A Novel Simultaneous Grey Model SGM(1, 2) and Its Applications in Prediction

Maolin CHENG; Bin LIU

doi:10.21078/JSSI-2022-466-18

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Journal of Systems Science and Information ›› 2022, Vol. 10 ›› Issue (5) : 466-483. DOI: 10.21078/JSSI-2022-466-18

A Novel Simultaneous Grey Model SGM(1, 2) and Its Applications in Prediction

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Abstract

The common models used for grey system predictions include the GM(1, 1), the GM(1, N), the GM(N, 1), and so on, but their whitening equations are all single ordinary differential equations. However, objects and factors generally form a whole through mutual restrictions and connections when the objective world is developing and changing continuously. In other words, variables affect each other. The relationship can't be properly reflected by the single differential equation. Therefore, the paper proposes a novel simultaneous grey model. The paper gives a modeling method of simultaneous grey model SGM(1, 2) with 2 interactive variables. The example proves that the simultaneous grey model has high precision and improves the precision significantly compared with the conventional single grey model. The new method proposed enriches the grey modeling method system and has important significance for the in-depth study, popularization and application of grey models.

Key words

simultaneous grey model / whitening equation / time response equation / prediction precision

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Maolin CHENG , Bin LIU. A Novel Simultaneous Grey Model SGM(1, 2) and Its Applications in Prediction. Journal of Systems Science and Information, 2022, 10(5): 466-483 https://doi.org/10.21078/JSSI-2022-466-18

1 Introduction

The grey prediction is a prediction method for systems with uncertain factors. The grey prediction identifies the degree of difference between factors of system in the development trend, i.e., making a correlation analysis, and generates original data to find the laws of system variation, generates the data sequence with strong regularities, and then builds corresponding differential equation models to predict the future development trend of objects. The grey prediction predicts a series of quantitative values of characteristics of object using the responses observed to construct a grey prediction model for predicting the characteristic quantity at a particular moment in the future or then moment when reaching some characteristic quantity. The grey system theory is used for the analysis, modeling, prediction, decision-making and control of abstract systems such as society, economy and so on, and it has become a new theoretical tool for people to understand and systematically improve objective systems.

Currently, the grey system theory has been applied to the prediction of engineering control, economic management, futurology study, ecological system and complex & variable systems successfully^[1-5]. However, in the actual predictions of models, there are generally big prediction errors, for which reason, many scholars have studied the model. The studies currently focus on the optimization and extension of background value^{[6, 7]}, grey derivative^{[8, 9]}, parameter optimization^{[10, 11]} and extrapolation^{[12, 13]} of grey model, and further promote modeling effect and application fields. The common models used in grey system predictions include the GM (1, 1), the GM

(1, N)

, the GM

(N, 1)

, and so on, but their whitening equations are all single ordinary differential equations. However, objects and factors generally form a whole through mutual restrictions and connections when the objective world is developing and changing continuously. In other words, variables affect each other. For instance, the investment and the GDP, the consumption and the GDP, the energy and the GDP, and so on. The relationship can't be properly reflected by the single differential equation sometimes. Therefore, the paper proposes a simultaneous grey model^[14-17]. The paper gives a modeling method of simultaneous grey model SGM

(1, 2)

with 2 interactive variables.

The paper first proposes the form of a simultaneous grey model SGM

(1, 2)

and the times response equation of the model for predictions, then gives the parameter estimation method of model and finally offers two application examples. The examples show that the simultaneous grey model has high prediction precision and the model's precision is significantly improved compared with that of the single grey model.

2 The Form of Simultaneous Grey Model SGM(1, 2)

Suppose the two interactive time sequences are

{x_{1}^{(0)} (k)}

and

{x_{2}^{(0)} (k)}

, i.e.,

x_{i}^{(0)} (k) = {x_{i}^{(0)} (1), x_{i}^{(0)} (2), \cdot \cdot \cdot, x_{i}^{(0)} (N)}, i = 1, 2.

(1)

Their first-order accumulated generating equation is

x_{i}^{(1)} (k) = \sum_{j = 1}^{k} x_{i}^{(0)} (j), i = 1, 2; k = 1, 2, \dots, N .

(2)

The whitening equation of simultaneous grey model SGM

(1, 2)

proposed is

{\begin{array}{l} \frac{d x_{1}^{(1)} (t)}{d t} = a_{0} + a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t), \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{0} + b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) . \end{array}

(3)

Let

X = [\begin{array}{l} x_{1}^{(1)} (t) \\ x_{2}^{(1)} (t) \end{array}], A = [\begin{array}{cc} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}], f (t) = [\begin{array}{l} a_{0} \\ b_{0} \end{array}],

and then

X^{'} = A X + f (t)

Generally,

f (t)

can be extended to be

f (t) = [\begin{array}{l} a_{0} + a_{3} t + a_{4} t^{2} + \dots + a_{p} t^{p - 2} \\ b_{0} + b_{3} t + b_{4} t^{2} + \dots + b_{p} t^{p - 2} \end{array}],

i.e.,

{\begin{array}{l} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2} + \dots + a_{p} t^{p - 2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2} + \dots + b_{p} t^{p - 2} . \end{array}

(4)

To improve simulation and prediction precision, the value of

p

should be neither too big nor too small. The case in the paper is

p = 4

, i.e.,

{\begin{array}{l} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2} . \end{array}

(5)

The equation above can be written to be

{\begin{array}{l} \frac{d x_{1}^{(1)} (t)}{d t} = a_{0} + a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{0} + b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{3} t + b_{4} t^{2} . \end{array}

(6)

Get the integrals of the equations from both sides:

{\begin{array}{l} \int_{k - 1}^{k} \frac{d x_{1}^{(1)} (t)}{d t} d t = a_{0} + a_{1} \int_{k - 1}^{k} x_{1}^{(1)} (t) d t + a_{2} \int_{k - 1}^{k} x_{2}^{(1)} (t) d t + a_{3} \int_{k - 1}^{k} t d t + a_{4} \int_{k - 1}^{k} t^{2} d t, \\ \int_{k - 1}^{k} \frac{d x_{2}^{(1)} (t)}{d t} d t = b_{0} + b_{1} \int_{k - 1}^{k} x_{1}^{(1)} (t) d t + b_{2} \int_{k - 1}^{k} x_{2}^{(1)} (t) d t + b_{3} \int_{k - 1}^{k} t d t + b_{4} \int_{k - 1}^{k} t^{2} d t . \end{array}

(7)

Let

z_{1} (k) = \int_{k - 1}^{k} x_{1}^{(1)} (t) d t, z_{2} (k) = \int_{k - 1}^{k} x_{2}^{(1)} (t) d t,

and call

z_{1}^{(1)} (k), z_{2}^{(1)} (k)

the background value. For convenience, Let

z_{3} (k) = \int_{k - 1}^{k} t d t = \frac{1}{2} [k^{2} - (k - 1)^{2}], z_{4} (k) = \int_{k - 1}^{k} t^{2} d t = \frac{1}{3} [k^{3} - (k - 1)^{3}] .

Then, get the grey differential equations of simultaneous grey model:

{\begin{array}{l} x_{1}^{(0)} (k) = a_{0} + a_{1} z_{1} (k) + a_{2} z_{2} (k) + a_{3} z_{3} (k) + a_{4} z_{4} (k), \\ x_{2}^{(0)} (k) = b_{0} + b_{1} z_{1} (k) + b_{2} z_{2} (k) + b_{3} z_{3} (k) + b_{4} z_{4} (k) . \end{array}

(8)

Apparently,

z_{i} (k) = α_{k} x_{i}^{(1)} (k - 1) + (1 - α_{k}) x_{i}^{(1)} (k)

(i = 1, 2; 0 \leq α_{k} \leq 1)

Take

α_{k} = 0.5

, i.e.,

z_{i} (k) = 0.5 x_{i}^{(1)} (k - 1) + 0.5 x_{i}^{(1)} (k)

, and then we have

{\begin{array}{l} x_{1}^{(0)} (k) = a_{0} + a_{1} z_{1} (k) + a_{2} z_{2} (k) + a_{3} z_{3} (k) + a_{4} z_{4} (k) + e_{1} (k), \\ x_{2}^{(0)} (k) = b_{0} + b_{1} z_{1} (k) + b_{2} z_{2} (k) + b_{3} z_{3} (k) + b_{4} z_{4} (k) + e_{2} (k), \end{array}

(9)

where

e_{1} (k)

and

e_{2} (k)

are the errors when

α_{k} = 0.5

3 Modeling Method of Simultaneous Grey Model SGM(1, 2)

3.1 Time Response Equation of Simultaneous Grey Model SGM(1, 2)

The whitening equation of simultaneous grey model SGM

(1, 2)

{\begin{cases} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2} . \end{cases}

(10)

The equations above form system of non-homogeneous linear equations, i.e.,

X^{'} = A X + f (t) .

(11)

where

X = [\begin{array}{l} x_{1}^{(1)} (t) \\ x_{2}^{(1)} (t) \end{array}], A = [\begin{array}{cc} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}], f (t) = [\begin{array}{l} a_{0} + a_{3} t + a_{4} t^{2} \\ b_{0} + b_{3} t + b_{4} t^{2} \end{array}] .

We solve the equations to get the solution meeting original condition

X (t_{0}) = η

Generally, there are three methods, including the method of variation of constant, the method of undetermined coefficients and the operator method, to solve the system of non-homogeneous linear equations above. The paper uses the variation-of-constant formula.

Suppose the standard basic solution matrix of corresponding system of homogeneous linear equations is

\exp (A t)

, and then, using the general method of variation of constant, get:

X (t) = \exp (A t) \cdot (\exp (A t_{0})^{- 1}) η + \exp (A t) \int_{t_{0}}^{t} (\exp {(A s)}^{- 1}) f (s) d s .

(12)

Then, according to the properties of matrix exponential, get the further simplified variation-of-constant formula as follows:

X (t) = \exp [(t - t_{0}) A] \cdot η + \int_{t_{0}}^{t} \exp [(t - s) A] f (s) d s .

(13)

Because

η = [\begin{array}{l} x_{1}^{(1)} (1) \\ x_{2}^{(1)} (1) \end{array}],

then have

X (t) = \exp [(t - 1) A] \cdot [\begin{array}{l} x_{1}^{(1)} (1) \\ x_{2}^{(1)} (1) \end{array}] + \int_{1}^{t} \exp [(t - s) A] [\begin{array}{l} a_{0} + a_{3} s + a_{4} s^{2} \\ b_{0} + b_{3} s + b_{4} s^{2} \end{array}] d s .

(14)

Get the time response equation of

x_{1}^{(1)} (t)

and

x_{2}^{(1)} (t)

using the equation above, which can be realized with software Matlab.

3.2 Parameter Estimation of Simultaneous Grey Model SGM(1, 2)

Write

{\begin{cases} x_{1}^{(0)} (k) = a_{0} + a_{1} z_{1} (k) + a_{2} z_{2} (k) + a_{3} z_{3} (k) + a_{4} z_{4} (k) + e_{1} (k), \\ x_{2}^{(0)} (k) = b_{0} + b_{1} z_{1} (k) + b_{2} z_{2} (k) + b_{3} z_{3} (k) + b_{4} z_{4} (k) + e_{2} (k), \end{cases}

in the form of matrix, and then have

[\begin{array}{l} x_{1}^{(0)} (k) \\ x_{2}^{(0)} (k) \end{array}] = [\begin{array}{ccccc} a_{0} & a_{1} & a_{2} & a_{3} & a_{4} \\ b_{0} & b_{1} & b_{2} & b_{3} & b_{4} \end{array}] \cdot [\begin{matrix} 1 \\ z_{1}^{(1)} (k) \\ \begin{array}{l} z_{2}^{(1)} (k) \\ z_{3}^{(1)} (k) \\ z_{4}^{(1)} (k) \end{array} \end{matrix}] + [\begin{array}{l} e_{1} \\ e_{2} \end{array}],

(15)

i.e.,

Y = (1_{n} X) β + e,

(16)

where

\begin{array}{rcl} Y = [\begin{array}{c} x_{1}^{(0)} (2) & x_{2}^{(0)} (2) \\ x_{1}^{(0)} (3) & x_{2}^{(0)} (3) \\ ⋮ & ⋮ \\ x_{1}^{(0)} (N) & x_{2}^{(0)} (N) \end{array}], \end{array}

(17)

\begin{array}{rcl} (1_{n} X) = [\begin{array}{c} 1 & z_{1} (2) & z_{2} (2) & z_{3} (2) & z_{4} (2) \\ 1 & z_{1} (3) & z_{2} (3) & z_{3} (3) & z_{4} (3) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & z_{1} (N) & z_{2} (N) & z_{3} (N) & z_{4} (N) \end{array}], \end{array}

(18)

\begin{array}{rcl} β = [\begin{array}{c} a_{0} & b_{0} \\ a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \\ a_{4} & b_{4} \end{array}] \begin{array}{c} , & e = [\begin{matrix} e_{1} (2) & e_{1} (3) & \dots & e_{1} (N) \\ e_{2} (2) & e_{2} (3) & \dots & e_{2} (N) \end{matrix}] \end{array} . \end{array}

(19)

From

Y = (1_{n} X) β + e

, get

e = Y - (1_{n} X) β

, and then

e^{'} e = (Y - (1_{n} X) β)^{'} (Y - (1_{n} X) β)

. According to the principles of least squares, make

e^{'} e

the minimum in the sense of matrix. The corresponding normal equation is

[\begin{matrix} {1^{'}}_{n} \\ X^{'} \end{matrix}] (1_{n} X) β = [\begin{matrix} {1^{'}}_{n} \\ X^{'} \end{matrix}] Y .

(20)

Then,

β

's least squares estimate is

\hat{β} = {([\begin{matrix} {1^{'}}_{n} \\ X^{'} \end{matrix}] (1_{n} X))}^{- 1} [\begin{matrix} {1^{'}}_{n} \\ X^{'} \end{matrix}] Y = {[\begin{matrix} n & {1^{'}}_{n} X \\ X^{'} 1_{n} & X^{'} X \end{matrix}]}^{- 1} [\begin{matrix} {1^{'}}_{n} Y \\ X^{'} Y \end{matrix}],

(21)

which can be realized with software Matlab.

Suppose there are the data of

x_{i}^{(0)}

n

years in which the data from year 1 to year

m

are used for modeling and the data from year

m + 1

to year

n

are used for prediction. In this case, after getting the estimates of parameters, get the simulation value and prediction value of

x_{i}^{(0)}

, which are

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = 2, 3, \dots, m)

and

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = m + 1, m + 2, \dots, n)

, respectively, through calculations with the time response equation.

4 Modeling Examples of Simultaneous Grey Model

4.1 Example 1

In a country, the GDP and the energy consumption affects each other. This part builds a simultaneous grey prediction model SGM

(1, 2)

with the data of GDP and energy consumption of China from year 2002 to year 2020 using the method proposed. The original data came from the official website of National Bureau of Statistics of China, which are shown in Table 1. In the table,

x_{1}^{(0)}

is China's GDP (￥0.1 billion) and

x_{2}^{(0)}

is the energy consumption of China (unit: 0.1 billion tons of standard coal).

Table 1 Related data of China's GDP & energy consumption and modeling results of GM(1, 1)

Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	GM (1, 1) Model
Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)
2002	1	121717.4	169577.0	-	-	-	-
2003	2	137422.0	197083.0	181946.65	32.4	244298.64	24.0
2004	3	161840.2	230281.0	203846.07	26.0	256604.56	11.4
2005	4	187318.9	261369.0	228381.34	21.9	269530.35	3.12
2006	5	219438.5	286467.0	255869.71	16.6	283107.25	1.17
2007	6	270092.3	311442.0	286666.64	6.14	297368.05	4.52
2008	7	319244.6	320611.0	321170.33	0.603	312347.21	2.58
2009	8	348517.7	336126.0	359826.95	3.24	328080.89	2.39
2010	9	412119.3	360648.0	403136.35	2.18	344607.13	4.45
2011	10	487940.2	387043.0	451658.54	7.44	361965.83	6.48
2012	11	538580.0	402138.0	506020.94	6.05	380198.93	5.46
2013	12	592963.2	416913.0	566926.5	4.39	399350.47	4.21
2014	13	643563.1	428333.99	635162.76	1.31	419466.72	2.07
2015	14	688858.2	434112.78	711612.06	3.3	440596.28	1.49
2016	15	746395.1	441491.81	797262.92	6.82	462790.19	4.82
2017	16	832035.9	455826.92	893222.87	7.35	486102.06	6.64
				Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)
2018	17	919281.1	471925.15	1000732.73	8.86	510588.2	8.19
2019	18	990865.1	487000.0	1121182.66	13.2	536307.77	10.1
2020	19	1015986.0	493000.0	1256130.15	23.6	563322.9	14.3
Average Simulation Relative Error(2002–2017)				-	9.71	-	5.65
Average Prediction Relative Error (2018–2020)				-	15.22	-	10.86
Average Relative Error (2002–2020)				-	10.63	-	6.52

Build a conventional grey GM

(1, 1)

model for original sequence

x_{1}^{(0)}

, and then get the following parameter estimates:

a = - 0 .1136, b = 157969 .81 .

(22)

The time response equation is

{\hat{x}}_{1}^{(1)} (t) = ({x_{1}}^{(1)} (1) - \frac{b}{a}) e^{- a (t - 1)} + \frac{b}{a} = 2405933 .83 e^{0.1136 (t - 1)} - 1389947 .83 .

(23)

Similarly, build the conventional grey GM

(1, 1)

model for original sequence

x_{2}^{(0)}

, then get the following parameter estimates:

a = - 0 .0491, b = 230010 .98 .

(24)

The time response equation is

{\hat{x}}_{2}^{(1)} (t) = ({x_{2}}^{(1)} (1) - \frac{b}{a}) e^{- a (t - 1)} + \frac{b}{a} = 4849848 .0 {e^{0.0491}}^{(t - 1)} - 4680271 .0 .

(25)

For

x_{1}^{(0)}

and

x_{2}^{(0)}

, the simulation value is

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = 2, 3, \dots, 16)

, the prediction value is

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = 17, 18, 19)

, the relative error is

{R E}_{i} (k) = | \frac{x_{i}^{(0)} (k) - {\hat{x}}_{i}^{(0)} (k)}{x_{i}^{(0)} (k)} | \times 100 %, i = 1, 2,

and the average relative error is

{M A P E}_{i} = \frac{1}{N - 1} \sum_{k = 2}^{N} | \frac{x_{i}^{(0)} (k) - {\hat{x}}_{i}^{(0)} (k)}{x_{i}^{(0)} (k)} | \times 100 %, i = 1, 2.

See Table 1 for relative calculation results.

Then, build a simultaneous grey model SGM

(1, 2)

using the method proposed. Suppose simultaneous grey model SGM

(1, 2)

's grey differential equation is

{\begin{cases} x_{1}^{(0)} (k) = a_{0} + a_{1} z_{1} (k) + a_{2} z_{2} (k) + a_{3} z_{3} (k) + a_{4} z_{4} (k), \\ x_{2}^{(0)} (k) = b_{0} + b_{1} z_{1} (k) + b_{2} z_{2} (k) + b_{3} z_{3} (k) + b_{4} z_{4} (k) . \end{cases}

(26)

The whitening equation is

{\begin{cases} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2} . \end{cases}

(27)

First, through calculation, get:

β = [\begin{matrix} a_{0} & b_{0} \\ a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \\ a_{4} & b_{4} \end{matrix}] = [\begin{matrix} 135524 .3 & 145430 .6 \\ - 0 .27811735 & 0 .10621089 \\ 0 .069578291 & 0 .10864369 \\ 13464 .454 & 9800 .3085 \\ 7750 .4202 & - 4486 .9271 \end{matrix}],

(28)

i.e.,

{\begin{aligned} \frac{d x_{1}^{(1)} (t)}{d t} = & - 0.278117 x_{1}^{(1)} (t) + 0.069578 x_{2}^{(1)} (t) \\ + 135524.3 + 13464.454 t + 7750.4202 t^{2}, R^{2} = 0.99920 \\ \frac{d x_{2}^{(1)} (t)}{d t} = & 0.106210 x_{1}^{(1)} (t) + 0.108644 x_{2}^{(1)} (t) \\ + 145430.6 + 9800.3085 t - 4486.9271 t^{2}, R^{2} = 0.99839 . \end{aligned}

(29)

The multiple correlation coefficient shows that the model has high fitting precision.

Then, get the following time response equation through calculation:

\begin{aligned} {\overset{⌢}{x}}_{1}^{(1)} (t) = & 0.17179505 * \exp (0.1268902 * t) * (142100.6 * \exp (- 0.1268902 * t) \\ + 191200.57 * t * \exp (- 0.1268902 * t) + 18509.024 * t^{2} \\ * \exp (- 0.1268902 * t) - 140052.07) + 3.8132391 * \exp (- 0.29636385 * t) \\ * (224234.26 * \exp (0.29636385 * t) - 38716.134 * t * \exp (0.29636385 * t) \\ + 7215.1628 * t^{2} * \exp (0.29636385 * t) - 227970.8) \end{aligned}

(30)

\begin{aligned} {\overset{⌢}{x}}_{2}^{(1)} (t) = & \exp (0.1268902 * t) * (142100.6 * \exp (- 0.1268902 * t) \\ + 191200.57 * t * \exp (- 0.1268902 * t) + 18509.024 * t^{2} \\ * \exp (- 0.1268902 * t) - 140052.07) - \exp (- 0.29636385 * t) \\ * (224234.26 * \exp (0.29636385 * t) - 38716.134 * t * \exp (0.29636385 * t) \\ + 7215.1628 * t^{2} * \exp (0.29636385 * t) - 227970.8) . \end{aligned}

(31)

From

{\hat{x}}_{i}^{(0} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1), (i = 1, 2)

, get the simulation values and prediction values of original sequence through calculation. See calculation results in Table 2. Table 2 shows relative errors and average relative errors in the periods.

Table 2 Modeling results of simultaneous grey model of China's GDP and energy consumption

Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simultaneous Grey Model
Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)
2002	1	121717.4	169577.0	-	-	-	-
2003	2	137422.0	197083.0	139373.05	1.42	198812.87	0.878
2004	3	161840.2	230281.0	157739.99	2.53	229640.58	0.278
2005	4	187318.9	261369.0	186944.67	0.2	257214.68	1.59
2006	5	219438.5	286467.0	224180.88	2.16	282215.39	1.48
2007	6	270092.3	311442.0	267358.58	1.01	305127.62	2.03
2008	7	319244.6	320611.0	314919.73	1.35	326288.21	1.77
2009	8	348517.7	336126.0	365701.26	4.93	345920.69	2.91
2010	9	412119.3	360648.0	418833.18	1.63	364160.74	0.974
2011	10	487940.2	387043.0	473662.68	2.93	381074.57	1.54
2012	11	538580.0	402138.0	529697.6	1.65	396672.02	1.36
2013	12	592963.2	416913.0	586564.36	1.08	410915.75	1.44
2014	13	643563.1	428333.99	643976.47	0.0642	423727.26	1.08
2015	14	688858.2	434112.78	701711.03	1.87	434990.57	0.202
2016	15	746395.1	441491.81	759591.07	1.77	444554.08	0.694
2017	16	832035.9	455826.92	817472.26	1.75	452230.82	0.789
				Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)
2018	17	919281.1	471925.15	875232.82	4.79	457797.53	2.99
2019	18	990865.1	487000.0	932765.74	5.86	460992.48	5.34
2020	19	1015986.0	493000.0	989972.83	2.56	461512.5	6.39
Average Simulation Relative Error (2002–2017)				-	1.75	-	1.26
Average Prediction Relative Error (2018–2020)				-	4.40	-	4.90
Average Relative Error (2002–2020)				-	2.19	-	1.87

To compare the model built with the method proposed with the grey models proposed by other reference documents in terms of precision, the paper makes a calculation.

Build an improved grey GM

(1, 1)

power model proposed by Ma and Wang^[18] for original sequence

x_{1}^{(0)}

, and then get the following parameter estimates:

(a, b, α) = (- 0 .022654679, 22760 .514, 0 .17693958) .

(32)

In this case, the time response equation is

\begin{aligned} {\hat{x}}_{1}^{(1)} (k) & = {\frac{b}{a} + [x^{(1)} (1)^{(1 - α)} - \frac{b}{a}] e^{a (α - 1) (k - 1)}}^{\frac{1}{α - 1}} \\ = {(1024812.9 e^{0.0186 (k - 1)} - 1004671.7)}^{1.2150} . \end{aligned}

(33)

For

x_{2}^{(0)}

, we get the following parameter estimates through calculation:

(a, b, α) = (- 0 .10626189, 56630 .898, 0 .079434514) .

(34)

In this case, the time response equation is

\begin{aligned} {\hat{x}}_{2}^{(1)} (k) & = {\frac{b}{a} + [x^{(1)} (1)^{(1 - α)} - \frac{b}{a}] e^{a (α - 1) (k - 1)}}^{\frac{1}{α - 1}} \\ = {(580954.64 e^{0.0978 (k - 1)} - 532937.07)}^{1.0863} . \end{aligned}

(35)

From

{\hat{x}}_{i}^{(0} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2)

, we get the simulation values and prediction values of original sequence through calculation. See calculation results in Table 3. Table 3 shows relative errors and average relative errors in the periods.

Table 3 Modeling results of Grey models proposed by other references

Year	No.	Grey Bernoulli Model Proposed by Ma and Wang^[18]				Grey Difference Equation Model Proposed by Cheng and Shi^[19]
Year	No.	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)
2002	1	-	-	-	-	-	-	-	-
2003	2	171051.9	24.5	213968.7	8.57	137422.0	0	183299.2	6.99
2004	3	198757.58	22.8	243352.57	5.68	171685.82	6.08	218522.96	5.11
2005	4	226961.01	21.2	266785.94	2.07	207737.06	10.9	249454.19	4.56
2006	5	257110.69	17.2	287285.85	0.286	245706.67	12.0	276764.3	3.39
2007	6	289932.32	7.35	306078.79	1.72	285734.16	5.79	301022.85	3.35
2008	7	325971.25	2.11	323796.04	0.993	327968.2	2.73	322713.0	0.656
2009	8	365726.81	4.94	340809.18	1.39	372567.28	6.9	342244.71	1.82
2010	9	409701.88	0.587	357357.32	0.912	419700.3	1.84	359965.83	0.189
2011	10	458426.9	6.05	373604.61	3.47	469547.35	3.77	376171.63	2.81
2012	11	512474.62	4.85	389669.48	3.1	522300.44	3.02	391112.76	2.74
2013	12	572471.07	3.46	405640.9	2.7	578164.26	2.5	405002.04	2.86
2014	13	639105.21	0.693	421587.99	1.57	637357.13	0.964	418020.26	2.41
2015	14	713138.23	3.52	437565.98	0.795	700111.83	1.63	430321.06	0.873
2016	15	795413.03	6.57	453620.16	2.75	766676.62	2.72	442035.01	0.123
2017	16	886864.4	6.59	469788.47	3.06	837316.29	0.635	453273.2	0.56
		Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)
2018	17	988529.91	7.53	486103.38	3.00	912313.27	0.758	464130.18	1.65
2019	18	1101562.0	11.2	502593.13	3.20	991968.8	0.111	474686.46	2.53
2020	19	1227241.0	20.8	519282.75	5.33	1076604.2	5.97	485010.71	1.62
Average Simulation Relative Error (2002–2017)		-	8.82	-	2.61	-	4.09	-	2.56
Average Prediction Relative Error (2018–2020)		-	13.17	-	3.85	-	2.28	-	1.93
Average Relative Error (2002–2020)		-	9.55	-	2.82	-	3.79	-	2.46

Next, build the following model using the grey difference equation model DDGM(2, 1) modeling method proposed by Cheng and Shi^[19]:

x^{(1)} (k) = \frac{2 + a_{1} - \frac{a_{2}}{2}}{1 + a_{1} + \frac{a_{2}}{2}} x^{(1)} (k - 1) - \frac{1}{1 + a_{1} + \frac{a_{2}}{2}} x^{(1)} (k - 2) + \frac{b}{1 + a_{1} + \frac{a_{2}}{2}} .

(36)

For

x_{1}^{(0)}

, through calculation get

a_{1} = - 0 .042578061, a_{2} = - 0 .0013306571, b = 26493 .922

Then, get the following time sequence response equation:

{\hat{x}}_{1}^{(1)} (k) = 14050580 .0 \times (0 .9793236)^{k} + 5877314 .4 0 \times 1 .067265 1^{k} - 19910687 .0 .

(37)

For

x_{2}^{(0)}

, through calculation get

a_{1} = 0 .16138637, a_{2} = - 0 .0029887633, b = 69111 .253

. And then, get the time sequence response equation:

{\hat{x}}_{2}^{(1)} (k) = 1471760 .8 \times (0 .84779632)^{k} + 21677825 .0 \times 1 .016929 7^{k} - 23123267 .0 .

(38)

From

{\hat{x}}_{i}^{(0} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2)

, get the simulation values and prediction values of original sequence through calculation. See calculation results in Table 3. Table 3 shows relative errors and average relative errors in the periods.

Table 1 gives the simulation values from year 2002 to year 2017 and the prediction values from year 2018 to year 2020 of conventional GM(1, 1) model of

x_{1}^{(0)}

and

x_{2}^{(0)}

, and corresponding simulation errors and prediction errors, and offers the average simulation errors and average prediction errors. It can be seen that the conventional GM(1, 1) models of

x_{1}^{(0)}

and

x_{2}^{(0)}

have big average simulation errors and average prediction errors. The main reason is that GDP and energy consumption grows too fast that the conventional GM(1, 1) model is not suitable.

Table 2 gives the simulation values from year 2002 to year 2017 and the prediction values from year 2018 to year 2020 of simultaneous grey model of

x_{1}^{(0)}

and

x_{2}^{(0)}

proposed, and corresponding simulation errors and prediction errors, and offers the average simulation errors and average prediction errors. It can be seen that the simultaneous grey models of

x_{1}^{(0)}

and

x_{2}^{(0)}

have the average simulation errors and average prediction errors significantly smaller than those of conventional models and show the high precision. The model of

x_{1}^{(0)}

has an average simulation error of 1.75%, an average prediction error of 4.40%, and an average total error of 2.19%. The model of

x_{2}^{(0)}

has an average simulation error of 1.26%, an average prediction error of 4.90%, and an average total error of 1.87%. All average errors are small.

Table 3 gives the modeling calculation results of grey models proposed by Ma and Wang^[18] and Cheng and Shi^[19]. It can be seen that the average simulation errors of and are both higher than those of simultaneous grey model proposed significantly. Although the average prediction errors are slightly lower than those of simultaneous model proposed, the average total error is higher than that of simultaneous grey model significantly.

Therefore, from Tables 1, 2 and 3 we can see the model built using the simultaneous grey SGM

(1, 2)

modeling method proposed has the total precision significantly higher than that of conventional grey GM

(1, 1)

model, and superior to the that of models built with the methods of Ma and Wang^[18] and Cheng and Shi^[19]. Figure 1 gives the histogram of average relative errors of four models built for

x_{1}^{(0)}

and Figure 2 gives the histogram of average relative errors of four models built for

x_{2}^{(0)}

. The figures show that the simultaneous grey model proposed has high reliability and effectiveness.

Figure 1 The histogram of average relative errors of four models built for x₁⁽⁰⁾

Full size|PPT slide

Figure 2 The histogram of average relative errors of four models built for x₂⁽⁰⁾

Full size|PPT slide

4.2 Example 2

The car industry, which is one of pillars of the national economy in China, develops rapidly and makes China the top car seller in the world, and has changed the Chinese's life style greatly. Many scholars at home and abroad have made a lot of studies on the influencing factors and development prospect of China's private car ownership. The paper builds a grey simultaneous prediction model for two interactive variables, the private car ownership and the transportation economy.

x_{1}^{(0)}

is China's private car ownership form year 2005 to year 2019 (unit: 10, 000 cars), and

x_{2}^{(0)}

is China's added value of transportation industry from year 2005 to year 2019 (unit: RMB 0.1 billion yuan). Data Source: China Statistical Yearbook. See Table 4 for the data.

Table 4 Modeling results of the grey model GM(1, 1) of China's private car ownership and the added value of the transportation industry

Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	GM (1, 1) Model
Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)
2005	1	1848.07	10668.8	-	-	-	-
2006	2	2333.32	12186.3	2931.301	25.6	13152.26	7.93
2007	3	2876.22	14605.1	3504.56	21.8	14455.08	1.03
2008	4	3501.39	16367.6	4189.929	19.7	15886.95	2.94
2009	5	4574.91	16522.4	5009.331	9.5	17460.66	5.68
2010	6	5938.71	18783.6	5988.98	0.846	19190.26	2.16
2011	7	7326.79	21842.0	7160.214	2.27	21091.18	3.44
2012	8	8838.6	23763.2	8560.499	3.15	23180.4	2.45
2013	9	10501.68	26042.7	10234.63	2.54	25476.57	2.17
2014	10	12339.36	28534.4	12236.17	0.836	28000.2	1.87
2015	11	14099.1	30519.5	14629.13	3.76	30773.8	0.833
2016	12	16330.2	33028.7	17490.07	7.1	33822.15	2.4
				Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)
2017	13	18515.1	37121.9	20910.51	12.9	37172.46	0.136
2018	14	20574.93	40337.2	24999.87	21.5	40854.64	1.28
2019	15	22508.99	42466.3	29888.97	32.8	44901.56	5.73
Average Relative Error of the Simulation (2005–2016)				-	8.83	-	2.99
Average Relative Error of the Prediction (2017–2019)				-	22.41	-	2.38
Average Relative Error (2005–2019)				-	11.74	-	2.86

Build a single grey GM

(1, 1)

model for original sequence

x_{1}^{(0)}

and get the estimates of parameters:

a = - 0 .1786, b = 2347 .1980 .

(39)

The time response equation is:

\begin{array}{rcl} {\hat{x}}_{1}^{(1)} (t) & = & ({x_{1}}^{(1)} (1) - \frac{b}{a}) e^{- a (t - 1)} + \frac{b}{a} \\ = & 14988 .9 e^{0.0491 (t - 1)} - 13140 .83 . \end{array}

(40)

Similarly, build a single grey model GM

(1, 1)

for original sequence

x_{2}^{(0)}

and get the estimates of parameters:

a = - 0 .09445, b = 11533 .218 .

(41)

The time response equation is:

\begin{array}{rcl} {\hat{x}}_{2}^{(1)} (t) & = & ({x_{2}}^{(1)} (1) - \frac{b}{a}) e^{- a (t - 1)} + \frac{b}{a} \\ = & 132775 .18 {e^{0.09445}}^{(t - 1)} - 122106 .38 . \end{array}

(42)

For

x_{1}^{(0)}

and

x_{2}^{(0)}

, the simulation value is

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = 2, 3, \dots, 12)

, the prediction value is

{\hat{x}}_{i}^{(0)} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2; k = 13, 14, 15)

, the relative error is

{R E}_{i} (k) = | \frac{x_{i}^{(0)} (k) - {\hat{x}}_{i}^{(0)} (k)}{x_{i}^{(0)} (k)} | \times 100 %, i = 1, 2,

and the average relative error is

{M A P E}_{i} = \frac{1}{N - 1} \sum_{k = 2}^{N} | \frac{x_{i}^{(0)} (k) - {\hat{x}}_{i}^{(0)} (k)}{x_{i}^{(0)} (k)} | \times 100 %, i = 1, 2.

See Table 4 for the calculation results.

Next, build the simultaneous grey model SGM

(1, 2)

using the method proposed. Suppose the simultaneous grey model SGM

(1, 2)

's grey differential equation is

{\begin{cases} x_{1}^{(0)} (k) = a_{0} + a_{1} z_{1} (k) + a_{2} z_{2} (k) + a_{3} z_{3} (k) + a_{4} z_{4} (k), \\ x_{2}^{(0)} (k) = b_{0} + b_{1} z_{1} (k) + b_{2} z_{2} (k) + b_{3} z_{3} (k) + b_{4} z_{4} (k) . \end{cases}

(43)

The whitening equation is

{\begin{cases} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2} . \end{cases}

(44)

First, calculate and get

β = [\begin{matrix} a_{0} & b_{0} \\ a_{1} & b_{1} \\ a_{2} & b_{2} \\ a_{3} & b_{3} \\ a_{4} & b_{4} \end{matrix}] = [\begin{matrix} 1814 .28 & 10751 .85 \\ 0 .06935741 & - 0 .01931713 \\ - 0 .232089 & - 0 .1180209 \\ 2371 .572 & 2289 .581 \\ 274 .6816 & 193 .0362 \end{matrix}],

(45)

i.e.,

{\begin{cases} \frac{d x_{1}^{(1)} (t)}{d t} = 0 .06935741 x_{1}^{(1)} (t) - 0 .232089 x_{2}^{(1)} (t) + 1814 .28 + 2371 .572 t + 274 .6816 t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = - 0 .01931713 x_{1}^{(1)} (t) - 0 .1180209 x_{2}^{(1)} (t) + 10751 .85 + 2289 .581 t + 193 .0362 t^{2} . \end{cases}

(46)

Then, calculate and get the time response equation:

\begin{aligned} {\overset{⌢}{x}}_{1}^{(1)} (t) = & 1.111296 * \exp (- 0.1394879 * t) * (100028.3 * \exp (0.1394879 * t) \\ - 4050.88 * t * \exp (0.1394879 * t) + 1420.067 * t^{2} * \exp (0.1394879 * t) \\ - 100675.8) - 10.8114 * \exp (0.09082446 * t) \\ * (2353.952 * \exp (- 0.09082446 * t) + 1063.79 * t * \exp (- 0.09082446 * t) \\ + 55.55703 * t^{2} * \exp (- 0.09082446 * t) - 2405.202), \end{aligned}

(47)

\begin{aligned} {\overset{⌢}{x}}_{2}^{(1)} (t) = & \exp (- 0.1394879 * t) * (100028.3 * \exp (0.1394879 * t) \\ - 4050.88 * t * \exp (0.1394879 * t) + 1420.067 * t^{2} * \exp (0.1394879 * t) \\ - 100675.8) + \exp (0.09082446 * t) * (2353.952 * \exp (- 0.09082446 * t) \\ + 1063.79 * t * \exp (- 0.09082446 * t) \\ + 55.55703 * t^{2} * \exp (- 0.09082446 * t) - 2405.202) \end{aligned}

(48)

Calculate and get the original sequence's simulation values and prediction values with

{\hat{x}}_{i}^{(0} (k) = {\hat{x}}_{i}^{(1)} (k) - {\hat{x}}_{i}^{(1)} (k - 1) (i = 1, 2)

. See Table 5 for the results. Table 5 gives the relative errors and average relative errors in the periods.

Table 5 Modeling results of simultaneous grey model of China's private car ownership and the added value of transportation industry

Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simultaneous Grey Model
Year	No.	x₁⁽⁰⁾	x₂⁽⁰⁾	Simulation Value of x₁⁽⁰⁾	Relative Error (%)	Simulation Value of x₂⁽⁰⁾	Relative Error (%)
2005	1	1848.07	10668.8	-	-	-	-
2006	2	2333.32	12186.3	2306.964	1.13	12590.42	3.32
2007	3	2876.22	14605.1	2869.723	0.226	14033.48	3.91
2008	4	3501.39	16367.6	3671.725	4.86	15667.54	4.28
2009	5	4574.91	16522.4	4687.336	2.46	17467.22	5.72
2010	6	5938.71	18783.6	5894.781	0.74	19410.4	3.34
2011	7	7326.79	21842.0	7275.696	0.697	21477.78	1.67
2012	8	8838.6	23763.2	8814.736	0.27	23652.52	0.466
2013	9	10501.68	26042.7	10499.24	0.0232	25919.92	0.471
2014	10	12339.36	28534.4	12318.96	0.165	28267.1	0.937
2015	11	14099.1	30519.5	14265.79	1.18	30682.8	0.535
2016	12	16330.2	33028.7	16333.59	0.0208	33157.13	0.389
				Prediction Value	Relative Error (%)	Prediction Value	Relative Error (%)
2017	13	18515.1	37121.9	18518.02	0.0158	35681.41	3.88
2018	14	20574.93	40337.2	20816.38	1.17	38247.98	5.18
2019	15	22508.99	42466.3	23227.5	3.19	40850.06	3.81
Average Simulation Relative Error (2005–2016)				-	1.07	-	2.27
Average Prediction Relative Error (2017–2019)				-	1.46	-	4.28
Average Relative Error (2005–2019)				-	1.15	-	2.70

Table 4 and Table 5 show that the single GM(1, 1) model built for

x_{1}^{(0)}

has comparatively bigger average simulation and prediction relative errors (8.83% and 22.41%, respectively), while the simultaneous grey model has significantly smaller average simulation and prediction relative errors (1.07% and 1.46%, respectively). As for the single GM(1, 1) model built for

x_{2}^{(0)}

, its average simulation and prediction relative errors (2.99% and 2.38%, respectively) are not big, but the simultaneous grey model's average simulation relative error is even smaller (2.27%). Although the simultaneous grey model has a big average prediction relative error (4.28%), it has the overall precision significantly higher than that of the single GM(1, 1) model.

5 Discussions

The paper gives the method to build the simultaneous grey model SGM

(1, 2)

. We can see from model's structure that for

x_{1}^{(1)} (t)

, the single GM

(1, 1)

model's whitening equation is

\frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{0} .

For

x_{2}^{(1)} (t)

, the single GM(1, 1) model's whitening equation is

\frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{2}^{(1)} (t) + b_{0} .

The simultaneous model SGM

(1, 2)

's whitening equation is

{\begin{cases} \frac{d x_{1}^{(1)} (t)}{d t} = a_{1} x_{1}^{(1)} (t) + a_{2} x_{2}^{(1)} (t) + a_{0} + a_{3} t + a_{4} t^{2}, \\ \frac{d x_{2}^{(1)} (t)}{d t} = b_{1} x_{1}^{(1)} (t) + b_{2} x_{2}^{(1)} (t) + b_{0} + b_{3} t + b_{4} t^{2}, \end{cases}

so the simultaneous grey model SGM

(1, 2)

built for

x^{(1)} (t), x^{(2)} (t)

is extended from the single GM

(1, 1)

model. The simultaneous grey model SGM

(1, 2)

built in this way has the precision significantly higher than that of the single GM

(1, 1)

model. It has been verified by the two examples.

In addition, for two interactive variables, if build a single model, the model is GM

(1, 2)

, i.e. and respectively. If the model is simultaneous, it overcomes the single GM

(1, 2)

model's defect of no exact solution (the single model only has the approximate solution).

Therefore, building a simultaneous grey model SGM

(1, 2)

has the following advantages: First, the model can predict two or more interactive variables simultaneously; second, the model has the modeling precision higher than that of the single GM

(1, 1)

model; third, the model solves the single GM

(1, 2)

model's problem of no exact solution for time response equation.

6 Conclusion

The grey prediction model has many forms but their whitening equations are generally single differential equations and corresponding grey differential equations are single equations. In consideration of the interactions of variables, the paper proposes the simultaneous grey model, gives a simultaneous grey model SGM

(1, 2)

with 2 interactive variables and its parameter estimation method and time response equation. The example shows that the simultaneous grey model has high precision and improves precision significantly compared with the conventional grey model. The method proposed can be further generalized and applied to the simultaneous grey models with more than 2 variables.

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Funding

the National Natural Science Foundation of China(11401418)

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Abstract
Key words
Cite this article
1 Introduction
2 The Form of Simultaneous Grey Model SGM(1, 2)
3 Modeling Method of Simultaneous Grey Model SGM(1, 2)
3.1 Time Response Equation of Simultaneous Grey Model SGM(1, 2)
3.2 Parameter Estimation of Simultaneous Grey Model SGM(1, 2)
4 Modeling Examples of Simultaneous Grey Model
4.1 Example 1
Table 1 Related data of China's GDP & energy consumption and modeling results of GM(1, 1)
Table 2 Modeling results of simultaneous grey model of China's GDP and energy consumption
Table 3 Modeling results of Grey models proposed by other references
Figure 1 The histogram of average relative errors of four models built for x1(0)
Figure 2 The histogram of average relative errors of four models built for x2(0)
4.2 Example 2
Table 4 Modeling results of the grey model GM(1, 1) of China's private car ownership and the added value of the transportation industry
Table 5 Modeling results of simultaneous grey model of China's private car ownership and the added value of transportation industry
5 Discussions
6 Conclusion
References
Funding

Received	Accepted	Published
2022-03-11	2022-05-17	2022-10-25
Issue Date
2022-11-04

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Abstract

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Cite this article

1 Introduction

2 The Form of Simultaneous Grey Model SGM(1, 2)

3 Modeling Method of Simultaneous Grey Model SGM(1, 2)

3.1 Time Response Equation of Simultaneous Grey Model SGM(1, 2)

3.2 Parameter Estimation of Simultaneous Grey Model SGM(1, 2)

4 Modeling Examples of Simultaneous Grey Model

4.1 Example 1

Table 1 Related data of China's GDP & energy consumption and modeling results of GM(1, 1)

Table 2 Modeling results of simultaneous grey model of China's GDP and energy consumption

Table 3 Modeling results of Grey models proposed by other references

Figure 1 The histogram of average relative errors of four models built for x₁⁽⁰⁾

Figure 2 The histogram of average relative errors of four models built for x₂⁽⁰⁾

4.2 Example 2

Table 4 Modeling results of the grey model GM(1, 1) of China's private car ownership and the added value of the transportation industry

Table 5 Modeling results of simultaneous grey model of China's private car ownership and the added value of transportation industry

5 Discussions

6 Conclusion

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References

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Footnotes

Funding

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Abstract

Key words

Cite this article

1 Introduction

2 The Form of Simultaneous Grey Model SGM(1, 2)

3 Modeling Method of Simultaneous Grey Model SGM(1, 2)

3.1 Time Response Equation of Simultaneous Grey Model SGM(1, 2)

3.2 Parameter Estimation of Simultaneous Grey Model SGM(1, 2)

4 Modeling Examples of Simultaneous Grey Model

4.1 Example 1

Table 1 Related data of China's GDP & energy consumption and modeling results of GM(1, 1)

Table 2 Modeling results of simultaneous grey model of China's GDP and energy consumption

Table 3 Modeling results of Grey models proposed by other references

Figure 1 The histogram of average relative errors of four models built for x1(0)

Figure 2 The histogram of average relative errors of four models built for x2(0)

4.2 Example 2

Table 4 Modeling results of the grey model GM(1, 1) of China's private car ownership and the added value of the transportation industry

Table 5 Modeling results of simultaneous grey model of China's private car ownership and the added value of transportation industry

5 Discussions

6 Conclusion

{{custom_sec.title}}

{{custom_sec.title}}

References

{{custom_fnGroup.title_en}}

Footnotes

Funding

Figure 1 The histogram of average relative errors of four models built for x₁⁽⁰⁾

Figure 2 The histogram of average relative errors of four models built for x₂⁽⁰⁾