## 2 Materials and Methods
### Methodologies
#### 2.2.2 Machine Learning  
We now come to the second methodology of our study. As already anticipated in Section 1, the rationale was to conduct an additional series of experiments that could confirm (or reject) our hypothesis, using a very different approach. Specifically, using machine learning (ML) algorithms, with which we have a strong experience [35; 36; 37]. Essentially, we devised a non-traditional procedure, resembling a kind of a ML (county) cross validation methodology, which went as follows: During the training activity, we let some ML algorithms be instructed with the daily values of the PM\({}_{2.5}\) (input) and the COVID-19 infections (output). The periods of these two series of daily data were those mentioned before: PM\({}_{2.5}\) (19 February-8 March) and COVID-19 infections (4-22 March). More precisely, the number of the COVID-19 infections for each given precise day, say \(X\), were put in relation with the amount of the values of the PM\({}_{2.5}\), registered in all those days included in the following time interval: [\(X\)-7, \(X\)-14]. The choice of these eight days, prior to \(X\), was taken depending on two different factors: (i) The need to be as close as possible to the correspondent lag value used in the Granger analysis (which was equal to 5), and (ii) as a result of the ML hyperparameters optimization process. After this learning phase, this procedure went through a kind of testing validation where the instructed algorithms had to predict if, in a given day, in a specific county, the number of infections either exceeded a predefined infections threshold or they did not. To summarize, the entire process worked as follows. With each round of our procedure, our ML algorithms were trained with the data (PM\({}_{2.5}\) vs. COVID-19 infections) relative to all the 12 studied counties, except for the one for which we asked our algorithms to predict the number of daily infections, given the concentrations of the PM\({}_{2.5}\) particulate occurred in previous days. This procedure was repeated, in turn, for all the counties under investigation. We developed 48 different experiments (4 ML algorithms and 12 counties), to predict if the number of infected persons has exceeded the infections threshold value. Obviously, the more accurate were the predictions on the infection threshold exceedances,for the counties subjected to our investigation, the more was confirmed the hypothesis of a correlation between PM\({}_{2.5}\) and the COVID-19 spread in those areas. As to the choice of the infection threshold, we had two different alternatives: Either computing an infection threshold for each different county or computing a unique threshold to be used for all the counties of interest. Even if the first choice could seem, at a first impression, as more accurate, nonetheless managing twelve different thresholds, with four different learning algorithms, would have resulted, with a high probability, in a kind of an overfitting problem with our ML-based procedure. Hence, the choice of a unique threshold. Moreover, to further simplify the problem, we resorted to a procedure to compute that threshold, which was very simple and effective. The idea was that of counting the number of daily infections registered per each county, in all the twelve counties of interest, during the four days that preceded the lockdown decision. Once we obtained those daily infection counts, we computed an average for all four days on a per-county basis. We then got 12 numbers that were definitely aggregated under the form of a final average count, thus yielding an infections threshold equal to 122.8. All the values used to compute our average are reported in Table 3.  
The rationale behind this procedure was the following. When the Governor of the State of New York opted for a lockdown decision on 20 March, the average state number of daily infections, on a per-county basis, had just reached that threshold of 122.8 infections. Consequently, we can use that number as a key to design the predictions scheme of our ML model.