our campaign was situated indoors, the electrostatic deposition rate coefficient \(\delta^{-}\) was also included). After expanding \(\beta Z\) term we obtain:  
\[n^{-}\approx\frac{q}{\sum_{i}Z_{i}+\delta^{-}} \tag{3}\]  
or more conveniently  
\[\sum_{i}Z_{i}+\delta^{-}\approx\frac{q}{n^{-}} \tag{4}\]  
The form of the above equation suggests linear regression is a justified modelling approach if we want to model the interdependence between the concentration of larger aerosol particles of various diameter and small ion concentration. The physical meaning of the coefficients in linear regression (\(\beta_{i}\)) are an ion-aerosol attachment coefficient and the intercept term corresponds to the electrostatic deposition rate coefficient of the air ions (\(\delta^{-}\)). Note, however, that in a non-laboratory type of campaign, one cannot precisely control the aerosol distribution and there may be a significant correlation between individual channels corresponding to different particle sizes, which makes calculation (and interpretation) of the regression coefficients as attachment coefficients largely approximate. Taking one more look into (4) brings up an important issue, that is taking a quotient of two noisy variables on the right-hand side of the Equation (4), that may also have values close to zero. The right hand side of (4) therefore may produce a quotient not suitable for further statistical modelling. Instead of using Equation (4) directly, we will do the following. By [PERSON] expanding expression (3) around some value of volumetric production rate \(q_{0}\) and some value of particle concentrations \(Z_{i0}\) we obtain  
\[ \begin{array}{l}n^{-}\approx\frac{q}{\sum_{i}Z_{i}+\delta^{-}}=f(q,Z_{i}) \approx\\ \approx f(q_{0},Z_{i0})+\frac{\partial f(q,Z_{i})}{\partial q}\big{|}_{q_{0},Z_{0}}(q-q_{0})+\frac{\partial f(q,Z_{i})}{\partial Z_{i}}\big{|}_{q_{0},Z_{ 0}}(Z_{i}-Z_{i0})\end{array}  \tag{5}\]  
The derivatives in (5) are given by  
\[