The Research On Optimization Of Grey NGM(1,1,K) Model And Its Improved Model

The grey system theory put forward by Professor Deng Ju long is mainly used to study the grey uncertainty problems such as "small sample,poor information".Grey GM series model is an important part of grey system theory.This model has a good simulation and prediction effect on small sample data in real life.There are some good results and low modeling accuracy in the simulation and prediction of small sample data.Therefore,many experts and scholars have improved and optimized the model.It mainly optimizes the grey model from the aspects of grey derivative,background value,initial condition of time response,parameter estimation,etc.In this paper,the following work has been done on the optimization of GM model:Firstly,the background value of grey NGM(1,1,K)model is optimized and the solution method of parameters is studied;Secondly,the grey NGM(1,1,K)model is modified to adapt to the case that the increment sequence of the original sequence is non-homogeneous exponential sequence,so it widens the scope of application of the model;the improved grey NGM(1,1,K)model is optimized by grey derivative and the solution method of parameters is studied.Finally,the properties of the two optimization models are studied.It is proved by theorem that the optimized model has the coincidence of whitening index,whitening coefficient and whitening translation constant.The main research contents are as follows:(1)Optimization of the grey NGM(1,1,K)model:First,the solution of whitening differential equation dy(t)/dt+ay(t)=bt+d is derived to obtain x(0)(t),letting the function value at t=k be the grey derivative term.Then letting the value of integral function B(t)in t=k of x(0)(t)about t be the background value.In this way,NGM(1,1,K)model with optimized background value is obtained.When solving parameters,the parameter a is obtained by the traditional least square method,and the parameter b is represented by the formula with parameter d.Then the obtained parameters are substituted into the solution y(t)of the whitening differential equation.In this paper,the solution of y(t)the whitening differential equation is no longer a primary accumulation function of the original sequence.Therefore,the simulation prediction formula should be obtained for y(t)derivation.The parameters C and d in the model try to make use of the two methods that all the known data decide together:the first is to find the objective function with the minimum sum of relative error absolute value and solve the parameters of the model through nonlinear search;The second is to derive the analytic expression of C?d under the objective function of seeking the minimum sum of relative error squares.Finally,the solved parameters are substituted into the simulation prediction formula.(2)Improve the grey NGM(1,1,K)model to make it suitable for the case that the increment sequence of the original sequence is non-homogeneous index,and then optimize the improved model.In this paper,the difference equation and the differential equation are matched by optimizing the grey derivative.x(1)(t)is obtained from the solution of whitening differential equation,and finding the derivative function[x(1)(t)]' about t for x(1)(t).Substituting the function value of derivative function at t=k into the improved grey NGM(1,1,K)model instead of x(0)(k)in the original grey differential equation to obtain a new model with optimized grey derivative.Finally,when determining the undetermined coefficient C,the objective function consistent with the model evaluation standard is established,using nonlinear search to find C,and improved modeling accuracy.