Two Order Grey Markov Prediction Modeling and Its Application

Abstract

This paper based on the Grey Theory and Markov Theory, the Two Order Grey Markov Model(GMM(2,1)) is established. The modeling process are analyzed firstly, instead of using only one theory model or only using GM(1,1) in the past, this article also through a real instance comparing GM(1, 1) ,GM (2, 1) and GMM(2,1). Experiments demonstrate that the GMM(2,1) gets the better result performance than that of the other models. Introduction Unknown or uncertain information is called black information. The known information is called white information. The information which contain unknown information and known information is called gray system. Since the grey model has been proposed, it has been widely employed in various fields, ranging from economics through agriculture to engineering, and demonstrated promising results. In general, people are highly interested in forecasting future tendency of some data series or event, such as investment in stock market or in the Agriculture trading. Because of the accuracy prediction can reduce the uncertain and investment risk in making decision. GM(1,1) is the most commonly employed grey model, such as predict chaotic time series[1], predict the price of the agriculture production[2], predict the price of the stock[3-4],compute weights of the criteria[5], evaluate restoration plans for power distribution systems [6],design the evaluation[7], estimation and decision of the mechanic motion project [8] , control road tunnel ventilation[9]and so on[10]. In spite of GM(1,1) or its variants can obtain good performance on the various applications, but for acquiring the best forecasting results, an effective method, combined the Grey Theory with Markov Theory(GMM(2,1)) is proposed in our research. Grey theory In 1982,Professor Deng Julong founded Grey System Theory[11-13] for dealing with poor information system, usually 3-5 points, the grey model can be constructed. A brief description of the procedure of GM(1,1) is given as follows. Step 1: for random sequence (0) (0) (0) (0) (0) { (1), (2), (3), , ( )} x x x x x n =  (1) where n is the sample size. By 1-AGO(one time Accumulated Generating Operation) (0) x ,the preprocessed series, (1) (1) (1) (1) (1) { (1), (2), (3), , ( )} x x x x x n =  (2) where (1) (0) 1 ( ) ( ) k i x k x i = =∑ , for i=1,2, ,n  . By mean operation on (1) x , the series (1) (1) (1) (1) (1) { (1), (2), (3), , ( )} z z z z z n =  (3) where (1) (1) (1) ( ) 0.5 ( ) 0.5 ( 1) z k x k x k = + − ,  (k= 2,3, ,n) Step 2: obtained the grey differential equation and its whiting equation of GM(1,1,), respectively, as follows: International Industrial Informatics and Computer Engineering Conference (IIICEC 2015) © 2015. The authors Published by Atlantis Press 1419 (0) (1) ( ) ( ) , x k az k b + = k=2,3, ,n  (4) (1) (1) , dx ax b dt + = (5) where a and b is the developing coefficient and grey input, respectively. Step 3:Let =( , ) a b θ Τ be the parameters vector. By using least square method (Hsia,1979), the parameters a and b can be obtain as -1 ˆ=( ) a X X X Y b θ Τ Τ   =     (6)