## 4 Discussion and Conclusions  
The potential role played by the exposure to particulate matter in the spread of the COVID-19 pandemic has attracted a lot of interest in the scientific community. Different techniques have been employed to analyze this potential relationship from several perspectives. This paper has provided a further contribution to this discussion, analyzing this association in the context of the first COVID-19 outbreak that hit New York City in March 2020. The analyzed data consisted of both the PM\({}_{2.5}\) daily levels and the daily number of infections, treated as time series. Two different methodologies (Granger causality and machine learning predictions) were used and extended to 12 different counties in the State of New York, thus including both densely populated and less populated districts. Both the employed methodologies returned results in favor of a possible association between the PM\({}_{2.5}\) particulate and the spread of this contagion in NYC, in the period of interest. However, it is important to conclude by discussing the potential limitations of this study. The first relevant issue is that a Granger causality approach, like any other statistical method in this context, cannot establish if a true causal link does exist between two phenomena under observation. Instead, it checks if a predictive causality holds, which is a stronger evidence than mere correlation. Nonetheless, this is still very far from the very complex concept of true causality. Much of this problem technically derives from the fact that the _Granger_ methodology is designed to manage pairs of time series. For this reason, it can produce erroneous results when this relationship engages more than two variables. In fact, if both \(X\) and \(Y\) were influenced by a third common phenomenon, say \(W\), a Granger causality test on \(X\) and \(Y\) might lead to accepting the alternative hypothesis (i.e., \(X\) Granger causes \(Y\)), even though the mutual correlation between \(X\) and \(Y\) were simply caused by \(W\)[33]. In our specific case, the role of \(W\) could be played by humans. In fact, one could argue that our observed phenomena (PM\({}_{2.5}\) particulate and COVID-19 infections) might be both due to the common daily human activities that have led to an increase both in the level of pollution and in the risk of being infected. However, this is exactly why we have structured our Granger-based experiments to include two different types of counties. In fact, those counties comprised in NYC \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**County** & **KNN** & **SVC** & **MLP** & **ET** & **Avg. per County** \\ \hline New York & 1 & 1 & 0.95 & 0.82 & 0.94 \\ Kings & 0.95 & 0.8 & 1 & 0.79 & 0.89 \\ Bronx & 0.85 & 1 & 0.95 & 0.82 & 0.91 \\ Queens & 0.9 & 0.89 & 0.89 & 0.89 & 0.89 \\ Richmond & 0.87 & 0.87 & 0.87 & 0.91 & 0.88 \\ Nassau & 0.8 & 0.7 & 0.95 & 0.89 & 0.84 \\ Rockland & 0.77 & 0.82 & 0.82 & 0.82 & 0.81 \\ Westchester & 0.95 & 0.83 & 0.76 & 0.76 & 0.83 \\ Suffolk & 0.9 & 0.85 & 0.85 & 0.9 & 0.88 \\ Rockland & 0.77 & 0.82 & 0.82 & 0.82 & 0.81 \\ \hline Avg per & & & & & \\ algorithm & 0.89 & 0.86 & 0.89 & 0.84 & \\ \hline Monroe & 0.85 & 0.85 & 0.88 & 0.91 & 0.87 \\ Onondaga & 0.85 & 0.88 & 0.91 & 1 & 0.91 \\ Oneida & 0.91 & 0.94 & 1 & 0.94 & 0.95 \\ \hline Avg per & & & & & \\ algorithm & 0.87 & 0.89 & 0.93 & 0.95 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: ML results: F1-score obtained with data from each county as a validation set. (or very close to NYC) could be, in theory, more subjected to the influence of the \(W\) factor (i.e., the humans). Instead, those counties that are less populated, and far away from the city, should not suffer from that influence. Another limitation is that we have extended our study to a limited number of counties (12) in the State of New York. Nonetheless, we judge this number of counties as not being a real limitation, based on the following reasoning. The State of New York comprises 62 different counties.