Ranking of functional data in application to worldwide PM ₁₀ data analysis

Ranking of functional data is important for conducting further rank-based analysis. The paper reviews several ranking methods, such as the principal component analysis/functional principal component analysis and the discrete rank method, and proposes a new method for functional data: the weighted local rank method. The proposed method allows the presence of missing values or values measured at unmatched time points. It also has physical interpretation. All the methods are compared through simulation and the proposed method is more robust in various scenarios. In real data analysis, the proposed method is applied to worldwide PM₁₀ data to generate ranks, then further analysis such as nonparametric rank sum test and linear regression based on ranks are performed to produce meaningful results.