# Machine Learning Approaches for Outdoor Air Quality Modelling: A Systematic Review
## 2 Method
### Synthesis of Results  
The results are synthetized according to descriptive statistics and a factorial analysis. First, we describe the most used algorithms over time in order to define the current tendencies. Second, we quantify the types of algorithms applied for the prediction of the principal contaminants. Third, we identify the evolution of the modelling performance for each pollutant over the last decade. Finally, since several parameters are considered for the description of the selected papers, we perform a factorial analysis to summarize the main outcomes of this systematic review. The most appropriate method to identify the relationships between the qualitative factors that characterize each study is the Multiple Correspondence Analysis (MCA). First, a data table is created from the parameters identified in Section 2.2 (see Figure A1 of Appendix A). Each row (i) corresponds to each paper and each column (j) corresponds to each qualitative variable (e.g., types of modelling, algorithms, etc.). The total number of papers and qualitative variables are identified as I and J, respectively. Next, this data table is fed to the R software, in order to proceed with the analysis through the MCA library. The initial table is transformed into an indicator matrix called the Complete Disjunctive Table, in which rows are still the papers, but the columns are now the categories (e.g., estimation model vs. forecasting model) of the qualitative variables. The total number of categories is identified as k\({}_{j}\). The entry of the intersection of the i-th row and the k-th column, which is called y\({}_{ik}\), is equal to 1 (or true) if the i-th paper has category k of the j-th variable and 0 (or false) otherwise. All the papers have the same weight, which is equal to 1/I. However, the method highlights papers that are characterized by rare categories by implementing Equation (5).  
\[\mathrm{x_{ik}}=\ \frac{\mathrm{y_{ik}}}{\mathrm{P_{k}}} \tag{5}\]  
where p\({}_{k}\) represents the proportion of papers in the category k. Then, the data are centralized by applying Equation (6).  
\[\mathrm{x_{ik}}=\frac{\mathrm{y_{ik}}}{\mathrm{P_{k}}}-1 \tag{6}\]The table of the \(x_{ik}\) is used to build the points cloud of papers and categories. Since, the variables are centered, the cloud has a center of gravity in the origin of the axes. The distance between a paper i and a paper i' is given by Equation (7).  
\[d_{i,i^{\prime}}^{2}=\sum_{k=1}^{K}\frac{p_{k}}{J}\left(x_{ik}-x_{i \prime k}\right)^{2}=\frac{1}{J}\sum_{k=1}^{K}\frac{1}{p_{k}}(y_{ik}-y_{i\prime k })^{2} \tag{7}\]  
where \(x_{ik}\) and \(x_{i^{\prime}k}\) are the coordinates of the papers i and i', respectively. This equation shows that the distance is 0 if two papers are in the same set of categories (or profile). If two papers share many categories, the distance will be small. However, if two papers share several categories except a rare one, the distance between them will be relatively large due to the \(p_{k}\) value. Next, the calculation of the distance of a point from the origin (O) is given by Equation (8).  
\[d(i,O)^{2}=\sum_{k=1}^{K}\frac{p_{k}}{J}(x_{ik})^{2}=\frac{1}{J}\sum_{k=1}^{K} \frac{y_{ij}}{P_{k}}-1 \tag{8}\]  
The equation shows that a distance gets larger when the categories of a paper are rarer, because its \(p_{k}\) are small. In other words, the more a paper possesses rare categories, the further it is from the origin of the plot axes. To conclude the process, the point cloud must be projected into a smaller-dimensional space, which is usually reduced to two dimensions. To do so, the cloud (N\({}_{I}\)) is projected onto a sequence of orthogonal axes with maximum inertia as calculated in Equation (9).  
\[\text{Inertia}(N_{I})=\max\sum_{i=1}^{I}\frac{1}{i}d^{2}(i,O) \tag{9}\]  
In terms of the first two dimensions obtained, we end up with a factor plane for the papers and categories that is the best bi-dimensional representation of them.