The Study On Fractional Order Grey Predicting Model Based On Fractional Order Summation And Caputo Definition

Grey model is an important part of grey theory.The grey model represented by GM(1,1)model has been widely used in the field of economy,control and managem.Based on the existing model,the researchers optimized the original model from two aspects:improving the precision of the model and extending the application range of the model.This paper mainly studies the fractional order grey model based on fractional order accumulation and the fractional order grey model defined by Caputo.The main contents of this paper are as follows:(1)The fractional order expressions of ODGM,EGM,DGM and EDGM are derived on the basis of fractional order accumulation operator.The order of fractional model is optimized by genetic algorithm,and the optimal order with minimum average error is obtained.(2)It has been a difficult problem to explore the rules of small samples and poor information objects with statistical rules.In this paper,TOFT toolbox and genetic algorithm are used to extend integral derivative to fractional derivative,and a grey model of fractional derivative defined by Caputo is obtained.The method of this model is simple,and examples are given in this paper to prove that the effect of this model is good.(3)The modeling object of the traditional grey model is the original sequence with equal spacing.The processing of non-equidistant sequence will be approximately calculated,and the non-equidistant sequence will be transformed into equidistant sequence.The fractional order grey model based on Caputo definition can be used to directly model the non-equidistant sequence.The results are good,and the modeling method and calculation examples are given.(4)The interface of fractional grey system in Matlab is designed.Using this interface,users can directly use the grey model to fit or predict the original data.The grey models that can be selected are GM(1,1)and grey model based on fractional order accumulation.The results are output in the form of data and images,which is convenient for users to record and observe.