Abstract: In this paper, we propose a test to check whether in motion, the point groups of high-dimensional longitudinal data keep a rigid structure and suggest a definition of transmutation coefficient of the groups. In terms of testing the rigid structure, the major characters of the motion of the groups can be handled so that the prediction for the motion in future can be expected. To demonstrate the rationale and potential use of the method, simulations and an analysis for the diversity of economic changes between the city group in the east China and the city group in the west China have been carried out.