灰色GM(1,1)派生模型病态性的差异分析

本文主要研究GM(1,1)六种不同派生模型的病态性.首先利用特征分析法和参数估计矩阵B~TB的条件数探讨了辨识参数系数矩阵的病态性情况,根据病态性程度对派生模型进行归类,进而分析了各类模型产生病态性的原因,并对具有病态性的派生模型采用向量变换法进行改进.在此基础上,比较了定义型与派生模型在建模机理、计算繁简程度等方面的差异.研究结果表明:1)派生模型b型、派生模型内涵型属于第一类即不存在病态性的模型;定义型、派生模型x~(1)型、离散型属于第二类即存在一般程度病态性的模型;派生模型x~(0)型、派生模型指数型属于第三类即具有严重病态的模型;2)与各类派生模型相比,定义型在建模机理、计算量的大小等方面有明显优势.最后,通过数值实例验证了分析结果的合理性.

Morbidity analysis of the grey GM(1,1) derived models

This paper mainly studied the morbidity problem of six different derived models of GM(1,1). Firstly, we discussed the morbidity of identification parameter coefficient matrix by using the condition number of parameter estimation matrix B^TB and eigenvalue analytical theorem. According to the degree of morbidity, the derived models were classified. Furthermore, the reasons for the morbidity of various models are analyzed, and then the vector transformation method was used to improve the morbidity models. On the basis of this, the differences between the defined model and the derived models are compared on the modeling mechanism and the calculated amount. The results show that:1) The derived model of b and connotative derived model belong to the first category, i.e., they are non-morbidity. The defined model, the derived model of x⁽¹⁾ and the discrete model belong to the second category, i.e., they have general morbidity. The exponential derived model and the derived model of x⁽⁰⁾ belong to the third category, i.e., they have very serious morbidity. 2) Compare with the derived models, the defined model has obvious advantages on the aspects of modeling mechanism and calculated amount. Finally, several numerical examples were used to verify the rationality of the classification and analysis results.