# The Morbidity of Multivariable Grey Model MGM(1,m)

Authors: Haixia Wang; Lingdi Zhao; Mingzhao Hu
Journal: International Journal of Differential Equations
 (2017)

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## Abstract

This paper proposes the morbidity of the multivariable grey prediction MGM(1,m) model. Based on the morbidity of the differential equations, properties of matrix, and Gerschgorin Panel Theorem, we analyze the factors that affect the morbidity of the multivariable grey model and give a criterion to justify the morbidity of MGM(1,m). Finally, an example is presented to illustrate the practicality of our results.


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## Body

## 1. Introduction

In recent decades, grey system theory, as well as fuzzy set theory [1] and rough set theory [2], is one of the most widely used theories to study uncertain problems. The grey system theory which was introduced by Deng [3], characterized by few data and poor information, has been successfully utilized in uncertain problems. On account of their enormous applications in agriculture, economics, management, and engineering, the grey system attracts many scientific research workers and scholars devoted to various aspect of those fields.Grey forecasting models, an important part of grey systems, have been widely adopted to predict practical problems due to their simple calculating process and higher forecasting accuracy [4, 5]. However, some researchers put forward that the tiny changes of the initial data can result in the estimation errors, which is called the morbidity of the grey models. The research on morbidity and stability problems occupies an important part in grey forecasting system. Zheng et al. [6] pointed out that there existed morbidity in grey prediction models and analyzed the reasons in earlier times. Dang et al. [7] showed the possibility of the morbidity problem could only exist in GM(1,1) when the first item of original sequence was unequal to zero while other items were equal to zero approximatively. Wei [8] resolved the morbidity problem for the grey model with the accumulating method based on the condition number theory. Xiao and Li [9] studied the effects of the multiple transformation to the condition number of the non-equigap GM(1,1) model.Except for the research on the morbidity of GM(1,1) model, there are also some studies concentrating on the morbidity of other grey models. Xiao and Guo [10] and Zeng and Xiao [11] researched on the morbidity problem of GM(2,1) which had two characteristic values. Wang et al. [12] summarized the main factor that affected the morbidity of GM(1,1,tα) and suggested that there existed morbidity in some cases. Cui et al. [13, 14] found that there was no morbidity in NGM(1,1,k) and grey Verhulst model; the solution of those models will not make significant drift for the original data series of systems if there exist minor errors in collecting process.Compared to the morbidity of grey models group, there is a little attention on the morbidity of multivariate grey prediction model MGM(1,m). The MGM(1,m) model was proposed by Zhai et al. [15] and has been developed rapidly and caught the attention of many researchers. Zou [16] applied a step by step optimum new information modeling method to build multivariable nonequidistance information grey model. Xiong et al. [17] optimized the background value and set the multiple linear regression model based on MGM in order to eliminate the fluctuations or random errors of the original data. Guo et al. [18] constructed SMGM(1,m) through coupling self-memory principle of dynamic system to MGM; examples showed that it had superior predictive performance over other traditional grey prediction models.Does the possibility of the morbidity in MGM exist? How to identify the morbidity of the multivariable grey model has become an important aspect in the process of constructing the MGM(1,m) model. This paper discusses the possibility of the MGM(1,m) model and the remainder of the paper is organized as follows: Section 2 introduces the morbidity of matrix equations and analyzes the factors that affect the condition number of special matrix. Section 3 provides a criterion to justify the morbidity of MGM(1,m). Section 4 gives an example to illustrate the practicality of our results. Some conclusions are presented in Section 5.


## 2. The Morbidity of Equations

Considering the differential equationAx=b, A is nonsingular matrix, b is the constant variable, and x is the solution of the equation.Definition 1 (see [19]).
IfA or b has a small change and causes a larger change in the solution of the equation Ax=b, the equation is said to be morbidity equation.Definition 2 (see [19]).
Suppose thatA is a square matrix with full rank. The condition number of A is (1)condAv=A-1v·Av,v=1,2,…,∞.If A is a real symmetric matrix, then the condition number of A is (2)condA=λmaxAλminA,where λmax is the maximal eigenvalue of the matrix and λmin is the minimal eigenvalue of matrix. If cond(A)∈(1,10), A is well conditioned. If cond(A)∈[10,100), A is slightly ill-conditioned. If cond(A)∈[100,1000), A is moderately ill-conditioned. If cond(A)∈[1000,∞), A is strongly ill-conditioned.
In the process of parameters identification of multivariable grey model, we usually use the least square method to estimate the parameters, so there exist least square problems in the parameters.
Assuming thatC∈Rm×n, y∈Rm, and C is the parameters matrix of the grey model. If there exists a vector x0∈Rn, making Cx-y2 achieve the minimum of the function, which is (3)Cx0-y2=minx∈RnCx-y2,then x0 is the solution of the linear equation Cx=y, which is the estimated parameter of the grey model.
Suppose thatf(x)=Cx-y2=(Cx-y)T(Cx-y)=xTCTCx-xTDTy-yTDx+yTy. By the extremum condition of the equation, we have (4)dfxdx=2CTCx-2CTy=0.Then we obtain the solution CTCx=CTy, which is also the least square solution of the equation Cx=y.
In the multivariable grey prediction models, the data matrixC is usually the long matrix; it is not easy to solve its condition number. It should be noted that CTC is a real symmetric matrix, the condition number is easy to obtain. Therefore, we often justify the morbidity of the multivariable grey model by the condition number of CTC.


## 3. The Morbidity of MGM(1,m)

### 3.1. Grey MGM(1,m) Model

The multiple variable grey prediction model abbreviated as MGM(1,m) is one of the frequently used grey forecasting models. The MGM(1,m) model constructing process is presented below.Definition 3.
Assume that the data sequence(5)Xj0=xj01,xj02,…,xj0mT,j=1,2,…,mis the original nonnegative data matrix. The data matrix (6)Xj1=xj11,xj12,…,xj1nT,j=1,2,…,mis the first-order accumulated generating matrix of X(0), where (7)xj1k=∑i=1kxj0i.The adjacent neighbour average sequence of X(1) is (8)Zj1=zj11,zj12,…,zj1n,where zj(1)(k)=0.5(xj(1)(k)+xj(1)(k-1)), k=2,3,…,n.The first-order differential equations of the multivariable grey model MGM(1,m) are as follows: (9)dx11dt=α11x11+α12x21+⋯+α1mxm1+β1dx21dt=α21x11+α22x22+⋯+α2mxm1+β2⋮dxm1dt=αm1x11+αm2x21+⋯+αmmxmm+βm.Note that (10)A=αijm×m,β=β1,β2,…,βmT,and (9) can be noted as (11)dX1tdt=AX1t+β.Applying the least square method to the first-order differential equation (12)dX1tdt=AZ1t+β,we obtain the estimated parameters (13)Q^=A^′β^′=PTP-1PTY1,Y2,…,Ym,where(14)Q^=α^11α^21⋯α^m1α^12α^22⋯α^m2⋮⋮⋱⋮α^1mα^2m⋯α^mmβ^1β^2⋯β^m,P=z112z212⋯zm121z113z213⋯zm131⋮⋮⋱⋮⋮z11nz21n⋯zm1n1,and Yj=(xj(0)(2),xj(0)(3),…,xj(0)(n))T,j=1,2,…,m.


### 3.2. The Morbidity of MGM

In this part, we give a criterion to justify the morbidity of MGM(1,m).Lemma 4 (Gerschgorin Panel Theorem).
IfA∈Cn×n and A=(aij), then every eigenvalue of A is contained in the plane, which is (15)λ∈⋃i=1nDi,where Di is the panel centred by aii in the complex plane and (16)Di=z∈C∣z-aii≤∑j=1,j≠inaij,i=1,2,…,n.Theorem 5.
Suppose thatXj(0)(1),Xj(0)(2),…,Xj(0)(n) are data vectors, and Xj(1)(n) is the first-order accumulated generating vector. If every consecutive neighbour zj(1)(k)≥1(j=1,2,…,m), then the multivariable grey model MGM(1,m) is morbidity.Proof.
In the process of estimating the parameters ofA,β, by least square method, we calculate the matrix of PTP, which is (17)PTP=∑k=2nz11k2⋯∑k=2nz11kzm1k∑k=2nz11k∑k=2nz11kz21k⋯∑k=2nz21kzm1k∑k=2nz21k⋮⋱⋮⋮∑k=2nzm1kz11k⋯∑k=2nzm1k2∑k=2nzm1k∑k=2nz11k⋯∑k=2nzm1kn-1.From (18)PTPT=PTP,we know PTP is a symmetric matrix; since P is invertible, we deduce that all the eigenvalues of the matrix PTP are positive real numbers and PTP is positive definite matrix. Therefore, the condition number of matrix PTP can be represented by the maximal eigenvalue and minimal eigenvalue of the matrix.
Setλ1,λ2,…,λn-1 as the eigenvalues of PTP. By Gerschgorin Panel Theorem, we have (19)D1=λ1∈R+∣λ1-∑k=2nz11k2≤∑j=2m∑k=2nz11kzj1k+∑k=2nz11k,D2=λ2∈R+∣λ2-∑k=2nz21k2≤∑j=1,j≠2m∑k=2nz21kzj1k+∑k=2nz21k,⋮Dn-1=λn-1∈R+∣λn-1-n-1≤∑j=1m∑k=2nzj1k.It is easy to see that all the eigenvalues of PTP are contained in the D1∪D2∪⋯∪Dn-1; that is to say, every eigenvalue of PTP is contained in the panel.
If all the adjacent neighbour average sequenceszi(1)(k)≥1 and the chosen sample is the minimal permitted data in grey system, then we conclude that ∑k=2n‍(zi(1)(k))2 is larger than n-1, and the maximal eigenvalue and minimal eigenvalue are contained in different circles, and the centres of circles are far from each other. Therefore, the maximal eigenvalue and minimal eigenvalue are far away from each other on the number line. From the definition of the ill-conditioned matrix, we deduce that the multivariable grey model MGM(1,m) is morbidity. This completes the proof.


## 3.1. Grey MGM(1,m) Model

The multiple variable grey prediction model abbreviated as MGM(1,m) is one of the frequently used grey forecasting models. The MGM(1,m) model constructing process is presented below.Definition 3.
Assume that the data sequence(5)Xj0=xj01,xj02,…,xj0mT,j=1,2,…,mis the original nonnegative data matrix. The data matrix (6)Xj1=xj11,xj12,…,xj1nT,j=1,2,…,mis the first-order accumulated generating matrix of X(0), where (7)xj1k=∑i=1kxj0i.The adjacent neighbour average sequence of X(1) is (8)Zj1=zj11,zj12,…,zj1n,where zj(1)(k)=0.5(xj(1)(k)+xj(1)(k-1)), k=2,3,…,n.The first-order differential equations of the multivariable grey model MGM(1,m) are as follows: (9)dx11dt=α11x11+α12x21+⋯+α1mxm1+β1dx21dt=α21x11+α22x22+⋯+α2mxm1+β2⋮dxm1dt=αm1x11+αm2x21+⋯+αmmxmm+βm.Note that (10)A=αijm×m,β=β1,β2,…,βmT,and (9) can be noted as (11)dX1tdt=AX1t+β.Applying the least square method to the first-order differential equation (12)dX1tdt=AZ1t+β,we obtain the estimated parameters (13)Q^=A^′β^′=PTP-1PTY1,Y2,…,Ym,where(14)Q^=α^11α^21⋯α^m1α^12α^22⋯α^m2⋮⋮⋱⋮α^1mα^2m⋯α^mmβ^1β^2⋯β^m,P=z112z212⋯zm121z113z213⋯zm131⋮⋮⋱⋮⋮z11nz21n⋯zm1n1,and Yj=(xj(0)(2),xj(0)(3),…,xj(0)(n))T,j=1,2,…,m.


## 3.2. The Morbidity of MGM

In this part, we give a criterion to justify the morbidity of MGM(1,m).Lemma 4 (Gerschgorin Panel Theorem).
IfA∈Cn×n and A=(aij), then every eigenvalue of A is contained in the plane, which is (15)λ∈⋃i=1nDi,where Di is the panel centred by aii in the complex plane and (16)Di=z∈C∣z-aii≤∑j=1,j≠inaij,i=1,2,…,n.Theorem 5.
Suppose thatXj(0)(1),Xj(0)(2),…,Xj(0)(n) are data vectors, and Xj(1)(n) is the first-order accumulated generating vector. If every consecutive neighbour zj(1)(k)≥1(j=1,2,…,m), then the multivariable grey model MGM(1,m) is morbidity.Proof.
In the process of estimating the parameters ofA,β, by least square method, we calculate the matrix of PTP, which is (17)PTP=∑k=2nz11k2⋯∑k=2nz11kzm1k∑k=2nz11k∑k=2nz11kz21k⋯∑k=2nz21kzm1k∑k=2nz21k⋮⋱⋮⋮∑k=2nzm1kz11k⋯∑k=2nzm1k2∑k=2nzm1k∑k=2nz11k⋯∑k=2nzm1kn-1.From (18)PTPT=PTP,we know PTP is a symmetric matrix; since P is invertible, we deduce that all the eigenvalues of the matrix PTP are positive real numbers and PTP is positive definite matrix. Therefore, the condition number of matrix PTP can be represented by the maximal eigenvalue and minimal eigenvalue of the matrix.
Setλ1,λ2,…,λn-1 as the eigenvalues of PTP. By Gerschgorin Panel Theorem, we have (19)D1=λ1∈R+∣λ1-∑k=2nz11k2≤∑j=2m∑k=2nz11kzj1k+∑k=2nz11k,D2=λ2∈R+∣λ2-∑k=2nz21k2≤∑j=1,j≠2m∑k=2nz21kzj1k+∑k=2nz21k,⋮Dn-1=λn-1∈R+∣λn-1-n-1≤∑j=1m∑k=2nzj1k.It is easy to see that all the eigenvalues of PTP are contained in the D1∪D2∪⋯∪Dn-1; that is to say, every eigenvalue of PTP is contained in the panel.
If all the adjacent neighbour average sequenceszi(1)(k)≥1 and the chosen sample is the minimal permitted data in grey system, then we conclude that ∑k=2n‍(zi(1)(k))2 is larger than n-1, and the maximal eigenvalue and minimal eigenvalue are contained in different circles, and the centres of circles are far from each other. Therefore, the maximal eigenvalue and minimal eigenvalue are far away from each other on the number line. From the definition of the ill-conditioned matrix, we deduce that the multivariable grey model MGM(1,m) is morbidity. This completes the proof.


## 4. Example

In what follows, we give an example to illustrate the practicality of our results. The data are the price indexes of financial intermediation and real estate in 1981–1984, and data resource is the China statistical yearbook. SetX1(0) and X2(0) as the price index of financial intermediation and price index of the real estate, respectively; the data are shown in Table 1. As usual, we chose 4 group samples which are the minimum permitted data in grey models.Table 1
The data vectors.
k
=
1
k
=
2
k
=
3
k
=
4
X
1
(
0
)
1.102
1.576
1.995
2.608
X
2
(
0
)
1.084
1.182
1.243
1.587We construct MGM(1,2) model to simulate and predict the data vectors. By the definition of P, we obtain (20)P=1.3391.13311.78551.212512.30151.4151,PTP=10.27786.93865.42606.93864.75613.76055.4263.76053.By Theorem 5, there exists morbidity in MGM(1,2) model. In fact, all the eigenvalues of PTP are (21)λ1=0.0011,λ2=0.1249,λ3=17.9079.It is clear that cond(A)≥1000, and there exists morbidity in the model. It proves that our criterion is a useful way to justify the morbidity of MGM(1,m) model.


## 5. Conclusions

This paper discusses the morbidity of the multivariable grey model. From the morbidity of the differential equations, we analyze the factors that affect the morbidity of MGM(1,m) model. By Gerschgorin Panel Theorem and the knowledge of matrix, we give a criterion to justify the morbidity of MGM(1,m). An example is given to illustrate the maneuverability of our results.



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