The monitoring and forecasting of particulate matter (e.g., PM2.5) and gaseous pollutants (e.g., NO, NO2, and SO2) is of significant importance, as they have adverse impacts on human health. However, model performance can easily degrade due to data noises, environmental and other factors. This paper proposes a general solution to analyse how the noise level of measurements and hyperparameters of a Gaussian process model affect the prediction accuracy and uncertainty, with a comparative case study of atmospheric pollutant concentrations prediction in Sheffield, UK, and Peshawar, Pakistan. The Neumann series is exploited to approximate the matrix inverse involved in the Gaussian process approach. This enables us to derive a theoretical relationship between any independent variable (e.g., measurement noise level, hyperparameters of Gaussian process methods), and the uncertainty and accuracy prediction. In addition, it helps us to discover insights on how these independent variables affect the algorithm evidence lower bound. The theoretical results are verified by applying a Gaussian processes approach and its sparse variants to air quality data forecasting.
