A multi-variable grey model with a self-memory component and its application on engineering prediction

Engineering Applications of Artificial Intelligence 42 (2015) 82–93

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

A multi-variable grey model with a self-memory component and its application on engineering prediction Xiaojun Guo a,b,n, Sifeng Liu c,b, Lifeng Wu b, Yanbo Gao a, Yingjie Yang c a

School of Science, Nantong University, Nantong 226019, China College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China c Centre for Computational Intelligence, De Montfort University, Leicester, LE1 9BH, UK b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 October 2014 Received in revised form 21 March 2015 Accepted 29 March 2015

This paper presents a novel multi-variable grey self-memory coupling prediction model (SMGM(1,m)) for use in multi-variable systems with interactional relationship under the condition of small sample size. The proposed model can uniformly describe the relationships among system variables and improve the modeling accuracy. The SMGM(1,m) model combines the advantages of the self-memory principle of dynamic system and traditional MGM(1,m) model through coupling of the above two prediction methods. The weakness of the traditional grey prediction model, i.e., being sensitive to initial value, can be overcome by using multi-time-point initial field instead of only single-time-point initial field in the system's self-memorization equation. As shown in the two case studies of engineering settlement deformation prediction, the novel SMGM(1,m) model can take full advantage of the system's multi-time historical monitoring data and accurately predict the system's evolutionary trend. Three popular accuracy test criteria are adopted to test and verify the reliability and stability of the SMGM(1,m) model, and its superior predictive performance over other traditional grey prediction models. The results show that the proposed SMGM(1,m) model enriches grey prediction theory, and can be applied to other similar multi-variable engineering systems. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Grey prediction theory Multi-variable system MGM(1,m) model Self-memory principle Subgrade settlement Foundation pit deformation

1. Introduction With rapid development of economy and urbanization in China, issues of subgrade settlement and foundation pit deformation are frequently observed in large-scale construction projects. These are important indicators influencing the stability and safety of engineering construction. Moreover, accurate estimation and effective control of engineering settlement deformation are key to the success of engineering construction. Existing methods for predicting settlement deformation usually adopt the time series analysis model by utilizing monitoring data (Zhou et al., 2010; Xu et al., 2012; Dai et al., 2013), which have effectively promoted the development of the settlement deformation prediction theory and its practice. However, since engineering construction is a complicated nonlinear dynamical system with multiple influencing factors, the issue of settlement deformation involves not only time influence but also spatial effect. Most of these current models are categorized as “parameter model”, which is often criticized that parameters lack of clear physical meaning and effective parts of

n Corresponding author at: School of Science, Nantong University, Nantong 226019, China. Tel.: þ 86 13862960631. E-mail address: [email protected] (X. Guo).

http://dx.doi.org/10.1016/j.engappai.2015.03.014 0952-1976/& 2015 Elsevier Ltd. All rights reserved.

monitoring data are lost. Consequently, it is of great importance to develop more effective techniques for predicting settlement deformation. Future development trends could be predicted on the basis of previously measured information of engineering systems. Taking into account polytropism and uncertainty of construction projects, the BP neural network model (Leu and Lo, 2004; Zhang et al., 2011; Ismail and Jeng, 2011) and the Grey systems model (Li and Li, 2009; Feng et al., 2010) are gradually introduced into engineering applications. Artificial intelligence analysis methods could help to effectively reveal the structure and trends of settlement deformation data, and improve prediction accuracy and reliability. The grey systems model is especially applicable for small samples and poor information emerging in engineering applications. Owing to increasing complexity, uncertainty and chaos of the system's structure, traditional statistical methods — such as autoregressive moving-average model (Wiesel et al., 2013), bilinear model (Matsubara and Morimoto, 2013), nonlinear autoregressive model (Li et al., 2011)—may not accurately approximate the evolutionary trend. To overcome this drawback, the grey systems theory was initially proposed by Deng to study the uncertainty of systems (Liu et al., 2013b). As an important theoretical component, the grey prediction approach represented by GM(1,1) model, can weaken the randomness of original

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

statistical data by means of accumulated generating operation (AGO) (Xie et al., 2013). The superiority of grey models over conventional statistical methods is that they only require a limited amount of statistical data without knowing their statistical distribution. And they have unique advantages for the short-term prediction of small sample sequence. So far, they have already been effectively utilized in numerous fields, such as social economy (Wu et al., 2013; Xia and Wong, 2014), geographical environment (Lin et al., 2012), energy system (Pao et al., 2012), transportation (Yin and Wang, 2013; Guo et al., 2013), and so on. Similarly, several grey models were developed for the prediction of engineering science (Lin and Lee, 2007; Peng and Dong, 2011). Nevertheless, correlative multiple variables are involved in numerous engineering and economic systems in practice, especially with the constraint of small samples, such as in the case of subgrade settlement and foundation pit deformation. Existing single-point prediction models cannot take into account the relationship among variables, and are unable to adequately reflect the integral laws of system evolution. The grey multi-variable MGM(1,m) model (Zhai and Sheng, 1997) is the extension of univariable GM(1,1) model in multi-dimensions. The multi-variable model can uniformly describe each variable from the systematic angle and better reflect the interactional relationship among every systematic variables, and thus are better suited for modeling and forecasting in multi-variable systems. So far, the grey multivariable MGM(1,m) model has been successfully applied to the prediction of subgrade settlement (Wang and Pan, 2005; Liu et al., 2013a), foundation pit deformation (Feng et al., 2007; Xiong et al., 2011a) and social economy (Xiong et al., 2011b), and obtained almost ideal simulation and prediction effects. Meanwhile, some background value optimization methods (Xiong et al., 2011a; Liu et al., 2013a) have been introduced into the MGM(1,m) model to improve modeling precision. The above research results have played an important role in promoting the development of the multi-variable MGM(1,m) model among complex systems. But the traditional multi-variable MGM(1,m) model essentially belongs to the initial value solving problem of differential equations which only meet the initial condition at one point, i.e., the observed values at one moment. Accordingly, the original dynamical differential equations have the limitation of being sensitive to initial values, and that becomes a disadvantage when historical information is not fully available. Based on inverse modeling, the self-memory principle of dynamic system was first developed by Cao (1993). As a mathematic realization of integrating deterministic and random theories, the principle is the statistical-dynamic method to solve problems of nonlinear dynamic systems (Phienwej et al., 2005). The selfmemory principle can retrieve ideal nonlinear dynamic models by using empirical data. It not only overcomes the weakness of being sensitive to the initial values of the initial-value problems of differential equations, but also is more relevant to mechanism modeling due to utilization of historical data. The method is a breakthrough for numerical solution of traditional initial-value problems and statistical approaches, and it has been used increasingly in time-series forecasting in fields including meteorology, hydrology, and engineering science (Liu et al., 2010; Wang et al., 2012). In recent years, self-memory principle has also been gradually used in simple grey prediction models. Fan and Zhang (2003) derived a self-memory numerical method for solving the GM(1,1) model and established the novel grey model recollecting the last data. Chen et al. (2009) established a coupled equation by combining DHGM(2,2) grey differential equation with the selfmemory principle to forecast floods. Guo et al. (2014) developed a GM(1,1) power pharmacokinetics model with self-memory component for serum concentration prediction. The above-mentioned research shows the advantages of self-memory techniques which

83

can overcome the sensitivity to initial values. Accordingly, to extend the applications of the grey prediction model and promote its predictive performance, the self-memory principle is introduced into a grey multi-variable MGM(1,m) model for the first time. The purpose of this paper is to construct the novel multi-variable grey self-memory coupling prediction model (SMGM(1,m)) appropriate for the phenomena of subgrade settlement and foundation pit deformation in the engineering science under the condition of small sample size. This novel prediction model combines the advantages of the self-memory principle and grey MGM(1,m) model through coupling the above two prediction methods. Its excellent predictive performance lies in that the weakness of traditional MGM(1,m) models, i.e., sensitivity to initial values, can be overcome by using multi-time-point initial field instead of only single-time-point initial field. The paper is organized as follows. Section 2 provides an overview of the relevant literature on the grey correlation analysis, the basic form and basic characteristics of traditional multivariable MGM(1,m) model. Section 3 presents the detailed algorithm of the novel SMGM(1,m) model and its accuracy test criterion. In Section 4, two case studies of subgrade settlement prediction and foundation pit deformation prediction are adopted to demonstrate the adaptability and effectiveness of the proposed novel coupling prediction model. Finally, some conclusions are drawn in Section 5.

2. Traditional multi-variable grey prediction model—MGM(1, m) 2.1. Grey correlation analysis among the variables of MGM(1,m) model When making predictions using the multi-variable MGM(1,m) model, correlation analysis should be first conducted among all system variables. The correlation analysis could test whether interactional relationship exists in every system variable so that data of every variable could be used for the establishment and prediction of the MGM(1,m) model. With regard to the m original data sequences, the grey correlation theory is employed to estimate whether the connection among different sequences is compact according to the similarity of sequence curve's geometrical shape. The more approximate the geometrical shape is, the greater the relational degree and relevancy among corresponding sequences are, and vice versa (Liu et al., 2013b). Assume that there are m original non-negative data sequences ð0Þ ð0Þ ð0Þ X ð0Þ is the observation sequence of the ith 1 ; X 2 ; …; X m , where X i variable at times 1; 2; …; n, and ð0Þ ð0Þ ð0Þ X ð0Þ i ¼ fxi ð1Þ; xi ð2Þ; …; xi ðnÞg; i ¼ 1; 2; …; m:

ð1Þ

ð0Þ ð0Þ X ð0Þ 1 ; X 2 ; …; X m

The sequences are considered as the systematic signature sequence χ 0 ðkÞ in turn, and the remainder m  1 sequences are considered as the relevant factor sequences χ h ðkÞ, where k ¼ 1; 2; …; n, h ¼ 1; 2; …; m  1 and n is the observation time points of corresponding sequence. Let χ 00 ðkÞ ¼ χ 0 ðkÞ=χ 0 ð1Þ and χ 0h ðkÞ ¼ χ h ðkÞ=χ h ð1Þ, then the grey correlation degree between sequences χ 0 and χ h is represented by



min min
χ 00 ðkÞ  χ 0h ðkÞ
þ ρmax maxk
χ 00 ðkÞ  χ 0h ðkÞ
h k h



γ 0h ðkÞ ¼
χ 0 ðkÞ χ 0 ðkÞ
þ ρmax max
χ 0 ðkÞ  χ 0 ðkÞ
; 0 0 h h h

ð2Þ

k

where γ 0h ðkÞ is called grey correlation coefficient at point k, ρ A ð0; 1Þ is called resolution coefficient and often set to 0.5. Averaging all grey correlation coefficients at every points, the grey correlation degree between relevant factor sequence χ h ðkÞ

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X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

and systematic signature sequence χ 0 ðkÞ is obtained by

γ 0h ¼

1 Xn γ ðkÞ; h ¼ 1; 2; …; m 1; k ¼ 1 0h n

ð3Þ

where γ 0h is considered as one correlational measurement among variables. Usually when γ 0h Z 0:5, the relevant factor sequence and systematic signature sequence could be regarded as relevant. And the greater γ 0h is, the more compact their relationship is.

2.2. Basic form of multi-variable MGM(1,m) model

Definition 2.1. Assume that the original data matrix is ð0Þ ð0Þ ð0Þ T is defined in Eq. (1). Then X ð0Þ ¼ ½X ð0Þ 1 ; X 2 ; …; X m  , where X i the matrix X

ð1Þ

ð1Þ ð1Þ T ¼ ½X ð1Þ 1 ; X 2 ; …; X m 

ð4Þ

is the first-order accumulated generation (1-AGO) matrix of X and ð1Þ ð1Þ ð1Þ X ð1Þ i ¼ fxi ð1Þ; xi ð2Þ; …; xi ðnÞg

ð0Þ

,

ð5Þ

is the 1-AGO sequence of X ð0Þ i , where xð1Þ i ðjÞ ¼

Xj k¼1

xð0Þ i ðkÞ; i ¼ 1; 2; …; m; j ¼ 1; 2; …; n:

ð6Þ

The sequence ð1Þ ð1Þ ð1Þ Z ð1Þ i ¼ fzi ð2Þ; zi ð3Þ; …; zi ðnÞg

ð7Þ

is the background value sequence taken to be the mean generation of consecutive neighbors of X ið1Þ , where ð1Þ ð1Þ 1 zð1Þ i ðkÞ ¼ 2 ðxi ðkÞ þ xi ðk  1ÞÞ; i ¼ 1; 2; …; m; k ¼ 2; 3; …; n:

ð8Þ

ð1Þ ð1Þ T where X ð1Þ ðtÞ ¼ ½xð1Þ 1 ðtÞ; x2 ðtÞ; …; xm ðtÞ . Meanwhile the parameter matrices A and B are called developing and grey input coefficients matrix, respectively.

2.3. Basic characteristics of multi-variable MGM(1,m) model ð0Þ ð0Þ Theorem 2.1. Assume that X ð0Þ 1 ; X 2 ; …; X m are the non-negative data sequences, X ð1Þ ði ¼ 1; 2; …; mÞ are the 1-AGO sequences of X ð0Þ i i , ð1Þ and Z i ði ¼ 1; 2; …; mÞ are the mean generation sequences of consecutive neighbors of X ð1Þ i , then

(1) Let sampling time Δt ¼ 1, by the least square method, the parameter matrices A and B of MGM model X ð0Þ ðkÞ þ AZ ð1Þ ðkÞ ¼ B can be obtained as ^ BÞ ^ ¼ ðA; ^ T ¼ ðP T PÞ  1 P T X 0 ; H where 2 a^ 11 6 a^ 6 12 6 ^ ¼6 ⋮ H 6 6 a^ 4 1m b^ 1 2

a^ 21 a^ 22

⋯

⋮

⋱

a^ 2m b^

⋯

2

xð0Þ 1 ð2Þ

6 6 xð0Þ ð3Þ 6 X0 ¼ 6 1 6 ⋮ 4 xð0Þ 1 ðnÞ 2

 z1ð1Þ ð2Þ 6 6  zð1Þ ð3Þ 6 1 P¼6 6 ⋮ 4  z1ð1Þ ðnÞ

ð1Þ ð1Þ Definition 2.2. The matrices X ð0Þ , X ð1Þ and sequences X ð0Þ i , Xi , Zi are defined as in Definition 2.1, then the equation Xm xð0Þ a zð1Þ ðkÞ ¼ bi ; i ¼ 1; 2; …; m; j ¼ 1; 2; …; n ð9Þ i ðkÞ þ l ¼ 1 il l

is called the basic form of the multi-variable MGM(1,m) model (MGM model), while the first-order differential system of m variables 8 ð1Þ dx1 ð1Þ ð1Þ ð1Þ > > > dt þa11 x1 þ a12 x2 þ … þ a1m xm ¼ b1 > > > < dxð1Þ ð1Þ 2 þa21 x1ð1Þ þ a22 xð1Þ 2 þ … þ a2m xm ¼ b2 dt ð10Þ > > ⋮ > > > ð1Þ > : dxm þam1 xð1Þ þ am2 xð1Þ þ … þamm xð1Þ ¼ bm dt

1

2

m

is called the whitenization equation system of the multi-variable MGM(1,m) model. Denote as 2 3 2 3 a11 a12 ⋯ a1m b1 6 7 6 a21 a22 ⋯ a2m 7 6 b2 7 6 7 7 A¼6 ð11Þ 7; B ¼ 6 6 ⋮ 7; ⋮ ⋱ ⋮ 5 4 ⋮ 4 5 am1 am2 ⋯ amm bm Eq. (9) can be rewritten as X ð0Þ ðkÞ þ AZ ð1Þ ðkÞ ¼ B;

ð12Þ

and Eq. (10) can be rewritten as dX ð1Þ ðtÞ þ AX ð1Þ ðtÞ ¼ B; dt

ð13Þ

⋯

⋯

3 a^ m1 a^ m2 7 7 7 ⋮ 7 7; a^ mm 7 5 b^ m

xð0Þ 2 ð2Þ

⋯

xð0Þ 2 ð3Þ ⋮

⋯ ⋱

xð0Þ 2 ðnÞ

ð14Þ

⋯

ð15Þ

xð0Þ m ð2Þ

3

7 7 xð0Þ m ð3Þ 7 7; ⋮ 7 5 xð0Þ m ðnÞ

 zð1Þ 2 ð2Þ

⋯

 zð1Þ m ð2Þ

 zð1Þ 2 ð3Þ

⋯

 zð1Þ m ð3Þ

⋮

⋱

 zð1Þ 2 ðnÞ

ð16Þ

⋯

⋮

 zð1Þ m ðnÞ

1

3

7 17 7 7: ⋮7 5 1

ð17Þ

ð1Þ ð0Þ (2) By setting the initial value x^ i ð1Þ ¼ x^ i ð1Þði ¼ 1; 2; …; mÞ, the time response function of the whitenization differential equað1Þ tion system dXdt ðtÞ þ AX ð1Þ ðtÞ ¼ B is given by

X ð1Þ ðtÞ ¼ e  At ðX ð1Þ ð1Þ  A  1 BÞ þ A  1 B:

ð18Þ

(3) By discretizing Eq. (18), the time response sequence of the MGM model X ð0Þ ðkÞ þ AZ ð1Þ ðkÞ ¼ B is given by 1 ð1Þ ^ ^ þ A^  1 B; ^ k ¼ 2; 3; …; n; ð19Þ X^ ðkÞ ¼ e  Aðk  1Þ ðX ð1Þ ð1Þ  A^ BÞ i P ^  1Þ 1 i A^  Aðk where e ¼ I  i ¼ 1 i! ðk 1Þ and I is the m-order unit matrix. Then the simulative value of matrix X ð1Þ can be obtained from Eq. (19) accordingly. In the end, consider the inverse accumulated generation ð0Þ ð1Þ ð1Þ X^ ðkÞ ¼ X^ ðkÞ  X^ ðk  1Þ; k ¼ 2; 3; …; n; ð0Þ

ð20Þ

then the simulative value of matrix X can be obtained. The multi-variable MGM(1,m) model is an extension and supplement of the uni-variable GM(1,1) model under multidimensional situations. It is not a simple combination of the GM (1,1) models and is different from the GM(1,m) model. In particular, when m ¼ 1, the MGM model degrades into uni-variable GM (1,1) model (GM model). However, the GM model cannot reflect the interactional and promotional relationship among all system variables only by means of unique sequence modeling and forecasting. Particularly when B ¼ 0, the MGM model is a combination of m units of GM(1,m) model. But GM(1,m) model reflects the influence of m  1 units of correlative factor sequences on system

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

feature sequence, and is appropriate for dynamic analysis among the variables. Therefore, the GM(1,m) model is a kind of state model which only reflects the change law of system feature sequence and cannot be used for prediction. From the viewpoint of a system, the MGM model is aimed at reflecting the interactional and promotional relationship among all variables. So it can be considered as not only a state model but also a prediction model. Numerous studies indicate that the MGM model can be more suitable for practical problems in engineering technology and social economy systems subjecting to multi-faceted factors. And the MGM model is much superior to other univariable grey models with respect to its applied range and prediction performance.

3. Multi-variable grey self-memory coupling prediction model—SMGM(1,m) 3.1. Fundamental principles of self-memory principle of dynamic system One of the developments in natural sciences is the study of chaotic phenomena and its theory. It has filled a gap between determinism and random theory, and questions the predictability of things. Chaotic dynamics tells us that even in a simple dynamic system, random behavior is exhibited due to nonlinearity. It implies that a prediction model combining dynamics with random theory is inevitable (Cao, 1993). By introducing the memory concept into physics, the selfmemory principle of dynamic system is proposed on the basis that natural and social phenomena are all irreversible. Historical information should be investigated fully if we want to realize present system and predict its future. Accordingly, the principle emphasizes the relationship between before and after of system status itself, particularly on the systematic evolution law per se. After the memory function, which contains historical information, is introduced into the system's dynamic differential equation, it can be transformed into an appropriate difference-integral equation which is called a self-memorization one by defining the inner product in Hilbert space. Because the systematic selfmemorization equation contains multiple time-point initial fields instead of the single time-point initial field, the weakness of being sensitive to the initial value of the original dynamic differential equation can be overcome. Then through studying systematic inner memorability, the systematic evolutionary trend can be modeled and predicted. The self-memory principle has superior utilization because systematic predictability can be improved by not only combining dynamics calculations and estimating parameters of historical data, but also extracting systematic information from historical data in statistics. 3.2. Coupling modeling steps of multi-variable MGM(1,m) model and self-memory principle Based on the above-mentioned literature analysis, the superior self-memory technique is introduced in this section to support the multi-variable MGM(1,m) model to produce a novel SMGM(1,m) model. Let the original data matrix and the 1-AGO matrix be ð0Þ ð0Þ T ð1Þ ð1Þ T X ð0Þ ¼ ½X ð0Þ ¼ ½X 1ð1Þ ; X ð1Þ 1 ; X 2 ; …; X m  and X 2 ; …; X m  respectively. Then, the step-by-step procedure of a novel SMGM(1,m) model is described as follows. Step 1: Determining the self-memory dynamic equation. If we let dX ð1Þ ðtÞ=dt in the whitenization differential equation system of the MGM model be FðX; tÞ, then FðX; tÞ ¼  AX ð1Þ ðtÞ þ B:

ð21Þ

85

The differential equation dX ð1Þ ðtÞ=dt, which has been determined by Eq. (21), is considered to be the systematic self-memory dynamic equation of the SMGM model: dx ¼ Fðx; λ; tÞ; dt

ð22Þ

where x is a variable, λ is a parameter, t is time interval series, and Fðx; λ; tÞ is the dynamic kernel. Meanwhile, introduce a memory function β ðtÞ, and define an inner product in the Hilbert space: Z ðf ; g Þ ¼

b0 a0

  f ðξÞgðξÞdξ f ; g A L2 :

ð23Þ

Step 2: Deducing the difference-integral equation. Let one time set T ¼ ½t  p ; t  p þ 1 ; …; t  1 ; t 0 ; t, where t  p ; t  p þ 1 ; …; t  1 ; t 0 is historical observation time, t 0 is predicted initial time, t is coming prediction time, the retrospective order of the equation is p and time sampling interval is Δt. After applying the above inner product operation into Eq. (22) and supposing that variables x,β are continuous, differentiable and integrable, the analytic formula of Eq. (22) is obtained as Z t Z t ∂x β ðτ Þ dτ ¼ βðτÞFðx; λ; τÞdτ; ð24Þ ∂τ t p tp that is Z t pþ1 t p

∂x

βðτÞ dτ þ ∂τ

Z Z

t pþ2 t pþ1 t

¼

tp

∂x

βðτÞ dτ þ ⋯ þ ∂τ

Z

t

t0

∂x

βðτÞ dτ ∂τ

βðτÞFðx; λ; τÞdτ:

ð25Þ

For every integral term on the left-hand side of Eq. (25), after integration by parts, applying the median theorem and performing algebra operation, a difference-integral equation is deduced as:

β t xt  β  p x  p 

0 X

xm i ðβ i þ 1  β i Þ 

i ¼ p

Z

t tp

βðτÞFðx; λ; τÞdτ ¼ 0;

ð26Þ

where β t  βðtÞ, xt  xðtÞ, βi  βðt i Þ, xi  xðt i Þ, i ¼  p;  p þ 1; …; 0, and mid-value xm i  xðt m Þ, t i ot m ot i þ 1 . Step 3: Discretizing the self-memory prediction equation. Let x  p  xm  p  1 and β  p  1  0, Eq. (26) can be converted into xt ¼

1

0 X

βt i ¼  p  1

xm i ðβ i þ 1  β i Þ þ

¼ S1 þ S2 ;

1

βt

Z

t t p

βðτÞFðx; λ; τÞdτ ð27Þ

which is called the self-memory equation with the retrospective order p. As the first term S1 in Eq. (27) denotes the relative contributions of historical data at p þ1 times to the value of variable xt , it is defined as the self-memory term. The second term S2 is the total contribution of the function Fðx; λ; tÞ in the retrospective time interval ½t  p ; t 0 , and it is defined as the exogenous effect term. Equation (27) emphasizes serial correlation of the system by itself, i.e., the self-memory characteristic of the system. Therefore, it is the self-memory prediction equation of the system. If integral operation is substituted by summation and differential is transformed into difference in Eq. (27), then the midvalue xm is replaced simply by two values of different times, i namely 1 xm i ¼ ðxi þ 1 þ xi Þ  yi : 2

ð28Þ

By taking equidistance time interval Δt i ¼ t i þ 1  t i ¼ 1, and merging β t and βi together, the self-memory equation of discrete

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X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

form is shown as follows: 1 X

xt ¼

0 X

α i yi þ

i ¼ p1

θi Fðx; λ; iÞ;

ð29Þ

i ¼ p

criteria of testing the average magnitude of the forecast errors. MSE can be used to ascertain the discrete degree of the predicted value and the monitoring value. AME can eliminate the influence of the extreme value in the positive or negative direction and the minus–plus offsetting of the error through absolute value calculation (Lin et al., 2012). The three test criteria are expressed as follows.

where αi ¼ ðβi þ 1  β i Þ=β t , θi ¼ βi =β t . αi and θi are called memory coefficients, and Fðx; λ; tÞ is determined by the dynamic kernel AX ð1Þ ðtÞ þ B of the MGM model. Step 4: Solving the self-memory prediction model. Assume that there are L items of historical data, the memory coefficients αi and θi can be estimated by the least square method. Let 2 3 2 3 y  p  1;1 y  p;1 ⋯ y  1;1 xt1 6y 7 6 xt2 7 6  p  1;2 y  p;2 ⋯ y  1;2 7 6 7 7; Xt ¼ 6 7; Y ¼ 6 6 ⋮ ⋮ ⋱ ⋮ 7 4 ⋮ 5 Lðp þ 1Þ 4 L1 5 y  p  1;L y  p;L ⋯ y  1;L xtL 2 3

^ the residual error sequence, in which ε^ i ðkÞ ¼ xð0Þ i ðkÞ  xi ðkÞ, k ¼ 1; 2; …; n. The Absolute Percentage Error (APE) at time k is denoted

n
xð0Þ ðkÞ  x^ ð0Þ ðkÞ
P
i ð0Þ i
 100%. So, by APE ¼
xi ðkÞ
k¼1 n 1X ð0Þ ðxð0Þ ðkÞ  x^ i ðkÞÞ2 ; ð35Þ MSE ¼ nk¼1 i

6 αp 6 A ¼6 4⋮ ðp þ 1Þ1

AME ¼

αp1

2

α1

7 7 7; 5

Fðx; λ; pÞ1 6 6 Fðx; λ; pÞ2 Γ ¼6 6 ⋮ Lðp þ 1Þ 4 Fðx; λ;  pÞL

Θ

ðp þ 1Þ1

2

3

¼6 4⋮

7; 5

Fðx; λ;  pþ 1Þ1 Fðx; λ;  pþ 1Þ2

⋯ ⋯

⋮

⋱

Fðx; λ; p þ 1ÞL

⋯

3 Fðx; λ; 0Þ1 7 Fðx; λ; 0Þ2 7 7; 7 ⋮ 5 Fðx; λ; 0ÞL

θ p 6 7 6 θ pþ1 7 6 7 θ0

ð0Þ ð0Þ The monitoring data sequence is X ð0Þ i ¼ fxi ð1Þ; xi ð2Þ; …;

xð0Þ i ðnÞg, and its corresponding ð0Þ ð0Þ ð1Þ; x^ i ð2Þ; …; x^ i ðnÞg. Thus ^ i ¼

ε

ð0Þ ð0Þ predicted sequence is X^ i ¼ fx^ i

fε^ i ð1Þ; ε^ i ð2Þ; …; ε^ i ðnÞg is said to be ð0Þ

n

1X
ð0Þ
ð0Þ
x ðkÞ  x^ i ðkÞ
; nk¼1 i



ð0Þ
n
ð0Þ 1X
xi ðkÞ  x^ i ðkÞ
MAPE ¼

 100%: ð0Þ
nk¼1
x ðkÞ

ð36Þ

ð37Þ

i

3.4. Programming procedure of Matlab software ð30Þ

The calculation was performed as mentioned above with the help of Matlab software. The programming procedure for the SMGM(1,m) model is shown in Fig. 1.

then Eq. (29) can be expressed in matrix form as follows: X t ¼ YAþ ΓΘ: Let Z ¼ ½Y; Γ , W ¼ X t ¼ ZW;



A

Θ

 , then Eq. (31) turns into

ð31Þ

ð32Þ

thereby W is obtained by the least square method: W ¼ ðZ T ZÞ  1 Z T X t :

ð33Þ

When the memory coefficients matrix W is obtained, the simulation and prediction of original data matrix X ð0Þ can be carried out. ð1Þ For the simulated and predicted value x^ ðtÞ of the 1-AGO matrix ð0Þ in SMGM model, its inverse accumulated value x^ ðtÞ can be obtained as follows: ð0Þ ð1Þ ð1Þ x^ ðtÞ ¼ x^ ðtÞ  x^ ðt  1Þ;

ð34Þ ð1Þ

where t ¼ 1; 2; …; n and x^ ð0Þ  0. 3.3. Modeling simulation and prediction accuracy test Simulation and prediction accuracy is an important criterion for evaluating prediction models. Accuracy test must be performed to evaluate the rationality and reliability of prediction models before extrapolation and application. Various accuracy test methods to determine whether it is reasonable or not can be used in practical situation. In this study, considering typical grey uncertainty characteristics in the engineering settlement deformation prediction, three popular test criteria such as Mean Squared Error (MSE), Absolute Mean Error (AME) and Mean Absolute Percentage Error (MAPE) have been used to compare the accuracy of different prediction models (Zhao et al., 2012; Lin et al., 2012; Wang et al., 2014). MAPE is a generally accepted criterion for testing prediction accuracy, as it can objectively provide the relevant difference of the monitoring value and the forecast value. MSE and AME are two

4. Case study Engineering settlement deformation is a complicated system changing process. We should take full advantage of the relevant information of multiple monitoring points rather than carry out local research only on a single point. In practice, settlement deformation of each monitoring point is not isolated because that one monitoring point could be influenced by other points, and vice versa. Therefore, we should uniformly describe the overall development trend and individual change rules of settlement deformation from the viewpoint of settlement deformation monitoring system. Also, because of uncertainty factors such as loss of effective monitoring data, the monitoring system of settlement deformation presents typical grey system characteristics of small sample size and poor information. In this section, subgrade settlement and foundation pit deformation are adopted as study objects. The corresponding SMGM(1,m) models will be compared with the traditional GM(1,1) model, the MGM(1,m) model and the optimized MGM(1,m) model (OMGM(1,m)). Three popular test criteria, i.e., MSE, AME and MAPE, are adopted to check the simulation and prediction accuracy of different prediction models. The practicability and effectiveness of the proposed SMGM(1,m) model and its superiority to interactional multi-variable system prediction problems could be confirmed. 4.1. Case study of subgrade settlement prediction Subgrade settlement is one important indicator affecting road safety because the major hidden danger could result in road traffic accidents. So subgrade settlement prediction is one of the major research topics in the field of geotechnical engineering. For example, a certain section of Beijing-Harbin (G102 line) freeway is an important arterial highway. Through arranging three

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

87

START

Obtain the engineering system raw data

F

Pass through correlation analysis? T Generate the 1-AGO matrix of original data matrix Establish the grey different equation system of MGM(1,m) model

Least square estimate Calculate the parameter matrices A , B Dynamic kernel F ( x, λ, t ) Update new raw data by measuring

Deduce the difference-integral equation Discretize the self-memory prediction equation

Retrospective order p

Calculate the memory coefficients matrix W

Least square estimate

Calculate the simulated values

F

Pass through accuracy check ? Engineering settlement deformation prediction

T STOP

Fig. 1. Programming procedure for the SMGM(1,m) model.

monitoring points (Points A, B and C) at certain roadbed sections, the method of single point extensometer was employed to monitor its subgrade settlement. The original monitoring data sequences of three monitoring points at the initial stage are selected for modeling analysis. Then the SMGM(1,3) model is established to predict the subgrade settlement values, and compared with the traditional GM(1,1) model (Wu et al., 2013) and the traditional MGM(1,3) model (Zhai and Sheng, 1997). And the accumulated settlement value sequences of 10 periods are chosen as the original monitoring data, where 15 days is regarded as one period. The monitoring data of the first eight periods are taken as the modeling samples, and the data of the latter two periods are chosen as testing samples for prediction test. Meanwhile, the accuracy check of simulation and prediction is performed to evaluate the reliability of prediction model before predicting the overall developing trend of subgrade settlement. Table 1 lists three groups of accumulated subgrade settlement data at different monitoring points (Liu et al., 2013a). Step 1: Performing the correlation analysis. At first, the grey correlation analysis is adopted to verify the correlation degree of three monitoring points at the same roadbed section. Table 2 lists the grey correlation degree between every

Table 1 The accumulated subgrade settlement data of monitoring points A, B and C (unit: mm). Period Number of days

Accumulated subgrade settlement value Monitoring point Monitoring point Monitoring A B point C

1 2 3 4 5 6 7 8 9 10

35 50 65 80 95 110 125 140 155 170

13.42 15.38 22.18 23.30 24.55 25.41 26.91 28.02 28.64 28.44

9.89 12.20 16.27 17.66 19.07 20.85 21.91 23.40 23.77 24.12

12.03 15.60 19.57 20.80 22.03 23.38 24.60 25.79 26.36 27.16

monitoring point obtained by Eqs. (2) and (3). It can be seen clearly from Table 2 that the grey correlation degree between two random monitoring points is far greater than 0.5. It shows that

88

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

Table 2 The grey correlation degrees of subgrade settlement data between different monitoring points. Monitoring point

Grey correlation degree

Point A Point B Point C

Table 3 The model values and APE of the SMGM(1,3) model and other grey prediction models at monitoring point A (unit: mm). Number Actual value

Point A

Point B

Point C

1.0000 0.5911 0.7896

0.5911 1.0000 0.6267

0.7355 0.5547 1.0000

there is coupling relationship and certain correlation exists at every monitoring point, so the multi-variable MGM(1,3) model could be adopted to model and predict the settlement values of three monitoring points. Step 2: Determining the self-memory dynamic equation. Based on the corresponding subgrade settlement monitoring

1 2 3 4 5 6 7 8 9 10

13.42 15.38 22.18 23.30 24.55 25.41 26.91 28.02 28.64 28.44

GM(1,1) model

MGM(1,3) model

SMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 18.931 20.333 21.839 23.456 25.193 27.059 29.063 31.216 30.298

– 23.088 8.327 6.270 4.456 0.854 0.554 3.722 8.994 6.533

– 16.283 21.182 23.276 24.601 25.715 26.775 27.824 29.057 30.340

– 5.871 4.500 0.103 0.208 1.200 0.502 0.700 1.456 6.681

– – 22.168 23.331 24.625 25.105 27.229 27.913 28.594 29.746

– – 0.055 0.132 0.307 1.201 1.184 0.381 0.161 4.591

ð1Þ

data, the whitenization differential equation system dXdt ðtÞ þ AX ð1Þ ðtÞ ¼ B of the traditional MGM(1,3) model is formulated as follows: 8 ð1Þ dx1 ð1Þ ð1Þ ð1Þ > > > dt þ4:0920x1 þ 2:8789x2  7:0870x3 ¼ 7:2671 > < ð1Þ dx2 ð1Þ ð1Þ : ð38Þ þ1:7787xð1Þ 1 þ 1:5361x2  3:3707x3 ¼ 7:4767 dt > > ð1Þ > dx > ð1Þ ð1Þ ð1Þ : 3 þ1:9224x þ 1:5285x  3:5145x ¼ 10:9312 1 2 3 dt Consequently, the differential equation dX ð1Þ ðtÞ=dt ¼  AX ð1Þ ðtÞ þ B, which has been determined by Eq. (38), is considered as the dynamic kernel Fðx; λ; tÞ of the self-memory equation of the SMGM (1,3) model. Step 3: Deducing the self-memory prediction equation system. The self-memorization equation system of the SMGM(1,3) model can be established for subgrade settlement forecasting. The value of retrospective order is determined as p ¼ 1 by trial calculation method under the principle of minimum error of fitting root-mean-square. After applying the inner product operation (23) into dx=dt ¼ Fðx; λ; tÞ, we obtained the analytic formula as Rt Rt R t0 ∂x ∂x t  1 β ðτ Þ∂τ dτ þ t 0 β ðτ Þ∂τ dτ ¼ t  1 β ðτ ÞFðx; λ; τ Þdτ . According to the modeling steps as mentioned above, a difference-integral equ0 Rt P ation is deduced as β t xt  β  1 x  1  xm i ðβ i þ 1  β i Þ  t  1

Table 4 The model values and APE of the SMGM(1,3) model and other grey prediction models at monitoring point B (unit: mm). Number Actual value

1 2 3 4 5 6 7 8 9 10

9.89 12.20 16.27 17.66 19.07 20.85 21.91 23.40 23.77 24.12

GM(1,1) model

MGM(1,3) model

SMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 14.171 15.484 16.920 18.488 20.202 22.075 24.121 26.357 26.624

– 16.156 4.831 4.190 3.052 3.108 0.753 3.081 10.883 10.381

– 12.625 15.789 17.713 19.256 20.662 22.001 23.300 24.654 25.131

– 3.484 2.956 0.300 0.975 0.902 0.415 0.427 3.719 4.192

– – 16.258 17.737 19.165 20.373 22.222 23.520 24.527 24.842

– – 0.075 0.434 0.496 2.288 1.423 0.512 3.186 2.993

Table 5 The model values and APE of the SMGM(1,3) model and other grey prediction models at monitoring point C (unit: mm).

i ¼ 1

βðτÞFðx; λ; τÞdτ ¼ 0, and then the self-memory equation is obtained as xt ¼ β1 t

0 P i ¼ 2

1 xm i ðβ i þ 1  β i Þ þ β

t

Rt

t1

βðτÞFðx; λ; τÞdτ. After integral

operation is substituted by summation and differential is transformed into difference, the discrete form of self-memory equation system for subgrade settlement forecasting can be expressed as 8 1 0 X X > > > α1i y1i þ θ1i F 1 ðx; λ; iÞ > x1t ¼ > > > i ¼ 2 i ¼ 1 > > > > 1 0 < X X α2i y2i þ θ2i F 2 ðx; λ; iÞ : x2t ¼ ð39Þ > > i ¼ 2 i ¼ 1 > > > > 1 0 X X > > > > α3i y3i þ θ3i F 3 ðx; λ; iÞ x ¼ > : 3t i ¼ 2

Number Actual value

1 2 3 4 5 6 7 8 9 10

12.03 15.60 19.57 20.80 22.03 23.38 24.60 25.79 26.36 27.16

GM(1,1) model

MGM(1,3) model

SMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 17.443 18.696 20.039 21.479 23.022 24.676 26.449 28.349 29.087

– 11.814 4.466 3.659 2.501 1.531 0.309 2.555 7.546 7.095

– 16.044 19.085 20.810 22.153 23.378 24.558 25.719 26.954 27.445

– 2.846 2.478 0.048 0.558 0.009 0.171 0.275 2.253 1.049

– – 19.564 20.836 22.107 23.084 24.771 25.886 26.813 27.412

– – 0.032 0.171 0.351 1.266 0.694 0.374 1.720 0.927

i ¼ 1

With the help of the least square estimation, the memory coefficients matrix can be obtained as 2 3

α1;  2 6 6 α1;  1 W ¼ ½W 1 ; W 2 ; W 3  ¼ 6 6 θ1;  1 4 θ1;0

α2;  2 α2;  1 θ2;  1 θ2;0

α3;  2 α3;  1 7 7 7 θ3;  1 7 5 θ3;0

2

 0:0520

6 1:0468 6 ¼6 4 0:2141

1:2761

0:0890

0:0260

3

0:3616

0:9712 7 7 7: 0:2931 5

1:2769

1:2702

0:9067

Step 4: Solving the self-memory prediction model.

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

After the memory coefficients matrix W has been obtained, the simulation and prediction of original subgrade settlement data matrix X ð0Þ can be carried out according to Eq. (39). Through calculations, the model values and their corresponding APE of three compared models, SMGM(1,3) model, GM(1,1) model and MGM(1,3) model, are presented in Tables 3–5 respectively. Among the three tables, there is no simulated value of the first time-point as a result of modeling mechanism in the GM(1,1) model and MGM (1,3) model. Similarly, there are no simulated values of the first two time-points owing to the retrospective order p ¼ 1 in SMGM (1,3) model. Step 5: Testing the modeling simulation and prediction accuracy. The accuracy test criteria values (MSE, AME and MAPE) of different subgrade settlement prediction models are listed in Table 6. From the viewpoint of error testing, the multi-variable MGM(1,3) model and SMGM(1,3) model always show lower error values than the uni-variable GM(1,1) model. It is shown that the multi-point prediction models can take the relationship among variables into account, and are able to adequately reflect the integral evolution laws of subgrade settlement system. The selfmemory technique helped SMGM(1,3) model to further reduce the modeling errors compared with the traditional MGM(1,3) model. Meanwhile, the SMGM(1,3) model has passed the modeling simulation and prediction accuracy test, and the single-step and two-step rolling prediction precisions are also generally superior than that of the other two grey models. In summary, the SMGM Table 6 Accuracy check of simulated values for different subgrade settlement prediction models. Monitoring point

Model

MSE

AME

MAPE (%)

Point A

GM(1,1) MGM(1,3) SMGM(1,3)

2.930 0.281 0.035

1.337 0.373 0.142

6.753 1.869 0.543

Point B

GM(1,1) MGM(1,3) SMGM(1,3)

0.908 0.072 0.059

0.802 0.218 0.182

5.024 1.351 0.871

Point C

GM(1,1) MGM(1,3) SMGM(1,3)

0.802 0.065 0.022

0.732 0.168 0.114

3.834 0.912 0.481

Subgrade system

GM(1,1) MGM(1,3) SMGM(1,3)

1.547 0.139 0.039

0.957 0.253 0.146

5.204 1.377 0.632

(1,3) model markedly promoted the predictive performance compared with other grey prediction models. In addition, Figs. 2–4 illustrate the fitting results of the simulative curves obtained by the three compared models with the monitoring curves at different monitoring points, and their corresponding comparison results of APE distribution. As can be seen from the given figures, the proposed SMGM(1,3) model possesses more stable and ideal simulation and prediction effects, and the single-point simulation and prediction errors are significantly reduced and stable. From the comparative analysis, the proposed SMGM(1,3) model can better catch the tendency of integral development and individual variation of original data, and is a reliable and stable prediction model for predicting the future evolutionary trend of subgrade settlement system. 4.2. Case study of foundation pit deformation prediction The retaining structure deformation in deep foundation pit engineering is one major factor leading to foundation pit engineering accidents. According to the measured information, predicting the new situation that may arise at the next construction stage can provide reliable information for optimal design and reasonable construction. For example, the Xiongao subway engineering of subway line 10 in Beijing possesses the characteristics of complicated geological conditions and deep excavation of foundation pit. The foundation pit deformation prediction should be conducted in order to ensure the security of its building envelope and nearby buildings. According to the real-time monitoring status of previous deep foundation pit construction, the preliminary measured deformation data of three neighboring monitoring points at the same fender post are selected. Through comparing and filtering, three groups of representative original data sequences are obtained, truly reflecting the foundation pit deformation trend. Then the SMGM(1,3) model is established to predict the foundation pit deformation values, and compared with the MGM(1,3) model (Zhai and Sheng, 1997) and the OMGM(1,3) model (Xiong et al., 2011a). And the accumulated displacement value sequences of nine periods are chosen as the original monitoring data, with four days being regarded as one period. The monitoring data of the first seven periods are taken as the modeling samples, and the data of the latter two periods are chosen as the testing samples for prediction test. Table 7 lists three groups of accumulated foundation pit deformation data at different monitoring points (Xiong et al., 2011a). Step 1: Performing the correlation analysis.

Comparison of simulative curves at point A

GM(1,1)

30

8

25

6

20

Actual GM(1,1)

15

MGM(1,3) SMGM(1,3)

10

Comparison of APE at point A

10

35 50 65 80 95 110 125 140 155 170 Day

APE / %

Settlement value / mm

35

89

MGM(1,3) SMGM(1,3)

4 2 0

65

80

95 110 125 140 155 170 Day

Fig. 2. Comparison among monitoring and prediction curves of different MGM models at monitoring point A.

90

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

Comparison of simulative curves at point B

GM(1,1)

10

25

MGM(1,3) SMGM(1,3)

8

20 15

Actual GM(1,1)

10 5

Comparison of APE at point B

12

APE / %

Settlement value / mm

30

MGM(1,3)

6 4 2

SMGM(1,3)

0

35 50 65 80 95 110 125 140 155 170 Day

65

80

95 110 125 140 155 170 Day

Fig. 3. Comparison among monitoring and prediction curves of different MGM models at monitoring point B.

30

Comparison of simulative curves at point C

Comparison of APE at point C

10

20 Actual GM(1,1)

15

10

MGM(1,3) SMGM(1,3)

8

25

MGM(1,3) SMGM(1,3) 35 50 65 80 95 110 125 140 155 170 Day

APE / %

Settlement value / mm

GM(1,1)

6 4 2 0

65

80

95 110 125 140 155 170 Day

Fig. 4. Comparison among monitoring and prediction curves of different MGM models at monitoring point C.

Table 7 The original foundation pit deformation data of monitoring points A, B and C (unit: mm). Period Number of days

Monitoring point

Foundation pit deformation value Monitoring point Monitoring point Monitoring A B point C

1 2 3 4 5 6 7 8 9

4 8 12 16 20 24 28 32 36

8.48 12.77 15.10 17.87 19.66 22.30 24.32 26.10 28.90

9.29 13.67 16.23 19.00 20.84 23.33 25.39 27.22 29.35

Table 8 The grey correlation degree of foundation pit deformation data between different monitoring points.

10.07 14.52 17.28 20.05 21.84 24.28 26.34 28.15 30.40

Similarly, the grey correlation analysis is adopted to verify the correlation degree of three neighboring monitoring points at the same fender post. The grey correlation degrees between every monitoring point, as listed in Table 8 below, show clearly that there is coupling relationship and certain correlation exists at every monitoring point. Consequently, the multi-variable MGM(1,3) model could be adopted to predict the deformation value of three monitoring points.

Point A Point B Point C

Grey correlation degree Point A

Point B

Point C

1.0000 0.7113 0.5936

0.6270 1.0000 0.6609

0.5936 0.7435 1.0000

Step 2: Determining the self-memory dynamic equation. Based on the corresponding foundation pit deformation monitorð1Þ ing data, the whitenization differential equation system dXdt ðtÞ þ ð1Þ AX ðtÞ ¼ B of traditional MGM(1,3) model is formulated as follows: 8 ð1Þ dx1 ð1Þ ð1Þ ð1Þ > > > dt  6:0461x1 þ 14:7183x2  8:6881x3 ¼ 9:6664 > < ð1Þ dx2 ð1Þ ð1Þ ð40Þ  9:1062xð1Þ 1 þ 21:2485x2  12:1549x3 ¼ 10:3281 dt > > ð1Þ > > dx : 3  12:1097xð1Þ þ27:5471xð1Þ  15:4514xð1Þ ¼ 11:0004 1 2 3 dt Consequently, the differential equation dX ð1Þ ðtÞ=dt ¼  AX ð1Þ ðtÞ þ B, which has been determined by Eq. (40), is considered as the dynamic kernel Fðx; λ; tÞ of the self-memory equation of the SMGM(1,3) model.

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

Table 9 The model values and APE of the SMGM(1,3) model and other MGM models at monitoring point A (unit: mm). Number Actual value

1 2 3 4 5 6 7 8 9

8.48 12.77 15.10 17.87 19.66 22.30 24.32 26.10 28.90

MGM(1,3) model

OMGM(1,3) model

SMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 8.586 13.983 16.428 18.863 21.186 23.405 25.639 28.084

– 32.764 7.397 7.761 4.052 4.996 3.762 1.765 2.825

– 9.324 14.353 16.410 18.608 20.925 23.337 25.812 28.315

– 26.989 4.946 7.861 5.353 6.166 4.044 1.105 2.026

– – 15.101 17.551 20.399 21.859 24.240 26.453 28.675

– – 0.005 1.786 3.759 1.979 0.329 1.354 0.778

Table 10 The model values and APE of the SMGM(1,3) model and other MGM models at monitoring point B (unit: mm). Number Actual value

1 2 3 4 5 6 7 8 9

9.29 13.67 16.23 19.00 20.84 23.33 25.39 27.22 29.35

OMGM(1,3) model

SMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 9.071 14.973 17.572 20.051 22.332 24.474 26.656 29.126

– 33.642 7.747 7.516 3.785 4.278 3.608 2.071 0.763

– 9.991 15.375 17.510 19.758 22.091 24.482 26.894 29.288

– 26.913 5.269 7.840 5.194 5.310 3.578 1.199 0.213

– – 16.229 18.628 21.705 22.817 25.292 27.438 29.682

– – 0.009 1.959 4.152 2.199 0.386 0.801 1.131

Table 11 The model values and APE of the SMGM(1,3) model and other MGM models at monitoring point C (unit: mm). Number Actual value

1 2 3 4 5 6 7 8 9

10.07 14.52 17.28 20.05 21.84 24.28 26.34 28.15 30.40

MGM(1,3) model

OMGM(1,3) model

Model value

APE (%)

Model value

APE (%)

Model value

APE (%)

– 9.525 15.902 18.607 21.102 23.339 25.424 27.587 30.114

– 34.401 7.974 7.200 3.379 3.878 3.480 2.001 0.942

– 10.599 16.313 18.483 20.748 23.079 25.444 27.805 30.120

– 27.003 5.597 7.814 5.001 4.948 3.404 1.227 0.921

– – 17.278 19.595 22.894 23.661 26.214 28.357 30.650

– – 0.012 2.268 4.824 2.547 0.480 0.737 0.821

SMGM(1,3) model

Step 3: Deducing the self-memory prediction equation system. Similarly, after the value of retrospective order was determined as p ¼ 1, the self-memorization prediction equation system of foundation pit deformation can be expressed as 8 1 0 X X > > > α1i y1i þ θ1i F 1 ðx; λ; iÞ > x1t ¼ > > > i ¼ 2 i ¼ 1 > > > > 1 0 < X X x2t ¼ α2i y2i þ θ2i F 2 ðx; λ; iÞ ; > > i ¼ 2 i ¼ 1 > > > > 1 0 X X > > > α3i y3i þ θ3i F 3 ðx; λ; iÞ x ¼ > > : 3t i ¼ 2

i ¼ 1

Table 12 Accuracy check of simulated values for different foundation pit deformation prediction models. Monitoring point

Model

MSE

AME

MAPE (%)

Point A

MGM(1,3) OMGM(1,3) SMGM(1,3)

3.924 3.088 0.170

1.595 1.511 0.316

10.174 9.277 1.572

Point B

MGM(1,3) OMGM(1,3) SMGM(1,3)

4.538 3.336 0.232

1.665 1.542 0.370

10.096 9.017 1.741

Point C

MGM(1,3) OMGM(1,3) SMGM(1,3)

5.200 3.700 0.343

1.735 1.607 0.451

10.051 8.961 2.026

Foundation pit system

MGM(1,3) OMGM(1,3) SMGM(1,3)

4.554 3.375 0.248

1.665 1.553 0.379

10.107 9.085 1.780

where the memory coefficients matrix is 2

MGM(1,3) model

ð41Þ

91

α1;  2 6 6 α1;  1 W ¼ ½W 1 ; W 2 ; W 3  ¼ 6 6 θ1;  1 4 θ1;0 2

α2;  2 α2;  1 θ2;  1 θ2;0

3

α3;  2 α3;  1 7 7 7 θ3;  1 7 5 θ3;0

0:0412 0:5346

0:9245 7 7 7: 0:5569 5

1:1359

1:1394

1:1271

0:9407

0:0589

3

0:0189 6 0:9612 6 ¼6 4 0:5270

Step 4: Solving the self-memory prediction model. After the memory coefficients matrix W has been introduced into Eq. (41), the model values and their corresponding APE of three compared models, SMGM(1,3) model, MGM(1,3) model and OMGM(1,3) model, have been obtained and are shown in Tables 9– 11, respectively. Similarly, there is no simulated value of the first time-point as a result of modeling mechanism in the MGM(1,3) model and OMGM(1,3) model. And there are no simulated values of the first two time-points owing to the retrospective order p ¼ 1 in SMGM(1,3) model. Step 5: Testing the modeling simulation and prediction accuracy. Three accuracy test criteria values (MSE, AME and MAPE) of different foundation pit deformation prediction models are all listed in Table 12. From the viewpoint of error testing, the SMGM (1,3) model possesses obvious low error values compared with MGM(1,3) model and OMGM(1,3) model. It is shown that the selfmemory technique helped SMGM(1,3) model to further reduce the modeling errors compared with multi-variable MGM(1,3) model. Meanwhile, SMGM(1,3) model has passed the modeling simulation and prediction accuracy test, and the single-step and two-step rolling prediction precision are also generally superior than that of the other two MGM(1,3) models. In summary, SMGM(1,3) model markedly promoted the predictive performance compared with other grey multi-variable prediction models. Moreover, Figs. 5–7 illustrate the fitting results of the simulative curves obtained by the three compared models with the monitoring curves at different monitoring points, and their corresponding comparison results of APE distribution. From the comparative analysis, the proposed SMGM(1,3) model can better follow the tendency of integral development and individual variation of original data, and is a reliable and stable prediction model for predicting the future evolutionary trend of foundation pit deformation system.

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X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

30

Comparison of simulative curves at point A

Comparison of APE at point A

10 8

20

6

15

Actual MGM(1,3)

10 5

APE / %

Deformation value / mm

MGM(1,3) 25

SMGM(1,3) 8

4 2

OMGM(1,3) 4

OMGM(1,3) SMGM(1,3)

0

12 16 20 24 28 32 36 Day

12

16

20

24 Day

28

32

36

Fig. 5. Comparison among monitoring and prediction curves of different MGM models at monitoring point A.

30

Comparison of simulative curves at point B

Comparison of APE at point B

10 8

20

6

15

Actual MGM(1,3)

10 5

APE / %

Deformation value / mm

MGM(1,3) 25

SMGM(1,3) 8

4 2

OMGM(1,3) 4

OMGM(1,3) SMGM(1,3)

0

12 16 20 24 28 32 36 Day

12

16

20

24 Day

28

32

36

Fig. 6. Comparison among monitoring and prediction curves of different MGM models at monitoring point B.

Comparison of simulative curves at point C

MGM(1,3)

30

OMGM(1,3) SMGM(1,3)

8

25 20 Actual

15

MGM(1,3) OMGM(1,3)

10 5

Comparison of APE at point C

10

SMGM(1,3) 4

8

12 16 20 24 28 32 36 Day

APE / %

Deformation value / mm

35

6 4 2 0

12

16

20

24 Day

28

32

36

Fig. 7. Comparison among monitoring and prediction curves of different MGM models at monitoring point C.

5. Conclusion In this paper, forecasting of the engineering settlement deformation with a limited amount of data has been studied. As engineering science problems are subject to many factors, such as correlative multiple variables, limited amount of monitoring data, traditional statistical methods cannot be applied to this kind of problems. The grey multi-variable MGM(1,m) model can exactly

overcome the disadvantage of local analysis in a single-variable model and can consider the interactional relationship among all systematic variables under the circumstances of small sample data. In order to further promote its prediction accuracy, the self-memory principle of dynamic system is introduced into the multi-variable grey prediction model MGM(1,m). Its excellent predictive performance lies in that the system's selfmemorization equation contains multiple-time-point initial field

X. Guo et al. / Engineering Applications of Artificial Intelligence 42 (2015) 82–93

instead of only single-time-point initial field. And it overcomes the weakness of being sensitive to initial values of the conventional MGM(1,m) model. As shown in the case studies of subgrade settlement and foundation pit deformation, the novel coupling prediction model can take full advantage of the systematic multitime historical monitoring data and accurately predict the system's evolutionary trend. To our knowledge, it is the first attempt to employ SMGM(1,m) model in the prediction of engineering settlement deformation, which provides an important starting point. It is worth popularizing and applying to other similar multivariable engineering systems. However, there are still many problems which should be solved in our future work. Firstly, we have not found an ideal algorithm for the optimal retrospective order, so the trial calculation method under the principle of minimum error was used instead. Therefore, certain intelligent optimization algorithms, such as nonlinear programming and particle swarm optimization, could be introduced into the coupling model needs further exploration. The other problem to be solved is how to integrate various kinds of optimization techniques with the self-memory principle for the purpose of further improving the prediction accuracy and stability in engineering systems. In addition, new monitoring data will continually enter into the original engineering systems as a result of the periodical monitoring of settlement deformation. With sample size increasing, cross validation method may be introduced into the SMGM(1,m) model to test and verify its reliability and stability. Then it would need to be decided which kind of proper statistical test should be considered to determine the significance of the results. Acknowledgments The authors are grateful to the editors and the anonymous referees for their helpful and constructive comments and suggestions on this paper. This work was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (No. FP7-PIIF-GA-2013-629051), the National Natural Science Foundation of China (Nos. 71111130211, 71171113, 71363046 and 71401051), the Humanistic and Social Science Youth Foundation of Ministry of Education of China (No. 13YJC790198) and Funding of Nantong Science and Technology Program (Nos. HS2013026 and BK2014030). Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.engappai.2015.03.014. References Cao, H.X., 1993. Self-memory equation for atmosphere motion. Sci. Chin. Chem. (Ser. B) 36 (7), 845–855. Chen, X.D., Xia, J., Xu, Q., 2009. Differential hydrological grey model (DHGM) with self-memory function and its application to flood forecasting. Sci. Chin. Technol. Sci. 39 (2), 341–350. Dai, W.J., Liu, B., Ding, X.L., Huang, D.W., 2013. Modeling dam deformation using independent componentregression method. Trans. Nonferr. Met. Soc. Chin. 23 (7), 2194–2200. Fan, X.H., Zhang, Y., 2003. A novel self-memory grey model. Syst. Eng. Theory Pract. 23 (8), 114–117. Feng, Q.G., Zhou, C.B., Fu, Z.F., Zhang, G.C., 2010. Grey fuzzy variable decisionmaking model of supporting schemes for foundation pit. Rock Soil Mech. 31 (7), 2226–2231.

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