Abstract: To address the issues in existing panel data-based grey relational analysis models (PD-GRA), such as the neglect of indicator’s time delay and data transformation considerations, a time-delay grey relational analysis model (TDGRA) is proposed. First, the concepts of the sample matrix, the submatrix under time delay, and the corresponding truncation matrix are introduced. Second, singular value analysis is performed on the relevant matrices, and the optimal rotation matrix is derived. Then, an optimization model is constructed with the goal of minimizing the Procrustes distance, through which the time delays between indicators are determined. And then, a PD-GRA model is proposed based on the optimal Procrustes distance with time delay. Finally, the TDGRA model is used to identify the driving factors of AQI in the Yellow River Basin, obtaining urbanization rate (UR), carbon emissions (CE), gross domestic product (GDP), and the growth of total investments in fixed assets (GIFA) as the main drivers. The results demonstrate the rationality and effectiveness of the proposed TDGRA model, and its advantages are further highlighted through comparative analysis.

Key words: Grey Relational Analysis, Procrustes Distance, Time Delay, Panel Data