Abstract

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.

1. 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 [17–19](b)Optimization of GM(1,1) structure: realizing the optimization of model structure from the single exponential form to intelligent variable structure [20–22](c)Extension of GM(1,1) modeling object: to achieve the expansion of modeling objects from real data to grey uncertain data [23–25](d)GM(1,1) combined forecasting model: the combination prediction technologies of GM(1,1) and other methods are studied, such as Grey neural network model [26–28], Grey Markov model [29, 30], Grey support vector machine [31, 32], and Grey deep learning [33, 34]

The above research results play an important role in improving the simulation and prediction performance and expanding the application scope of GM(1,1). However, GM(1,1) is a grey model with incomplete structural information (the absence of independent variables). According to the “Nonuniqueness Principle” of grey theory [35], the solution with incomplete and uncertain information is not unique. Therefore, the simulation or prediction results of GM(1,1) should be nonunique. On the contrary, the current GM(1,1) model’s simulation or prediction results are unique [36]. This is mainly because the GM(1,1) model does not consider the “grey” uncertainty of grey action quantity and simplifies it to a real number. However, the real number means that the GM(1,1) model is a time sequence prediction model with deterministic structure, so its simulation and prediction results are unique.

It can be seen that the existing GM(1,1) model is a simplified model, its research process ignores the uncertainty characteristics of grey action quantity, and its prediction results also violate the “nonuniqueness principle of solution” of grey theory. For this reason, starting from the network model of GM(1,1), the interval uncertainty form of grey action quantity is restored. On this basis, a new GM(1,1) model is established. The simulation and prediction results of the new model are both interval grey numbers with known probability functions.

The remainder of this paper is organized as follows. In Section 2, we analyzed the essence and connotation of the grey action quantity “b” of GM(1,1). In Section 3, we proposed and deduced the new GM(1,1) model with an interval grey action quantity. In Section 4, we employed the new model to simulate and predict the nature gas total consumption in China and compared and analyzed the reasonableness of the results. Our conclusions are presented in Section 5.

2. 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. They are all real numbers. The influencing factors of system development are “cause.” The 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.” Therefore, parameter “b” is called the grey action quantity and represents all grey uncertainty information (Grey Information Coverage) [37].

In GM(1,1), the relationship between grey action quantity “b” and system output (system characteristic variable) [36] is shown in Figure 1.

Figure 1 

The network model diagram of GM(1,1) [37].

In Figure 1, the input variable “b” represents all the uncertain factors (Grey factors) affecting the system development and the output variable is the characteristic variable (White result) of the system. adjusts the size of parameter “b” by AGO (Accumulation Generation Operator, weakening randomness) and MEAN (MEAN generation of consecutive neighbors sequence, improving smoothness). The 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 .

According to the relationship between input, output, and feedback of the system in Figure 1, can be obtained, which is the basic form of the classical GM(1,1) model. The 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. This 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.

The GM(1,1) model is a grey model with incomplete structural information. The 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. Therefore, 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.

3. 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.

3.1. Basic Concepts of the GM(1,1) Model

Definition 1. (see [35]). Assume that is a nonnegative sequence, where , . Then, is called the 1-AGO (Accumulating Generation Operator) sequence of , whereand is called the mean generation of consecutive neighbors sequence of , where

Definition 2. (see [1]). Let , , and be the same as in Definition 1; then,is called the basic form of GM(1,1), which is derived from Figure 1, that is,

Theorem 1 (see [1]). Let , , and be the same as in Definition 1, be a sequence of parameters, and

Then, the least square estimate sequence of grey differential equation satisfies

Detailed proof can be found in reference [38].

3.2. 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, are not equal, that is, .

Figure 2 

Corresponding relationship between input and output of GM(1,1).

According to Theorem 1, the parameters are estimated by the least square method under the condition of minimizing the sum of squares of simulation errors of , . In other words, the parameters “b” in Theorem 2 is an approximate value, which is used to represent all the grey action quantities , , , and of each input. Then, the information difference between grey action quantities is completely ignored. Therefore, the simulated and predicted data based on parameter “b” in Theorem 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 can be calculated, as follows:

Then, we call is the sequence of grey action quantity of GM(1,1). The maximum value and minimum value of 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 is positively correlated with . That is, the bigger the is, the bigger the is. The parameter “b” in GM(1,1) is estimated by the least square method, which is a compromise value between and . Obviously, , that is, . On the other hand, under the existing conditions, the maximum possible value of interval grey number is neither or not , but “b.” The parameter “b” is the real number most likely to represent the whitening value of interval grey number , that is, . According to the definition of probability function [35], can be expressed as in Figure 3.

Figure 3 

Interval grey number form of Grey action quantity and its possibility function.

3.3. New GM(1,1) Model with Interval Grey Action Quantity

Definition 3. Let , , , and be the same as in Definition 1 and Theorem 1. Then, is called the sequence of grey parameters, and is named as the development coefficient, and is called the interval grey action quantity.

Definition 4. Let , , , and 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 , for short. Andis called the whitinization (or image) equation of .

Theorem 2. Let , , , and be the same as in Definitions 1 and 3; then,(i)The solution (or called a time response function) of is given by(ii)The time response sequence of is given by(iii)The restored values can be given by

According to Theorem 2, when the grey action quantity of GM(1,1) is expanded from real number b to interval grey number , the GM (1,1) model evolves into the new model, and the simulation or predicted results of have the following characteristics:(1)The simulated or predicted result of is an interval grey number (2)The interval grey number has the definite lower and upper bounds , that is, (3)The possibility function of the interval grey number is a triangle, and its maximum possible value is , that is

The schematic diagram of the interval grey number 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 , the simulation and prediction data of the model are also interval grey numbers. In the case of uncertain system structure information, the grey number form of simulation or prediction results conforms to the Nonuniqueness Principle of Grey theory solution. Meanwhile, the real number form of simulation and prediction results of the traditional GM(1,1) model is retained in the results. Compared with the traditional GM(1,1) model, which simplifies the grey action quantity b excessively and the reasonable prediction results may be lost, the proposed new model extends the effective range of the simulation and prediction results of GM(1,1) to the greatest extent.

Figure 4 

The possibility function of the simulated or predicted result of .

4. 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 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 model and the last three data are used as the reserved data to test the prediction performance of the model.

Then, the modeling data is as follows:

Step 1. Generating new sequences and :
According to Definition 1, and are be obtained, as follows:

Step 2. Constructing Matrices Y and B and computing parameters and :
According to Theorem 1, Matrices Y and B can be constructed, as follows:Then,

Step 3. Constructing the interval grey action quantity :
According to Definition 1 and the development coefficient , the known data and , , the grey action quantity at time point can be computed, as follows:Then,So, the interval grey action quantity is as follows:and the possibility function of is shown in Figure 5.
The 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. Therefore, 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.

Figure 5 

The possibility function of .

Figure 6 

Grey action quantity at different time point.

Step 4. Computing the simulation and prediction data: , , and :
According to Theorem 2 and , when , the simulated data , , and can be computed, as follows:Then,Similarly, when , the predicted data , , and can be computed, as follows: