An Optimized Grey GM(2,1) Model and Forecasting of Highway Subgrade Settlement

Abstract

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.

Ning Xu 1 and Yao-guo Dang 1, 2, 3

Academic Editor:Haipeng Peng

1, College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China
2, Institute of Grey System Study, Nanjing 211100, China
3, Jiangsu Center of Post Evaluation, Nanjing 211100, China

Received 27 July 2014; Accepted 28 December 2014; 8 January 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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 [1, 2]. This theory employs generated data to construct model not directly raw data and utilizes whitenization equation (or reflection equation) to investigate development tendency. Many applications in engineering field have demonstrated its effectiveness and high precision. Due to the simplicity comparing with other models, grey models are more practicable. Over two decades, this prediction theory has caught much attention of scholars all over the world.

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 (1-AGO) and the first order inverse AGO (1-IAGO) sequence as input. Comparing with [figure omitted; refer to PDF] generation revealing system's global function, [figure omitted; refer to PDF] generation unveils specific information and variation trend. GM(2,1) model combines both and employs a second order reflection equation to restore the uncertain information.

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 [figure omitted; refer to PDF] data as independent variable to simplify solving process. For IAGO operator affects parameter estimation, GM(2,1) is sensitive to random variation of system development.

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).

2. Literature Review

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 [3]. Further, based on the optimized GM(1,1) model he proposed another approach taking the periodic characteristic into account. The new model with trigonometric parameters was used to simulate the periodic trend [4]. Improvement of time response function is another way to optimize GM(1,1); Tien studied derivation of time response function of GM(1,1) by matrix tool and put forward a method to optimize the establishment method of initial value in time response function [5]. Besides, background value is reconstructed by some new method to improve the grey model. Li et al. tried to improve the adaptive ability for more complex sequences by reconstructing background formula. In the study, the reflection function was combined with new background formula for extracting development information from raw data, and the new model was used to forecast electricity consumption of Asian countries [6].

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 [7]. Grey power models are derived from GM(1,1) model, which is used to simulate growth process of the uncertain systems; Wang analyzed the property of grey power GM(1,1) model and suggested an optimization method based on error vector's linear programming with unbiased constrain [8]. Evans constructed a combining model consisting of grey Verhulst model and statistic model, and the new Verhulst model was used to study the tendency of steel production intensity of UK [9].

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 [10]. Wang et al. proposed a computing formula of window length for GM(2,1) modeling and simulated the deformation in landslide process by improved GM(2,1) model [11]. Zhao and Chen indicated that the GM(2,1) equation setting was unreasonable, and his further study analyzed the cause of systemic error in GM(2,1) [12]. Liu and Zhang studied the optimization method of GM(2,1), in which a linear combination parameter [figure omitted; refer to PDF] and an accumulation adjust parameter [figure omitted; refer to PDF] were used to improve grey equation of GM(2,1) model [13]. Shen and Zhao tried to simulate oscillatory sequence with optimized GM(2,1), and its computational results showed that the optimized GM(2,1) model could fit a certain scope of oscillatory systems action with obvious trend.

3. Methodology

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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from original data [figure omitted; refer to PDF] , and the background sequence is constructed for smoothing the increasing rates. [figure omitted; refer to PDF] denotes nonnegative sequence of system action. [figure omitted; refer to PDF] is the sequence by AGO operator, defined as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Similarly, generate [figure omitted; refer to PDF] by IAGO and denote it as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Besides, construct the background sequence [figure omitted; refer to PDF] , which is generated by mean generation of adjoining neighbors based on [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

The new model is stated stepwise as follows. Construct the grey differential equation of SOGM(2,1) model: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are the low order parameter and high order parameter, respectively. [figure omitted; refer to PDF] denotes the control variable and [figure omitted; refer to PDF] is the background sequence as mentioned earlier.

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 (1). And background value and IAGO data are used to construct the grey differential equation.

Then, estimate parameters of the grey differential equation. Let [figure omitted; refer to PDF] And [figure omitted; refer to PDF] is parameter vector. After that, estimate the parameter vector by least error square method, and computation formula is [figure omitted; refer to PDF] .

Construct the grey reflection equation (or grey whitenization equation): [figure omitted; refer to PDF] Then, solve the grey reflection equation and obtain time response function. The homogenous differential equation of (3) is constructed as follows: [figure omitted; refer to PDF] And the characteristic equation is [figure omitted; refer to PDF] For analyzing the simulation function, let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is used to check the pattern of system development. For [figure omitted; refer to PDF] , solution of SOGM(2,1) has a pair of conjugate complex roots [figure omitted; refer to PDF] ; for [figure omitted; refer to PDF] , SOGM(2,1) model has a pair of equal real roots, [figure omitted; refer to PDF] ; for [figure omitted; refer to PDF] , SOGM(2,1) model has a pair of unequal real roots.

According to the roots of (5), the form of time response function can be determined. First, while [figure omitted; refer to PDF] , the time response function will be [figure omitted; refer to PDF] Second, while [figure omitted; refer to PDF] , the time response function will be [figure omitted; refer to PDF] Third, while [figure omitted; refer to PDF] , the function will be [figure omitted; refer to PDF]

The SOGM(2,1) model can simulate a certain pattern of actual system running. If [figure omitted; refer to PDF] , the model reveals a nonlinear development combining linear growth and exponential growth; if [figure omitted; refer to PDF] , it reveals a periodic trend combining a with power growth; if [figure omitted; refer to PDF] , SOGM(2,1) model contains the scope of quasiexponential law with a certain deviation.

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.

Criteria 1 (priority of latest information).

Suppose time response function could be described as [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Restraint conditions for the function satisfying priority of new information should be [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , the conditions are [figure omitted; refer to PDF]

Theorem 1.

For [figure omitted; refer to PDF] , values of [figure omitted; refer to PDF] satisfying Criteria 1 should be [figure omitted; refer to PDF]

Proof.

Using Criteria 1 as constraint condition, equations system (10) with substitution of original sequence can be presented as matrix form as follows: [figure omitted; refer to PDF] According to the Cramer rule, solution of the equation is presented as [figure omitted; refer to PDF]

As for the integral modeling algorithm, Figure 1 shows the stepwise programs.

Figure 1: Modeling steps of SOGM(2,1).

[figure omitted; refer to PDF]

4. Case Study

4.1. Model Test

The modeling precision could be tested by the following criteria, and we investigate the forecasting performance by these methods.

(1) Relative percentage error (PRE) compares the original data and the simulation values to evaluate the precision of a specific point; [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF]

(2) Average relative percentage error (APRE) can evaluate the model's total precision, and computation method can be shown as follows: [figure omitted; refer to PDF]

(3) Forecasting ability can be evaluated mainly by ARPE. Table 1 shows the levels for testing a model forecasting ability.

Table 1: Evaluation for model precision.

APRE (%)	Forecasting ability
<5	Highly accurate predictability
5â[euro]"10	Good predictability
10â[euro]"20	Reasonable predictability
>20	Weak and inaccurate predictability

4.2. Modeling Process of the Numerical Case

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 2 and the development trend is presented in Figure 2.

Table 2: Measurements of highway settlement (unit: mm).

Time	1	2	3	4	5
Measurement	1.9892	2.1702	2.3266	2.4332	2.4525

Figure 2: Scattering of original data.

[figure omitted; refer to PDF]

Analyze the trend of the measurements; Figure 1 indicates that the highway subgrade settlement has a saturating tendency, in which the increase rate decreases. Since the data size is few and cannot reach statistical requirement, grey model is a reasonable alternative.

Brief description presents the modeling process. Let actual data in Table 1 be the original sequence [figure omitted; refer to PDF] . [figure omitted; refer to PDF] is the AGO sequence and [figure omitted; refer to PDF] the IAGO sequence as introduced earlier. Construct SOGM(2,1) model and calculate the values of parameters by LS method.

Construct the grey reflection equation of SOGM(2,1) and estimated values of parameter are substituted in [figure omitted; refer to PDF] Characteristic equation is [figure omitted; refer to PDF] . Solve the equation and acquire the solution as [figure omitted; refer to PDF] . After that, we could get [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , time response function should be [figure omitted; refer to PDF] .

Establish the initial constants [figure omitted; refer to PDF] . Calculated as Theorem 1, the solution is [figure omitted; refer to PDF] . Then, the time response function can be achieved as [figure omitted; refer to PDF]

The simulation values can be calculated from (17). For [figure omitted; refer to PDF] , calculated value will be treated as prediction one. Table 3 shows the results of SOGM(2,1) model, GM(2,1) model, and GM(1,1) model as comparison.

Table 3: 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	â[euro][per thousand]	1.05%	â[euro][per thousand]	20.14%	â[euro][per thousand]	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 1. The result shows that SOGM(2,1) is better than traditional GM(2,1) model.

From Figures 3, 4, and 5, we could conclude that SOGM(2,1) model has a highly accurate predictability. The APRE of SOGM(2,1) and GM(1,1) is 1.05% and 1.23%, respectively. However, SOGM(2,1) simulates a similar trend of actual data but the simulation of GM(1,1) has an ascending trend.

Figure 3: Simulation of SOGM(2,1) model.

[figure omitted; refer to PDF]

Figure 4: Simulation of GM(2,1) model.

[figure omitted; refer to PDF]

Figure 5: Simulation of GM(1,1) model.

[figure omitted; refer to PDF]

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.

5. Conclusion

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.

Nomenclature

SOGM(2,1):

Structure optimized GM(2,1) model

AGO:

Accumulated generating operation

IAGO:

Inverse accumulated generating operation

[figure omitted; refer to PDF] :

Development coefficient

[figure omitted; refer to PDF] :

Grey control variable

[figure omitted; refer to PDF] :

Length of sequence in model

[figure omitted; refer to PDF] :

Length of forecasting horizon

[figure omitted; refer to PDF] :

Original sequence

[figure omitted; refer to PDF] :

First-order generated sequence by AGO

[figure omitted; refer to PDF] :

First-order generated sequence by IAGO

[figure omitted; refer to PDF] :

Constants of the time response function

[figure omitted; refer to PDF] :

Background value sequence

[figure omitted; refer to PDF] :

Relative error of [figure omitted; refer to PDF] point

TRF:

Time response function

PRE:

Relative percentage error

APRE:

Average of relative percentage error.

Acknowledgments

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).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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