Environmental pollution is harmful to human health, as it can lead to chronic respiratory diseases. In particular, fine particles suspended in the air (PM2.5) count among the most aggressive air pollutants. PM2.5 levels vary depending on local conditions. The goal of this work was to compare year-round airborne PM2.5 readings from three air quality surveillance stations in Cali (Colombia) to determine whether these show significant spatial and temporal variation. We subjected the obtained PM2.5 dataset to a functional analysis of variance. We observed that PM2.5 levels vary significantly among the three measurement sites on a temporal scale. Whereas in the morning hours PM2.5 levels among the three sites differed most, in the afternoon and evening hours, the corresponding PM2.5 levels were not significantly different.

Se discute la estimación de la fracción PM2.5/PM10, que algunos autores reportan como la razón PM2.5/PM10. Estudios previos destacan la importancia de esta fracción en el estudio del impacto de las partículas en el aire sobre la salud de las personas y su uso potencial en la reconstrucción y en la imputación de datos de los niveles de partículas finas PM2.5 a partir de los niveles de partículas gruesas PM10. Se sugiere adaptar la estimación, mediante el uso de diferentes estrategias para situaciones particulares, incluyendo el modelo lineal general (GLM), la regresión a través del origen (RTO), los modelos aditivos generalizados (GAM) y los modelos de regresión usando mínimos cuadrados ponderados (WLS).

In Functional Data Analysis (FDA), the existing summary statistics so far are elements in the Hilbert space $L_2$ of square-integrable functions. These elements do not constitute an ordered set; therefore, they are not sufficient to solve problems related to comparability such as obtaining a correlation measurement or comparing the variability between two sets of curves, determining the efficiency and consistency of a functional estimator, among other things. Consequently, we present an approach of coherent redefinition of some common summary statistics such as sample variance, sample covariance and correlation in Functional Data Analysis (FDA). Regarding variance, covariance and correlation between functional data, our summary statistics lead to numbers instead of functions which is helpful for solving the aforementioned problems. Furthermore, we briefly discuss the functional forms coherence of some statistics already present in the FDA. We formally enumerate and demonstrate some properties of our functional summary statistics. Then, a simulation study is presented briefly, with evidence of the consistency of the proposed variance. Finally, we present the implementation of our statistics through two application examples.