Accumulating Method GM Model And Research On Its Ill-conditioned Problem

The gray system had been suffering an extensive concern since the beginning of 80's in the 20 centuries. It has so many advantages that it needs less sample data and it carries to calculate with convenience. Moreover, it has a high accuracy in short-term estimate. Consequently, the grey system has been applied extensively. But, in hereafter of continuous study, we discover that the morbidity has been existed in gray expectation in different scale. The trivial changes of the data and numerical round-off can result in the estimation and parameter errors. In the document [1] we can notice the problem first. How to overcome the morbidity and guarantee the reliability and stability of the model has become an important aspect in the reconstruction and improvement of the gray model. Accumulating method is a new kind of curve fitting technical established by Italy mathematician P.E.Marchesi with which we make use of it in the parameter estimation of the gray model. This paper has analyzed the morbidity problem in accumulating method GM(l,l),GM(2,l),GM(0,2), and GM(0,3) model and the influence of the multiple transformation on the morbidity existed in the accumulating method GM(n,h) model. It found that the calculation perplexity and morbidity of the model have been greatly descended after introducing the accumulating method to the GM model. Moreover, the multiple transformation can relieve the morbidity further.Firstly, this paper presents the racial theories of the accumulating method and the accumulating method GM models which have existed. Secondly, it introduces the improvement of the research on the morbidity problem.GM(1,1) model is the core of the gray model and applied most extensively in the field of gray prediction and gray control. This paper has analyzed the morbidity problem in accumulating method GM(1,1) model basing on the condition number theorem, it proves that we can easily resolve the morbidity problems only if we make a multiple transformation on the primitive data and as a result this kind of the data changes would not affect on the accuracy of the model. As a whole, the calculationquantities of matrix are reduced and it is more convenient comparing to the other method. Therefore, the newly accumulating method GM(1,1) model with multiple transformation has more advantages than the traditional one and it is deserved to be introduced.GM(2,1) model has an extensive application due to it has two eigenvalues and can reflect the trait of monotony and non- monotony (sway). This paper introduces the accumulating method into the parameter estimation of GM(2,1) model and gets the new parameter estimation formula whose calculation is more simple than before. Meanwhile, the condition number of the new formula is mostly well-conditioned. But, some examples are still ill-conditioned. So, it analyses the morbidity of the accumulating method GM(2,1) model and proves that the morbidity can be descended with a multiple transformation on the initial data and the transformation doesn't affect the development parameter of the model and the precision of the system.This paper has some preliminary research on the morbidity problem of the gray model and gets some innovative results. But, it still has many problems and doesn't resolve the morbidity problem utterly. And it need further probe the demonstration about the reason why the condition number of accumulating method GM model is relatively small. We shall continue researching on these aspects.