Research On Optimization Method Of Non-homogeneous GM(2,1) And Its Direct Discrete Model

Grey system theory is a theory that studies a few uncertainties.On the basis of analyzing the characteristics of the few data,understanding the behavior of the few data,discussing the potential mechanism of the few data,and synthesizing the phenomenon of the few data,grey theory reveals the evolution law of things under the background of the few data and the few information.Grey prediction model is an important part of grey system theory.It can reliably simulate and predict the future development trend of the system by processing the original data and establishing the grey prediction model.In this paper,the non-homogeneous GM(2,1)model is optimized and improved by means of minimizing the sum of squared relative errors,direct modeling and discretization,so as to obtain higher accuracy of simulation prediction and broaden the scope of application of the model.The specific work is as follows:(1)Taking(?)as the target function.However,in order to avoid cumulative errors,all the calculated or uncalculated coefficients or constants in the time response formula are regarded as undetermined coefficients.,and the formulas of time response coefficients under different conditions are derived.(2)Based on the idea of direct modeling without the participation of one cumulative sequence and the idea of discrete modeling without the participation of whitening differential equation,a direct discrete model of non-homogeneous GM(2,1)is proposed,and the simulation prediction formula and its properties are studied.Examples show that the model is operable and has a high accuracy of simulation prediction.At the same time,starting from the recursive form of the model simulation prediction formula,the general term of the simulation prediction formula is derived by reducing the order(second order is reduced to first order),homogeneous(non-homogeneous is transformed to homogeneous)and other methods,the formula shows the intuitive apply basic forms:the sequence of the model with exponential sequence,linear and parabolic sequences,three curve type of four types of basic sequences and the index of two different base sequence and linear sequence and poor combination,an index sequence and combination of parabolic sequence and difference,an index sequence and linear sequence and poor combination,linear sequence and exponential sequence to the product of the combination with another type of linear sequence and poor combination of four types of combination sequence.It is proved theoretically that this model can realize complete simulation when the sequence strictly follows these basic forms,and thus has higher simulation prediction accuracy when these basic forms are approximately followed.