Fractional Order Grey Forecasting Models And Their Application

Since grey system theory is put forward by Deng Julong, grey modeling technique has obtained many inspiring research achievements. However, as an emerging discipline, its base needs to perfect. Follow the way of “question raising, question sovling, and real case test’, the idea of “fractional order” is introduced into grey system theory. Aim to improve and to extend grey system theory, the main achievements of this study are as follows.(1) The perturbation theory of least squares method is applied to prove that the perturbation bound is larger when the size of sample is bigger, and vice versa in the case of the disturbance is identical. From the pointview of stability, the grey forecasting model is stable when the sample size is small. The fractional order sequence accumulation is put forward in order to reduce the perturbation bound. Furthermore, traditional grey forecasting model and fractional order accumulation grey model are compared on the stability, the initial data, the fitted error and monotone.(2) How to make use of the law in forecasting system with small sample and lack of statistics properties is a problem. The integer order derivative of grey prediction model is extended to fractional order derivative based on fractional calculus in order to use the memory property of fractional order derivative to describe the forecasting system with small sample, and we prove that grey prediction model with fractional order derivative is better than that of grey prediction model with integer order derivative for Principle of new information priority, effect of initial value, stability and so on. The effectiveness and practicability of proposed model is validated by real example.(3) The principle of new information priority of traditional weakening buffer operator, the buffer operator with variable weights, and a kind of strengthening buffer operator are proved by matrix perturbation theory. The relationship between the buffer effect and the sample size is analysed.Due to traditional buffer operators can not realize fine tuning of effect intensity, which leads to problems that the effectiveness of buffer action may be strong or too weak. The number of buffer order is extended to fractional order by the method of matrix calculation. The buffer effect is in line with the number of buffer order.Multivariate weakening buffer model is developed to study multivariate buffer operators. Based on matrix perturbation theory, we prove that multivariate weakening buffer model consider the priority of each period data. That is perturbing the newer data, the bigger perturbation bound under the equal perturbation, and the buffer effect is obvious when the sample size is smaller. The result of energy forecasting shows multivariate weakening buffer model is effective.(4) The suitability of common grey relational analysis is discussed. From different viewpoints, fractonal order grey relational degree for time series, the grey similarity degree for cross-sectional data, and grey convex relational degree for panel data are proposed respectively. These properties are also discussed. The effectiveness and practicability of proposed models are validated by real examples.(5) The regression model with similar information priority is set up in order to predict the complicated equipment costs, whose index is more relevant to cost; GM(0,N) model based on grey similarity degree is suitable to the complicated equipment costs, whose index is less relevant to cost, and the principle of GM(0,N) model is proved theoretically; The results of practical examples demonstrate that the new proposed GM(1,1) is effective to predict the maintenance cost of complex equipment.