《Applied Mechanics amp; Materials》 , 2013 , 256-259 (1) :1721-1725

A Multivariable Grey Model Based on Optimized Background Value and Its Application to Subgrade Settlement Prediction

Hanbing Liu1, a , Yiming Xiang 1, b , and Nguyen Huu Hung 1,c 1College of Transportation, Jilin University, Changchun, Jilin, 130022, China a liuhb57@sina.com, b elesun@126.com, c nhhungct@yahoo.com

Keywords: Subgrade, Settlement prediction, GM(1,1), Multivariable Grey Model.

Abstract. Subgrade settlement is a complex process system. Commonly used single-point prediction models canrsquo;t consider the correlation between the discrete deformation monitoring points, which doesnrsquo;t adequately reflect the overall deformation law of subgrade. A multivariable grey model (MGM(1,n)), which is an expansion of the single-point GM(1,1) model for multiple variables, is introduced to resolve the above problem. Aiming at the drawback of background value in the traditional MGM(1,n) model, the functions with non-homogeneous exponential law are used to fit the accumulated sequences for every variable, reconstruct the calculating formula of background value, and gets a new MGM(1,n) model based on optimized background value (OMGM(1,n)). A case study shows that the forecast result of the proposed model is more precise and effective than these of the single-point GM(1,1) model and the traditional MGM(1,n) model for predicting subgrade settlement.

Introduction

Subgrade settlement is a great hidden danger to create the road traffic accident, and an important index related to road safety, so subgrade settlement prediction is one of the most important topics in the geotechnical engineering field [1-4]. In order to improve forecast accuracy and reliability of subgrade settlement, domestic and foreign scholars have proposed a lot of prediction methods. However, these methods are mostly limited to the local modeling and prediction of single monitoring point. In the deformation monitoring network composed of multiple monitoring points, the deformation of a monitoring point is actually subject to other monitoring points, and it also affects the deformation of the other monitoring points, which illustrates that the deformation of monitoring points is correlative in the subgrade settlement process. Therefore, a multivariable grey model (MGM(1,n)), which is an expansion of the single-point GM(1,1) model for multiple variables, is proposed to achieve modeling and prediction of the mutual influence of multiple monitoring points in literature [5]. As the same as that of the single-point GM(1,1) model, the calculating formula of background value in MGM(1,n) model has certain shortcoming, so this paper uses the functions with non-homogeneous exponential law to fit the accumulated sequences for every variable, reconstruct the calculating formula of background value, and gets a new multivariable grey model of optimized background value. An engineering example shows that the prediction accuracy of MGM(1,n) model based on optimized background value (OMGM(1,n)) is greatly improved compared with the single-point GM(1,1) model, and higher than the non-optimized MGM (1, n) model.

MGM(1,n) model with optimized background value Establishment of Original

Establishment of Original MGM(1,n) Model. Suppose that there are n interrelated monitoring points in a certain subgrade section by which the m-cycle observational data is obtained, the original observation data sequence is:{}(k=1, 2, hellip;, m; i=1, 2,hellip;, n), and a first-order accumulated generating sequence is: (k=1, 2, hellip;, m; i=1, 2, hellip;, n).Considering the interrelated and mutually influence of n points [6, 7], the first-order ordinary differential equations based on the accumulated generating sequence is established:

(1)

where (t)=,A=,B=.

Then the time response formula of Eq. 1 is :

(t)= ( B)-B (2)

Eq. 1 is discretized, and a grey differential equation is got:

= ,(i=1,2,hellip;,n ; k=2,3,hellip;,m) (3)

where =[] .

Based on the least square method, the parameter sequences are obtained:

==,(i=1,2,hellip;,n) (4)

where

,.

The identification values of the model parameters A and B can be got:

=,= (5)

The time response formula of the MGM(1,n) model is:

= = ((1) )- (6)

By the inverse accumulated generating operation, Eq. 6 is reduced:

=-,(k=2,3,hellip;,m) (7)

MGM(1,n) Model Based on Optimized Background Value. Both sides of the n whitening equations in (1) are integrated in the interval [k-1, k], then

= (8)

By simplifying, we get

= ,(i=1,2,hellip;,n;k=1,2,hellip;,m) (9)

Comparing Eq. 3 with Eq. 9, it is seen that the calculating formula of the traditional background value is approximately replacing with , namely =[] . The difference of the above two equations is the error source of traditional background value.

According to the non-homogeneous index vector form and quasi-exponential law of the first-order accumulated generating sequence of MGM(1,n) model, we may assume:

= ,(i=1,2,hellip;,n) (10)

where , and are undetermined constants. The optimized background value is denoted by

= ,(i=1,2,hellip;,n) (11)

Through calculating and deducing, the final formula of optimized background value is obtained

(12)

Substituting for and Taking Eq. 12 into Eq. 4, the parameter sequences = , = can be gained, which are taken into Eq. 6 and Eq. 7 to get the simulation and prediction values of MGM(1,n) model.

Application example

The subgrade settlement monitoring data were collected from a certain section of the national highway 102 from Changchun city to Dehui city in Jilin Province of China. The three monitoring points A and B

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基于优化背景值的多变量灰色模型及其在路基沉降预测中的应用

Hanbing Liu1, a , Yiming Xiang 1, b , and Nguyen Huu Hung 1,c 1College of Transportation, Jilin University, Changchun, Jilin, 130022, China a liuhb57@sina.com, b elesun@126.com,

cnhhungct@yahoo.com

《Applied Mechanics amp; Materials》 , 2013 , 256-259 (1) :1721-1725

关键词：路基，结算预测，GM(1,1)，多变量灰色模型

概要：路基沉降是一个复杂的系统，常用的单点预测模型不能考虑离散形变的监测点之间的相关性，因此不足以反映路基整体变形规律。为了解决上述问题，引入了多变量灰色模型（MGM（1，n）），该模型是多变量单点GM（1,1）模型的扩展。针对传统MGM（1，n）模型中背景值的缺点，采用非齐次指数规律函数拟合每个变量的累加序列，重构背景值的计算公式，得到一个新的MGM （1，n）模型基于优化背景值（OMGM（1，n））。实例研究表明，该模型的预测结果比单点GM（1,1）模型和传统的MGM（1，n）预测路基沉降模型的预测结果更加精确和有效。

介绍：路基沉降是造成道路交通事故的重大隐患，也是与道路安全的重要指标，因此路基沉降预测是岩土工程领域最重要的课题之一。为提高路基沉降预测的准确性和可靠性，国内外学者提出了大量的预测方法。但是，这些方法大多局限于单个监测点的局部建模和预测。在由多个监测点组成的变形监测网络中，监测点的变形实际上受到其他监测点的影响，同时也影响其他监测点的变形，说明监测点变形与路基相关结算流程有关你。因此，提出了多变量灰色模型（MGM（1，n）），该模型是多变量单点GM（1,1）模型的扩展，用于实现多监测点相互影响的建模和预测[5]。与单点GM（1,1）模型相同，MGM（1，n）模型背景值的计算公式存在一定的缺陷，因此本文采用非齐次指数规律函数拟合重构每个变量的累积序列，重构背景值的计算公式，得到一个新的优化背景值的多变量灰色模型。一个工程实例表明，与单点GM（1,1）模型相比，基于优化背景值（OMGM（1，n））的MGM（1，n）模型的预测精度大大提高。

具有优化背景值的MGM（1，n）模型建立原始

原MGM（1，n）模型的建立。

假设在获得m周期观测数据的某路基段有n个相互关联的监测点，原始观测数据序列为：{}(k=1, 2, hellip;, m; i=1, 2,hellip;, n)，并且一阶累积生成序列是：，(k=1, 2, hellip;, m; i=1, 2, hellip;, n)。考虑到n点的相互关联和相互影响[6,7]，建立了基于累积生成序列的一阶常微分方程：

(1)

其中 (t)=,A=,B=。

然后，公式的时间响应方程（1）是：

(t)= ( B)-B (2)

方程（1）离散化，得到灰色微分方程：

= ,(i=1,2,hellip;,n ; k=2,3,hellip;,m) (3)

其中=[]。
基于最小二乘法，获得参数序列：

==,(i=1,2,hellip;,n) (4)

其中 ,.

可以得到模型参数A和B的识别值为：

=,= (5)

MGM（1，n）模型的时间响应公式为：

= = ((1) )- (6)

通过反向累积生成操作，方程（6）缩略为

=-,(k=2,3,hellip;,m) (7)

基于优化背景值的MGM（1，n）模型

中的n个白化方程的两侧被集中在区间[k-1，k]中，得

= (8)

通过简化，我们得到了

= ,(i=1,2,hellip;,n;k=1,2,hellip;,m) (9)

比较方程（3）与方程（9）可以看出，传统背景值的计算公式大致是取代，即=[]。上述两个方程的差异是传统背景值的误差来源。

根据MGM（1，n）模型的一阶累积生成序列的非齐次指标向量形式和准指数规律，我们可以假设：

= ,(i=1,2,hellip;,n) (10)

当是 , 和不确定的常量。优化的背景值表示为：

= ,(i=1,2,hellip;,n) (11)

通过计算和推导，得到优化背景值的最终公式：

(12)

将带入方程（4）就可以得到参数序列= , =，然后带入方程（6）从而得到MGM（1，n）模型的模拟和预测值。

应用示例

路基沉降监测数据来自长春市至吉林省德惠市的102国道某段。三个监测点A，B和C分别位于剖面的左半部分，即多点变形数n = 3，如图1所示。通过选择一组三个原始数据序列监测点，建立MGM（1,3）模型和GM（1,1）模型，分别用于预测路基沉降变形。从同样定期的结算监控数据中取得12组数据。根据各监测点前8组数据序列，对OMGM（1,3）模型，MGM（1,3）模型和GM（1,1）模型进行建模，分别预测最后4组数据用于测试预测值准确度的序列，预测结果示于表1至3中，并且三种不同模型的预测误差比较如表4所示。从表1至表3可以看出， OMGM（1,3）模型和MGM（1,3）模型的模拟和预测值比GM（1,1）模型更接近实测值，说明多点模型比单一模型更加准确和高效点模型。通过查表4可以看出，与GM（1,1）模型相比，OMGM（1,3）模型的预测精度大大提高，且高于非优化MGM（1，n）模型。

Fig.1.监测点示意图（单位：cm）

表1 A点不同模型的预测（单位：cm）

表2 B点不同模型的预测（单位：mm）

表 3 C点不同模型的预测(单位：mm)

表4 三种不同模型的预测误差比较（单位：mm）

结论

路基沉降是一个复杂的非线性系统，受多种因素影响。基于上述工程实例，可以看出多元灰色模型克服了许多路基沉降预测模型缺乏单点局部分析的问题，考虑了路基各监测点之间的相关性和相互影响，达到了多点监测变形的总体预测。在分析了传统MGM（1，n）模型的背景值误差后，利用非齐次指数规律函数建立了基于优化背景值的MGM（1，n）模型，拟合出一阶累积生成序列MGM（1，n）模型。路基沉降预测的实际应用表明，OMGM（1，n）模型的预测精度比MGM（1，n）模型有所提高。由于路基沉降监测数据不断补充，可以对最新的实时数据进行建模，排除旧信息的干扰，进一步提高OMGM（1，n）模型的精度，使预测值与实际值一致，得到有价值的下一时间的可靠预测。

Acknowledgements

The authors gratefully acknowledge the support of the National High Technology Research and Development Program ('863 Program') of China (Project No. 2009AA11Z104), “985 Project” of Jilin University and the innovation team program of Jilin University.

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