Stability of Time Series Models Based on Fractional-Order Weakening Buffer Operators

Abstract

Different weakening buffer operators in a time-series model analysis usually result in different model sensitivities, which sometimes affect the effectiveness of relevant operator-based methods. In this paper, the stability of two classic fractional-order weakening buffer operator-based series models is studied; then, a new data preprocessing method based on a novel fractional-order bidirectional weakening buffer operator is provided, whose effect in improving the model’s stability is tested and utilized in prediction problems. Practical examples are employed to demonstrate the efficiency of the proposed method in improving the model’s stability in noise scenarios. The comparison indicates that the proposed method overcomes the disadvantage of many weakening buffer operators in the subjectively biased weighting of the new or old information in forecasting. These expand the application of the proposed method in time series analysis.

1. Introduction

Time series prediction models play important roles in many real-world applications, including economic and social forecasting problems [1,2]. A literature survey reveals that fractional-order dynamical models sometimes provide more valuable insights to portray complex natural phenomena than their integer-order counterparts [3,4]. However, data noise and missing data are often encountered in model applications, and they affect the effectiveness of prediction results. The existence and stability of stationary solutions are basic problems for models subject to exogenous random perturbations [5,6]. Researchers have proposed ways and methods to improve the accuracy of model solutions [7,8,9,10].

Sequence operators play an important role in reducing the interference information in collected data and highlight the potential development process of analyzed objects. Among them are the widely used exponential smoothing operators [11], especially the seasonal exponential smoothing method in the application of forecasting [12,13]. The buffer operators in the grey systems theory also exhibit similar features [14]. They are based on the three axioms of the buffer operator and are suitable for small sample analyses. They greatly improve the performance of grey prediction models in real applications [15]. These operators can be further subdivided into weakening operators and strengthening operators [16,17]. When it comes to system development trend analysis or series prediction, weakening buffer operators (WBO) are preferred. These types of operators include the average weakening buffer operator (AWBO), the geometric averageweakening buffer operator (GAWBO), and the weighted WBOs (WAWBO, WGAWBO), etc., see [18,19]. The main disadvantage of these operators is their subjective determination of the weight coefficients of data [20]. This will undoubtedly miss some of the useful information under certain conditions and limit their applications.

Wu et al. first studied the essence of WBOs in grey prediction models [21,22,23]. Based on the perturbation theory, they found that series prediction models are more sensitive to earlier data than newer data in a given sequence. Their findings reveal that the classic integral accumulated generating operator, 1-AGO, and the extended fractional accumulated generating operator attach more importance to the earlier data than to recent data, and both operators donot satisfy the priority theory of new information in the grey systems theory. To solve this problem, they later proposed some reverse fractional-order operators in Wu et al. [23] and Wu et al. [21], which attach more importance to the morerecent data and thus are consistent with the priority theory of new information. Recent studies show that these new fractional-order accumulation and inverse accumulation operators could improve the prediction performance of time series models [24,25,26]. However, these new fractional-order operators emphasize either the new or old data and thus are only suitable for series analyses with special time preferences.

To solve the above problems, this paper applies a data preprocessing method based on a novel fractional-order bidirectional weakening buffer operator, which was first proposed by Li et al. [27]. Recent studies have demonstrated the good performance of this fractional-order weakening buffer operator in predictions against data noise interference [28]. The effect of this adopted operator, together with two other widely used fractional-order operators, on improving the stability of time series models is examined and compared. When applying the perturbation theory, this study shows that: (1) the disadvantage of the two classic fractional-order weakening buffer operators lies in the subjectively biased weighting of either the new or old information in forecasting; (2) the proposed data preprocessing method, based on a fractional bidirectional weakening buffer operator, improves the stability of time series models; (3) this newly adopted fractional-order weakening buffer operator-based method can more objectively deal with both the old and new data noise in samples and thereby improve the accuracy of model predictions. Both theoretical investigations and numerical experiments show favorable effects for the new fractional-order bidirectional weakening buffer operator in improving the performance of time series models. These enlarge the application of series prediction models.

This paper is structured as follows: Section 2 introduces three important fractional-order weakening buffer operators used in the series models. Section 3 analyzes the performance of the proposed operator-based data preprocessing method on the sensitivity of a fractional-order accumulate discrete grey model $DG{M}^p(1,1)$. Section 4 shows the model perturbation analysis result of the proposed method and its comparison operator on another model—the new information prior to the grey model $NIGM(p,1)$. The case studiesin Section 5 demonstrate the effect of the new method in improving the model’s stability. Finally, conclusions are drawn in Section 6.

2. Fractional-Order Weakening Buffer Operators

Weakening buffer operators are commonly used in series prediction models for finding the implicit pattern in samples. In order to show the advantage of the adopted fractional-order bidirectional weakening buffer operator in the proposed data preprocessing method, two other widely used classic fractional-order weakening buffer operators are introduced in Definition 1 and Definition 2. They are chosen as the comparative objects of the proposed one.

 Definition 1

[22]. Let  $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ be the original sequence, and $D^p{X}^{(0)}=X^p=\{x^p(1),x^p(2),\dots ,x^p(n)\}\hspace{0.17em}$ be the fractional $p\text{-}\mathrm{order}\hspace{0.17em}\hspace{0.17em}(0<p<1)$ accumulated generating operator sequence of non-negative $X^{(0)}$, with operator $D^p$ satisfying:

$$x^p(k)={\displaystyle \sum _{i=1}^k{C}_{k-i+p-1}^{k-i}}{x}^{(0)}(i),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(k=1,2,\dots n\right)$$

(1)

where $C_{p-1}^0=1,\hspace{0.17em}\hspace{0.17em}{C}_{k-1}^k=0,\hspace{0.17em}\hspace{0.17em}{C}_{k-i+p-1}^{k-i}=\frac{(k-i+p-1)(k-i+p-2)\dots (p+1)(p)}{(k-i)!}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}$

 Definition 2

[23]. Let $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ be the original sequence, and $D^r{X}^{(0)}=X_{(r)}=\{x_{(r)}(1),x_{(r)}(2),\dots ,x_{(r)}(n)\}$ be defined as the reverse fractional $r\text{-}\mathrm{order}$ accumulated generating operator sequence of non-negative $X^{(0)}$, with operator $D^r$ satisfying:

$$x_{(r)}(k)={\displaystyle \sum _{i=k}^n{C}_{i-k+r-1}^{i-k}{x}^{(0)}(i)},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(k=1,2,\dots n\right)$$

(2)

where $C_{r-1}^0=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{n-1}^n=1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_{i-k+r-1}^{i-k}=\frac{(i-k+r-1)(i-k+r-2)\cdots (r+1)r}{(i-k)!}$.

If order parameters $p$ in Definition 1 and $r$ in Definition 2 are limited to positive integers, the p-AGO and the r-IAGO are obtained. The important properties of these operators have been intensively studied [23,29]. In this study, the fractional-order bidirectional weakening buffer operator adopted in the proposed data preprocessing method of the time series prediction models is provided in the following definition:

 Definition 3

[27]. For the original sequence $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$, an operator sequence $X_v=\{x_v(1),x_v(2),\dots ,x_v(n)\}$ is obtained by $D_v{X}^{(0)}=X_v$, where the time series operator $D_v$$(v\in R^{+})$ is a fractional-order bidirectional weakening buffer operator, and elements in sequence $X_v$ satisfy:

$$\begin{array}{l}{x}_v(i)=\left({\displaystyle \sum _{j=i-\alpha (i)}^{i}w(i-j)x^{(0)}(j)}+{\displaystyle \sum _{j=i+1}^{i+\alpha (i)}w(j-i)x^{(0)}(j)}\right)/\left({\displaystyle \sum _{j=i-\alpha (i)}^{i}w(i-j)}+{\displaystyle \sum _{j=i+1}^{i+\alpha (i)}w(j-i)}\right),\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots n\end{array}$$

(3)

where $\alpha (i)=\min \left(i-1, n-i\right)$, and

$$w(i)=\{\begin{array}{l}\left(n^v(v+1-n)+{(n-1)}^{v+1}\right)/\Gamma (v+2),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=n\\ \left({(i+1)}^{v+1}-2i^{v+1}+{(i-1)}^{v+1}\right)/\Gamma (v+2),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1,2,\dots ,(n-1)\\ \hspace{0.17em}1/\Gamma (v+2),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=0\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.$$

(4)

It has been proved by Li et al. [27] that the fractional-order bidirectional weakening buffer operator $D_v$ defined in (3) is a weakening buffer operator. According to Definition 3, the value adjustment of an element in a given series is determined by both the older and newer data, while it is exclusively determined by the old or new data if applying Definition 1 or Definition 2. In the following sections, we will test the performance of operator $D_v$ in improving the stability of two grey prediction models, which are based on the operators presented in Definition 1 and Definition 2, respectively.

3. Perturbation Analysis of Model 1: The Fractional-Order Accumulate Discrete Grey Model in [22]

Whether a time series model is sensitive to the data samples plays different functions in different applications. In this section, the perturbation theory is employed to test the effect of the fractional-order bidirectional weakening buffer operator $D_v$ on the sensitivity of the fractional-order accumulate discrete grey model $DG{M}^p(1,1)$ designed in [22]. In this paper, we denote the consistent matrix norm by $\Vert .\Vert$ and denote the spectral matrix norm by ${\Vert .\Vert }_2$.

 Lemma 1

[30]. Let matrices $A\in C^{n\times n},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}F\in C^{n\times n}$, and vectors $b\in C^n,\hspace{0.17em}c\in C^n$. Assume that thematrix or vector norm $\Vert .\Vert$ is consistent, $rank(A)=rank(A+F)=n$ and the matrix norm $\Vert A^{-1}\Vert \Vert F\Vert <1$ is true, then the least squares estimation solution $x$ of the linear system models $Ax=b$ and the least squares estimation solution $x+h$ of the model $(A+F)(x+h)=b+c$ satisfy:

$$\Vert h\Vert \le \frac{{\kappa }_{†}}{{\gamma }_{†}}\left(\frac{{\Vert F\Vert }_2\Vert x\Vert }{\Vert A\Vert }+\frac{\Vert c\Vert }{\Vert A\Vert }+\frac{{\kappa }_{†}}{{\gamma }_{†}}\frac{{\Vert F\Vert }_2\Vert {\gamma }_x\Vert }{{\Vert A\Vert }^2}\right)$$

(5)

where ${\kappa }_{†}=\left|\right|A^{-1}\left|{|}_2\right|\left|A\right||$, ${\gamma }_{†}=1-\Vert A^{-1}\Vert {}_2\left|\right|F|{|}_2$, ${\gamma }_x=b-Ax$.

To keep the consistency of the parameters, the fractional-order parameter $p/q$ used in [22] is replaced by parameter $p$ in the rest of this paper.

 Definition 4

[22]. Let $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ be the original sequence, and $X^p=\{x^p(1),x^p(2),\dots ,x^p(n)\}$ be its fractional p-order accumulated generating operator sequence based on Definition 1, then the following time series model is called a fractional-order accumulate discrete grey model $DG{M}^p(1,1)$:

$$x^p(k+1)={\beta }_1{x}^p(k)+{\beta }_2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1,2,\dots ,n-1$$

(6)

The least squares estimation of model parameters ${\beta }_1$ and ${\beta }_2$ in model (6) are:

$$\left[\begin{array}{c}{\beta }_2\\ {\beta }_1\end{array}\right]={(B^{T}B)}^{-1}{B}^{T}Y,$$

(7)

where

$$B=\left[\begin{array}{cc}1& x^p(1)\\ 1& x^p(2)\\ ⋮& ⋮\\ 1& x^p(n-2)\\ 1& x^p(n-1)\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Y=\left[\begin{array}{c}{x}^p(2)\\ x^p(3)\\ ⋮\\ x^p(n-1)\\ x^p(n)\end{array}\right].$$

For simplicity, let $\beta ={\left[{\beta }_2,\hspace{0.17em}{\beta }_1\right]}^T$. The perturbation analysis result of model (6) in [22] is summarized in the following Theorems 1–3.

 Theorem 1

[22]. For the $DG{M}^p(1,1)$ model with original data $X^{(0)}=\{x^{(0)}(1),\dots ,x^{(0)}(n)\}$, let $\beta$ be its least squares estimation. When the raw data is disturbed, $DG{M}^p(1,1)$ becomes $(B+\Delta B)\tilde{\beta }=(Y+\Delta Y)$, where $\Delta B$ and $\Delta Y$ are determined by the disturbance item $\epsilon$. Assume that the matrixnorm condition $\left|\right|B^{-1}\left|{|}_2\right||\Delta B|{|}_2<1$ holds for $DG{M}^p(1,1)$. Let ${\kappa }_{†}=\left|\right|B^{-1}\left|{|}_2\right|\left|B\right||$, ${\gamma }_{†}=1-\left|\right|B^{-1}\left|{|}_2\right||\Delta B|{|}_2$, ${\gamma }_{\beta }=Y-B\beta$. If the disturbance occurs in the first data of the original sequence, that is ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, then the difference between solutions $\tilde{\beta }$ and $\beta$ is denoted by $\Vert \delta (x^{(0)}(1))\Vert$ and it satisfies:

$$\Vert \delta (x^{(0)}(1))\Vert \le \left|\epsilon \right|\frac{{\kappa }_{†}}{{\gamma }_{†}}\left(\frac{\sqrt{{\displaystyle {\sum }_{k=1}^{n-1}{\left(C_{k+p-2}^{k-1}\right)}^2}}\Vert \beta \Vert }{\Vert B\Vert }+\frac{\sqrt{{\displaystyle {\sum }_{k=2}^n{\left(C_{k+p-2}^{k-1}\right)}^2}}}{\Vert B\Vert }+\frac{{\kappa }_{†}}{{\gamma }_{†}}\frac{\sqrt{{\displaystyle {\sum }_{k=1}^{n-1}{\left(C_{k+p-2}^{k-1}\right)}^2}}\Vert {\gamma }_{\beta }\Vert }{{\Vert B\Vert }^2}\right).$$

(8)

 Theorem 2

[22]. For the same model and parameters as those in Theorem 1, if the disturbance occurs in the non-boundary nodes, that is ${\tilde{x}}^{(0)}(k)=x^{(0)}(k)+\epsilon$, $(k=2,3,\dots ,n-1)$, then the difference between solutions of model $DG{M}^p(1,1)$ is denoted by $\Vert \delta (x^{(0)}(k))\Vert$, and it satisfies:

$$\Vert \delta (x^{(0)}(k))\Vert \le \left|\epsilon \right|\frac{{\kappa }_{†}}{{\gamma }_{†}}\left(\frac{\sqrt{{\displaystyle {\sum }_{i=1}^{n-k}{\left(C_{i+p-2}^{i-1}\right)}^2}}\Vert \beta \Vert }{\Vert B\Vert }+\frac{\sqrt{{\displaystyle {\sum }_{i=1}^{n-k+1}{\left(C_{i+p-2}^{i-1}\right)}^2}}}{\Vert B\Vert }+\frac{{\kappa }_{†}}{{\gamma }_{†}}\frac{\sqrt{{\displaystyle {\sum }_{i=1}^{n-k}{\left(C_{i+p-2}^{i-1}\right)}^2}}\Vert {\gamma }_{\beta }\Vert }{{\Vert B\Vert }^2}\right).$$

(9)

 Theorem 3

[22]. For the same model and parameters as those in Theorem 1, if the disturbance occurs in the last data of the original sequence, that is ${\tilde{x}}^{(0)}(n)=x^{(0)}(n)+\epsilon$, then the difference between solutions of model $DG{M}^p(1,1)$ is denoted by $\Vert \delta (x^{(0)}(n))\Vert$, and it satisfies:

$$\Vert \delta (x^{(0)}(n))\Vert \le \frac{{\kappa }_{†}}{{\gamma }_{†}}\frac{\left|\epsilon \right|}{\Vert B\Vert }$$

(10)

Based on the above theorems, Wu et al. [22] demonstrate that the proposed $DG{M}^p(1,1)$ has better solution stability than the classic $GM(1,1)$ model. Now, to test the advantageof the fractional-order bidirectional weakening buffer operator $D_v$ in improving the model’s stability, we first apply a preprocessing step to the original time series based on operator $D_v$ and then carry out the $DG{M}^p(1,1)$ analysis. The perturbation theory is employed to test the effect of operator $D_v$ on the solution stability of $DG{M}^p(1,1)$.

Let $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ be the original sequence, $X_v=\{x_v(1),x_v(2),\dots ,x_v(n)\}$ be its weakening buffer operator sequence based on operator $D_v$. Apply the $DG{M}^p(1,1)$ model to sequence $X_v$ and let ${\beta }_v$ be the new model solution values of parameters ${\left[{\beta }_2,{\beta }_1\right]}^T$; then we have

$$\hspace{0.17em}{\beta }_v={(B_v{}^T{B}_v)}^{-1}{B}_v{}^T{Y}_v,$$

(11)

where $x_v^p(k)={\displaystyle {\sum }_{i=1}^k{C}_{k-i+p-1}^{k-i}{x}_v(i)}\hspace{0.17em}$, and

$$B_v=\left[\begin{array}{cc}1& x_v^p(1)\\ 1& x_v^p(2)\\ ⋮& ⋮\\ 1& x_v^p(n-2)\\ 1& x_v^p(n-1)\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Y}_v=\left[\begin{array}{c}{x}_v^p(2)\\ x_v^p(3)\\ ⋮\\ x_v^p(n-1)\\ x_v^p(n)\end{array}\right].$$

When the raw data is disturbed, let $\Delta B_v$ and $\Delta Y_v$ be the disturbance items of $B_v$ and $Y_v$, respectively, and ${\tilde{\beta }}_v$ be the new model solution. Assume that $\left|\right|B_v^{-1}\left|{|}_2\right||\Delta B_v|{|}_2<1$ and set ${\kappa }_{v†}=\left|\right|B_v^{-1}\left|{|}_2\right|\left|B_v\right||$, ${\gamma }_{v†}=1-\left|\right|B_v^{-1}\left|{|}_2\right||\Delta B_v|{|}_2$ and ${\gamma }_{v\beta }=Y_v-B_v{\beta }_v$. The parameters $\alpha (i)$ and $w(i)$ used in the remaining part of this section are the same as those defined in model (3). The main results are stated in the following theorems.

 Theorem 4.

If a raw data disturbance occurs in the first data of the original sequence, that is ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, then the difference between ${\tilde{\beta }}_v$ and ${\beta }_v$ satisfies:

$$\Vert \delta (x^{(0)}(1))\Vert \le \left|\epsilon \right|\frac{{\kappa }_{v†}}{{\gamma }_{v†}}\left(\frac{M_v\Vert {\beta }_v\Vert }{\Vert B_v\Vert }+\frac{N_v}{\Vert B_v\Vert }+\frac{{\kappa }_{v†}}{{\gamma }_{v†}}\frac{M_v\Vert {\gamma }_{v\beta }\Vert }{{\Vert B_v\Vert }^2}\right),$$

(12)

where: (i). If sequence size $n$ is even, then:

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2}{\left({\displaystyle \sum _{j=1}^i{C}_{j+p-2}^{j-1}\frac{w(i-j)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i-j+1)}w(k)}}}\right)}^2}+{\displaystyle \sum _{i=n/2+1}^{n-1}{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{i+p-j-1}^{i-j}\frac{w(j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (j)}w(k)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=2}^{n/2}{\left({\displaystyle \sum _{j=1}^i{C}_{j+p-2}^{j-1}\frac{w(i-j)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i-j+1)}w(k)}}}\right)}^2}+{\displaystyle \sum _{i=n/2+1}^n{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{i+p-j-1}^{i-j}\frac{w(j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (j)}w(k)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(13)

(ii). If sequence size $n$ is odd, then:

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=1}^{(n+1)/2}{\left({\displaystyle \sum _{j=1}^i{C}_{j+p-2}^{j-1}\frac{w(i-j)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i-j+1)}w(k)}}}\right)}^2}+{\displaystyle \sum _{i=(n+3)/2}^{n-1}{\left({\displaystyle \sum _{j=1}^{(n+1)/2}{C}_{i+p-j-1}^{i-j}\frac{w(j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (j)}w(k)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=2}^{(n+1)/2}{\left({\displaystyle \sum _{j=1}^i{C}_{j+p-2}^{j-1}\frac{w(i-j)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i-j+1)}w(k)}}}\right)}^2}+{\displaystyle \sum _{i=(n+3)/2}^n{\left({\displaystyle \sum _{j=1}^{(n+1)/2}{C}_{i+p-j-1}^{i-j}\frac{w(j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (j)}w(k)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(14)

 Proof of Theorem 4.

(i). Sequence size $n$ is even. When ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, according to the logic of $DG{M}^p(1,1)$, we have

$$\Delta B_v=\left[\begin{array}{cc}0& \epsilon \\ 0& C_p^1\epsilon +\frac{w(1)\epsilon }{w(0)+2w(1)}\\ ⋮& ⋮\\ 0& C_{p+n/2-2}^{n/2-1}\epsilon +C_{p+n/2-3}^{n/2-2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots \frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ 0& C_{p+n/2-1}^{n/2}\epsilon +C_{p+n/2-2}^{n/2-1}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_p^1\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ 0& C_{p+n/2}^{n/2+1}\epsilon +C_{p+n/2-1}^{n/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+1}^2\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ ⋮& ⋮\\ 0& C_{p+n-3}^{n-2}\epsilon +C_{p+n-4}^{n-3}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+n/2-2}^{n/2-1}\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\end{array}\right],$$

$$\Delta Y_v=\left[\begin{array}{c}{C}_p^1\epsilon +\frac{w(1)\epsilon }{w(0)+2w(1)}\\ C_{p+1}^2\epsilon +C_p^1\frac{w(1)\epsilon }{w(0)+2w(1)}+\frac{w(2)\epsilon }{w(0)+2\left(w(1)+w(2)\right)}\\ ⋮\\ C_{p+n/2-1}^{n/2}\epsilon +C_{p+n/2-2}^{n/2-1}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_p^1\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ C_{p+n/2}^{n/2+1}\epsilon +C_{p+n/2-1}^{n/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+1}^2\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ C_{p+n/2+1}^{n/2+2}\epsilon +C_{p+n/2}^{n/2+1}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+2}^3\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\\ ⋮\\ C_{p+n-2}^{n-1}\epsilon +C_{p+n-3}^{n-2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+n/2-1}^{n/2}\frac{w(n/2-1)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha (n/2)}w(k)}}\end{array}\right].$$

Now, calculate the norms of the matrices $\Delta B_v$ and $\Delta Y_v$, extract parameter $\epsilon$ and apply Lemma 1; parameter expressions in Equation (13) are then obtained.

(ii). When sequence size $n$ is odd, we have:

$$\Delta B_v=\left[\begin{array}{cc}0& \epsilon \\ 0& C_p^1\epsilon +\frac{w(1)\epsilon }{w(0)+2w(1)}\\ ⋮& ⋮\\ 0& C_{p+(n-3)/2}^{(n-1)/2}\epsilon +C_{p+(n-5)/2}^{(n-3)/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots \frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\\ 0& C_{p+(n-1)/2}^{(n+1)/2}\epsilon +C_{p+(n-3)/2}^{(n-1)/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_p^1\frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\\ ⋮& ⋮\\ 0& C_{p+n-3}^{n-2}\epsilon +C_{p+n-4}^{n-3}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+(n-5)/2}^{(n-3)/2}\frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\end{array}\right],\hspace{0.17em}$$

$$\Delta Y_v=\left[\begin{array}{c}{C}_p^1\epsilon +\frac{w(1)\epsilon }{w(0)+2w(1)}\\ C_{p+1}^2\epsilon +C_p^1\frac{w(1)\epsilon }{w(0)+2w(1)}+\frac{w(2)\epsilon }{w(0)+2w(1)+2w(2)}\\ ⋮\\ C_{p+(n-1)/2}^{(n+1)/2}\epsilon +C_{p+(n-3)/2}^{(n-1)/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_p^1\frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\\ C_{p+(n+1)/2}^{(n+3)/2}\epsilon +C_{p+(n-1)/2}^{(n+1)/2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+1}^2\frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\\ ⋮\\ C_{p+n-2}^{n-1}\epsilon +C_{p+n-3}^{n-2}\frac{w(1)\epsilon }{w(0)+2w(1)}+\cdots C_{p+(n-3)/2}^{(n-1)/2}\frac{w((n-1)/2)\epsilon }{w(0)+2{\displaystyle \sum _{k=1}^{\alpha ((n+1)/2)}w(k)}}\end{array}\right].$$

Then, calculate the matrix norm $\left|\right|\Delta B_v|{|}_2$ and vector norm $\left|\right|\Delta Y_v\left|\right|$, extract parameter $\epsilon$ and apply Lemma 1; parameter expressions in Equation (14) are then obtained. □

 Theorem 5.

If a raw data disturbance occurs in the non-boundary nodes of the first half of the original sequence, that is ${\tilde{x}}^{(0)}(k)=x^{(0)}(k)+\epsilon ,\hspace{0.17em}\hspace{0.17em}$ then the difference between ${\tilde{\beta }}_v$ and ${\beta }_v$ satisfies expression (12); however, the values of parameters $M_v$ and $N_v$ are determined by the following scenarios. Let ${\phi }_1$ be the largest integer less than $k/2$, ${\phi }_2$ be the largest integer less than $(k-1)/2$, then $M_v$ and $N_v$ are:

(i). If sequence size $n$ is even, then $\hspace{0.17em}k=2,3,\dots ,n/2\hspace{0.17em}$ and

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i={\phi }_1+1}^{n/2+{\phi }_1}{\left({\displaystyle \sum _{j=1}^{i-{\phi }_1}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n/2+{\phi }_1+1}^{n-1}{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{p+i+j-{\phi }_1-2-n/2}^{i+j-{\phi }_1-1-n/2}\frac{w(\left|k+j-{\phi }_1-1-n/2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+{\phi }_1+1-j)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i={\phi }_1+1}^{n/2+{\phi }_1}{\left({\displaystyle \sum _{j=1}^{i-{\phi }_1}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n/2+{\phi }_1+1}^n{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{p+i+j-{\phi }_1-2-n/2}^{i+j-{\phi }_1-1-n/2}\frac{w(\left|k+j-{\phi }_1-1-n/2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+{\phi }_1+1-j)}w(h)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(15)

(ii). If sequence size $n$ is odd, then $\hspace{0.17em}k=2,3,\dots ,(n+1)/2\hspace{0.17em}$ and

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i={\phi }_1+1}^{(n+1)/2+{\phi }_2}{\left({\displaystyle \sum _{j=1}^{i-{\phi }_1}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=(n+1)/2+{\phi }_2+1}^{n-1}{\left({\displaystyle \sum _{j=1}^{(n+1)/2+{\phi }_2-{\phi }_1}{C}_{p+i+j-{\phi }_2-2-(n+1)/2}^{i+j-{\phi }_2-1-(n+1)/2}\frac{w(\left|k+j-{\phi }_2-1-(n+1)/2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+{\phi }_2+1-j)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i={\phi }_1+1}^{(n+1)/2+{\phi }_2}{\left({\displaystyle \sum _{j=1}^{i-{\phi }_1}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=(n+1)/2+{\phi }_2+1}^n{\left({\displaystyle \sum _{j=1}^{(n+1)/2+{\phi }_2-{\phi }_1}{C}_{p+i+j-{\phi }_2-2-(n+1)/2}^{i+j-{\phi }_2-1-(n+1)/2}\frac{w(\left|k+j-{\phi }_2-1-(n+1)/2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+{\phi }_2+1-j)}w(h)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(16)

 Proof of Theorem 5.

Please see Appendix A.1. □

 Theorem 6.

If a raw data disturbance occurs in the second half of the original sequence, excluding the last data, then the difference between ${\tilde{\beta }}_v$ and ${\beta }_v$ satisfies expression (12). Let ${\phi }_3$ be the largest integer less than $(n-k)/2$, ${\phi }_4$ be the largest integer less than $(n-k+1)/2$, then $M_v$ and $N_v$ are:

(i). If sequence size $n$ is even, then $\hspace{0.17em}k=n/2+1,\hspace{0.17em}n/2+2,\dots ,n-1$ and

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=n/2+1-{\phi }_4}^{n-{\phi }_4}{\left({\displaystyle \sum _{j=1}^{i-n/2+{\phi }_4}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n+1-{\phi }_4}^{n-1}{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{p+i+j+{\phi }_4-2-n}^{i+j+{\phi }_4-1-n}\frac{w(\left|k+j+{\phi }_4-1-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n+1-{\phi }_4-j)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=n/2+1-{\phi }_4}^{n-{\phi }_4}{\left({\displaystyle \sum _{j=1}^{i-n/2+{\phi }_4}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n+1-{\phi }_4}^n{\left({\displaystyle \sum _{j=1}^{n/2}{C}_{p+i+j+{\phi }_4-2-n}^{i+j+{\phi }_4-1-n}\frac{w(\left|k+j+{\phi }_4-1-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n+1-{\phi }_4-j)}w(h)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(17)

(ii). If the sequence size n is odd, then $k=(n+1)/2+1,\hspace{0.17em}(n+1)/2+2,\dots ,n-1$ and

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=(n+1)/2-{\phi }_3}^{n-{\phi }_4}{\left({\displaystyle \sum _{j=1}^{i+{\phi }_3-(n-1)/2}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1-j)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n-{\phi }_4+1}^{n-1}{\left({\displaystyle \sum _{j=1}^{(n+1)/2+{\phi }_3-{\phi }_4}{C}_{p+i+j+{\phi }_4-2-n}^{i+j+{\phi }_4-1-n}\frac{w(\left|k+j+{\phi }_4-1-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n+1-{\phi }_4-j)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=(n+1)/2-{\phi }_3}^{n-{\phi }_4}{\left({\displaystyle \sum _{j=1}^{i+{\phi }_3-(n-1)/2}{C}_{j+p-2}^{j-1}\frac{w(\left|k+j-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i-j+1)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n-{\phi }_4+1}^n{\left({\displaystyle \sum _{j=1}^{(n+1)/2+{\phi }_3-{\phi }_4}{C}_{p+i+j+{\phi }_4-2-n}^{i+j+{\phi }_4-1-n}\frac{w(\left|k+j+{\phi }_4-1-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n+1-{\phi }_4-j)}w(h)}}}\right)}^2}}.\end{array}$$

(18)

 Proof of Theorem 6.

Please see Appendix A.2. □

 Theorem 7.

If a raw data disturbance occurs in the last data of the original sequence, that is ${\tilde{x}}^{(0)}(n)=x^{(0)}(n)+\epsilon$, then the difference between ${\tilde{\beta }}_v$ and ${\beta }_v$ satisfies expression (12); however, the values of parameters $M_v$ and $N_v$ are determined by the size of samples.

(i). If sequence size $n$ is even, then

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2-1}{\left({\displaystyle \sum _{j=1}^i{C}_{i+p-j-1}^{i-j}\frac{w(n/2-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+j)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2}{\left({\displaystyle \sum _{j=1}^i{C}_{i+p-j-1}^{i-j}\frac{w(n/2-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+j)}w(h)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(19)

(ii). If sequence size $n$ is odd, then

$$\begin{array}{l}{M}_v=\sqrt{{\displaystyle \sum _{i=1}^{(n-1)/2}{\left({\displaystyle \sum _{j=1}^i{C}_{i+p-j-1}^{i-j}\frac{w((n+1)/2-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+j-1)}w(h)}}}\right)}^2}},\\ N_v=\sqrt{{\displaystyle \sum _{i=1}^{(n+1)/2}{\left({\displaystyle \sum _{j=1}^i{C}_{i+p-j-1}^{i-j}\frac{w((n+1)/2-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+j-1)}w(h)}}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(20)

 Proof of Theorem 7.

Please see Appendix A.3. □

4. Perturbation Analysis of Model 2: The $\mathit{N}\mathit{I}\mathit{G}\mathit{M}(\mathit{p}\mathbf{,}\mathbf{1})$ Model in [23]

The fractional-order accumulate discrete grey model in Section 3 emphasizes old data in the modeling samples. To provide more weight to the new information, Wu et al. [23] provide the following model.

 Definition 5

[23]. Let $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ be the original sequence, and $X_{(1-p)}=\{x_{(1-p)}(1),x_{(1-p)}(2),\dots ,x_{(1-p)}(n)\}\hspace{0.17em}(0<p<1)$ be the fractional (1 − p)-order reverse accumulated generating operator sequence of $X^{(0)}$ (see Definition 2). Set mean formula $z^{(0)}(k)=\left(x^{(0)}(k)+x^{(0)}(k-1)\right)/2$ and formula $d_{(1)}{x}_{(1-p)}(k)=x_{(1-p)}(k)-x_{(1-p)}(k-1)$, then the new information prior grey model $NIGM(p,1)$ is defined by:

$$d_{(1)}{x}_{(1-p)}(k)+{\tau }_1{z}^{(0)}(k)={\tau }_2$$

(21)

The least squares estimation of model parameters ${\tau }_1$ and ${\tau }_2$ in (21) can be obtained by:

$$\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{\tau }_1\\ {\tau }_2\end{array}\right]={(E^{T}E)}^{-1}{E}^{T}U\hspace{0.17em}$$

(22)

where

$$E=\left[\begin{array}{cc}-z^{(0)}(2)& 1\\ -z^{(0)}(3)& 1\\ ⋮& ⋮\\ -z^{(0)}(n)& 1\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}U=\left[\begin{array}{c}{d}_{(1)}{x}_{(1-p)}(2)\\ d_{(1)}{x}_{(1-p)}(3)\\ ⋮\\ d_{(1)}{x}_{(1-p)}(n)\end{array}\right]$$

Let $\tau ={\left[{\tau }_1,{\tau }_2\right]}^T$, $\Delta E$ and $\Delta U$ be the disturbance items of model parameters $E$ and $U$, respectively, and $\epsilon$ be the disturbance item on raw data. Assume that the matrix norm condition $\left|\right|E^{-1}\left|{|}_2\right||\Delta E|{|}_2<1$ holds for model $NIGM(p,1)$. Let ${\mu }_{†}=\left|\right|E^{-1}\left|{|}_2\right|\left|E\right||$, ${\eta }_{†}=1-\left|\right|E^{-1}\left|{|}_2\right||\Delta E|{|}_2$, ${\eta }_{\tau }=U-E\tau$. Based on Lemma 1, the perturbation analysis results of $NIGM(p,1)$ are summarized in the following Theorems 8 to 10. Theorem 8 directlycomes from Wu et al. [23], while Theorems 9 to 10 are derived from the corrected parameters.

 Theorem 8

[23]. If the disturbance $\epsilon$ occurs in the first data of the original sequence, that is ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, then the difference between solutions $\tilde{\tau }$ and $\tau$ is denoted by $\Vert \delta (x^{(0)}(1))\Vert$ and it satisfies:

$$\Vert \delta (x^{(0)}(1))\Vert \le \left|\epsilon \right|\frac{{\mu }_{†}}{{\eta }_{†}}\left(\frac{\Vert \tau \Vert }{2\Vert E\Vert }+\frac{1}{\Vert E\Vert }+\frac{{\mu }_{†}}{{\eta }_{†}}\frac{\Vert {\eta }_{\tau }\Vert }{2{\Vert E\Vert }^2}\right).$$

(23)

 Theorem 9.

If the disturbance occurs in the non-boundary nodes, that is ${\tilde{x}}^{(0)}(i)=x^{(0)}(i)+\epsilon$, $(i=2,3,\dots ,n-1)$, then the difference between the solutions of model $NIGM(p,1)$ with and without data disturbance is denoted by $\Vert \delta (x^{(0)}(i))\Vert$, and it satisfies:

$$\Vert \delta (x^{(0)}(i))\Vert \le \left|\epsilon \right|\frac{{\mu }_{†}}{{\eta }_{†}}\left(\frac{\sqrt{2}\Vert \tau \Vert }{2\Vert E\Vert }+\frac{\sqrt{{\displaystyle {\sum }_{j=2}^i{(C_{j-2-p}^{j-1})}^2}+1}}{\Vert E\Vert }+\frac{{\mu }_{†}}{{\eta }_{†}}\frac{\sqrt{2}\Vert {\eta }_{\tau }\Vert }{2{\Vert E\Vert }^2}\right).$$

(24)

 Proof of Theorem 9.

(i). When ${\tilde{x}}^{(0)}(2)=x^{(0)}(2)+\epsilon$, the disturbance items of parameters $\Delta E$ and $\Delta U$ are:

$$\Delta E=\left[\begin{array}{cc}-\epsilon /2& 0\\ -\epsilon /2& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta U=\left[\begin{array}{c}p\epsilon \\ -\epsilon \\ 0\\ \hspace{0.17em}⋮\\ 0\end{array}\right].$$

Then, $\left|\right|\Delta E|{|}_2=\sqrt{2}|\epsilon |/2,\hspace{0.17em}||\Delta U||=|\epsilon |\sqrt{p^2+1}$.

(ii). When ${\tilde{x}}^{(0)}(3)=x^{(0)}(3)+\epsilon$, the disturbance items of model parameters $\Delta E$ and $\Delta U$ are:

$$\Delta E=\left[\begin{array}{cc}0& 0\\ -\epsilon /2& 0\\ -\epsilon /2& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta U=\left[\begin{array}{c}-C_{1-p}^2\epsilon \\ p\epsilon \\ -\epsilon \\ 0\\ ⋮\\ 0\end{array}\right].$$

Then, $\left|\right|\Delta E|{|}_2=\sqrt{2}|\epsilon |/2,\hspace{0.17em}||\Delta U||=|\epsilon |\sqrt{(p^4-2p^3+5p^2)/4+1}$.

Likewise, (iii) when ${\tilde{x}}^{(0)}(i)=x^{(0)}(i)+\epsilon ,\hspace{0.17em}(i=4,\dots n-1)$, the disturbance items of model parameters $\Delta E$ and $\Delta U$ are:

$$\Delta E=\left[\begin{array}{cc}0& 0\\ ⋮& ⋮\\ 0& 0\\ -\epsilon /2& 0\\ -\epsilon /2& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta U=\left[\begin{array}{c}-C_{i-p-2}^{i-1}\epsilon \\ -C_{i-p-3}^{i-2}\epsilon \\ ⋮\\ p\epsilon \\ -\epsilon \\ 0\\ ⋮\\ 0\end{array}\right].$$

Then, $||\Delta E|{|}_2=\sqrt{2}|\epsilon |/2,\hspace{0.17em}||\Delta U||=|\epsilon |\sqrt{{\displaystyle {\sum }_{j=2}^i{(C_{j-2-p}^{j-1})}^2}+1}$.

Summarizing the above three scenarios, the perturbation result (24) is obtained. □

 Theorem 10.

If the disturbance occurs in the last sample point, that is ${\tilde{x}}^{(0)}(n)=x^{(0)}(n)+\epsilon$, then the difference between the solutions of model $NIGM(p,1)$ is denoted by $\Vert \delta (x^{(0)}(n))\Vert$, and it satisfies:

$$\Vert \delta (x^{(0)}(n))\Vert \le \left|\epsilon \right|\frac{{\mu }_{†}}{{\eta }_{†}}\left(\frac{\Vert \tau \Vert }{2\Vert E\Vert }+\frac{\sqrt{{\displaystyle {\sum }_{j=2}^n{(C_{j-2-p}^{j-1})}^2}}}{\Vert E\Vert }+\frac{{\mu }_{†}}{{\eta }_{†}}\frac{\Vert {\eta }_{\tau }\Vert }{2{\Vert E\Vert }^2}\right).$$

(25)

 Proof of Theorem 10.

When ${\tilde{x}}^{(0)}(n)=x^{(0)}(n)+\epsilon$, the disturbance items of parameters $\Delta E$ and $\Delta U$ are:

$$\Delta E=\left[\begin{array}{cc}0& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\\ -\epsilon /2& 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta U=\left[\begin{array}{c}-C_{n-p-2}^{n-1}\epsilon \\ -C_{n-p-3}^{n-2}\epsilon \\ ⋮\\ -C_{1-p}^2\epsilon \\ p\epsilon \end{array}\right].$$

Then, $||\Delta E|{|}_2=|\epsilon |/2,\hspace{0.17em}||\Delta U||=|\epsilon |\sqrt{{\displaystyle {\sum }_{j=2}^n{(C_{j-2-p}^{j-1})}^2}}$. Apply these norms to Lemma 1; perturbation bound (25) is proved. □

Now, we consider the effect of operator $D_v$ on the stability of model $NIGM(p,1)$. $X^{(0)}=\{x^{(0)}(1),x^{(0)}(2),\dots ,x^{(0)}(n)\}$ and $X_v=\{x_v(1),x_v(2),\dots ,x_v(n)\}$ are the same as those defined in Section 3. Set $x_v^{(1-p)}(k)={\displaystyle {\sum }_{i=k}^n{C}_{i-k-p}^{i-k}{x}_v(i)}$, $\hspace{0.17em}{d}_{(1)}{x}_v^{(1-p)}(k)=x_v^{(1-p)}(k)-x_v^{(1-p)}(k-1)$ and $z_v^{(0)}(k)=\left(x_v(k)+x_v(k-1)\right)/2$. Let ${\tau }_v$ be the least squares solution of model $NIGM(p,1)$ based on sequence $X_v$. Then we have

$$\hspace{0.17em}{\tau }_v={(E_v^T{E}_v)}^{-1}{E}_v^T{U}_v\hspace{0.17em}$$

(26)

where

$$E_v=\left[\begin{array}{cc}-z_v^{(0)}(2)& 1\\ -z_v^{(0)}(3)& 1\\ ⋮& ⋮\\ -z_v^{(0)}(n)& 1\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{U}_v=\left[\begin{array}{c}{d}_{(1)}{x}_v^{(1-p)}(2)\\ d_{(1)}{x}_v^{(1-p)}(3)\\ ⋮\\ d_{(1)}{x}_v^{(1-p)}(n)\end{array}\right].$$

Let $\Delta E_v$ and $\Delta U_v$ be the disturbance items of $E_v$ and $U_v$, respectively, and ${\tilde{\tau }}_v$ be the new model solution. Assume that $\left|\right|E_v^{-1}\left|{|}_2\right||\Delta E_v|{|}_2<1$ holds and let ${\eta }_{v†}=1-\left|\right|E_v^{-1}\left|{|}_2\right||\Delta E_v|{|}_2$, ${\mu }_{v†}=\left|\right|E_v^{-1}\left|{|}_2\right|\left|E_v\right||$, ${\eta }_{v\tau }=U_v-E_v{\tau }_v$. The perturbation analysis results of model $NIGM(p,1)$ with operator $D_v$ are stated in the following theorems.

 Theorem 11.

If a raw data disturbance occurs in the first data of the original sequence, that is ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, then the difference between ${\tilde{\tau }}_v$ and ${\tau }_v$ satisfies:

$$\Vert \delta (x^{(0)}(1))\Vert \le \left|\epsilon \right|\frac{{\mu }_{v†}}{{\eta }_{v†}}\left(\frac{P_v\Vert {\tau }_v\Vert }{\Vert E_v\Vert }+\frac{Q_v}{\Vert E_v\Vert }+\frac{{\mu }_{v†}}{{\eta }_{v†}}\frac{P_v\Vert {\eta }_{v\tau }\Vert }{{\Vert E_v\Vert }^2}\right)$$

(27)

where (i). If sequence size $n$ is even, then:

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\displaystyle \sum _{i=1}^{n/2-1}{\left(\frac{w(i-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i)}w(k)}}+\frac{w(i)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i+1)}w(k)}}\right)}^2}+{\left(\frac{w(n/2-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (n/2)}w(k)}}\right)}^2},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2-1}{\left(\frac{w(i-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i)}w(k)}}+{\displaystyle \sum _{j=1}^{n/2-i}{C}_{j-p-1}^j\frac{w(i+j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i+j)}w(k)}}}\right)}^2}+{\left(\frac{w(n/2-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (n/2)}w(k)}}\right)}^2}.\end{array}$$

(28)

(ii). If sequence size $n$ is odd, then:

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\displaystyle \sum _{i=1}^{(n-1)/2}{\left(\frac{w(i-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i)}w(k)}}+\frac{w(i)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i+1)}w(k)}}\right)}^2}+{\left(\frac{w((n-1)/2)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha ((n+1)/2)}w(k)}}\right)}^2},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{(n-1)/2}{\left(\frac{w(i-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i)}w(k)}}+{\displaystyle \sum _{j=1}^{(n+1)/2-i}{C}_{j-p-1}^j\frac{w(i+j-1)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha (i+j)}w(k)}}}\right)}^2}+{\left(\frac{w((n-1)/2)}{w(0)+2{\displaystyle {\sum }_{k=1}^{\alpha ((n+1)/2)}w(k)}}\right)}^2}\hspace{0.17em}\hspace{0.17em}.\end{array}$$

(29)

 Proof of Theorem 11.

(i). The sequence size $n$ is even. When ${\tilde{x}}^{(0)}(1)=x^{(0)}(1)+\epsilon$, according to the logic of $NIGM(p,1)$, we have

$$\Delta E_v=\left[\begin{array}{cc}-\frac{\epsilon }{2}\left(1+\frac{w(1)}{w(0)+2w(1)}\right)& 0\\ -\frac{\epsilon }{2}\left(\frac{w(1)}{w(0)+2w(1)}+\frac{w(2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (3)}w(h)}}\right)& 0\\ ⋮& ⋮\\ -\frac{\epsilon }{2}\left(\frac{w(n/2-2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2-1)}w(h)}}+\frac{w(n/2-1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2)}w(h)}}\right)& 0\\ -\frac{\epsilon }{2}\frac{w(n/2-1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2)}w(h)}}& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Delta U_v=\left[\begin{array}{c}-\epsilon -\epsilon {\displaystyle \sum _{i=1}^{n/2-1}\left(C_{i-p-1}^i\right)\frac{w(i)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (i+1)}w(h)}}}\\ -\epsilon \frac{w(1)}{w(0)+2w(1)}-\epsilon {\displaystyle \sum _{i=1}^{n/2-2}\left(C_{i-p-1}^i\right)\frac{w(i+1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (i+2)}w(h)}}}\\ ⋮\\ -\epsilon \frac{w(n/2-2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2-1)}w(h)}}+\epsilon p\frac{w(n/2-1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2)}w(h)}}\\ -\epsilon \frac{w(n/2-1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (n/2)}w(h)}}\\ 0\\ ⋮\\ 0\end{array}\right].$$

Then, calculate the matrix norm $\left|\right|\Delta E_v|{|}_2$ and vector norm $\left|\right|\Delta U_v\left|\right|$, extract parameter $\epsilon$ and apply Lemma 1; then, the parameter expressions in Equation (28) are obtained.

(ii). When sequence size $n$ is odd, we obtain:

$$\Delta E_v=\left[\begin{array}{cc}-\frac{\epsilon }{2}\left(1+\frac{w(1)}{w(0)+2w(1)}\right)& 0\\ -\frac{\epsilon }{2}\left(\frac{w(1)}{w(0)+2w(1)}+\frac{w(2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (3)}w(h)}}\right)& 0\\ ⋮& ⋮\\ -\frac{\epsilon }{2}\left(\frac{w((n-3)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n-1)/2)}w(h)}}+\frac{w((n-1)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n+1)/2)}w(h)}}\right)& 0\\ -\frac{\epsilon }{2}\frac{w((n-1)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n+1)/2)}w(h)}}& 0\\ 0& 0\\ ⋮& ⋮\\ 0& 0\end{array}\right],\hspace{0.17em}{U}_v=\left[\begin{array}{c}-\epsilon -\epsilon {\displaystyle \sum _{i=1}^{(n-1)/2}\left(C_{i-p-1}^i\right)\frac{w(i)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (i+1)}w(h)}}}\\ -\epsilon \frac{w(1)}{w(0)+2w(1)}-\epsilon {\displaystyle \sum _{i=1}^{(n-3)/2}\left(C_{i-p-1}^i\right)\frac{w(i+1)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha (i+2)}w(h)}}}\\ ⋮\\ -\epsilon \frac{w((n-3)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n-1)/2)}w(h)}}+\epsilon p\frac{w((n-1)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n+1)/2)}w(h)}}\\ -\epsilon \frac{w((n-1)/2)}{w(0)+2{\displaystyle \sum _{h=1}^{\alpha ((n+1)/2)}w(h)}}\\ 0\\ ⋮\\ 0\end{array}\right].\hspace{0.17em}$$

Then, calculate the matrix norm $\left|\right|\Delta E_v|{|}_2$ and vector norm $\left|\right|\Delta U_v\left|\right|$, extract parameter $\epsilon$ and apply Lemma 1; then, the parameter expressions in Equation (29) are obtained. □

 Theorem 12.

If a raw data disturbance occurs in the non-boundary nodes of the first half of the original sequence, that is ${\tilde{x}}^{(0)}(k)=x^{(0)}(k)+\epsilon ,\hspace{0.17em}\hspace{0.17em}$ then the difference between ${\tilde{\tau }}_v$ and ${\tau }_v$ satisfies expression (27); however, the values of parameters $P_v$ and $Q_v$ are determined by the following scenarios. Let parameters ${\phi }_1$ and ${\phi }_2$ be the same as those defined in Theorem 5, then $P_v$ and $Q_v$ are:

(i). If sequence size $n$ is even, then $\hspace{0.17em}k=2,3,\dots ,n/2\hspace{0.17em}$ and

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\left(\frac{w(\left|k-{\phi }_1-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ({\phi }_1+1)}w(h)}}\right)}^2+{\displaystyle \sum _{i={\phi }_1+1}^{n/2+{\phi }_1-1}{\left(\frac{w(\left|k-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}+{\left(\frac{w(\left|k-n/2-{\phi }_1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+{\phi }_1)}w(h)}}\right)}^2},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{{\phi }_1}{\left({\displaystyle \sum _{j={\phi }_1+1}^{n/2+{\phi }_1}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i={\phi }_1+1}^{n/2+{\phi }_1-1}{\left({\displaystyle \sum _{j=i+1}^{n/2+{\phi }_1}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\left(\frac{w(\left|k-n/2-{\phi }_1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+{\phi }_1)}w(h)}}\right)}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.\end{array}$$

(30)

(ii). If sequence size $n$ is odd, then $k=2,3,\dots ,(n+1)/2\hspace{0.17em}$ and

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\left(\frac{w(\left|k-{\phi }_1-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ({\phi }_1+1)}w(h)}}\right)}^2+{\displaystyle \sum _{i={\phi }_1+1}^{(n+1)/2+{\phi }_2-1}{\left(\frac{w(\left|k-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}+{\left(\frac{w(\left|k-(n+1)/2-{\phi }_2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+{\phi }_2)}w(h)}}\right)}^2},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{{\phi }_1}{\left({\displaystyle \sum _{j={\phi }_1+1}^{(n+1)/2+{\phi }_2}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i={\phi }_1+1}^{(n+1)/2+{\phi }_2-1}{\left({\displaystyle \sum _{j=i+1}^{(n+1)/2+{\phi }_2}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\left(\frac{w(\left|k-(n+1)/2-{\phi }_2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2+{\phi }_2)}w(h)}}\right)}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.\end{array}$$

(31)

 Proof of Theorem 12.

The proof is similar to that of Theorem 5, and we omit the details here. □

 Theorem 13.

If a raw data disturbance occurs in the second half of the original sequence, excluding the last data, let ${\phi }_3$ and ${\phi }_4$ be the same as those defined in Theorem 6, then the difference between ${\tilde{\tau }}_v$ and ${\tau }_v$ satisfies expression (27), where parameters $P_v$ and $Q_v$ are determined by:

(i). If sequence size $n$ is even, then $k=n/2+1,\hspace{0.17em}n/2+2,\dots ,n-1$ and

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\left(\frac{w(\left|k+{\phi }_4-n/2-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+1-{\phi }_4)}w(h)}}\right)}^2+{\displaystyle \sum _{i=n/2+1-{\phi }_4}^{n-{\phi }_4-1}{\left(\frac{w(\left|k-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}+{\left(\frac{w(\left|k+{\phi }_4-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n-{\phi }_4)}w(h)}}\right)}^2},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2-{\phi }_4}{\left({\displaystyle \sum _{j=n/2+1-{\phi }_4}^{n-{\phi }_4}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=n/2+1-{\phi }_4}^{n-{\phi }_4-1}{\left({\displaystyle \sum _{j=i+1}^{n-{\phi }_4}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\left(\frac{w(\left|k+{\phi }_4-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n-{\phi }_4)}w(h)}}\right)}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.\end{array}$$

(32)

(ii). If sequence size $n$ is odd, then $k=(n+1)/2+1,\hspace{0.17em}(n+1)/2+2,\dots ,n-1$ and

$$P_v=\frac{1}{2}\sqrt{{\left(\frac{w(\left|k+{\phi }_3-(n+1)/2\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2-{\phi }_3)}w(h)}}\right)}^2+{\displaystyle \sum _{i=(n+1)/2-{\phi }_3}^{n-{\phi }_4-1}{\left(\frac{w(\left|k-i-1\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}+{\left(\frac{w(\left|k+{\phi }_4-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n-{\phi }_4)}w(h)}}\right)}^2},$$

$$\begin{array}{l}{Q}_v=\sqrt{{\displaystyle \sum _{i=1}^{(n-1)/2-{\phi }_3}{\left({\displaystyle \sum _{j=(n+1)/2-{\phi }_3}^{n-{\phi }_4}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}+{\displaystyle \sum _{i=(n+1)/2-{\phi }_3}^{n-{\phi }_4-1}{\left({\displaystyle \sum _{j=i+1}^{n-{\phi }_4}{C}_{j-p-2}^{j-1}\frac{w(\left|k-j\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}+\frac{w(\left|k-i\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\overline{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\left(\frac{w(\left|k+{\phi }_4-n\right|)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n-{\phi }_4)}w(h)}}\right)}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}.\end{array}$$

(33)

 Proof of Theorem 13.

The proof is similar to that of Theorem 6, and we omit the details here. □

 Theorem 14.

If a disturbance occurs in the last data of the original sequence: ${\tilde{x}}^{(0)}(n)=x^{(0)}(n)+\epsilon$, then the difference between ${\tilde{\tau }}_v$ and ${\tau }_v$ satisfies expression (27) with the following values of parameters $P_v$ and $Q_v$:

(i). If sequence size $n$ is even, then

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\left(\frac{w(n/2-1)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (n/2+1)}w(h)}}\right)}^2+{\displaystyle \sum _{i=n/2+1}^{n-1}{\left(\frac{w(n-i-1)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(n-i)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{n/2}{\left({\displaystyle \sum _{j=n/2+1}^n{C}_{j-p-2}^{j-1}\frac{w(n-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2+{\displaystyle \sum _{i=n/2+1}^{n-1}{\left(\frac{w(n-i)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}+{\displaystyle \sum _{j=i+1}^n{C}_{j-p-2}^{j-1}\frac{w(n-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}}}\hspace{0.17em}.\end{array}$$

(34)

(ii). If sequence size$n$is odd, then

$$\begin{array}{l}{P}_v=\frac{1}{2}\sqrt{{\left(\frac{w((n-1)/2)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha ((n+1)/2)}w(h)}}\right)}^2+{\displaystyle \sum _{i=(n+1)/2}^{n-1}{\left(\frac{w(n-i-1)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i+1)}w(h)}}+\frac{w(n-i)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}\right)}^2}},\\ Q_v=\sqrt{{\displaystyle \sum _{i=1}^{(n-1)/2}{\left({\displaystyle \sum _{j=(n+1)/2}^n{C}_{j-p-2}^{j-1}\frac{w(n-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2+{\displaystyle \sum _{i=(n+1)/2}^{n-1}{\left(\frac{w(n-i)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (i)}w(h)}}+{\displaystyle \sum _{j=i+1}^n{C}_{j-p-2}^{j-1}\frac{w(n-j)}{w(0)+2{\displaystyle {\sum }_{h=1}^{\alpha (j)}w(h)}}}\right)}^2}}}\hspace{0.17em}.\end{array}$$

(35)

 Proof of Theorem 14.

The proof is similar to that of Theorem 7, and we omit the details here. □

5. Numerical Performance of Operator ${\mathit{D}}_{\mathit{v}}$ in Improving Model Stability

5.1. Numerical Study on Model $DG{M}^p(1,1)$

To evaluate the effect of the fractional-order bidirectional weakening buffer operator $D_v$ in improving the stability of model $DG{M}^p(1,1)$, we consider a real case presented in [22]. This example predicts theannual cargo turnover in Jiangsu, a large coastal province in China. The original data are shown in

Table 1. The originaldata of the annual cargo turnover used in [22]. (Unit: 100 million ton-km).

	Modeling Period						Forecasting Period
	2003	2004	2005	2006	2007	2008	2009
Freight Ton-Kilometers	1817.44	2398.13	3068.3	3644.14	4098.42	4707.5	5154.46

, and the forecasting values of four compared models are presented in

Table 2. The forecastingresults of four different models.

	$\mathit{D}\mathit{G}{\mathit{M}}^{1/2}\left(1,1\right)$	${\mathit{D}}_{\mathit{v}}+\mathit{D}\mathit{G}{\mathit{M}}^{1/2}\left(1,1\right)$	$\mathit{D}\mathit{G}{\mathit{M}}^{2/3}\left(1,1\right)$	${\mathit{D}}_{\mathit{v}}+\mathit{D}\mathit{G}{\mathit{M}}^{2/3}\left(1,1\right)$
Freight Ton-Kilometers (2009)	5236.45 *	5238.69	5303.33 *	5305.58
MAPE (%)	1.59 *	1.63	2.89 *	2.93

* values obtained from [22].

.

Table 2 reveals that the forecasting results of the $DG{M}^p(1,1)$ model with data preprocessing based on $D_v\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(v=0.02)$ are small, which means the validity of the predicted values is maintained.

Then, data noise interferences with different noise amplitudes (ranging from −10% to 10% of the original value) are studied. The effectiveness of operator $D_v$ on the prediction model is evaluated from two aspects: model stability and model accuracy. The model stability is measured by the perturbation bounds of the model parameter valueswith and without noise interference. Based on the theorems in Section 3, the perturbation bounds of the prediction models are calculated and summarized in

Table 3. The perturbation bounds of four models in different noise scenarios.

Noise Position	Model Index *	Noise Amplitude
		−10%	−8%	−6%	−4%	−2%	2%	4%	6%	8%	10%
2003	A	90.3	72.5	54.6	36.5	18.3	18.5	37.1	55.8	74.7	93.7
	B	87.2	70.1	52.7	35.3	17.7	17.8	35.8	54.0	72.2	90.6
2004	A	110.7	90.2	68.9	46.8	23.8	24.7	50.2	76.5	103.7	131.8
	B	109.3	89.0	67.8	46.0	23.4	24.1	49.0	74.6	100.9	128.1
2005	A	128.5	104.5	79.7	54.0	27.4	28.2	57.2	87.0	117.5	148.7
	B	127.0	103.0	78.3	52.9	26.8	27.5	55.6	84.3	113.6	143.6
2006	A	113.6	92.9	71.1	48.4	24.6	25.5	51.8	78.9	106.7	135.1
	B	112.3	91.4	69.7	47.2	24.0	24.6	49.9	75.7	102.2	129.1
2007	A	26.9	25.1	21.4	15.9	8.8	10.3	22.1	35.2	49.7	65.4
	B	33.5	29.9	24.6	17.9	9.6	10.9	23.1	36.5	51.0	66.6
2008	A	198.0	158.4	118.8	79.2	39.6	39.6	79.2	118.8	158.4	198.0
	B	195.3	156.2	117.2	78.1	39.0	39.0	78.1	117.1	156.1	195.2
2003	C	68.0	54.5	41.0	27.4	13.8	13.8	27.7	41.7	55.8	70.0
	D	64.8	52.0	39.1	26.2	13.1	13.2	26.5	39.8	53.2	66.7
2004	C	115.8	93.8	71.2	48.0	24.3	24.8	50.2	76.2	102.7	129.8
	D	113.8	92.0	69.8	47.0	23.8	24.3	49.0	74.3	100.0	126.2
2005	C	131.8	106.5	80.6	54.2	27.3	27.8	56.1	84.8	113.9	143.4
	D	129.7	104.5	79.0	53.1	26.7	27.1	54.6	82.4	110.6	139.1
2006	C	110.8	89.8	68.2	46.0	23.3	23.8	48.0	72.7	97.9	123.5
	D	108.8	87.9	66.6	44.8	22.6	23.0	46.4	70.2	94.2	118.7
2007	C	21.8	19.7	16.4	12.0	6.5	7.5	16.0	25.5	35.8	47.0
	D	27.4	23.9	19.3	13.8	7.4	8.2	17.3	27.2	37.8	49.2
2008	C	175.3	140.2	105.2	70.1	35.1	35.1	70.1	105.2	140.2	175.3
	D	173.0	138.4	103.8	69.2	34.6	34.6	69.2	103.7	138.3	172.9

* Model index:A—$DG{M}^{1/2}(1,1)$; B—$D_v+DG{M}^{1/2}(1,1)$; C—$DG{M}^{2/3}(1,1)$; D—$D_v+DG{M}^{2/3}\left(1,1\right)$.

. The model accuracy is measured by the variationsin the predicted values (VP). Let $y_0(i) (i=1,\dots n)$ be the predicted value based on the original modeling data without noise interference and $y_1(i) (i=1,\dots n)$ be the predicted value with data noise. We define $VP=\left({\displaystyle {\sum }_{i=1}^n\left|y_1(i)-y_0(i)\right|}\right)/n$ and $\Delta VP=VP(DG{M}^p(1,1))-VP(D_v+DG{M}^p(1,1))$. If $\Delta VP$ falls into the positive domain, it means $D_v$ improves the stability of the model $DG{M}^p(1,1)$. The results are shown in

Table 4. The $\Delta VP$ values of $DG{M}^p(1,1)$ in different noise scenarios.

Model Parameter	Noise Position	Noise Amplitude
		−10%	−8%	−6%	−4%	−2%	2%	4%	6%	8%	10%
$p=0.5$	2003	2.6628	2.1243	1.5884	1.0552	0.5247	0.5285	1.0513	1.5715	2.0892	2.6044
	2004	1.7216	1.5510	1.2953	0.9532	0.5230	0.6081	1.3126	2.1123	3.0087	4.0037
	2005	0.7736	0.9582	0.9699	0.8125	0.4891	0.6420	1.4430	2.3964	3.4982	4.7444
	2006	0.2291	0.6812	0.8729	0.8164	0.5230	0.7329	1.6756	2.8156	4.1435	5.6502
	2007	12.633	9.5106	6.7037	4.1937	1.9632	1.7191	3.1968	4.4470	5.4795	6.3032
	2008	−3.4280	−2.7186	−2.0199	−1.3334	−0.6607	−0.6377	−1.2610	−1.8653	−2.4499	−3.0140
$p=2/3$	2003	2.8174	2.2531	1.6890	1.1250	0.5612	0.5660	1.1295	1.6928	2.2561	2.8194
	2004	2.7897	2.3443	1.8434	1.2866	0.6732	0.7262	1.5134	2.3598	3.2661	4.2327
	2005	1.5322	1.4354	1.2316	0.9234	0.5129	0.6054	1.3085	2.1042	2.9900	3.9635
	2006	0.6858	0.8631	0.8772	0.7345	0.4410	0.5755	1.2875	2.1284	3.0930	4.1764
	2007	12.560	9.6861	6.9995	4.4937	2.1617	2.0047	3.8503	5.5440	7.0902	8.4931
	2008	−4.5834	−3.6746	−2.7613	−1.8443	−0.9244	−0.9208	−1.8446	−2.7685	−3.6919	−4.6142

.

Table 3 shows that the $DG{M}^p(1,1)$ models with the fractional-order bidirectional weakening buffer operator $D_v$ have smaller perturbation bounds in almost all the noise scenarios except for when the data noise occurred in 2007. The results indicate that the data preprocessing method based on the fractional-order bidirectional weakening buffer operator improves the stability of the $DG{M}^p(1,1)$ models. Table 4 shows that operator $D_v$ improves the accuracy of prediction performance in most noise samples, except for the noises in the last modeling time point (in 2008). The variations in the predicted values increase with increasing noise amplitudes; however, the data preprocessing based on $D_v$ effectively reduced the noise effect on the prediction results. Though $D_v$ fails when the noise occurs in the last modeling time point, the data interference occurring in other time points is well controlled.

5.2. Numerical Study on Model $NIGM(p,1)$

We now consider the effect of operator $D_v(v=0.06)$ in improving the stability of the model $NIGM(p,1)$ in forecasting the annual electricity consumption in Russia [23]. The forecasting results ofthe four comparedmodels without any data noise interference are shown in

Table 5. The forecastingresults of $NIGM(p,1)$ models with and without operator $D_v$.

	Modeling Period				Forecasting Period
	2000	2001	2002	2003	2004	2005	2006	2007
Actual value	52333	53151	53168	54372	55516	55898	58600	60281
$DGM(1,1)$*	52333	52953	53561	54176	54798	55428	56065	56709
$GM(0.98,1)$*	52333	53704	54858	55849	56704	57445	58087	58645
$NIGM(0.997,1)$*	52333	52627	53051	53666	54559	55855	57736	60464
$D_v+NIGM(0.997,1)$	52333	52701	53193	53854	54742	55934	57534	59684

* values obtained from [23].

. It is clear that operator $D_v$ improves the short-term (2004–2005) forecasting of $NIGM(p,1)$. Though its long-term (2006–2007) prediction performance is worse than those from $NIGM(p,1)$, its prediction accuracy is still acceptable and higher than the other two models ($DGM(1,1)$ and $GM(0.98,1)$).

Table 6. The perturbation bounds of two models in different noise scenarios.

Noise Position	Model Index *	Noise Amplitude (%)
		−0.03	−0.028	−0.026	−0.024	−0.022	−0.020	−0.018	−0.016	−0.014	−0.012	−0.010	−0.008	−0.006	−0.004	−0.002
2000	E	1183	1104	1026	947	869	790	711	633	554	475	396	317	238	159	79
	F	962	898	834	771	707	643	579	515	450	386	322	258	193	129	65
2001	E	680	635	591	546	501	456	411	366	320	275	229	184	138	92	46
	F	640	598	555	513	471	428	386	343	301	258	215	172	129	86	43
2002	E	1424	1328	1232	1136	1041	945	850	755	660	565	471	376	282	188	94
	F	1076	1003	931	859	787	715	643	571	499	428	356	285	214	142	71
2003	E	923	861	799	737	675	613	552	490	428	367	306	244	183	122	61
	F	757	706	655	604	554	503	452	402	351	301	251	201	150	100	50
		0.002	0.004	0.006	0.008	0.010	0.012	0.014	0.016	0.018	0.020	0.022	0.024	0.026	0.028	0.030
2000	E	79	159	238	318	397	477	557	637	716	796	876	956	1037	1117	1197
	F	65	129	194	259	323	388	453	518	583	648	713	778	844	909	974
2001	E	46	93	139	186	232	279	326	373	420	468	515	563	610	658	706
	F	43	87	130	174	217	261	304	348	392	436	480	524	568	613	657
2002	E	94	187	280	374	467	559	652	744	837	929	1021	1113	1204	1296	1387
	F	71	142	213	283	354	424	495	565	635	706	776	846	915	985	1055
2003	E	61	122	183	243	304	364	425	485	546	606	666	726	786	846	906
	F	50	100	150	200	249	299	349	398	448	497	547	596	645	694	744

* Model index:E—$NIGM(p,1)$; F—$D_v+NIGM(p,1)$.

shows the perturbation bounds of the $NIGM(p,1)$ models with and without operator $D_v$ in different noise scenarios. It is clear that the operator $D_v$ results in smaller perturbation values in all noise scenarios and thereby improves the stability of the $NIGM(p,1)$ model in this numerical study. When applying the indicator $VP$ defined in Section 5.1, the difference between $NIGM(p,1)$ with and without $D_v$ can be expressed by $\Delta VP=VP(NIGM(p,1))-VP(D_v+NIGM(p,1))$.

Table 7. The $\Delta VP$ values of $NIGM(p,1)$ in different noise scenarios.

Noise Position	Noise Amplitude (%)
	−0.03	−0.028	−0.026	−0.024	−0.022	−0.020	−0.018	−0.016	−0.014	−0.012	−0.010	−0.008	−0.006	−0.004	−0.002
2000	26.90	36.95	46.95	56.91	66.82	76.69	86.51	96.28	106.00	115.66	125.28	134.84	144.35	153.80	163.20
2001	162.80	163.51	164.21	164.91	165.59	166.27	166.94	167.60	168.25	168.89	169.52	170.14	170.75	171.35	171.95
2002	143.20	144.05	145.33	148.15	150.84	153.40	155.83	158.14	160.33	162.40	164.36	166.21	167.94	169.58	171.10
2003	−85.52	−68.22	−50.94	−33.66	−16.40	0.85	18.09	35.31	52.52	69.71	86.89	104.05	121.19	138.32	155.44
	0.002	0.004	0.006	0.008	0.010	0.012	0.014	0.016	0.018	0.020	0.022	0.024	0.026	0.028	0.030
2000	170.73	168.87	166.94	164.95	162.89	160.77	158.57	156.31	153.98	151.58	149.10	146.55	143.93	141.23	138.45
2001	162.74	152.94	143.13	133.30	123.47	113.62	103.76	93.89	84.01	74.12	64.21	54.29	44.37	34.42	24.47
2002	158.24	143.88	129.45	114.96	100.40	85.78	71.10	56.37	41.58	26.74	11.86	−3.07	−18.04	−33.06	−43.95
2003	172.27	171.98	171.66	171.31	170.94	170.53	170.10	169.64	169.15	168.63	168.08	167.50	166.89	166.24	165.57

shows the $\Delta VP$ results in different noise amplitude scenarios. We can see that when a data disturbance occurs in the early modeling period (2000 or 2001 in this case), operator $D_v$ improves the accuracy of the prediction results in all thirty noise amplitude conditions. When noise occurs in the later modeling time point in 2002 (or 2003), the prediction without $D_v$ outperforms that of with $D_v$ in only four (or five) out of thirty noise perturbations. These confirm the effectiveness of operator $D_v$ in improving the stability of the prediction model $NIGM(p,1)$.

The numerical cases in this section demonstrate that the fractional-order bidirectional weakening buffer operator $D_v$ reduces the one-sided reaction to sample disturbances, and on the other hand, it can effectively improve the stability of time series prediction models by taking into account both the old and new information.

6. Conclusions

The performance of sequence operators in the stability of operator-based time series models has a direct impact on the validity of the model findings. Without a specific requirement for a particular part of the information in a series object, all data elements in this series should be treated equally. However, some widely used fractional-order weakening buffer operators are too subjectively biased in dealing with samples and sample disturbances. In contrast, the fractional-order bidirectional weakening buffer operator $D_v$ presents a more objective approach to data processing. Both theoretical investigations and numerical experiments, based on real cases, show the favorable effects of operator $D_v$. This study reveals that the proposed fractional-order weakening buffer operator-based data preprocessing method reduces the perturbation bounds of models in data noise scenarios, thereby improving the stability and prediction accuracy of time series models. Other features of the various fractional-order operators and their effects in time series models are worth further studies.