Grey GRM ( 1 , 1 ) Model Based on Reciprocal Accumulated Generating and Its Application

Aiming the problem of low accuracy during establishing grey model in which monotonically decreasing sequence data and traditional modeling methods are used, this paper applied the reciprocal accumulated generating and the approach optimizing grey derivative which is based on three points to deduce the calculation formulas for model parameters, established grey GRM(1, 1) model based on reciprocal accumulated generating. It provides a new method for the grey modeling. The example validates the practicability and reliability of the proposed model.


Introduction
The main characteristic of grey system theory is the research about small data and uncertainty, and the basic tool is grey generation.Behavioral data of the system may be chaotic and complex, but there is always some kind of law among them.Grey generation is to find the law from these behavioral data, and establish grey model according to the law, further predict the system by solving the model [1].So the grey generation is the basis establishing grey model.The most commonly method used is accumulated or inverse accumulated generating operation on in the process of modeling.The accumulated generation is able to inverse with the inverse accumulated generation, that is, the one-time accumulated generation sequence can be reverted to the original sequence by the inverse accumulated generation one time.For non-negative discrete sequence (0) X , the one-time accumulated generation sequence (1)

X
is monotonically increasing.When a curve fits X is monotonically increasing and then the model value (1) X is also increased.When (1) X is inverse accumulated generated to the predicted value of the originnal sequence (0) X , there will produce an unreasonable calculation errors.Backward accumulated generation was put forward and GOM(1, 1) based on backward accumulated generation was established [2].GRM(1, 1) based on reciprocal generation was built after proposing reciprocal generation [3].GRM(1, 1) was improved to establish the improved grey model CGRM(1, 1) based on reciprocal accumulated generation with better modeling accuracy [4].Grey models based on reciprocal generation and opposite-direction accumulated generation make the generation sequence (1)

X
also monotone decreasing, and then fitted X .In this case, the reduction process from (0) X to (0) X will not produce the unreasonable error and it improves modeling accuracy.In the paper, the grey derivative was optimized by using three-point grey derivative, and the calculation formulas for model parameters were deduced in the condition that the first component of (1)

X
was taken as initial condition of grey differential equation in this model on the basis of Ref [3,4].Grey GRM(1, 1) model based on reciprocal accumulated generating was established.This model with high precision has better practical and theoretical significance.The example validates the practicability and reliability of the proposed model.

Grey GRM(1, 1) Model Based on
Reciprocal Accumulated Generating Definition 1. Supposed the original sequence x n is called as one-time reciprocal accumulated generation of (00) X .
Definition 3. Supposed the original sequence , where ( 1) . The inverse accumulated generation is the inverse of accumulated generation, and they meet that It is known that the solution of equation (1) at x t ce    f.When this curve is used to fit (1) X , the key is how to deal with the derivative signal of discrete points.We take three points , k A k x k and in the expo-  nential curve with monotone decreasing and up-concave   (1) at x t ce f    .It is known easily that the slope of the curve at the point (1) (1) The albino equation of grey differential equation where,  is the related coefficient with a, a is devel- opment coefficient and b is the control coefficient.
The following equation can be obtained by using the least squares method: When the first component of (1) X is taken as initial condition of grey differential equation, the continuous solution of albino differential equation in the initial conditions is: Its discrete solution is: The model value of the originnal sequence can be obtained by regressive generation.
Then the model value of the original sequence by using Definition 1 is obtained.
Presumed that (1) X is in the exponential curve   (1) at x t ce  f   , the accurate conditions during modeling is that two equations between Equation (3) and Equation ( 7) are satisfied at the same time.Equation (3) substituted by Equation ( 7) is simplificated, and then a relationship between  and a can be established as: Since that Y is the function of  in Equation (4) and  is the function of a in Equation ( 8), as long as giving an initial value 0 of a, a  can be obtained in Equation (8).Substituting again into Equation ( 8) will obtain  and into Equation (4) calculate a.After iterating sev- eral times the exact value will be found.After defining the absolute error , the rela- adaptability and the scientific of the proposed model.ree

Conclusion
This paper applied the reciprocal accumulated generating and the approach optimizing grey derivative which is based on th points to deduce the calculation formulas for model parameters in the condition that the first component of (1)  X was taken as initial condition of grey differential equation, established homogeneous GRM ( of the proposed model.

Example REFERENCES
[1] Y. X. Luo, L. T. Zhang and M. Li, "Grey Systems an There are the fatigue experimental data (Mpa) in [5]: (00)  The mean of the relative error is 0.38636%.This model has high precision.