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Integral matching-based nonlinear grey Bernoulli model for forecasting the coal consumption in China

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Abstract

Nonlinear grey Bernoulli model, abbreviated as NGBM model, has been validly used in real applications due to its high accuracy in nonlinear time series forecasting. However, there remain technical challenges to explain the mechanism of the accumulative sum operator in nonlinear grey modelling process and estimate structural parameters independent from the initial values. This paper aims to reconstruct the modelling process of the NGBM model so as to explain the modelling mechanism better by utilizing the integral matching approach, which consists of an integral formula and the numerical discretization-based least squares. First, the integral formula is employed to investigate the accumulative sum operator and further reconstruct the NGBM model to a generalized form, referred as to INGBM model. Then, a novel parameter estimation strategy, estimating structure parameters and initial values simultaneously, is developed by utilizing the numerical discretization-based least squares approach. Next, Monte Carlo simulation studies are designed to evaluate the finite sample performance of both models. Comparisons show that the INGBM model outperforms to the original one in terms of parameter estimation accuracy, forecasting accuracy and robustness to noise. Finally, we apply the INGBM model at a coal consumption in China study to further illustrate the usefulness of this model.

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Funding

This work was funded by National Natural Science Foundation of China (Grant No. 71671090), Joint research project of National Natural Science Foundation of China and Royal Society of UK (Grant No. 71811530338), the Fundamental Research Funds for the Central Universities (Grant No. NP2018466) and Qinglan Project for excellent youth or middle-aged academic leaders in Jiangsu Province (China).

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Authors and Affiliations

  1. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Lu Yang & Naiming Xie

Authors
  1. Lu Yang
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  2. Naiming Xie
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Correspondence to Naiming Xie.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by V. Loia.

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This work was supported by National Natural Science Foundation of China (71671090, 71871117); Joint Research Project of National Natural Science Foundation of China and Royal Society of UK (71811530338); Fundamental Research Funds for Central Universities of China (NP2018466); Qinglan Project for excellent youth or middle-aged academic leaders in Jiangsu Province, China.

Appendix

Appendix

The ode45 method, where 4 represents the fourth-order Runge-Kutta algorithm and 5 represents the fifth-order truncation error, is a variable-step numerical method to solve differential equations (Dormand and Prince 1980).

In general, the nonlinear ordinary differential equation can be written as

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}x(t) = f(t,x(t)), ~x(t_0) = x_0, ~a \le t \le b. \end{aligned}$$

The main equations of fourth-order Runge–Kutta algorithm are

$$\begin{aligned} {\left\{ \begin{array}{ll} t_{i+1}=t_i +h\\ x_{i+1} = x_i + h \frac{(k_1 +2k_2 + 2k_3 + k_4)}{6} \\ k_1 = f(t_i,x_i)\\ k_2 = f\left( t_i+\frac{h}{2}, x_i+\frac{k_1}{2}\right) \\ k_3 = f\left( t_i+\frac{h}{2}, x_i+\frac{k_2}{2}\right) \\ k_4 = f(t_i+h, x_i+k_3)\\ \end{array}\right. } \end{aligned}$$

where h is the step size, \( k_1,~k_2,~k_3,k_4\) are the slopes at points over the interval \([t_i, t_{i+1}]\). The accuracy of ode45 is determined by the fifth-order truncation error \({\mathcal {O}}(h^5)\).

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Yang, L., Xie, N. Integral matching-based nonlinear grey Bernoulli model for forecasting the coal consumption in China. Soft Comput 25, 5209–5223 (2021). https://doi.org/10.1007/s00500-020-05521-3

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