On the improved GM(1, 1) model based on concave sequences

Abstract

Grey forecasting model is an important part of grey theory, and it is an important tool to deal with small samples and poor information. The grey GM(1,1) model is one of the most important forecasting models. In recent years, the GM(1,1) model has been widely used in industrial production, social science and other fields. However, the GM(1,1) model has a large error in the practical application process, so many scholars have proposed a series of improved methods. For instance, the accurate calculating formula of the GM(1,1) model's background value was derived through the whitening equation. The new calculation formula of the GM(1,1) model's background value was deduced by using the non-homogeneous exponential function to fit the accumulated generation sequence, and it breaks through the restrictions of development coefficient. The weighted least square method and the total least squares method were used to improve the GM(1,1) model's parameter estimation values. The GM(1,1) model was also improved by optimizing the initial and boundary values in the whitening equation. However, no matter which of the above methods is used, the restored sequence of the GM(1,1) model can be proved to be a convex sequence, so it is infeasible to establish directly the GM(1,1) model for a concave sequence. If the GM(1,1) model is established based on a concave sequence, it will lead to the inconsistency between the forecasting sequence and the original sequence. The definition of sequence convexity is given as follows. Let x(0) ={x(0)(1), x(0)(2), 1 (0) (k) = 2x(0) (n) − x(0) (k), then x 1 (0) is a monotone decreasing convex sequence; if x(0) is a monotone decreasing sequence, let x 1 (0) (k) = 2x(0) (1) − x(0) (k), then x 1 (0) is a monotone increasing convex sequence. Therefore, first transform the concave sequence, and then establish the GM(1,1) model based on its transformation sequence. Finally, carry out the inverse transformation to get the restored values. The specific algorithm flow as follows. Step1: for a concave sequence x(0) = {x(0)(1), x(0) (2), 1 (0)(k) = f(x(0)(k)), and then get the symmetric sequence x 1 (0) = {x 1 (0) (1), x 1 (0) (2), 1 (0) (n)}. Step2: establish GM(1,1) model based on the symmetric sequence, and then obtain the fitting and forecasting sequence 1 (0). Step3: carry out the inverse transformation 1 (0) (k)) (k = 2, 3,<md<md<md), and then get the restored sequence <xC(0). Finally, two cases are used to illustrate the feasibility of this improved method. Case A is to forecast China's per capita energy consumption. From 2003 to 2008, the data is x(0)={1427, 1647, 1810, 1973, 2128, 2200}, unit: kg standard coal. This is a monotone increasing concave sequence. By establishing the GM(1,1) model based on the original sequence and its symmetric sequence respectively, the average relative error of the direct GM(1,1) modeling is 1.2931%, while the average relative error of the GM(1,1) modeling based on its symmetric sequence is 0.6981%. The forecasting value of the direct GM(1,1) modeling is 2409.6, while the forecasting value of the GM(1,1) modeling based on its symmetric sequence is 2353.5, which is closer to the actual value (2303.2) in 2009.