# A Novel Discrete DGM (2, 1,t2) Model Based on Stochastic Oscillation Sequence and Its Application

Authors: Jun-Lin Xu; You-Jun Chen
Journal: Journal of Mathematics
 (2022)

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## Abstract

This article aims to improve the prediction accuracy of the grey model for stochastic oscillation sequence, to overcome the defect of the direct span from difference to differential and to better depict the growth trend of data series. Firstly, the accelerated translation smoothing transformation is constructed, which transforms the stochastic oscillation sequence into the monotonic increasing sequence that is suitable for modeling. Secondly, the quadratic time power term is introduced to establish a novel discrete DGM (2, 1,t2) model, then derive a novel simulation prediction formula, and analyze properties of this novel model; it was concluded that its application scope is expanded. Finally, a novel discrete DGM (2, 1, t2) model based on a stochastic oscillation sequence is established to forecast a group of stochastic oscillation sequences with different amplitudes and two different city traffic flow series and compare the prediction accuracy with other grey models; the results show that the feasibility of this model is verified and has good simulated prediction effect in the field of traffic flow.


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## Body

## 1. Introduction

The grey prediction model is an important component of the grey system theory [1], and the DGM (2, 1) model has been widely used in various fields [2–6]. Moreover, many scholars have performed different aspects of research and improvement for the model: optimizing the grey derivative of the DGM (2, 1) model [7], the parameter optimization for the DGM (2, 1) model [8], and studying the predictive performance of the DGM (2, 1) model and its improved models [9, 10].For the traditional grey model, system parameters are estimated by the differential equation; however, the time response is estimated by the whitening differential equation; this leap from difference directly to differential will produce large errors. With the deepening of research, some scholars have found that discrete models can overcome this defect, and applied the discrete ideas to various grey models [11–14].Owing to a wide variety of factors, the sequences are mostly stochastic oscillation sequences in some practical applications, and the simulated prediction accuracy for those nonmonotonic sequences is not ideal. Therefore, in order to make up for this defect, scholars have researched both data and models, respectively.


### 1.1. Data Aspects

Scholars have performed data treatment, which improves the smoothness of the stochastic oscillation sequence and presented various data transformations: a weighting mean value generating transformation, a translation transformation, and a geometric mean transformation, which maintains the monotony, weakens the randomness, and improves the smoothness of the original sequence [15–17]; a smoothness operator can compress the amplitude of the stochastic oscillation sequence [18–20]; an empirical mode decomposition (EDM) algorithm is introduced to decompose the original stochastic oscillation sequences data information [21].


### 1.2. Model Aspects

Some scholars make up for this deficiency with other different modeling ideas: a new grey interval model is constructed by building envelope lines of the original oscillation sequence [22–24], the structure of the grey prediction model is improved by introducing periodic operator of triangular function [25], and an optimizing AF-MNGBM(1, 1) model is established by combining a self-adaptive artificial fish swarm algorithm (SAFSA) and the metabolic method, to overcome the randomness of the stochastic oscillation sequences [26].The DGM (2, 1) model is a class of monotonic second-order linear dynamic models that will fit better to data with strongly changing trends. So, in this article, the DGM (2,1) model is used to simulate the stochastic oscillation sequence. Firstly, this article applies an accelerated translation smoothing transformation to the original stochastic oscillation sequence, turning it into a monotonic increasing sequence. Then, in view of the error problem of the transition from difference to the differential of the model and the advantage of introducing the time power in the model to characterize the increasing trend of its data series [27], a model’s structure is an important factor affecting the performance of the model. The structure of the existing DGM (2, 1) grey prediction models is simple, which leads to poor simulation and prediction performance of the model [28]; the discrete DGM (2, 1, t2) model is constructed. Furthermore, this article derives the simulation prediction formula of the model and analyzes the property of this model, and the scope of application is expanded for this novel model. Finally, this novel model is built based on a batch of stochastic oscillation sequences with different amplitudes and two city traffic flow, the prediction effect of this novel model and the other four grey models is compared, and the result indicated that this model has the feasibility and good simulated prediction effect.


## 1.1. Data Aspects

Scholars have performed data treatment, which improves the smoothness of the stochastic oscillation sequence and presented various data transformations: a weighting mean value generating transformation, a translation transformation, and a geometric mean transformation, which maintains the monotony, weakens the randomness, and improves the smoothness of the original sequence [15–17]; a smoothness operator can compress the amplitude of the stochastic oscillation sequence [18–20]; an empirical mode decomposition (EDM) algorithm is introduced to decompose the original stochastic oscillation sequences data information [21].


## 1.2. Model Aspects

Some scholars make up for this deficiency with other different modeling ideas: a new grey interval model is constructed by building envelope lines of the original oscillation sequence [22–24], the structure of the grey prediction model is improved by introducing periodic operator of triangular function [25], and an optimizing AF-MNGBM(1, 1) model is established by combining a self-adaptive artificial fish swarm algorithm (SAFSA) and the metabolic method, to overcome the randomness of the stochastic oscillation sequences [26].The DGM (2, 1) model is a class of monotonic second-order linear dynamic models that will fit better to data with strongly changing trends. So, in this article, the DGM (2,1) model is used to simulate the stochastic oscillation sequence. Firstly, this article applies an accelerated translation smoothing transformation to the original stochastic oscillation sequence, turning it into a monotonic increasing sequence. Then, in view of the error problem of the transition from difference to the differential of the model and the advantage of introducing the time power in the model to characterize the increasing trend of its data series [27], a model’s structure is an important factor affecting the performance of the model. The structure of the existing DGM (2, 1) grey prediction models is simple, which leads to poor simulation and prediction performance of the model [28]; the discrete DGM (2, 1, t2) model is constructed. Furthermore, this article derives the simulation prediction formula of the model and analyzes the property of this model, and the scope of application is expanded for this novel model. Finally, this novel model is built based on a batch of stochastic oscillation sequences with different amplitudes and two city traffic flow, the prediction effect of this novel model and the other four grey models is compared, and the result indicated that this model has the feasibility and good simulated prediction effect.


## 2. Principle of the Stochastic Oscillation Sequence and the Data Transformation

### 2.1. Principle of the Stochastic Oscillation Sequence

Definition 1.
(see [1]). Assume that a sequence Y=y1,y2,⋯,yn is given, then(1)
If for∀k∈2,3,⋯,n, yk−yk−1>0, then Y is called a monotonic increasing sequence.(2)
If for∀k∈2,3,⋯,n, yk−yk−1<0, then Y is called a monotonic decreasing sequence.(3)
If for∃k,k′∈2,3,⋯,n, yk−yk−1>0 and yk′−yk′−1<0, then Y is called a stochastic oscillation sequence. AndM=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, then M−m is called the amplitude of Y.


### 2.2. Accelerated Translation Smoothing Transformation

Definition 2.
(see [15]). Assume that a sequence Y=y1,y2,⋯,yn, T is the amplitude ofY, then(1)f1yk=yk+k−1T,is called accelerated translation transformation, where(2)T=M−m,M=maxyk|k=1,2,⋯,n,m=minyk|k=1,2,⋯,n.Definition 3.
(see [18]). Let X=y1,y2,⋯,yn, yk>0, k=1,2,⋯,n−1 be a sequence, and another sequence(3)YD=y1d,y2d,⋯,ynd,k=1,2,⋯,n−1,where(4)ykd=yk+T+yk+1+T4,where T=M−m is the amplitude ofY, M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, D is a sequence operator and called a (first-order) smoothness operator ofY, and YD is called a smoothness sequence of Y.Definition 4.
Assume that a sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1, and T=M−mT>1, where M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, then(5)f2yk=yk+k−1T+yk+1+kT4,is called accelerated translation smoothing transformation.
And(6)f2yk=yk+k−1T+yk+1+kT4=14yk+yk+1+2k−1T,where 4>0; the discrete grey model will not change the relative error of the simulation and will not change the simulation and prediction effect of the model [29]. The following calculation example analysis also proves this.Theorem 1.
For any sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1, the sequence obtained after the accelerated translation smoothing transformation is monotonic increasing ones, namely,(7)xk<xk+1,k=1,2,⋯,n−1,where X=x1,x2,⋯,xn is transformed sequence, and(8)xk=f2yk=yk+k−1T+yk+1+kT4.Proof.
Assume that a sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1T=M−m, where M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n; according to (5), we get(9)xk−xk+1=f2yk−f2yk+1=yk+yk+1+2k−1T4−yk+1+yk+2+2k+1T4=yk−yk+2−2T4,and yk−yk+2≤T, therefore xk<xk+1, k=1,2,⋯,n−1.


## 2.1. Principle of the Stochastic Oscillation Sequence

Definition 1.
(see [1]). Assume that a sequence Y=y1,y2,⋯,yn is given, then(1)
If for∀k∈2,3,⋯,n, yk−yk−1>0, then Y is called a monotonic increasing sequence.(2)
If for∀k∈2,3,⋯,n, yk−yk−1<0, then Y is called a monotonic decreasing sequence.(3)
If for∃k,k′∈2,3,⋯,n, yk−yk−1>0 and yk′−yk′−1<0, then Y is called a stochastic oscillation sequence. AndM=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, then M−m is called the amplitude of Y.


## 2.2. Accelerated Translation Smoothing Transformation

Definition 2.
(see [15]). Assume that a sequence Y=y1,y2,⋯,yn, T is the amplitude ofY, then(1)f1yk=yk+k−1T,is called accelerated translation transformation, where(2)T=M−m,M=maxyk|k=1,2,⋯,n,m=minyk|k=1,2,⋯,n.Definition 3.
(see [18]). Let X=y1,y2,⋯,yn, yk>0, k=1,2,⋯,n−1 be a sequence, and another sequence(3)YD=y1d,y2d,⋯,ynd,k=1,2,⋯,n−1,where(4)ykd=yk+T+yk+1+T4,where T=M−m is the amplitude ofY, M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, D is a sequence operator and called a (first-order) smoothness operator ofY, and YD is called a smoothness sequence of Y.Definition 4.
Assume that a sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1, and T=M−mT>1, where M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n, then(5)f2yk=yk+k−1T+yk+1+kT4,is called accelerated translation smoothing transformation.
And(6)f2yk=yk+k−1T+yk+1+kT4=14yk+yk+1+2k−1T,where 4>0; the discrete grey model will not change the relative error of the simulation and will not change the simulation and prediction effect of the model [29]. The following calculation example analysis also proves this.Theorem 1.
For any sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1, the sequence obtained after the accelerated translation smoothing transformation is monotonic increasing ones, namely,(7)xk<xk+1,k=1,2,⋯,n−1,where X=x1,x2,⋯,xn is transformed sequence, and(8)xk=f2yk=yk+k−1T+yk+1+kT4.Proof.
Assume that a sequenceY=y1,y2,⋯,yn, yk>0k=1,2,⋯,n−1T=M−m, where M=maxyk|k=1,2,⋯,n, m=minyk|k=1,2,⋯,n; according to (5), we get(9)xk−xk+1=f2yk−f2yk+1=yk+yk+1+2k−1T4−yk+1+yk+2+2k+1T4=yk−yk+2−2T4,and yk−yk+2≤T, therefore xk<xk+1, k=1,2,⋯,n−1.


## 3. A New Discrete DGM (2, 1,t2) Model Based on Stochastic Oscillation Sequence

### 3.1. The Construction of the Discrete DGM (2, 1,t2) Model Based on Stochastic Oscillation Sequence

The system parameters are estimated by the difference equations, and the time response is estimated by the whitening differential equations in the original DGM (2, 1) model; crossing directly from the differential equation to the differential equation will cause errors. Moreover, there is the problem of a constant growth rate of the simulated values arises for the original DGM(2, 1) model [30], and according to the modeling mechanism in literature [31], the time power is introduced to establish a discrete DGM (2, 1, t2) model to achieve the unification of the estimated system parameters and the time response formula. Moreover, modeled with a nonnegative smoothing sequence X0=x01,x02,⋯,x0n, a sequence is formed by an accelerated translation smoothing transformation for a stochastic oscillation sequence.Definition 5.
LetX0=x01,x02,⋯,x0n be a transformed nonnegative smoothing sequence. Its 1-AGO (accumulating generation operator) sequence is X1=x11,x12,⋯,x1n, where x1k=∑i=1kx0i, k=1,2,⋯,n, then(10)d2x1dt2+adx1dt=bk2+ck+d,is called the whitening differential equation of the DGM(2, 1 t2) model.
According to the discrete whitening differential equation by the second-order difference of the 1-AGO sequenceX1 ofX0 [32] and according to the modeling mechanism of literature [33], the derivative in (10) can be approximated by(11)dx1dt|t=k≈limΔtΔx1tΔt|t=k=x1k−x1k−1k−k−1=x1k−x1k−1,d2x1dt2|t=k≈dx1kdt−dx1k−1dt.
Then, substituting them into (10), we get(12)x1k=β1x1k−1+β2x1k−2+β3k2+β4k+β5,which is called discrete DGM(2, 1, t2) model where(13)β1=2+a1+a,β2=−11+a,β3=b1+a,β4=c1+a,β5=d1+a.Theorem 2.
Assume that a nonnegative sequenceX0=x01,x02,⋯,x0n, then another sequence X1=x11,x12,⋯,x1n is the 1-AGO series of X0, where x1k=∑i=1kx0i, k=1,2,⋯,n and then the parameter column β^=β1,β2,β3,β4,β5T, and(14)B=x12x113231x13x124241⋮⋮⋮⋮⋮x1n−1x1n−2n2n1,Y=x13x14⋮x1n,then from the least square estimation, parameters of the discrete DGM (2, 1 t2) model are satisfied:(15)β^=β1,β2,β3,β4,β5T=BTB−1BTY.Proof.
According to (12) in Definition 5, we get(16)x13=β1x12+β2x11+β332+β43+β5x14=β1x13+β2x12+β342+β44+β5⋯x1n=β1x1n−1=+β2x1n−2+β3n2+β4n+β5,then Y=Bβ^=Bβ1,β2,β3,β4,β5T, error sequence ε=Y−Bβ^ can be obtained, and then assuming(17)s=εTε=Y−Bβ^TY−Bβ^=∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β52,we get(18)∂s∂β1=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5x1k−1=0,∂s∂β2=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5x1k−2=0,∂s∂β3=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5k2=0,∂s∂β4=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5k=0,∂s∂β5=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5=0,and by means of the above system of homogeneous linear equations, we can get the parameter column β^=β1,β2,β3,β4,β5T; then, from Y=Bβ^=Bβ1,β2,β3,β4,β5T, we can get β^=BTB−1BTY.


### 3.2. Simulation Prediction Formula in the General Term Form of the Discrete DGM (2, 1,t2) Model

The discrete model in this paper is a recursive form of the simulation prediction formula, and it can be directly used for simulation prediction. However, such formulations are more complex and may bring relatively large cumulative errors for medium- and long-term predictions. In addition, the simulation prediction formula in the recursive form will directly mask the formal features of the sequence itself [32]. If medium- and long-term prediction is needed in the case, it is necessary to derive the simulation prediction formula in the form of model general term which overcomes the above defects and is completely equivalent to the discrete model.Theorem 3.
Assume that a nonnegative sequenceX0=x01,x02,⋯,x0n, its 1-AGO is X1=x11,x12,⋯,x1n, where x1k=∑i=1kx0i, k=1,2,⋯,n, and the parameter column β^=β1,β2,β3,β4,β5T as shown above, if parametersβ1,β2 satisfy λ2−β1λ−β2=0 and β12+4β2≥0, this equation has two real number roots λ1 and λ2. The simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model is as follows:(1)
Whenλ1, λ2≠1, the simulation prediction formula for the general term form is(19)x1k=μ1k2+μ2k+μ3+μ4λ2k+μ5λ2k,k=3,4,⋯,n,where(20)μ1=β31−β1−β2,μ2=−2β3β1+2β2+β41−β1−β21−β1−β22,μ3=β1μ1−μ2+2β22μ1−μ2+β51−β1−β22,μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.(2)
When one of the values is 1 inλ1, λ2, suppose that λ1=1, so according to the Vieta theorem λ1+λ2=β1 andλ1λ2=−β2, we get λ2=−β2=β1−1, and then the simulation prediction formula for the general term form is(21)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n,where(22)μ1=β36−3β1,μ2=β33β1−44−6β12−β1,μ3=β1μ2−μ1+4β4μ2−2μ1+β5β1+2β2,μ4=x12−x11β1−1−−β1+9μ1−−β1+5μ2−−β1+3μ32−β1,μ5=x12−x11−3μ1−μ21−β12−β1.(3)
Whenλ1=λ2=1, according to the Vieta theorem, we get β1=2 and β2=−1; then, the simulation prediction formula for the general term form is(23)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n,where(24)μ1=β312,μ2=β33,μ3=β1μ1−μ2+8β22μ1−μ2+β5−β1−4β2,μ4=x11−x12−15μ1−7μ2−3μ3,μ5=2x11−x12+14μ1+6μ2+2μ3.Proof.
According to Definition5 mentioned above and the discrete DGM(2, 1 t2) model as (12), suppose that the concomitant homogeneous recursive equations arex1t=β1x1t−1+β2x1t−2, then Ft=β3t2+β4t+β5 is the free term. And parameters β1 and β1 satisfy (β1)2 + 4β2 ≥ 0 ; thus, the characteristic equation λ2−β1λ−β1 = 0 has two real number roots λ1 and λ2, and according to the Vieta theorem, the two roots satisfy λ1+λ2 = β1 and λ1λ2 = β2.(1)
Whenλ1, λ2≠1, then special solution form of the model equation is(25)x∗t=μ1t2+μ2t+μ3.(i)
Taking the abovex∗t into the original model equation, we get(26)μ1t2+μ2t+μ3=β1μ1t−12+μ2t−1+μ3+β2μ1t−22+μ2t−2+μ3+β3t2+β4t+β5.So,(27)μ1=β31−β1−β2,μ2=−2β3β1+2β2+β41−β1−β21−β1−β22,μ3=β1μ1−μ2+2β22μ1−μ2+β51−β1−β22.(ii)
Two real number roots of the characteristic equationλ2−β1λ−β2=0 are as follows:(28)λ1=β1−β12+4β22.Then, the solution of the model equation may be expressed as follows:(29)x1t=μ1t2+μ2t+μ3+μ4λ1t+μ5λ2t.(iii)
Taking the initial valuex11 and x12 into the abovex1t, we get(30)x11=μ4λ1+μ5λ2+μ1+μ2+μ3,x12=μ4λ12+μ5λ22+4μ1+2μ2+μ3.We get(31)μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.So, the simulation prediction formula for the general term form of the discrete DGM(2,1t2) model can be represented as follows:(32)x1k=μ1k2+μ2k+μ3+μ4λ1k+μ5λ2k,k=3,4,⋯,n.(3)
When one of these values is 1 inλ1, λ2, suppose that λ1=1, according to the Vieta theorem λ1+λ2=β1 andλ1λ2=−β2, we get λ2=−β2=β1−1 and then special solution form of the model equation is(33)x∗t=μ1t2+μ2t+μ3t.According to the calculation process (i) of the abovementioned (1), then getμ1, μ2, and μ3.The solution of the model equation may be expressed as follows:(34)x1t=μ1t2+μ2t+μ3t+μ4+μ5β1−1t.According to the calculation process (ii) and (iii) of the abovementioned (1), we getμ4 and μ5, so the simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model can be represented as follows:(35)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n.(4)
Whenλ1=λ2=1, according to the Vieta theorem, get β1=2 and β2=−1, then the special solution form of the model equation is(36)x∗t=μ1t2+μ2t+μ3t2.
According to the calculation process (i) of the abovementioned (1), then we getμ1, μ2, and μ3.
The solution of the model equation may be expressed as follows:(37)x1t=μ1t2+μ2t+μ3t2+μ4t+μ5.
According to the calculation process (ii) and (iii) of the abovementioned (1), getμ4 and μ5, so the simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model can be represented as(38)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n.
Note: as long asa≠−1, it must exist β12+4β2≥0; then, for this situation, β12+4β2<0 is left in another article for discussion.Inference 1.
If the sequencex1k=α1k2+α2k+α3 is that of quadratic functional type, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1 t2) model has whiteness coincidence for a sequence that x0k=ξ1k+ξ2, k=1,2,⋯,n, namely, x^0k=x0k.Proof.
Letx0k=ξ1k+ξ2, k=1,2,⋯,n, be the original sequence, then its 1-AGO sequence is x1k=α1k2+α2k+α3, k=3,4,⋯,n, where α1=ξ1/2, α2=ξ1/2+ξ2, and α3=0; according to the process of (1) in Theorem 3, we can get both roots of the characteristic equation as 0, so the simulation prediction formula for the general term form is x1k=α1k2+α2k+α3, k=3,4,⋯,n. Finally, the inverse accumulating restored value is(39)x^0k=x^1k−x^1k−1=ξ1k+ξ2=x0k.Inference 2.
Ifx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,nλ1, λ2≠0,1 is the original sequence, then the simulation prediction formula of the general term form of this discrete DGM (2, 1, t2) model has whiteness coincidence for sequences that strictly obey this functional form.Proof.
Letx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n, λ1, λ2≠0,1, be the original sequence, then its 1-AGO sequence still satisfies x1k=α1′λ1k+α2′λ2k+α3′k2+α4′k+α5′, k=1,2,⋯,n. According to the process of (1) in Theorem 3, it is easy to get the simulation value obtained from the simulation prediction formula in the form of the discrete DGM (2, 1, t2) model which highly coincides with the actual value. Therefore, there is high simulation prediction accuracy for sequences approximating this form.Inference 3.
Ifx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n, β≠0,1 is the original sequence, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1, t2) model has whiteness coincidence for sequences that strictly obey this functional form.
The proof process is the same as in (2) in Theorem3 and inference 2.Inference 4.
If the sequencex0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n is the original sequence, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1 t2) model has whiteness coincidence for sequences that strictly obey this functional form.
The proof process is the same as in (3) in Theorem3 and inference 2.


### 3.3. The Property of the Discrete DGM (2, 1,t2) Model

Property 1.
discrete DGM (2, 1,t2) model is suitable for sequence modeling with approximate homogeneous exponential laws, namely, x0t=meθt.Property 2.
The discrete DGM (2, 1,t2) model is suitable for sequence modeling with approximate in-homogeneous exponential laws, namely, x0t=meθt+c.
According to properties1 and 2 mentioned above, it can be shown that the discrete model is equally applicable to high-growth sequences.Property 3.
Whena=−1, then the discrete DGM (2, 1, t2) model is x1k=x1k−1+η1k2+η2k+η3; hence, the simulation prediction formula for the general term form is(40)x1k=x11+η1k+1k+22k+36−1+η2k1+k2−1+k−1η3.Property 4.
Ifa=−2, then β1=0 and β2=1, and the discrete DGM (2, 1, t2) model is x1k=x1k−2+β3k2+β4; according to Theorem 3, one of the two real number roots λ1 andλ2 in the concomitant homogeneous recursive equations these values is 1; hence, this is a special circumstance for (2) in Theorem 3.Property 5.
Ifc=0, then β4=0, and the discrete DGM (2, 1, t2) model is x1k=β1x1k−1+β2x1k−2+β3k2+β5; according to Theorem 3, we can get the following.
Whenλ1, λ2≠1, the simulation prediction formula for the general term form is(41)x1k=μ1k2+μ2k+μ3+μ4λ1k+μ5λ2k,k=3,4,⋯,n,where(42)μ1=β31−β1−β2,μ2=2β3β1+2β21−β1−β22,μ3=β3β1+4β21−β1−β2+2β3β1+2β22+β51−β1−β221−β1−β23,μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.
When one of the values is 1 inλ1, λ2, then the simulation prediction formula for the general term form is(43)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n,where(44)μ1=β36−3β1,μ2=β33β1−44−6β12−β1,μ3=−β3−7β1+82−3β1−β33β1−42+2β52−3β12−β122−3β12−β12,μ4=x12−x11β1−1−−β1+9μ1−−β1+5μ2−−β1+3μ32−β1,μ5=x12−x11−3μ1−μ21−β12−β1.
Whenλ1=λ2=1, then the simulation prediction formula for the general term form is(45)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n,where(46)μ1=β312,μ2=β33,μ3=5β3+6β512,μ4=x11−x12−15μ1−7μ2−3μ3,μ5=2x11−x12+14μ1+6μ2+2μ3.Property 6.
Whena=−1 and c=0, then the discrete DGM (2, 1, t2) model is x1k=x1k−1+η1k+12+η2; hence, the simulation prediction formula for the general term form is(47)x1k=x11+η1k+1k+22k+36−5+k−1η2.


## 3.1. The Construction of the Discrete DGM (2, 1,t2) Model Based on Stochastic Oscillation Sequence

The system parameters are estimated by the difference equations, and the time response is estimated by the whitening differential equations in the original DGM (2, 1) model; crossing directly from the differential equation to the differential equation will cause errors. Moreover, there is the problem of a constant growth rate of the simulated values arises for the original DGM(2, 1) model [30], and according to the modeling mechanism in literature [31], the time power is introduced to establish a discrete DGM (2, 1, t2) model to achieve the unification of the estimated system parameters and the time response formula. Moreover, modeled with a nonnegative smoothing sequence X0=x01,x02,⋯,x0n, a sequence is formed by an accelerated translation smoothing transformation for a stochastic oscillation sequence.Definition 5.
LetX0=x01,x02,⋯,x0n be a transformed nonnegative smoothing sequence. Its 1-AGO (accumulating generation operator) sequence is X1=x11,x12,⋯,x1n, where x1k=∑i=1kx0i, k=1,2,⋯,n, then(10)d2x1dt2+adx1dt=bk2+ck+d,is called the whitening differential equation of the DGM(2, 1 t2) model.
According to the discrete whitening differential equation by the second-order difference of the 1-AGO sequenceX1 ofX0 [32] and according to the modeling mechanism of literature [33], the derivative in (10) can be approximated by(11)dx1dt|t=k≈limΔtΔx1tΔt|t=k=x1k−x1k−1k−k−1=x1k−x1k−1,d2x1dt2|t=k≈dx1kdt−dx1k−1dt.
Then, substituting them into (10), we get(12)x1k=β1x1k−1+β2x1k−2+β3k2+β4k+β5,which is called discrete DGM(2, 1, t2) model where(13)β1=2+a1+a,β2=−11+a,β3=b1+a,β4=c1+a,β5=d1+a.Theorem 2.
Assume that a nonnegative sequenceX0=x01,x02,⋯,x0n, then another sequence X1=x11,x12,⋯,x1n is the 1-AGO series of X0, where x1k=∑i=1kx0i, k=1,2,⋯,n and then the parameter column β^=β1,β2,β3,β4,β5T, and(14)B=x12x113231x13x124241⋮⋮⋮⋮⋮x1n−1x1n−2n2n1,Y=x13x14⋮x1n,then from the least square estimation, parameters of the discrete DGM (2, 1 t2) model are satisfied:(15)β^=β1,β2,β3,β4,β5T=BTB−1BTY.Proof.
According to (12) in Definition 5, we get(16)x13=β1x12+β2x11+β332+β43+β5x14=β1x13+β2x12+β342+β44+β5⋯x1n=β1x1n−1=+β2x1n−2+β3n2+β4n+β5,then Y=Bβ^=Bβ1,β2,β3,β4,β5T, error sequence ε=Y−Bβ^ can be obtained, and then assuming(17)s=εTε=Y−Bβ^TY−Bβ^=∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β52,we get(18)∂s∂β1=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5x1k−1=0,∂s∂β2=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5x1k−2=0,∂s∂β3=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5k2=0,∂s∂β4=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5k=0,∂s∂β5=−2∑k=3nx1k−β1x1k−1−β2x1k−2−β3k2−β4k−β5=0,and by means of the above system of homogeneous linear equations, we can get the parameter column β^=β1,β2,β3,β4,β5T; then, from Y=Bβ^=Bβ1,β2,β3,β4,β5T, we can get β^=BTB−1BTY.


## 3.2. Simulation Prediction Formula in the General Term Form of the Discrete DGM (2, 1,t2) Model

The discrete model in this paper is a recursive form of the simulation prediction formula, and it can be directly used for simulation prediction. However, such formulations are more complex and may bring relatively large cumulative errors for medium- and long-term predictions. In addition, the simulation prediction formula in the recursive form will directly mask the formal features of the sequence itself [32]. If medium- and long-term prediction is needed in the case, it is necessary to derive the simulation prediction formula in the form of model general term which overcomes the above defects and is completely equivalent to the discrete model.Theorem 3.
Assume that a nonnegative sequenceX0=x01,x02,⋯,x0n, its 1-AGO is X1=x11,x12,⋯,x1n, where x1k=∑i=1kx0i, k=1,2,⋯,n, and the parameter column β^=β1,β2,β3,β4,β5T as shown above, if parametersβ1,β2 satisfy λ2−β1λ−β2=0 and β12+4β2≥0, this equation has two real number roots λ1 and λ2. The simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model is as follows:(1)
Whenλ1, λ2≠1, the simulation prediction formula for the general term form is(19)x1k=μ1k2+μ2k+μ3+μ4λ2k+μ5λ2k,k=3,4,⋯,n,where(20)μ1=β31−β1−β2,μ2=−2β3β1+2β2+β41−β1−β21−β1−β22,μ3=β1μ1−μ2+2β22μ1−μ2+β51−β1−β22,μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.(2)
When one of the values is 1 inλ1, λ2, suppose that λ1=1, so according to the Vieta theorem λ1+λ2=β1 andλ1λ2=−β2, we get λ2=−β2=β1−1, and then the simulation prediction formula for the general term form is(21)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n,where(22)μ1=β36−3β1,μ2=β33β1−44−6β12−β1,μ3=β1μ2−μ1+4β4μ2−2μ1+β5β1+2β2,μ4=x12−x11β1−1−−β1+9μ1−−β1+5μ2−−β1+3μ32−β1,μ5=x12−x11−3μ1−μ21−β12−β1.(3)
Whenλ1=λ2=1, according to the Vieta theorem, we get β1=2 and β2=−1; then, the simulation prediction formula for the general term form is(23)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n,where(24)μ1=β312,μ2=β33,μ3=β1μ1−μ2+8β22μ1−μ2+β5−β1−4β2,μ4=x11−x12−15μ1−7μ2−3μ3,μ5=2x11−x12+14μ1+6μ2+2μ3.Proof.
According to Definition5 mentioned above and the discrete DGM(2, 1 t2) model as (12), suppose that the concomitant homogeneous recursive equations arex1t=β1x1t−1+β2x1t−2, then Ft=β3t2+β4t+β5 is the free term. And parameters β1 and β1 satisfy (β1)2 + 4β2 ≥ 0 ; thus, the characteristic equation λ2−β1λ−β1 = 0 has two real number roots λ1 and λ2, and according to the Vieta theorem, the two roots satisfy λ1+λ2 = β1 and λ1λ2 = β2.(1)
Whenλ1, λ2≠1, then special solution form of the model equation is(25)x∗t=μ1t2+μ2t+μ3.(i)
Taking the abovex∗t into the original model equation, we get(26)μ1t2+μ2t+μ3=β1μ1t−12+μ2t−1+μ3+β2μ1t−22+μ2t−2+μ3+β3t2+β4t+β5.So,(27)μ1=β31−β1−β2,μ2=−2β3β1+2β2+β41−β1−β21−β1−β22,μ3=β1μ1−μ2+2β22μ1−μ2+β51−β1−β22.(ii)
Two real number roots of the characteristic equationλ2−β1λ−β2=0 are as follows:(28)λ1=β1−β12+4β22.Then, the solution of the model equation may be expressed as follows:(29)x1t=μ1t2+μ2t+μ3+μ4λ1t+μ5λ2t.(iii)
Taking the initial valuex11 and x12 into the abovex1t, we get(30)x11=μ4λ1+μ5λ2+μ1+μ2+μ3,x12=μ4λ12+μ5λ22+4μ1+2μ2+μ3.We get(31)μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.So, the simulation prediction formula for the general term form of the discrete DGM(2,1t2) model can be represented as follows:(32)x1k=μ1k2+μ2k+μ3+μ4λ1k+μ5λ2k,k=3,4,⋯,n.(3)
When one of these values is 1 inλ1, λ2, suppose that λ1=1, according to the Vieta theorem λ1+λ2=β1 andλ1λ2=−β2, we get λ2=−β2=β1−1 and then special solution form of the model equation is(33)x∗t=μ1t2+μ2t+μ3t.According to the calculation process (i) of the abovementioned (1), then getμ1, μ2, and μ3.The solution of the model equation may be expressed as follows:(34)x1t=μ1t2+μ2t+μ3t+μ4+μ5β1−1t.According to the calculation process (ii) and (iii) of the abovementioned (1), we getμ4 and μ5, so the simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model can be represented as follows:(35)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n.(4)
Whenλ1=λ2=1, according to the Vieta theorem, get β1=2 and β2=−1, then the special solution form of the model equation is(36)x∗t=μ1t2+μ2t+μ3t2.
According to the calculation process (i) of the abovementioned (1), then we getμ1, μ2, and μ3.
The solution of the model equation may be expressed as follows:(37)x1t=μ1t2+μ2t+μ3t2+μ4t+μ5.
According to the calculation process (ii) and (iii) of the abovementioned (1), getμ4 and μ5, so the simulation prediction formula for the general term form of the discrete DGM(2, 1 t2) model can be represented as(38)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n.
Note: as long asa≠−1, it must exist β12+4β2≥0; then, for this situation, β12+4β2<0 is left in another article for discussion.Inference 1.
If the sequencex1k=α1k2+α2k+α3 is that of quadratic functional type, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1 t2) model has whiteness coincidence for a sequence that x0k=ξ1k+ξ2, k=1,2,⋯,n, namely, x^0k=x0k.Proof.
Letx0k=ξ1k+ξ2, k=1,2,⋯,n, be the original sequence, then its 1-AGO sequence is x1k=α1k2+α2k+α3, k=3,4,⋯,n, where α1=ξ1/2, α2=ξ1/2+ξ2, and α3=0; according to the process of (1) in Theorem 3, we can get both roots of the characteristic equation as 0, so the simulation prediction formula for the general term form is x1k=α1k2+α2k+α3, k=3,4,⋯,n. Finally, the inverse accumulating restored value is(39)x^0k=x^1k−x^1k−1=ξ1k+ξ2=x0k.Inference 2.
Ifx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,nλ1, λ2≠0,1 is the original sequence, then the simulation prediction formula of the general term form of this discrete DGM (2, 1, t2) model has whiteness coincidence for sequences that strictly obey this functional form.Proof.
Letx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n, λ1, λ2≠0,1, be the original sequence, then its 1-AGO sequence still satisfies x1k=α1′λ1k+α2′λ2k+α3′k2+α4′k+α5′, k=1,2,⋯,n. According to the process of (1) in Theorem 3, it is easy to get the simulation value obtained from the simulation prediction formula in the form of the discrete DGM (2, 1, t2) model which highly coincides with the actual value. Therefore, there is high simulation prediction accuracy for sequences approximating this form.Inference 3.
Ifx0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n, β≠0,1 is the original sequence, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1, t2) model has whiteness coincidence for sequences that strictly obey this functional form.
The proof process is the same as in (2) in Theorem3 and inference 2.Inference 4.
If the sequencex0k=α1λ1k+α2λ2k+α3k2+α4k+α5, k=1,2,⋯,n is the original sequence, then the simulation prediction formula of the general term form of this the discrete DGM (2, 1 t2) model has whiteness coincidence for sequences that strictly obey this functional form.
The proof process is the same as in (3) in Theorem3 and inference 2.


## 3.3. The Property of the Discrete DGM (2, 1,t2) Model

Property 1.
discrete DGM (2, 1,t2) model is suitable for sequence modeling with approximate homogeneous exponential laws, namely, x0t=meθt.Property 2.
The discrete DGM (2, 1,t2) model is suitable for sequence modeling with approximate in-homogeneous exponential laws, namely, x0t=meθt+c.
According to properties1 and 2 mentioned above, it can be shown that the discrete model is equally applicable to high-growth sequences.Property 3.
Whena=−1, then the discrete DGM (2, 1, t2) model is x1k=x1k−1+η1k2+η2k+η3; hence, the simulation prediction formula for the general term form is(40)x1k=x11+η1k+1k+22k+36−1+η2k1+k2−1+k−1η3.Property 4.
Ifa=−2, then β1=0 and β2=1, and the discrete DGM (2, 1, t2) model is x1k=x1k−2+β3k2+β4; according to Theorem 3, one of the two real number roots λ1 andλ2 in the concomitant homogeneous recursive equations these values is 1; hence, this is a special circumstance for (2) in Theorem 3.Property 5.
Ifc=0, then β4=0, and the discrete DGM (2, 1, t2) model is x1k=β1x1k−1+β2x1k−2+β3k2+β5; according to Theorem 3, we can get the following.
Whenλ1, λ2≠1, the simulation prediction formula for the general term form is(41)x1k=μ1k2+μ2k+μ3+μ4λ1k+μ5λ2k,k=3,4,⋯,n,where(42)μ1=β31−β1−β2,μ2=2β3β1+2β21−β1−β22,μ3=β3β1+4β21−β1−β2+2β3β1+2β22+β51−β1−β221−β1−β23,μ4=λ2x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ12,μ5=λ1x11−μ1−μ2−μ3−x12+4μ1+2μ2+μ3−β2−λ22.
When one of the values is 1 inλ1, λ2, then the simulation prediction formula for the general term form is(43)x1k=μ1k3+μ2k2+μ3k+μ4+μ5β1−1k,k=3,4,⋯,n,where(44)μ1=β36−3β1,μ2=β33β1−44−6β12−β1,μ3=−β3−7β1+82−3β1−β33β1−42+2β52−3β12−β122−3β12−β12,μ4=x12−x11β1−1−−β1+9μ1−−β1+5μ2−−β1+3μ32−β1,μ5=x12−x11−3μ1−μ21−β12−β1.
Whenλ1=λ2=1, then the simulation prediction formula for the general term form is(45)x1k=μ1k4+μ2k3+μ3k2+μ4k+μ5,k=3,4,⋯,n,where(46)μ1=β312,μ2=β33,μ3=5β3+6β512,μ4=x11−x12−15μ1−7μ2−3μ3,μ5=2x11−x12+14μ1+6μ2+2μ3.Property 6.
Whena=−1 and c=0, then the discrete DGM (2, 1, t2) model is x1k=x1k−1+η1k+12+η2; hence, the simulation prediction formula for the general term form is(47)x1k=x11+η1k+1k+22k+36−5+k−1η2.


## 4. Algorithmic Steps for the Novel Discrete DGM (2, 1t2) Model Based on Stochastic Oscillation Sequence

Assume thatY=y1,y2,⋯,yn is the original stochastic oscillation sequence, where yk>0, k=1,2,⋯,n, the algorithmic procedure is shown in Figure 1.Figure 1
Flow chart of the algorithmic steps.


## 5. The Instance Analysis

### 5.1. Example Analysis

According to literature [8], assume that the data sequence of the annual revenue in 2005–2011 years is X where(48)X=0.874,1.278,1.390,1.679,1.770,1.800,1.915.Based on the sequenceX, there are four stochastic oscillation sequences Xk, k=1,2,3,4, with amplitudes of 1, 2, 3, and 4 where(49)X1=0.874,1.874,1.390,1.679,1.770,1.800,1.915,X2=0.874,1.278,1.390,2.874,1.770,1.800,1.915,X3=0.874,1.278,3.874,1.679,1.770,1.800,1.915,X4=0.874,1.278,1.390,1.679,4.874,1.800,1.915.Based on the four stochastic oscillation sequences described in this paper and modeling them based on the first six data, the last one data is predicted, the idea of controlling for different combinations of parameters is predicted according to literature [34], the prediction model of stochastic oscillation sequence based on amplitude compression (model 1) is established, and thus the following four models are established separately: the grey model of stochastic oscillation sequence based on amplitude compression (model 1), the discrete DGM (2, 1, t2) model based on the first-order smoothing operator (model 2), the DGM (1,1) model based on the accelerated translation smoothing transformation (model 3), and the discrete DGM (2, 1, t2) model based on stochastic oscillation sequence (model 4). According to the idea of special situation case analysis in literature [35], this paper takes the special cases as presented here (c=0 ), respectively. Both model 1 and model 2 are based on the same first-order smoothing operator data transformation, and model 3 and model 4 are modeled based on the same accelerated translation smoothing transformation. Model 1 and model 3 are DGM (1, 1) models based on the first-order smoothing operator and the accelerated translation smoothing transformation, respectively. Model 2 and model 4 are the discrete DGM (2, 1, t2) models in this paper based on the above two data transformations. The simulated prediction results of four models are shown in Table 1. Schematic diagrams are shown in Figures 2–5.Table 1
Comparison of modeling accuracy of four models based on four stochastic oscillation sequences with different amplitudes.
Sequence data (2005–2010)ModelSimulation valueMean relative error (%)Prediction value (2011)Mean prediction error (%)X110.874, 1.874, 1.262, 2.009, 1.400, 2.15211.5581.54719.23220.874, 1.874, 1.390, 1.679, 1.770, 1.8000.0001.9140.00130.874, 1.874, 1.802, 1.382, 1.650, 1.64810.4162.42926.81740.874, 1.874, 1.390, 1.679, 1.770, 1.8000.0001.9150.000X210.874, 1.278, 2.257, 1.395, 2.376, 1.51527.3192.49830.47120.874, 1.278, 1.390, 2.874, 1.770, 1.8000.0001.9140.00130.874, 1.278, 3.688, 0.015, 3.109, 0.28970.7384.451132.42340.874, 1.278, 1.390, 2.874, 1.770, 1.8000.0001.9150.000X310.874, 1.278, 4.033, 0.741, 3.521, 0.52340.8063.05759.64420.874, 1.278, 3.874, 1.679, 1.770, 1.8000.0001.9140.00130.874, 1.278, 5.446, −0.919, 4.168, −1.05581.5825.451184.62140.874, 1.278, 3.874, 1.679, 1.770, 1.8000.0001.9150.000X410.874, 1.278, 2.323, 1.731, 2.794, 2.22122.7713.30272.43920.874, 1.278, 1.390, 1.679, 4.874, 1.8000.0001.9160.00130.874, 1.278, 5.788, −1.364, 4.652, −0.572105.6627.914313.31040.874, 1.278, 1.390, 1.679, 4.874, 1.8000.0001.9150.000Figure 2
Schematic diagram of the predicted results for stochastic oscillation sequenceX1.Figure 3
Schematic diagram of the predicted results for stochastic oscillation sequenceX2.Figure 4
Schematic diagram of the predicted results for stochastic oscillation sequenceX3.Figure 5
Schematic diagram of the predicted results for stochastic oscillation sequenceX4.Table1 and Figures 2–5 show that the discrete DGM (2, 1, t2) model presented here is even more obviously effective based on the same data transformation, and the data transformation modeling presented in this paper with the discrete DGM (2, 1, t2) model simulated predictions accuracy is better. Although the simulated prediction accuracy of model 3 is not as good as model 1, as a whole, the simulation effect of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence is also more ideal for the other three types of models.


### 5.2. Applied Analysis

In recent years, with the development of the global economy, the expansion of urban scale, and the population explosion, there is the rise of traffic demand resulting in traffic congestion; this problem has become one of the most important problems faced by large- and medium-sized cities in the world. For various reasons, most of the sequence of traffic flow data is a stochastic oscillation sequence.


#### 5.2.1. Applied Analysis 1

According to literature [36], we can get the short-time traffic flow data from 7:00–7:30 am of the east straight road section in Nan tong city, Jiangsu Province from August 12 to August 25, 2018. The data are shown in Table 2. Using this data sequence as the original stochastic oscillation sequence and modeling it based on the first ten data, the last four data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 6 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions result of four models is shown in Table 3. Schematic diagrams of the predicted value of the four models are shown in Figure 7.Table 2
Traffic flow data veh/h.
TimeReal value8.121388.132938.142668.152058.162578.172708.181828.191368.201828.212278.222358.232308.242318.25183Figure 6
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 3
Comparison of the simulated predictions of the four models on short-time traffic flow.
Real value (8.12–8.21)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)138.00138.000.00138.000.00138.000.00138.000.00293.00293.000.00293.000.00293.000.00293.000.00266.00226.5914.81266.000.00434.3663.29266.000.00205.00233.0113.66216.225.47152.5725.57216.215.47257.00273.946.59228.6511.30319.0624.15228.6511.03270.00202.3625.05282.054.4666.0575.54282.054.46182.00170.276.44205.3612.84265.4945.88205.3612.83136.00239.6576.21100.7525.9150.1963.09100.7625.91182.00267.4946.97207.6914.12292.8360.89207.6914.11227.00203.7710.22237.034.42126.9644.07237.034.41Mean relative simulation error (%)20.007.8340.257.82Real value (8.22–8.25)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)235.00141.5039.78180.5123.19426.1981.36180.5023.18230.00116.6549.2863.3072.48325.1341.3617.0692.58231.00105.2254.45-40.26117.43698.53202.3960.5473.79183.0088.1951.80274.8450.19682.39272.89236.2829.11Figure 7
Schematic diagram of the predicted value of the four models on short-time traffic flow.According to the data of Table3 and Figure 7, overall, the simulation of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence is much smaller than the average relative error of the other three types of models; that is, the simulation effect will be more ideal. In conclusion, according to the simulation and prediction analysis of road short-time traffic flow in Nan tong city, Jiangsu Province, this new model is feasible and effective in practice.


#### 5.2.2. Applied Analysis 2

According to literature [37], the data of road traffic flow in New York City from 6 : 00 to 18 : 00 (New York time, with an interval of one hour) are shown in Table 4. Using this data sequence as the original sequence and modeling it based on the first eight data, the last five data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 8 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions results of four models are shown in Table 5. Schematic diagrams of the predicted value of the four models are shown in Figure 9.Table 4
One day 6 : 00–18 : 00 New York City road traffic flow veh/h.
Time pointData6 : 00379087 : 00712498 : 00769759 : 006828510 : 006331511 : 005678612 : 005364213 : 005317914 : 005419415 : 005861516 : 006459317 : 007007618 : 0074693Figure 8
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 5
Comparison of the simulated predictions of the four models on city traffic flow.
Real value (6 : 00–13 : 00)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)37908.0037908.000.0037908.000.0037908.000.0037908.000.0071249.0071249.000.0071249.000.0071249.000.0071249.000.0076975.0077695.740.9476975.000.00106884.3838.8676975.000.0068285.0062471.988.5168309.180.0338615.5441.9868309.170.0363315.0069257.989.3963364.370.0882572.4630.4263364.360.0756786.0054360.354.2756140.901.1423777.9258.1356140.861.1353642.0061459.8914.5755451.323.3776543.4442.6955451.323.3753179.0046863.6611.8850353.735.3129101.8745.2850353.725.31Mean relative simulation error (%)6.191.2432.171.23Real value (14 : 00–18 : 00)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)54194.0054252.940.1157429.006.0095007.8775.3157428.955.9658615.0039935.2931.8757023.302.7262775.777.0957148.192.5064593.0047592.3626.3172244.7411.84146285.98126.4772151.9011.7070076.0033532.1652.1481025.4315.63134429.8891.8381152.0415.80274693.0041436.7244.52107448.1143.85241524.31223.35107361.8643.73Figure 9
Schematic diagram of the predicted value of the four models on city traffic flow.According to the data of Table5 and Figure 9, in general, the simulation predictions of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence would be better than the other three types of models; that is, according to the simulation and prediction analysis of road traffic flow in New York City, this model is feasible and effective in practice.


## 5.1. Example Analysis

According to literature [8], assume that the data sequence of the annual revenue in 2005–2011 years is X where(48)X=0.874,1.278,1.390,1.679,1.770,1.800,1.915.Based on the sequenceX, there are four stochastic oscillation sequences Xk, k=1,2,3,4, with amplitudes of 1, 2, 3, and 4 where(49)X1=0.874,1.874,1.390,1.679,1.770,1.800,1.915,X2=0.874,1.278,1.390,2.874,1.770,1.800,1.915,X3=0.874,1.278,3.874,1.679,1.770,1.800,1.915,X4=0.874,1.278,1.390,1.679,4.874,1.800,1.915.Based on the four stochastic oscillation sequences described in this paper and modeling them based on the first six data, the last one data is predicted, the idea of controlling for different combinations of parameters is predicted according to literature [34], the prediction model of stochastic oscillation sequence based on amplitude compression (model 1) is established, and thus the following four models are established separately: the grey model of stochastic oscillation sequence based on amplitude compression (model 1), the discrete DGM (2, 1, t2) model based on the first-order smoothing operator (model 2), the DGM (1,1) model based on the accelerated translation smoothing transformation (model 3), and the discrete DGM (2, 1, t2) model based on stochastic oscillation sequence (model 4). According to the idea of special situation case analysis in literature [35], this paper takes the special cases as presented here (c=0 ), respectively. Both model 1 and model 2 are based on the same first-order smoothing operator data transformation, and model 3 and model 4 are modeled based on the same accelerated translation smoothing transformation. Model 1 and model 3 are DGM (1, 1) models based on the first-order smoothing operator and the accelerated translation smoothing transformation, respectively. Model 2 and model 4 are the discrete DGM (2, 1, t2) models in this paper based on the above two data transformations. The simulated prediction results of four models are shown in Table 1. Schematic diagrams are shown in Figures 2–5.Table 1
Comparison of modeling accuracy of four models based on four stochastic oscillation sequences with different amplitudes.
Sequence data (2005–2010)ModelSimulation valueMean relative error (%)Prediction value (2011)Mean prediction error (%)X110.874, 1.874, 1.262, 2.009, 1.400, 2.15211.5581.54719.23220.874, 1.874, 1.390, 1.679, 1.770, 1.8000.0001.9140.00130.874, 1.874, 1.802, 1.382, 1.650, 1.64810.4162.42926.81740.874, 1.874, 1.390, 1.679, 1.770, 1.8000.0001.9150.000X210.874, 1.278, 2.257, 1.395, 2.376, 1.51527.3192.49830.47120.874, 1.278, 1.390, 2.874, 1.770, 1.8000.0001.9140.00130.874, 1.278, 3.688, 0.015, 3.109, 0.28970.7384.451132.42340.874, 1.278, 1.390, 2.874, 1.770, 1.8000.0001.9150.000X310.874, 1.278, 4.033, 0.741, 3.521, 0.52340.8063.05759.64420.874, 1.278, 3.874, 1.679, 1.770, 1.8000.0001.9140.00130.874, 1.278, 5.446, −0.919, 4.168, −1.05581.5825.451184.62140.874, 1.278, 3.874, 1.679, 1.770, 1.8000.0001.9150.000X410.874, 1.278, 2.323, 1.731, 2.794, 2.22122.7713.30272.43920.874, 1.278, 1.390, 1.679, 4.874, 1.8000.0001.9160.00130.874, 1.278, 5.788, −1.364, 4.652, −0.572105.6627.914313.31040.874, 1.278, 1.390, 1.679, 4.874, 1.8000.0001.9150.000Figure 2
Schematic diagram of the predicted results for stochastic oscillation sequenceX1.Figure 3
Schematic diagram of the predicted results for stochastic oscillation sequenceX2.Figure 4
Schematic diagram of the predicted results for stochastic oscillation sequenceX3.Figure 5
Schematic diagram of the predicted results for stochastic oscillation sequenceX4.Table1 and Figures 2–5 show that the discrete DGM (2, 1, t2) model presented here is even more obviously effective based on the same data transformation, and the data transformation modeling presented in this paper with the discrete DGM (2, 1, t2) model simulated predictions accuracy is better. Although the simulated prediction accuracy of model 3 is not as good as model 1, as a whole, the simulation effect of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence is also more ideal for the other three types of models.


## 5.2. Applied Analysis

In recent years, with the development of the global economy, the expansion of urban scale, and the population explosion, there is the rise of traffic demand resulting in traffic congestion; this problem has become one of the most important problems faced by large- and medium-sized cities in the world. For various reasons, most of the sequence of traffic flow data is a stochastic oscillation sequence.


### 5.2.1. Applied Analysis 1

According to literature [36], we can get the short-time traffic flow data from 7:00–7:30 am of the east straight road section in Nan tong city, Jiangsu Province from August 12 to August 25, 2018. The data are shown in Table 2. Using this data sequence as the original stochastic oscillation sequence and modeling it based on the first ten data, the last four data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 6 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions result of four models is shown in Table 3. Schematic diagrams of the predicted value of the four models are shown in Figure 7.Table 2
Traffic flow data veh/h.
TimeReal value8.121388.132938.142668.152058.162578.172708.181828.191368.201828.212278.222358.232308.242318.25183Figure 6
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 3
Comparison of the simulated predictions of the four models on short-time traffic flow.
Real value (8.12–8.21)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)138.00138.000.00138.000.00138.000.00138.000.00293.00293.000.00293.000.00293.000.00293.000.00266.00226.5914.81266.000.00434.3663.29266.000.00205.00233.0113.66216.225.47152.5725.57216.215.47257.00273.946.59228.6511.30319.0624.15228.6511.03270.00202.3625.05282.054.4666.0575.54282.054.46182.00170.276.44205.3612.84265.4945.88205.3612.83136.00239.6576.21100.7525.9150.1963.09100.7625.91182.00267.4946.97207.6914.12292.8360.89207.6914.11227.00203.7710.22237.034.42126.9644.07237.034.41Mean relative simulation error (%)20.007.8340.257.82Real value (8.22–8.25)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)235.00141.5039.78180.5123.19426.1981.36180.5023.18230.00116.6549.2863.3072.48325.1341.3617.0692.58231.00105.2254.45-40.26117.43698.53202.3960.5473.79183.0088.1951.80274.8450.19682.39272.89236.2829.11Figure 7
Schematic diagram of the predicted value of the four models on short-time traffic flow.According to the data of Table3 and Figure 7, overall, the simulation of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence is much smaller than the average relative error of the other three types of models; that is, the simulation effect will be more ideal. In conclusion, according to the simulation and prediction analysis of road short-time traffic flow in Nan tong city, Jiangsu Province, this new model is feasible and effective in practice.


### 5.2.2. Applied Analysis 2

According to literature [37], the data of road traffic flow in New York City from 6 : 00 to 18 : 00 (New York time, with an interval of one hour) are shown in Table 4. Using this data sequence as the original sequence and modeling it based on the first eight data, the last five data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 8 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions results of four models are shown in Table 5. Schematic diagrams of the predicted value of the four models are shown in Figure 9.Table 4
One day 6 : 00–18 : 00 New York City road traffic flow veh/h.
Time pointData6 : 00379087 : 00712498 : 00769759 : 006828510 : 006331511 : 005678612 : 005364213 : 005317914 : 005419415 : 005861516 : 006459317 : 007007618 : 0074693Figure 8
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 5
Comparison of the simulated predictions of the four models on city traffic flow.
Real value (6 : 00–13 : 00)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)37908.0037908.000.0037908.000.0037908.000.0037908.000.0071249.0071249.000.0071249.000.0071249.000.0071249.000.0076975.0077695.740.9476975.000.00106884.3838.8676975.000.0068285.0062471.988.5168309.180.0338615.5441.9868309.170.0363315.0069257.989.3963364.370.0882572.4630.4263364.360.0756786.0054360.354.2756140.901.1423777.9258.1356140.861.1353642.0061459.8914.5755451.323.3776543.4442.6955451.323.3753179.0046863.6611.8850353.735.3129101.8745.2850353.725.31Mean relative simulation error (%)6.191.2432.171.23Real value (14 : 00–18 : 00)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)54194.0054252.940.1157429.006.0095007.8775.3157428.955.9658615.0039935.2931.8757023.302.7262775.777.0957148.192.5064593.0047592.3626.3172244.7411.84146285.98126.4772151.9011.7070076.0033532.1652.1481025.4315.63134429.8891.8381152.0415.80274693.0041436.7244.52107448.1143.85241524.31223.35107361.8643.73Figure 9
Schematic diagram of the predicted value of the four models on city traffic flow.According to the data of Table5 and Figure 9, in general, the simulation predictions of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence would be better than the other three types of models; that is, according to the simulation and prediction analysis of road traffic flow in New York City, this model is feasible and effective in practice.


## 5.2.1. Applied Analysis 1

According to literature [36], we can get the short-time traffic flow data from 7:00–7:30 am of the east straight road section in Nan tong city, Jiangsu Province from August 12 to August 25, 2018. The data are shown in Table 2. Using this data sequence as the original stochastic oscillation sequence and modeling it based on the first ten data, the last four data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 6 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions result of four models is shown in Table 3. Schematic diagrams of the predicted value of the four models are shown in Figure 7.Table 2
Traffic flow data veh/h.
TimeReal value8.121388.132938.142668.152058.162578.172708.181828.191368.201828.212278.222358.232308.242318.25183Figure 6
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 3
Comparison of the simulated predictions of the four models on short-time traffic flow.
Real value (8.12–8.21)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)138.00138.000.00138.000.00138.000.00138.000.00293.00293.000.00293.000.00293.000.00293.000.00266.00226.5914.81266.000.00434.3663.29266.000.00205.00233.0113.66216.225.47152.5725.57216.215.47257.00273.946.59228.6511.30319.0624.15228.6511.03270.00202.3625.05282.054.4666.0575.54282.054.46182.00170.276.44205.3612.84265.4945.88205.3612.83136.00239.6576.21100.7525.9150.1963.09100.7625.91182.00267.4946.97207.6914.12292.8360.89207.6914.11227.00203.7710.22237.034.42126.9644.07237.034.41Mean relative simulation error (%)20.007.8340.257.82Real value (8.22–8.25)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)235.00141.5039.78180.5123.19426.1981.36180.5023.18230.00116.6549.2863.3072.48325.1341.3617.0692.58231.00105.2254.45-40.26117.43698.53202.3960.5473.79183.0088.1951.80274.8450.19682.39272.89236.2829.11Figure 7
Schematic diagram of the predicted value of the four models on short-time traffic flow.According to the data of Table3 and Figure 7, overall, the simulation of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence is much smaller than the average relative error of the other three types of models; that is, the simulation effect will be more ideal. In conclusion, according to the simulation and prediction analysis of road short-time traffic flow in Nan tong city, Jiangsu Province, this new model is feasible and effective in practice.


## 5.2.2. Applied Analysis 2

According to literature [37], the data of road traffic flow in New York City from 6 : 00 to 18 : 00 (New York time, with an interval of one hour) are shown in Table 4. Using this data sequence as the original sequence and modeling it based on the first eight data, the last five data are predicted; the four models in the above example analysis were established separately. Moreover, Figure 8 shows that the transformed sequence is smoother than that of the original stochastic oscillation sequence. The simulated predictions results of four models are shown in Table 5. Schematic diagrams of the predicted value of the four models are shown in Figure 9.Table 4
One day 6 : 00–18 : 00 New York City road traffic flow veh/h.
Time pointData6 : 00379087 : 00712498 : 00769759 : 006828510 : 006331511 : 005678612 : 005364213 : 005317914 : 005419415 : 005861516 : 006459317 : 007007618 : 0074693Figure 8
Schematic diagram of the original stochastic oscillation sequence and transformed sequence.Table 5
Comparison of the simulated predictions of the four models on city traffic flow.
Real value (6 : 00–13 : 00)Model 1Model 2Model 3Model 4Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)Simulation valueSimulation error (%)37908.0037908.000.0037908.000.0037908.000.0037908.000.0071249.0071249.000.0071249.000.0071249.000.0071249.000.0076975.0077695.740.9476975.000.00106884.3838.8676975.000.0068285.0062471.988.5168309.180.0338615.5441.9868309.170.0363315.0069257.989.3963364.370.0882572.4630.4263364.360.0756786.0054360.354.2756140.901.1423777.9258.1356140.861.1353642.0061459.8914.5755451.323.3776543.4442.6955451.323.3753179.0046863.6611.8850353.735.3129101.8745.2850353.725.31Mean relative simulation error (%)6.191.2432.171.23Real value (14 : 00–18 : 00)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)Prediction valuePrediction error (%)54194.0054252.940.1157429.006.0095007.8775.3157428.955.9658615.0039935.2931.8757023.302.7262775.777.0957148.192.5064593.0047592.3626.3172244.7411.84146285.98126.4772151.9011.7070076.0033532.1652.1481025.4315.63134429.8891.8381152.0415.80274693.0041436.7244.52107448.1143.85241524.31223.35107361.8643.73Figure 9
Schematic diagram of the predicted value of the four models on city traffic flow.According to the data of Table5 and Figure 9, in general, the simulation predictions of the discrete DGM (2, 1, t2) model based on the stochastic oscillation sequence would be better than the other three types of models; that is, according to the simulation and prediction analysis of road traffic flow in New York City, this model is feasible and effective in practice.


## 6. Conclusions

In this article, we first transform the stochastic oscillation sequence into new ones suitable for modeling by establishing an accelerated translation smoothing transformation. Then, in order to avoid errors of the mismatch between the grey differential equation and whitening differential equation and the constant parameters, a quadratic time power termbk2+ck+d is introduced, establishing a discrete DGM (2, 1, t2) model. Therefore, it not only overcomes the defect of the span from difference directly to the differential but also achieves the purpose of depicting the growth trend of its data sequence.In this article, there is the simulation prediction formula of the recurrence form and the pass-term form, which can be selected in the practical application modeling. Moreover, according to the analysis of the discrete DGM (2, 1,t2) model properties, it is known that the scope of the new model of both the simulation prediction formula and the general term form is extended. Moreover, according to examples, verification concluded that models are based on the same data transformation, and the simulation and short-term prediction effect of the discrete DGM (2, 1, t2) model are more significant. Moreover, in the establishment of the same model based on the two different data transformations, the simulation and short-term prediction accuracy proposed in this paper are generally better. In brief, the novel discrete DGM (2, 1, t2) model based on stochastic oscillatory sequence achieves better short-term prediction effects in example analysis and in the field of traffic flow.



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*Source: 5241737-2022-11-07.xml*