Modeling Method of the Grey GM(1,1) Model with Interval Grey Action Quantity and Its Application

GM(1,1) is a univariate grey prediction model with incomplete structural information, in which the real number form of the simulation or prediction data does not conform to the Nonuniqueness Principle of Grey theoretical solution. In light of the network model of GM(1,1), the connotation of grey action quantity is systematically analyzed and the interval grey number form of grey action quantity is restored under uncertain inﬂuencing factors. A novel GM(1,1) model is then constructed. The new model has the basic characteristics of the grey model under incomplete information. Moreover, it can be fully compatible with the traditional GM(1,1) model. The developed model is employed to the natural gas consumption prediction in China, showing that its predicting rationality is much better than that of the traditional GM(1,1) model. It is worth mentioning that, for the ﬁrst time, the grey property of GM(1,1) has been restored in structure, which is of signiﬁcance for both academia and industry.


Introduction
In 1982, Professor Deng proposed the GM(1,1) model [1] with predictive function based on cybernetics. GM(1,1) is a single-variable grey prediction model with a first-order difference equation [2]. Its greatest feature is that GM(1,1) has only a dependent variable but no independent variables [3,4]. Grey theory holds that the development and evolution of a system are influenced by many uncertain external environments and internal factors (Grey causes) [5]. Under such circumstances, it is difficult to establish a definite functional relationship between dependent variables and independent variables to analyze and predict the future development trend of the system [6,7]. However, under the influence and restriction of many factors, the operation results of the system are determined (White results) [8]. In other words, the results of system operation are the final manifestation of the system under the influence of many factors, which can comprehensively reflect the evolution trend and development law of the system under the combined action of these factors [9,10].
GM(1,1) has many advantages [5,11], such as small amount of data needed, simple modeling process, and easy to learn and use. It has been widely used to solve various prediction problems in production and life [12]. With the deepening of application, the theoretical system of GM(1,1) has been enriched and improved, and a lot of research results have been produced. Generally speaking, these achievements mainly include the following four aspects: (a) Optimization of GM(1,1) parameters: such as initial condition optimization [13,14], background value optimization [15,16], and accumulation order optimization [17][18][19] (b) Optimization of GM(1,1) structure: realizing the optimization of model structure from the single exponential form to intelligent variable structure [20][21][22] (c) Extension of GM(1,1) modeling object: to achieve the expansion of modeling objects from real data to grey uncertain data [23][24][25]

Essence and Connotation of Grey Action Quantity
In the univariate grey system, system characteristic variables describe the evolution law of the system, which is the result of the interaction of many complex external factors. ey are all real numbers. e influencing factors of system development are "cause." e result of change embodied in the system is "result." In cybernetics, the former is called input, and the latter is called output. In a single-variable grey system, because the independent variables are unknown, the comprehensive effect of many uncertain and complex factors on the development of the system is expressed by parameter "b." erefore, parameter "b" is called the grey action quantity and represents all grey uncertainty information (Grey Information Coverage) [37].
In Figure 1, the input variable "b" represents all the uncertain factors (Grey factors) affecting the system development and the output variable x (0) (k) is the characteristic variable (White result) of the system. x (0) (k) adjusts the size of parameter "b" by AGO (Accumulation Generation Operator, weakening randomness) and MEAN (MEAN generation of consecutive neighbors sequence, improving smoothness). e main purpose of AGO [35] and MEAN [35] is to weaken the influence of extreme values in raw data on input variable "b." In Figure 1, the feedback coefficient "a" is called the development coefficient and its size and symbols reflect the development trend of x (0) (k).
According to the relationship between input, output, and feedback of the system in Figure 1 (1) (k) � b can be obtained, which is the basic form of the classical GM(1,1) model. e parameters "a" and "b" are estimated by the least square method, which are all real numbers. Because grey action quantity "b" represents the influence of all external factors on the development trend of the system, it is essentially uncertain (Grey factors), and its form should be grey number. However, in the modeling process of the GM(1,1), the grey attribute of "b" is not taken into account which is estimated and modeled with a real number. is obviously does not agree with the actual meaning of "b," which leads to the poor reliability of the prediction results of the GM(1,1) model. e GM(1,1) model is a grey model with incomplete structural information. e uncertainty and complexity of the influencing factors are caused by incomplete structural information. However, the simulation and prediction results of the current GM(1,1) model are determined as real numbers, which is totally inconsistent with the nonuniqueness principle of the grey theory solution. erefore, it is necessary to restore the "grey" uncertainty characteristics of grey action quantity "b" and build a new GM(1,1) model on this basis.

New GM(1,1) Model
In this section, the interval grey number form of grey action quantity "b" will be restored under the uncertainty of influencing factors. On this basis, a new GM(1,1) model is constructed. Because the grey action quantity "b" is an interval grey number, the simulation and prediction results of GM(1,1) are also interval grey numbers, which satisfies the nonuniqueness of GM(1,1) prediction results under uncertain conditions.

Interval Grey Number Form of Grey Action Quantity.
In cybernetics, there is a corresponding relationship between each input and output. Grey action quantity covers all unascertained information and has different sizes at different time points (Figure 2). Usually, b 2 , b 3 , . . . , b n are not equal, that is, b 2 ≠ b 3 ≠ · · · ≠ b n . According to eorem 1, the parameters a � (a, b) T are estimated by the least square method under the condition of minimizing the sum of squares of simulation errors of x (0) (k), k � 2, 3, . . . , n. In other words, the parameters "b" in eorem 2 is an approximate value, which is used to represent all the grey action quantities b 2 , b 3 , . . ., and b n of each input. en, the information difference between grey action quantities is completely ignored. erefore, the simulated and predicted data based on parameter "b" in eorem 2 are only an approximate solution. It can be seen that the traditional GM(1,1) model violates the nonuniqueness principle of the solution of grey theory under incomplete information.
In this section, according to the relationship between each input and output of the system, the uncertain information contained in grey action quantity is fully excavated, and the interval grey number form of grey action quantity is restored. On this basis, a new GM(1,1) model is constructed.
According to equation (3), the grey action quantity with different values of k(k � 2, 3, . . . , n) can be calculated, as follows: en, we call Bs � b 2 , b 3 , . . . , b n is the sequence of grey action quantity of GM(1,1). e maximum value b max and minimum value b min of Bs � b 2 , b 3 , . . . , b n can be obtained, as follows: After this, grey action quantity of GM(1,1) can be expressed as the interval grey number form, that is, According to equation (3), the grey action quantity b k is positively correlated with x (0) (k). at is, the bigger the b k is, the bigger the x (0) (k) is. e parameter "b" in GM(1,1) is estimated by the least square method, which is a compromise value between b min and b max .
On the other hand, under the existing conditions, the maximum possible value of interval grey number ⊗ b is neither b min or not b max , but "b." e parameter "b" is the real number most likely to represent the whitening value of interval grey number According to the definition of probability function [35], can be expressed as in Figure 3.

New GM(1,1) Model with Interval Grey Action Quantity
Definition 3. Let X (0) , X (1) , Z (1) , and a be the same as in Definition 1 and eorem 1. en, P � (a, ⊗ b ) T is called the sequence of grey parameters, and a is named as the Complexity is called the interval grey action quantity. (1) , and P be the same as in Definitions 1 and 3; then, is called the GM(1,1) model in which grey action quantity is the interval grey number ⊗ b , GM(1, 1, ⊗ b ) for short. And is called the whitinization (or image) equation of (1) , and P be the same as in Definitions 1 and 3; then, (ii) e time response sequence of (dx (1) (iii) e restored values can be given by According to eorem 2, when the grey action quantity of GM(1,1) is expanded from real number b to interval grey number ⊗ b , the GM (1,1) model evolves into the new GM(1, 1, ⊗ b ) model, and the simulation or predicted results of GM(1, 1, ⊗ b ) have the following characteristics: (1) e simulated or predicted result of GM(1, 1, ⊗ b ) is an interval grey number ⊗(k) (2) e interval grey number ⊗(k) has the definite lower e possibility function of the interval grey number ⊗(k) is a triangle, and its maximum possible value mid (k) e schematic diagram of the interval grey number ⊗(k) and its probability function is shown in Figure 4.
It can be seen that when the grey action quantity "b" is restored to an interval grey number ⊗ b ∈ [b min , b max ], the simulation and prediction data of the GM(1, 1, ⊗ b ) model are also interval grey numbers. In the case of uncertain system Grey factors White result

Grey factors
White result

Model Application and Rationality Analysis
With the increasing demand for natural gas in China's civil and industrial sectors, China has surpassed Japan to become the world's largest importer of natural gas and also the world's most heavily dependent importer of natural gas. In 2018 alone, China imported 125.4 billion cubic meters of natural gas, a growth rate of 31.7%. Under the background of the international trade rule of "take or pay" of natural gas and the rapid increase of China's demand for natural gas, the stable and orderly supply of natural gas has become an important factor threatening China's energy security. According to China's Statistical Yearbook (data.stats.gov.cn/easyquery.htm?cn�C01), China's total natural gas consumption (ten thousand tons of standard coal) in 2009-2018 is shown in Table 1.
In order to test the comprehensive performance of the GM(1, 1, ⊗ b ) model, it is necessary to test the simulation and prediction results of the model at the same time. In this paper, the first seven data in Table 1 are used as the raw data to build the GM(1, 1, ⊗ b ) model and the last three data are used as the reserved data to test the prediction performance of the GM(1, 1, ⊗ b ) model. en, the modeling data X (0) is as follows: Step 1. Generating new sequences X (1) and Z (1) : According to Definition 1, X (1) and Z (1) are be obtained, as follows: Step 3. Constructing the interval grey action quantity According to Definition 1 and the development coefficient a, the known data x (0) (k) and z (1) (k), (k � 2, 3, . . . , 7), the grey action quantity b k at time point k can be computed, as follows: en, So, the interval grey action quantity ⊗ b ∈ [b min , b max ] is as follows: and the possibility function of ⊗ b ∈ [b min , b max ] is shown in Figure 5. e relationship between the grey action quantity at different time points and the grey action quantity b of the traditional GM (1,1) model is shown in Figure 6.
According to Figure 6, we can see that the grey action quantity b of the traditional GM(1,1) model is a compromise value and the size of b is estimated under the condition of minimizing the sum of squares of residual errors of the simulated data. erefore, the process conceals the difference of grey action quantity at different points and loses some known information, which is the main reason why the simulation and prediction results of the traditional GM(1,1) model are unstable.

(22)
Similarly, when k � 8, 9, 10, the predicted data x (0) min (k), , and x (0) mid (k) can be computed, as follows: The value grey action quantity at different time points Step 5. Analyzing the rationality of simulation and prediction data: Based on the above calculation results, the original data and various simulation and prediction data curves are drawn, as shown in Figure 7.
According to Figure 7, before analyzing the rationality of the proposed GM(1, 1, ⊗ b ) model in this paper, we first analyze the irrationality of the traditional GM(1,1) model: (a) e overall trend of China's total natural gas consumption is increasing year by year, but it is not balanced, such as the rapid growth in 2012-2014 and the slowdown in 2014-2015. However, the traditional GM(1,1) model is an exponential model with a constant growth rate, so it is difficult for the GM(1,1) model to achieve unbiased simulation of China's total natural gas consumption. It can be found from Figure 7 that there are obvious deviations between curves ① and ②.
(b) In the traditional GM(1,1) model, the grey action quantity b represents the influence of all external factors on the development trend of the system. It is essentially uncertain, and its form should be grey number. However, in the modeling process of GM(1,1), the size of b is estimated by the least squares method, which is a real number. is completely ignores the uncertainty characteristics of grey action quantity and leads to the poor reliability of the simulation and prediction results of the traditional GM(1,1) model (see curves ② and ⑤).
(c) e GM(1,1) model is a grey model with incomplete structural information which mainly reflects in the uncertainty and complexity of the influencing factors. According to the "Nonuniqueness Principle" of Grey theory, solutions with incomplete and uncertain information show nonuniqueness. erefore, the simulation and prediction results of GM(1,1) should be nonunique. However, the GM(1,1) model is a time sequence prediction model with deterministic structure, and its simulation and prediction results are unique (see curves ② and ⑤), which does  1, ⊗ b ) is an interval grey number (see curves ⑥ and ⑦), which enables the decision maker to clearly understand the future change range of the research object. However, the prediction result of GM(1,1) is a determined real number (see curve ⑤), which usually has some errors; it leads decision makers to question its reliability. In this case, a certain interval is often more valuable than an uncertain real number.

Conclusions
e single variable grey prediction model represented by GM(1,1) simply uses a real number (grey action quantity) "b" to express the comprehensive effect of many uncertain and complex factors on the system development because the factors affecting the system (independent variables) are unknown. In other words, grey action quantity "b" represents the influence of all external factors on the system development trend. Hence, the parameter "b" is essentially uncertain and should be in the form of grey number. However, in the traditional GM(1,1) modeling process, the grey attribute of "b" is not taken into account, which is estimated and modeled according to the real number, which is obviously inconsistent with the actual meaning of "b".
On the other hand, the GM(1,1) model is a grey model with incomplete structural information (the absence of independent variables). According to the "Nonuniqueness Principle" of Grey theory, the solution with incomplete and uncertain information is not unique. erefore, the simulation and prediction results of GM(1,1) should be nonunique. However, the current GM(1,1) model is a time sequence prediction model with deterministic structure, so its simulation and prediction results are unique, which obviously violates the "Nonuniqueness Principle" of Grey theory.
Starting from the origin of the grey prediction model, this paper analyses the defects of the traditional GM(1,1) model. en, according to the Nonuniqueness Principle and Minimum Information Principle of Grey theory, the interval grey number form of grey action quantity b is restored and the new GM(1, 1, ⊗ b ) model is put forward. e new GM(1, 1, ⊗ b ) model is applied to simulate and forecast China's natural gas consumption, and the rationality of the simulation and prediction results of GM(1, 1, ⊗ b ) and GM(1,1) is analyzed. e results show that the prediction results of GM(1, 1, ⊗ b ) have more reference values.
Although this paper only extends grey action quantity b from real number to interval grey number ⊗ b ∈ [b min , b max ], it is no exaggeration to say that the proposed GM(1, 1, ⊗ b ) model makes the classical grey prediction model really to have the "grey" attribute. At present, there are many kinds of 8 Complexity grey prediction models, and GM(1,1) is only one of the most primitive grey models. erefore, how to use GM(1, 1, ⊗ b ) as the basis to carry out in-depth research on the "grey" attributes of other grey models, so as to build the new grey prediction model with stronger modeling ability, is the next work of our team.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper