Abstract: Grey relational analysis is a key part of grey system theory and the basis of grey system modeling, grey decision-making, and grey control model. The basic idea of grey relational analysis is to judge the degree of relation between different sequences from the geometric point of view by comparing the similarity of curve features. Because the grey relational analysis model has “small computation and low data quantity requirement”, it has been successfully applied in economic management, geological and environmental protection, biological science, etc. Although grey relational analysis has achieved remarkable achievements in practical application, it still has some shortcomings. For example, different construction methods of surface clusters may produce the unique result of the relational degree, and changing the order of indicators often changes the degree of relation. It can be seen that the traditional grey relational analysis model may be easily affected by the construction method of surface clusters or the order of indicators.
Aiming to the problems mentioned above, a new spatial projection method for panel data is proposed, and a grey barycentric triangular grid relational analysis model based on panel data is constructed in the paper. Firstly, based on the permutation and combination principle, the sample matrix is decomposed into the binary index sub-matrix, which is projected as points in the three-dimensional space, and the spatial tetrahedron is obtained by connecting the adjacent four points in pairs. Secondly, the barycenter of the tetrahedron is given and associated with the four vertices, the two adjacent vertices are also connected, and the barycentric triangular surface is established, thus the barycentric triangular grid of the binary index sub-matrix is obtained. Thirdly, based on the volume of the curved top cylinder of the barycentric triangular surface, the formula of the relational coefficient is constructed, and a grey barycentric triangular grid relational analysis model of panel data is proposed. The proposed model overcomes the shortcomings of the existing grey relational analysis of panel data, is easier toapply in practice, and has good properties, such as normalization, symmetry, and invariance of the relational degree for the translation transformation.
The proposed model has been applied to the air quality assessment for the six cities along the east-west route of Shandong province, including Jinan, Weifang, Zibo, Liaocheng, Yantai, and Weihai. According to Ambient Air Quality Standards, AQI, PM_2.5, PM₁₀, SO₂, CO, NO₂, and O₃ are selected as evaluation indicators. The range transform is used to normalize the raw data, and the characteristic behavior matrix of the system is determined based on the processed data of six cities along the east-west route of Shandong province. Using the proposed model to calculate the relational degree between the sample matrix of the six cities and the characteristic behavior matrix, we obtain that Weihai and Yantai rank first, followed by Weifang, Jinan, Liaocheng, and Zibo. By contrasting and analysis, the rationality and effectiveness of the model are verified. Furthermore, the results indicate that the proposed model can measure the degree of relation between panel data, confirming the objectivity and practicality of the proposed model.
For panel data, the paper presents a new spatial projection method and proposes a grey barycentric triangular grid relational analysis model. The proposed model can measure the relationship and influence between panel data, and complement and perfect the theory of the grey relational analysis for panel data. It is worth pointing out that the focus of the grey relational analysis model is mainly on the order of relationships rather than on the size of the relational values, which measures the interrelationships and influences between sample data.

Key words: grey relational analysis, panel data, grey system, barycenter

摘要： 针对现有面板数据曲面簇的构造以及指标排列顺序改变引起关联度结果不唯一问题,提出了基于面板数据的灰色重心三角网格关联分析模型。首先,将样本矩阵分解为二元指标子矩阵,并将其投射为空间四面体网格。然后,利用空间几何学给出四面体的重心,并与四顶点相连得到重心三角曲面。其次,基于重心三角曲面为曲顶的柱体体积构建了灰色关联系数公式,并得到了面板数据的灰色重心三角网格关联分析模型。最后,所提出的模型成功应用到山东省东西沿线6城市的空气质量评估,验证了模型的合理性和有效性,结果表明所提出模型能够测度面板数据间的相关程度。

关键词: 灰色关联分析, 面板数据, 灰色系统, 重心