\textit{Focus on What's Important!}\\ Inspecting Variational Distributions for \\ Gaussian Processes for better \textit{AQ} Station Deployment

In urban locales, the intricate dynamics of air quality indicators such as Particulate Matter (PM2.5) and Carbon Monoxide (CO) necessitate sophisticated modeling for precise prediction and monitoring. However, monitoring stations are sparse, and effective placement is a key problem in the domain. This study explores a novel approach utilizing Variational Multi-Task Gaussian Processes (VMTGP) endowed with a Spectral Mixture (SM) kernel to model the spatiotemporal distribution of these pollutants in Beijing, which beats the state-of-the-art Gaussian Process techniques on this dataset in the exact MTGP case. However, our innovation lies in an in-depth examination of the variational distribution of the inducing points, which are critical for scalability and accurate approximations in GP models. Through an empirical lens, we observe a pronounced clustering of inducing points around certain monitoring stations, hinting at a higher information content in these locales. Our findings underscore the inherent value in exploiting the clustering phenomenon of inducing points, opening up new vistas for enhancing the efficacy and interpretability of multi-task learning paradigms in air quality forecasting. This insight holds promise for developing more robust and localized air quality prediction models, crucial for urban planning and public health policy formulations, and adaptively deciding the most effective locations for placing AQ monitoring stations.