Optimizing the Grey GM(1,N) Model by Rebuilding All the Back Ground Values

1 Introduction

The prediction models[1] are important parts of the grey system models. Being easy to implement and simply structured, the grey prediction models have been applied to a range of areas, varying in economics[2], industry[3, 4], aerography[5] and so on. But there still exist some defects in the grey prediction models, which appeals a plenty of researches.

Tan’s[6] research revealed that it was the structure of back ground value of the GM(1,1) model effects the precision of the model, and also for the generalized models[7]. Luo, Liu, et al.[8] pointed out that the back ground value was the integration of the 1-AGO sequence, and induced the formulation of the back ground value. Based on the research of Luo, researchers employed a plenty of different numeric integral formulas to structure the back ground value, such as Gauss-Legendre formula[9, 10], Simpson formula[11], Newtown-Cortes[12], and the revised models all performed well than the original GM(1,1). Other researchers such as Li[13], Li[14] and Wang[15], structured a lot of formulations to compute the back ground value and improved the precision of the GM(1,1) model and its generalized models. These researches proved that restructuring of the back ground value was capable to improve the precision of the GM(1,1) model.

As an extension model of GM(1,1), the GM(1,N) is more complex and more useful in the multiple regression problems. It was also used in varied of areas, such as industry[16], economics[17], biology[18] and so on. But the GM(1,N) is still not available in a plenty of situations, in which the error is very large. Several works have been employed to try to solve this problem. As is sharing a lot of similarities to GM(1,1), researchers tried to restructure the back ground value of GM(1,N). Liu[19] structured a formulation of the background value for GM(1,N), which was proved to perform better in predicting the upside-down of the road. Shen, etc.[20] used the Newton-Cotes formula and Gauss-Legendre formula to structure the back ground value, which was proved to be better performing in prediction of the transportation noise. But as its complexity, restructuring the back ground value is still limited, which will be shown in this paper, in some cases, the revised back ground value is still not available to improve the precision of the GM(1,N).

In order to extend the validation of the GM(1,N), Zhai, etc.[21] introduced the MGM(1,N) model, which analysed the grey relationship value of the input data and the output data, and then built N GM(1,1) models. Some improving works have also been done on this model[22–24], which indeed presented higher precision and robustness. However, these methods still failed to revise the GM(1,N) itself, the stationed defects still exist, of which the essence has still not been revealed. And obviously, there has been no way to overcome the defects.

The rest of this paper is organized as follows, Sec.2 introduces the principles of the GM(1,N) model; Sec.3 analyses the essence of the defects of GM(1,N); Sec.4 gives the method to improve the precision and robustness of GM(1,N); Sec.5 presents some simulation tests and some conclusions are drawn in Sec.6.

2 The principles of GM(1, N)

3 The analysis on the source of error for GM(1,N)

Consider the integration of the equation (3), which is

As x1(0)(k)=x1(1)(k)−x1(1)(k), we have

To the Definition 3, equation (8) is equivalent to the original GM(1,N) model, i.e. the equation (1).

It is obvious that the ∫k−1kx1(1)(t)dt has been taken place by z1(1)(k)=0.5(x1(1)(k)+x1(1) (k − 1)) and the ∫k−1kxi(1)(t)dt has been taken place by xi(1) (k). The z1(1)(k) is also called the back ground value. In this paper, we call the zi(1)(k) as the i^th back ground value, thus the z1(1)(k) will be called as the first back ground value in the rest of this paper.

3.1 Errors from the first back ground value

According to the research of Tan[6], the z1(1)(k)=0.5(x1(1)(k)+x1(1)(k − 1)) is the 2-point trapezoid formula, which performs a very little algebra precision in integration. And researches proved that, a higher precision integration formula could overcome this defect. Fig.1 shows the reason why the low precision integration formula will cause a higher error.

Figure 1

The indication of error which caused by the structure of the first back ground value

3.2 Errors from the second to the N^th back ground value

However, no research has paid attention to the right side of the equation (8). The original theorem of GM(1,N) made an assumption “the xi(1) (k) does not fluctuate violently”, and then the ∫k−1kxi(1)(t)dt(i = 2, 3, ⋯, N) was taken by xi(1) (k) itself. But researches have never shown that in what kind of conditions the sequence could be regarded as “not fluctuating violently”, and how close is the ∫k−1kxi(1)(t)dt(i = 2, 3, ⋯, N) to the xi(1) (k). Thus, it is still difficult to explain why the GM(1,N) model could not perform well in a plenty of cases.

Firstly, we consider the geometrical meaning of the i^th back ground value zi(1)(k) = ∫k−1kxi(1)(t)dt (i = 2, 3, ⋯, N). As is shown in Fig.2, the ∫k−1kxi(1)(t)dt is the area of the trapezoid with curve side, of which the vertices are k, k−1, xi(1)(k) and xi(1) (k − 1), and xi(1)(k) is the area of the rectangle, of which the length is xi(1)(k) and width is 1. The shadow area is the redundancy when the ∫k−1kxi(1)(t)dt is taken place by xi(1)(k). This is obviously another reason which cause some significant error to GM(1,N) model.

Figure 2

The indication of error which caused by the structure of the i^th(> 1) back ground value

According to the geometrical analysis, we could draw some results, which reflect the relationship between the error and the original sequence. To prove these results, we need to overview the definition of the convex and concave function and the Hadamard theorem.

Definition 5

Iff(x1)+f(x2)2≥f(x1+x22)orf(x1)+f(x2)2≤f(x1+x22),then f(x) is a convex or concave function, respectively.

Theorem 3

(Hadmard Theorem) If f(x) is a convex or concave function, there must be

f(a+b2)≤1b−a∫abf(x)dx≤f(a)+f(b)2

or

f(a+b2)≥1b−a∫abf(x)dx≥f(a)+f(b)2

respectively.

Remark 4

If the 1-AGO sequence Xi(1)=(xi(1)(1),xi(1)(2),⋯,xi(1)(n)) is a linear sequence, i.e. xi(1) (k) = αk + β (i = 1, 2, ⋯, N, k = 1, 2, ⋯, n), then xi(1)(k)−∫k−1kxi(1)(t)dt = 0.5xi(0)(k).

Proof

As xi(1)(k) = αk + β, thus

xi(1)(k)−∫k−1kxi(1)(t)dt=(αk+β)−{α[k2−(k−1)2]2+β}=0.5α=0.5xi(0)(k).

Remark 5

If the 1-AGO sequence Xi(1)=(xi(1)(1),xi(1)(2),⋯,xi(1)(n)) is a concave sequence, then xi(1)(k)−∫k−1kxi(1)(t)dt<0.5xi(0)(k).

Proof

As Xi(1)=(xi(1)(1),xi(1)(2),⋯,xi(1)(n)) is a strict concave sequence, xi(1)(t) approaches to a concave function. To the Hadamard theorem and differential mean value theorem, we have

∫k−1kxi(1)(t)dt=xi(1)(ξk)>xi(1)(k)+xi(1)(k−1)2, where k−1<ξk<k.

Then

xi(1)(k)−∫k−1kxi(1)(t)dt<xi(1)(k)−xi(1)(k)+xi(1)(k−1)2=0.5xi(0)(k).

Remark 6

If the 1-AGO sequence Xi(1)=(xi(1)(1),xi(1)(2),⋯,xi(1)(n)) is a convex sequence, then xi(1)(k)−∫k−1kxi(1)(t)dt>0.5xi(0)(k).

Proof

In a similar way to the Remark 5, we have

xi(1)(k)−∫k−1kxi(1)(t)dt>xi(1)(k)−xi(1)(k)+xi(1)(k−1)2=0.5xi(0)(k).

It could be easily seen based on the above three results that only when the 1-AGO sequence Xi(1) is a strict concave sequence and the original sequence Xi(0) is very small, the ∫k−1kxi(1)(t)dt could be taken place by xi(1)(k), and then the least squares estimation of the Eq.(1) is close enough to its real value. Thus, the assumption in Theorem 2(b), which says “the xi(1)(k) does not fluctuate violently” could be translated to “ the xi(1)(k) is a concave sequence and not too large”. But if the Xi(1) is linear or convex, the difference between xi(1) (k) and ∫k−1kxi(1)(t)dt could not be ignored anymore, and the original operations of GM(1,N) may cause very large error, and this is exactly the inherent defect of the original GM(1,N) theory. Hence, the assumption in Theorem 2(b) is a too strong condition, and could not be satisfied in a lot of situations. But we want the GM(1,N) to be available in more cases, such as when the xi(1)(k) is concave or linear sequence. To extend the GM(1,N) and overcome its inherent defect original GM(1,N), we need to consider the following theorem.

Theorem 7[1]

If the original sequenceXi(0)is a non-negative quasi-smooth sequence, its 1-AGO sequenceXi(1)satisfies the approximation exponential law.

A so-called “non-negative quasi-smooth sequence” is used most frequently in the grey modeling. Within the above discussion and Theorem 7, we could see that for any non-negative quasi-smooth sequence Xi(0), the original theory of GM(1,N) is not available at all. However, as the 1-AGO sequence Xi(1) satisfies the approximation exponential law, it is reasonable to set xi(1)(t) = B_ie^A_it, and then the i^th background value is ∫k−1kBieAitdt. Now, the i^th background value is an integration of the exponential function. If an appropriate numeric formula is employed to compute its real value, the precision and applicability of GM(1,N) could be improved. The following section will present the details of rebuilding all the background values using different numeric formula.

4 Methods to improve the GM(1,N)

According to the analysis, it is the back ground values that effect the error of the GM(1,N) model. Thus, to improve the GM(1,N) model, we need to choose better ways to compute all the back ground values. Being similar to the first back ground value, the second to N^th back ground values are essentially integrations. Thus, using appropriate numeric integral formula could reduce the error of GM(1,N).

5 Simulation test

The raw data NO.1 to NO.7 is taken from reference [25], and raw data NO.8 is taken from reference [1]. All the raw data is shown in Table 1 to Table 8 below.

In Table 5, the growing speed is computed as x⁽⁰⁾(k)−x⁽⁰⁾(k − 1), and growing rate is x(0)(k)−x(0)(k−1)x(0)(k)×100%, respectively. The raw data of which both the growing speed is larger than 50 and growing rate is larger than 10% is NO.2, 3, 4, 6, 7. According to analysis, the error may be very large using the GM(1,N) model.

Table 10 shows the simulation results, which are the mean percentage error (MPE) of each model. In Table 10, 1st BGV-EF indicates the GM(1,N) model of which the 1st back ground value is computed by the exponential form(Eq.(13)), and the 1st BGV-GL indicates the GM(1,N) model of which the 1st back ground value is computed by the Gauss-Legendre formula (Eq.(16)), respectively. The results shows that the models with revised 1st back ground value perform better than the original model in most cases. However, the original model and the models with revised 1st back ground value all shows very large errors for raw data NO.2, NO.3, NO.6 and NO.7, even the revised models show smaller errors. For raw data NO.2 and NO.7, especially, the models are totally invalid, of which the errors are extremely large. This indicates, the models with revised 1st back ground value doesn’t overcome the inherent defect of GM(1,N) model.

Also in Table 10, the A-BGV-TF, A-BGV-EF and A-BGV-GL indicate the GM(1,N) model with all revised back ground value, which are computed by trapezoid formula (Eq.(9)), exponential form (Eq.(13)) and Gauss-Legendre formula (Eq.(16)), respectively. Compared with the original model and the models with revised 1st back ground value, the models with all revised back ground value perform much better in almost all the raw data. Especially for the raw data that the original model and models with revised 1st back ground value are invalid, the models with all revised back ground values are still valid and perform a very high precision. This indicates that the models with all revised back ground values have a higher precision and robustness than the original model and models with revised 1st back ground value.

However, the simulation results present another attribute of the models with all revised back ground values. For these models, the differences of errors are not significant, which means the new models are not sensitive to the form of the integration formulas.

6 Conclusions

The GM(1,N) model has some inherent defect which may cause significant error or even make the GM(1,N) model invalid. In this paper, we analyze the source of the error of GM(1,N) model, and indicate that it is the form of all the back ground values that effect the error of GM(1,N) model, and also point out it is “dangerous” to use the original GM(1,N)when the reliance sequences are non-negative quasi-smooth sequences.

To overcome this defect, we employ three types of numeric integral formula to compute all the back ground values. The simulation test indicates that the methods of this paper are valid, even in some extreme cases, in which the GM(1,N) is invalid, the revised models still perform well. Thus the methods of this paper enhance the precision and robustness of the GM(1,N) model.