Study On Several Kinds Of Developed Models Of GM(1,1)

Grey models are applicable to the uncertain system with less sample and poor infor-mation. However, many typical forecasting models based on statistical theory and ma-chine learning, such as exponential smoothing, autoregressive moving average (ARMA), generalized autoregressive conditional heteroscedasticity model (GARCH), artificial neu-ral network model(ANN), and support vector machine (SVM), need large sample. Thus, grey models are widely applied in engineering technology and economic management. GM(1,1) is one of core models of grey models. In this paper, the accumulation method is introduced into the parameter estimation of the model. The prediction precision is en-hanced. The applicable series of the model is extended to the interval series. Several kinds of developed models of GM(1,1) are proposed in this paper:accumulation method GM(1,1), accumulation method non-equidistant GM(1,1), interval sequence GM(1,1) based on sequence transformation, interval sequence GM(1,1) based on parameter trans-formation, and the prediction model applied to the fluctuant interval series based on Markov and GM(1,1) models. The main works are as follows:1. The parameter estimation and the predictor formula of GM(1,1) are devel-oped. Accumulation method is applied to the parameter estimation of GM(1,1) and non-equidistant GM(1,1). The connotation predictor formula is deduced from the definition equation of GM(1,1) directly and takes the place of the white response of GM(1,1). Thus, accumulation method GM(1,1) and accumulation method non-equidistant GM(1,1) are proposed. Then the characters of the two models are analyzed. Firstly, the matrix form of the parameter estimation based on accumulation method is deduced and can reflect the direct relation between the parameter estimation and the raw series. Then some characters of two models are proved from the relation.2. Based on the sequence transformation, GM(1,1) can be applied to the forecasting of binary or ternary interval series. The interval series is transformed into several equiv-alent precise number series. The transformation process ensures the relative positions of the bounds of the interval number in the reduction process. GM(1,1) can be built on these transformed series firstly, and then the interval prediction values are restored.3. The form of the parameters in GM(1,1)is developed.One of parameters of the definition equation of GM(1,1)-integral development coefficient, is taken as the weighted mean of the development coefficients of the several boundary series of the interval series. The other parameter-grey input, is taken as an interval number, which is amended after the integral development coefficient obtained. By this method, the applicable range of GM(1,1) is extended to the binary or ternary interval number series essentially, while the interval series need not to be transformed into precise number series. Then, BIGM(1,1) (Binary Interval GM(1,1)) and TIGM(1,1) (Trinary Interval GM(1,1)) are proposed.4.Markov forecasting method is used to amend the prediction values of BIGM(1,1). In order to maintain the relative positions of the bounds of the interval number in the amendment, the above method of the sequence transformation is introduced. Thus, the applicable range of GM(1,1) is extended to the forecasting of fluctuant interval series.