Understanding cumulative sum operator in grey prediction model with integral matching

doi:10.1016/j.cnsns.2019.105076

Communications in Nonlinear Science and Numerical Simulation

Volume 82, March 2020, 105076

https://doi.org/10.1016/j.cnsns.2019.105076 Get rights and content

Highlights

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We investigate the mechanism of cumulative sum operator in grey prediction model by using integral transformation.
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We provide a novel paradigm for estimating the structural parameters and the initial value simultaneously.
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We analyze the influence of sample size and noise level on modelling performance by using large-scale simulation studies.

Abstract

Grey prediction models have been widely used in various fields and disciplines. Cumulative sum operator, also called accumulative generation operator, is an essential step in grey modelling, but until now relatively limited attention has been paid to its mechanism of action. In this paper, we introduce the integral matching to explain it. By using the integral transformation, the grey prediction model whose nature is modelling the cumulative sum series with a differential equation proves to be equivalent to that modelling the original series with a reduced differential equation. The cumulative sum operator is the discretization and approximation of the definite integral terms by using the piecewise constant integral and thus can be improved by using the piecewise linear integral. Simulation studies detail the advantages in terms of the stability and robustness to noise.

Graphical abstract

Introduction

Time series prediction has been an important topic in various applications ranging from science through technology to engineering. In the past years many prediction methods have been developed, such as exponential smoothing [1], Box-Jenkins models [2], support vector regression [3], neural network [4], etc. See [5] for a review. Each model has its own modelling assumptions and application scopes. Grey prediction model, as one of the time series analysis and prediction approaches, assumes that the original time series is partially quasi-exponential. Since the cumulation and release in many generalized energy system usually conform to the quasi-exponential law [6], research has been consistently shown that grey modelling approach is promising in various fields and disciplines.

Grey model GM(1,1), first order (the first ‘1’) ordinary differential equation with one variable (the second ‘1’), was first proposed to modelling and forecasting the small-sample quasi-exponential time series in socioeconomic system [7]. In order to improve the accuracy, a large number of studies have been conducted. These include the optimization methods and the extension models. The optimization work is mainly focused on improving the performance without changing the structure of GM(1,1) model, such as the background coefficient optimization [8], [9], [10], the initial value improvement [11], [12], [13], the introduction of rolling and moving mechanism [14], [15], [16], and the combination with other models [17], [18], [19]. The extension studies are focused on extending the application scopes by changing the forcing term of the differential equation in GM(1,1) model from a constant to a time-varying term. These include NGM(1,1) model using the linear function of time as the forcing term [20] for the quasi non-homogeneous exponential series, and GPM(1,1,N) model using the polynomial function of time as the forcing term for the time series coupled with quasi-exponential and time-varying polynomial characteristics [21], [22], as well as KRNGM(1,1) model using the kernel function as the forcing term for more complex nonlinear time series [23]. In addition, these grey prediction models have been successfully used in both traditional and emerging fields. See [24] for a review.

In the above optimized and novel models, cumulative sum operator is not only an essential step but also one of the major feature distinguishing grey prediction models from other methods. But by now it still lacks the mechanism analysis especially from a mathematical analysis perspective, although reference [6] claims that the cumulative sum operator is able to recognize the pattern hidden in the time series. Furthermore, the structural parameters and the initial value are estimated by using a two-step method in the classical models, which may introduce extra error in the second step. Therefore, it is also necessary to analyze the mechanism of cumulative sum operator from a statistical perspective and then propose a simultaneous parameter estimation method.

The remainder of this paper is organized as follows: we give a brief introduction to GM(1,1) model in Section 2, and elaborate the modelling mechanism by introducing the integral matching in Section 3; the large-scale simulation studies are conducted in Section 4; the conclusion and future work are present in Section 5.

Section snippets

Basic concepts of GM(1,1) model

Definition 1

For a time series $X (t) = {x (t_{1}), x (t_{2}), \dots, x (t_{n})},$ the cumulative sum series is defined as $Y (t) = {y (t_{1}), y (t_{2}), \dots, y (t_{n})},$ where $y (t_{k}) = \sum_{i = 1}^{k} h_{i} x (t_{i}), h_{k} = {\begin{matrix} 1, & k = 1, \\ t_{k} - t_{k - 1}, & k = 2, 3, \dots, n . \end{matrix}$ Then the coupled equation ${\begin{matrix} \frac{d}{d t} y (t) = a y (t) + b, y (t_{1}) = ξ, t \geq t_{1} (2 a) \\ x (t_{k}) = a (\frac{1}{2} y (t_{k - 1}) + \frac{1}{2} y (t_{k})) + b, k = 2, 3, \dots, n (2 b) \end{matrix}$ is referred to as GM(1,1) model [6], and equations (2a) and (2b) are called the continuous-time and discrete-time forms, respectively.

Remark 1

The discrete-time equation (2b) is the approximation of equation (2a) by using the trapezoid rule $x (t$

Integral matching for reduced differential equation

Lemma 1

Letting $y (t) = x (t_{1}) + \int_{t_{1}}^{t} x (τ) d τ, t \geq t_{1},$ then the differential equation (2a) is equivalent to the reduced differential equation $\frac{d}{d t} x (t) = α x (t), x (t_{1}) = β, t \geq t_{1}$ where $α = a$ and $β = \frac{b}{1 - a}$ .

Proof

It can be seen from Eq. (6) that the initial values of differential equations (2a) and (7) are both equal to $y (t_{1}) = ξ = x (t_{1}) = β = \frac{b}{1 - a}$ .

Let us first prove that Eq. (7) can be derived from equation (2a). By substituting Eq. (6) into the left side and right side of equation (2a), respectively, it follows that $\frac{d}{d t} y (t) = \frac{d}{d t} (x (t_{1}) + \int_{t_{1}}^{t} x (τ) d τ) = x$

Data generation process

In this section, we analyze the performance of the above approaches in Monte Carlo simulation studies. The following data generation process is considered: $\tilde{x} (t) = x (t) + ε (t)$ where x(t) is obtained by explicitly solving the differential Eq. (7); $ε (t) \sim N (0, σ^{2})$ is the additive Gaussian noise where the noise level is measured by the signal to noise ratio $SNR = \frac{Var [x (t)]}{Var [ε (t)]} = \frac{1}{σ^{2}} Var [x (t)]$ .

The observations are sampled from $\tilde{x} (t)$ at every time interval of h in the range of t ∈ [0, 5], which generates $\frac{5}{h} + 1$

Conclusion and future work

The cumulative sum operator is an essential step in the grey modelling process, and also the main feature distinguishing grey prediction models from other methods. We explain the mechanism of cumulative sum operator in the modelling process from both mathematical analysis and parameter estimation viewpoints. The integral matching provides a novel paradigm for simultaneous estimation of structural parameter and initial value in grey prediction models. Large-scale simulation experiments are

Declaration of Competing Interest

The authors declare that there are no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors appreciate the anonymous referees for the insightful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grants 71671090 and 71871117), Joint Research Project of National Natural Science Foundation of China and Royal Society of UK (Grant 71811530338), Fundamental Research Funds for the Central Universities of China (Grant NP2018466), Qinglan Project for excellent youth or middle-aged academic leaders in Jiangsu Province, China.

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Short communicationUnderstanding cumulative sum operator in grey prediction model with integral matching

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Basic concepts of GM(1,1) model

Integral matching for reduced differential equation

Data generation process

Conclusion and future work

Declaration of Competing Interest

Acknowledgments

Int J Prod Econ

Automatica

Pattern Recognit

Int J Forecast

Appl Math Comput

Eng Appl Artif Intell

Math Comput Model

Expert Syst Appl

Omega

Omega

Mech Syst Signal Process

Expert Syst Appl

Appl Math Model

Appl Math Model

Commun Nonlinear Sci Numer Simul

Short communication
Understanding cumulative sum operator in grey prediction model with integral matching