Interval-valued functional clustering based on the improved Euclidean distance with application to air quality index

Abstract

Interval function clustering is a statistical method used to classify functional data based on interval number similarity measurements. However, existing similarity measurements focus on measuring the similarity of the curves in terms of numerical distance. This ignores the changing characteristics of the curve shape, which may lead to unreasonable clustering results when clustering interval-valued functional data. To address this issue, an improved Euclidean-distance-based interval-valued functional clustering method is proposed in this study. By deducing a specific calculation formula for the Euclidean distance of the interval function under the basis function and derivative information, the absolute difference in the numerical value was reflected based on the distance of the basis function, and the curve shape difference was reflected based on the distance of the derivative function information. Furthermore, a similarity measurement method combining numerical distance and curve shape was constructed to reflect the change characteristics of the functional data more completely and improve the K-means clustering process. Finally, the air quality index of different cities is taken as an example to cluster, and the differences and variation characteristics of different types of air quality are analyzed, which verifies the effectiveness of the proposed method.