复数阶累加的灰色系统模型

Grey system model with complex order accumulation

在整数(分数)累加的经典GM(1,1)模型的基础上,提出了复数累加的灰色模型(对于某个z∈X,记为CAGMz(1,1))。该公式将灰色模型参数的选择范围从实轴扩展到复平面。我们认为所有复数累加生成算子的集合具有一维可加复李群的结构,它与X同构。这一构造首次将李群理论引入灰色模型理论,为在灰色系统中引入李群的工具奠定了基础。指数n的幂零矩阵E1和泰勒级数的方法避免了使用现有流行的Γ函数,也为计算机编程提供了一种有效的途径。作为一种新的方法,复数阶累加生成算子可以同时调整新信息和旧信息之间、实部和虚部之间的权重,可以得到更好的仿真和预测结果。通过实例讨论了CAGMz(1,1)模型的优点,给出了较好的仿真和预测结果。

Based on the classical GM(1,1) model of integer(fractional) accumulation, we propose Grey model of complex accumulation(which is denoted by CAGMz(1,1) for some complex number z∈X). This formulation extends the choice of Grey model’s parameter from the real axis to the complex plane. We claim the set of all complex accumulated generating operators admits a 1-dimensional additive complex Lie group’s structure,which is isomorphic to X. This construction brings Lie group’s theory in Grey model theory for the first time,and lays a foundation for introducing Lie group’s tools in Grey system. The method of nilpotent matrix E1 of indexnand Taylor series could avoid any usage of existing popular Γ functions, which also provides an efficient way for computer programming. As a novel method, accumulated generating operators of complex order could adjust weights between old information and new information, between the real part and the imaginary part simultaneously, better simulation and prediction results could be expected. The advantages of CAGMz(1,1)model are discussed with several cases, better simulation and prediction results are presented.