## 2 Methods
### Input Selector: Mutual Information  
The mutual information (MI) of two random variables is a measure of the mutual dependence between these two variables. MI is thus a method for measuring the degree of relationship between data sets. This method has been widely used in various modern disciplines of science and technology, such as medical imaging [33], search engine [34] and DNA sequencing [35]. In the field related to atmospheric and environmental sciences, MI has been introduced to large data sets to detect nonlinear relationship between new-particle formation and other ambient variables in Hyytiala Forest, Finland, as described in our previous work [36]. In this case, we adopt the same strategy for finding the relationship between a pollutant proxy and other measured variables.  
The concept of MI is depicted in Figure 2. Suppose \(X\) and \(Y\) are the measured variables and the variable of interest. The area covered by red and orange is the entropy of \(X\), \(H(X)\), whereas the area covered by yellow and orange represents the entropy of \(Y\), \(H(Y)\). The red area is the conditional entropy of \(X\) given by \(Y\), \(H(X|Y)\), whereas the yellow area is the conditional entropy of \(Y\) given by \(X\), \(H(Y|X)\). The area contained by both circles is the joint entropy \(H(X,Y)\) and the orange area is the mutual information between \(X\) and \(Y\), \(I(X;Y)\). Therefore, MI can be found by calculating  
\[I(X;Y) =H(X,Y)-H(X|Y)-H(Y|X) \tag{1}\] \[=H(X)+H(Y)-H(X,Y) \tag{2}\]  
As used in [PERSON] et al. [36], we also adopt a nearest-neighbour MI implementation based on [PERSON] et al. [37]. The nearest neighbour concept assumes the space \(Z=(X,Y)\). We can rank for each point \(z_{i}=(x_{i},y_{i})\), its neighbours by distance \(d_{i,j}=\parallel z_{i}-z_{j}\parallel\) and similar rankings can also be done for the subspaces \(X\) and \(Y\). So, the distance can then be found by finding the maximum norm: \(\parallel z-z^{\prime}\parallel=\max\{\parallel x-x^{\prime}\parallel, \parallel y-y^{\prime}\parallel\}\) where the distance from \(z_{i}\) to its \(k\)th neighbour is denoted by \(\epsilon(i)/2\). The MI estimation is then given by  
\[I(X;Y)=\psi(k)-\langle\psi(n_{x}+1)+\psi(n_{y}+1)\rangle+\psi(N) \tag{3}\]  
where the symbol \(\psi(.)\) represents the digamma function. \(N\) is the number of all data points and \(k\) constitutes the number of nearest neighbours (the user choice). Parameters \(n_{x}\) is the number of point \(x_{j}\) whose distance from \(x_{i}\) is strictly less than \(\epsilon(i)/2\), and similarly for \(y\) instead of \(x\). The symbol \(\langle.\rangle\) denotes the averages both over all data points \(N\). The \(I(X;Y)\) can then be normalised between 0 and 1 by a scaling factor [38], so the final MI formulation is given by  
\[\hat{I}(X;Y)=\text{sign}[I(X;Y)]\sqrt{1-\exp(-2|I(X;Y)|)} \tag{4}\]  
Figure 2: Venn diagram of entropy properties. \(I(X,Y)\) is the mutual information between entropy \(X\) and \(Y\), symbolised by \(H(X)\) and \(H(Y)\), respectively.  
where symbol sign represents a signum function and notation \(|.|\) is the absolute value. The mathematical details of the used MI method can be found in [PERSON] et al. [37], and its implementation in this field can be found in [PERSON] et al. [36].