Grey prediction technique is a useful tool for few data analysis and short term forecasting. GM(2,1) model is one of the most important grey models. For improving the precision and prediction ability, we proposed a structure optimized GM(2,1) model, namely, SOGM(2,1) model. This study contributes grey prediction theory on three points. First, SOGM(2,1) model utilizes background sequence and inverse accumulating generated sequence to construct new grey equation with optimized structure, and then estimation of parameters is derived based on least errors. Second, reflection equation is constructed and the solving process is derived with the time response function acquired. Third, we put forward a new method for establishing initial values of time response function. After that, the new model is used to predict highway settlement of an engineering assessment. Comparing with other models, the results show that SOGM(2,1) is effective and practicable to forecast.
Grey prediction is an important theory as part of grey system theory proposed by Professor Deng. It is dedicated to forecasting uncertain system with imperfect information or few data [
GM(2,1) is proposed to change the linear structure of GM(1,1) model and expands application scope of grey prediction theory. It is an important model among the group of grey prediction models. Modeling method of GM(2,1) is derived from GM(1,1) and satisfies a similar mechanism. GM(1,1) model employs accumulating generation operator (AGO) for weakening randomness of raw data and extracting useful information, while GM(2,1) uses the first order AGO sequence (1AGO) and the first order inverse AGO (1IAGO) sequence as input. Comparing with
However, the structure of GM(2,1) model and solving process have limited its application. In GM(2,1) model, grey differential equation is constructed with
The present study proposed a structure optimized GM(2,1) model, namely, SOGM(2,1), to improve precision based on traditional GM(2,1) model. The information contained is believed to be useful for forecasting in engineering problems. Structure of the following content is divided into three parts: the first section is a brief literature review; the second section described the approach and modeling program of SOGM(2,1); in the last section, an application is studied to forecast a highway settlement by SOGM(2,1) model, and computational results indicated that the new one has a higher prediction than traditional GM(2,1) and GM(1,1).
In recent years, many researchers have promoted the development of grey prediction theory and many optimized algorithms have emerged expanding greatly the application scope of grey prediction theory. A brief summary of literatures is presented as follows.
First, some new methods have been suggested to improve the flexibility of grey prediction model for dealing with more complex system sequences. Professor Deng, originator of grey systems theory, suggested a method with new parameter in the grey equation of GM(1,1) model, and an improved model was constructed to simulate oscillation sequences [
Based on the improvement method for original GM(1,1), many new models were derived and constructed. Qian et al. expanded the grey function variable in basic grey differential equation, adding time power items to simulate a certain kind of systems [
Second order grey model is an important expansion from GM(1,1) model group. As for the second order grey model, Zeng and Xiao considered the mobility of GM(2,1) with matrix tool and put forward an algorithm by accumulating product for original sequences; the new method improved prediction accuracy and decreased the mobility of parameters evaluation process [
In this section, we suggest an optimized method for GM(2,1) and propose a new algorithm to achieve high prediction ability. A new grey equation is constructed, and the original sequence is set as independent variable. By the grey equation, we derived the parameter estimation, solving process and time response function. Besides, the latest two points are employed to establish the initial terms of simulation function and keep the simulation function with a high precision and a similar trend as actual data series.
Operators AGO and IAGO are used to generate new sequences
The new model is stated stepwise as follows. Construct the grey differential equation of SOGM(2,1) model:
Structure of grey differential equation of SOGM(1,1) is different from that of traditional GM(2,1) model. Comparing with traditional GM(2,1) model, original sequence is an explained variable in (
Then, estimate parameters of the grey differential equation. Let
Construct the grey reflection equation (or grey whitenization equation):
According to the roots of (
The SOGM(2,1) model can simulate a certain pattern of actual system running. If
The constants of TRF could be established by the latest actual data according to priority of new information.
After the algorithm program, an analysis of the TRF is presented, and the initial terms will be established, which is used to acquire discrete simulation values. Unlike GM(1,1) model, the TRF needs two different initial terms. Grey systems theory suggests a priority of the latest information in few data modeling.
Suppose time response function could be described as
For
Using Criteria
As for the integral modeling algorithm, Figure
Modeling steps of SOGM(2,1).
The modeling precision could be tested by the following criteria, and we investigate the forecasting performance by these methods.
Relative percentage error (PRE) compares the original data and the simulation values to evaluate the precision of a specific point;
Average relative percentage error (APRE) can evaluate the model’s total precision, and computation method can be shown as follows:
Forecasting ability can be evaluated mainly by ARPE. Table
Evaluation for model precision.
APRE (%)  Forecasting ability 


<5  Highly accurate predictability 
5–10  Good predictability 
10–20  Reasonable predictability 
>20  Weak and inaccurate predictability 
Grey prediction technique is useful in engineering. The new model SOGM(2,1) is used to forecast a highway subgrade settlement as one part of a project postassessment. The data is measured in a highway construction project in Jiangsu, one of China’s provinces. Measurements have equal time intervals. Measurements are listed in Table
Measurements of highway settlement (unit: mm).
Time  1  2  3  4  5 

Measurement  1.9892  2.1702  2.3266  2.4332  2.4525 
Scattering of original data.
Analyze the trend of the measurements; Figure
Brief description presents the modeling process. Let actual data in Table
Construct the grey reflection equation of SOGM(2,1) and estimated values of parameter are substituted in
Establish the initial constants
The simulation values can be calculated from (
Calculations and comparison of SOGM(2,1) and GM(2,1).
Time  Original data  SOGM(2,1)  GM(2,1)  GM(1,1)  

Fitting value  RPE  Fitting value  RPE  Fitting value  RPE  
1  1.9892  2.0460  2.85%  1.9892  0.00%  1.9892  0 
2  2.1702  2.2071  1.70%  2.2504  1.93%  2.2060  1.65% 
3  2.3266  2.3426  0.69%  3.1339  13.68%  2.2964  1.30% 
4  2.4332  2.4332  0.00%  4.3049  30.94%  2.3905  1.75% 
5  2.4525  2.4525  0.00%  5.8496  54.14%  2.4885  1.47% 


ARPE  1.05%  20.14%  1.23% 
From the accuracy point of view, APRE of SOGM(2,1) model is 1.05% and is lower than that of GM(2,1) 20.14% and GM(1,1) 1.23%. The error series of SOGM(2,1) has a convergence trend, which is indicated in Figure
From Figures
Simulation of SOGM(2,1) model.
Simulation of GM(2,1) model.
Simulation of GM(1,1) model.
The original sequence describes a system tending to saturation, and SOGM(2,1) has extracted the tendency information from AGO sequence and IAGO sequence, simulating precisely. However, time response function of GM(2,1) has a disparity from original sequence. So the simulation indicates that the new model has a better information processing ability and is more flexible to fit larger scope of system characteristic than traditional grey model method.
We proposed the algorithm of SOGM(2,1) model for few data problems; its main objective is to forecast in short terms of engineering project. The new model can achieve high precision comparing with traditional GM(1,1) and GM(2,1) models. And it is believed to be adaptive to many uncertain system problems.
Structure optimized GM(2,1) model
Accumulated generating operation
Inverse accumulated generating operation
Development coefficient
Grey control variable
Length of sequence in model
Length of forecasting horizon
Original sequence
Firstorder generated sequence by AGO
Firstorder generated sequence by IAGO
Constants of the time response function
Background value sequence
Relative error of
Time response function
Relative percentage error
Average of relative percentage error.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is financially supported by the National Natural Science Foundation (nos. 71371098 and 71071077) and the Major Project of Key Research Base of Philosophy and Social Science in Jiangsu Colleges (no. 2012JDXM005).