**4. Conclusions**

In this paper, three different analytical methods were adopted and compared to determine their effectiveness to identify pollution episodes and outliers. The data used come from a sub-urban air quality monitoring site in Dublin, Ireland, and cover the whole year 2013 with hourly measurements. Firstly, a classical vectorial approach was applied by analysing the data through time series, boxplots and frequency plots. Secondly, a statistical process control was adopted to study the data grouped by days or hours, and with different control charts (Xbar-chart, S-chart). Finally, to identify air pollution episodes and outliers, a functional data analysis approach was adopted to analyse the daily patterns of NO2 concentrations.

To effectively support the mitigation of air pollution and provide true air quality conditions, a new approach and set of tools are required to effectively assess local air pollution. With this in mind, the classical or vectorial approach is too simplistic, despite providing significant information for decision making. It has several weaknesses related to the time correlation structure of the data, and the inability to detect real outliers, days with behaviours far from the standard, just because they do not exceed limit values. As such, more advanced and modern techniques can provide deeper insights to support controlling air pollution episodes. Statistical process control presents similar problems. Even though it manages to take into account the correlation between data, the lack of normality causes a significant increase in false alarms for days that are within the legal limits; this method labels them as outliers. This is because it works with discrete observations and it is not able to extract continuous information from the data.

Alternately, the functional data analysis method, adapted to study pollution episodes, as shown in this paper, presents important improvements as a method that can be extrapolated to any city in the world. In short, it is not restricted to certain characteristics of the data distribution; takes into account complete time units; works with the entire time spectrum of a continuous mode; and the detection of outliers is more effective, which is very important in pollution issues. On the contrary, working from a functional point of view can also have its drawbacks, such as the need for large amounts of data (it is not always possible to ge<sup>t</sup> them), or other data constraints, such as continuity, positivity and monotonicity. But as explained, the fact that all the hourly values of a day are not outliers does not mean that that the day is not an outlier. The outlier search with the FDA allows one to detect days that, for some reason, have different sets of measurements of NO2 than the rest (usually higher or with strange peaks). Being able to study the reasons that make these daily functions behave differently from others makes it possible to prevent them; to try to reduce the variability in emissions; and in short, to improve air pollution control. With a classical method there would be no possibility of relating certain events to lost hours that are labelled as outliers.

The FDA's approach is presented in this document as a methodology for more effective assessment of air pollution, which is hazardous to the health of living beings, and to inform effective mitigation measures in the future.

**Author Contributions:** The tasks of this paper were divided as follows: on the one hand, J.P.P., J.S.V. and M.M.C. focused on the development of the mathematical methodology; on the other hand, J.M.T., A.M. and J.G. focused on the environmental part of the work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This paper was funded by the Spanish Government (Industry and Competitiveness Ministry) under the project RTI2018-096296-B-C21.

**Acknowledgments:** This paper was supported by the Spanish Government (Industry and Competitiveness Ministry) under the project RTI2018-096296-B-C21.

**Conflicts of Interest:** The authors declare no conflict of interest.
