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In Europe since 2017 (and what CAMS use for Euronews AQI products), the following source of AQI is used:

http://airindex.eea.europa.eu/ 

The computation of AQI is explained under "About European Air Quality Index" link (see in the Country and Stations selection box on the right-hand side at the top of the page).

In the United States, the U.S. Environmental Protection Agency (EPA) has published a document on the reporting of daily air quality:
https://www3.epa.gov/airnow/aqi-technical-assistance-document-sept2018.pdf

but there are almost as many computation formulas as countries!

https://en.wikipedia.org/wiki/Air_quality_index