A Grey Relational Analysis Model Based on a Vector-Valued Function and Its Application

To address the issue that the ordering of sample indicators may affect the results of the grey relational analysis model (GRA) based on panel data, a novel GRA model based on vector-valued function is proposed. The model provides a more robust characterization of the relationships among different sequences by reducing sensitivity to the indicator arrangement. First, the behavior matrix is regarded as a realization of a vector-valued function. This allows the matrix data to be reduced to the functional value of this vector-valued mapping. Next, the derivative of the vector-valued function is defined to capture the dynamic variation of the sample across time. Based on the differences in the derivative change rates between vector-valued functions corresponding to different samples, a new GRA model is constructed. Furthermore, the properties of the model including normativity, proximity, symmetry are discussed. Finally, the proposed model is successfully used to analyze the new quality productivity, identifying its primary and secondary influencing factors. The results further verify the rationality and effectiveness of the model and provide theoretical support for the regional development of new quality productivity.