Page 1 of 12

Archives of Business Research – Vol. 12, No. 12

Publication Date: December 25, 2024

DOI:10.14738/abr.1212.17968.

Phuong, N. T. T., Nhan, V. T., & Phan, V. T. (2024). An Improved the Prediction Accuracy of the Nonlinear Grey Bernoulli Model by

Fourier Series and Its Application in Container Throughput Forecasting in Danang Port. Archives of Business Research, 12(12). 01-

12.

Services for Science and Education – United Kingdom

An Improved the Prediction Accuracy of the Nonlinear Grey

Bernoulli Model by Fourier Series and Its Application in

Container Throughput Forecasting in Danang Port

Nguyen Thi Thuy Phuong

Central Region College of Technology,

Economics and Water Resources

Vu Thanh Nhan

Vietnam-Korea University of Information and

Communications Technology – University of Danang

Van Thanh Phan

Vietnam-Korea University of Information and

Communications Technology – University of Danang

ABSTRACT

In order to improve the prediction accuracy of Nonlinear Grey Bernoulli Model

NGBM (1,1), this study using Fourier series to modify their residual error of this

model. To verify the effectiveness of the proposed approach, the annual water

consumption in Wuhan from 2005 to 2012 is used for the modeling to forecast the

annual water consumption demand from 2013 to May 2020, and the forecasting

results proved that the Fourier- NGBM (1,1) is a better than the among forecasting

model used in this situation. Furthermore, this proposed approach is applied the

real case in forecasting the Container Throughput Forecasting in Danang Port. The

empirical results show that the proposed model will get a higher accuracy

performance with the lowest MAPE =1.93%. This result is not only show the

effectiveness of proposed model but also offers valuable insights for Danang

policymakers in orientation and planning management agency so as to boost the

development of upcoming port activities.

Keywords: Nonlinear Grey Bernoulli model, Fourier series, Forecasting, Accuracy,

Container, Danang Port.

INTRODUCTION

Grey forecasting is one of main part of Grey system theory, an effective method for modeling

and forecasting small sample time series. In the early 1980s, Deng [1, 2] proposed the grey

model GM (1,1) based on control theory, which is the core model used in the grey forecasting

model. This model utilizes an operator obtained by first -order accumulation to operate on the

non-negative original sequence. It demonstrates the approximate exponential growth laws and

achieves short-term forecasting accuracy. With Its advantages in dealing with uncertain

information and using as few as four data points [3,5], The GM (1,1) has been validated and

widely used in various fields such as tourism [6, 7], transportation [8- 10], financial and

economic [11- 13], integrated circuit industry [14-17], energy industry [18-20] etc...

Page 2 of 12

2

Archives of Business Research (ABR) Vol. 12, Issue 12, December-2024

Services for Science and Education – United Kingdom

In the recent years, there are many scholars propose new procedures with different ways to

improve the precision accuracy of GM (1,1) model. For instant, Lin et al. [21] and Wang et al.

[22] used different methods to calculate new background values to improve the background

values. Hsu [17] and Wang et al. [23] used different methods to modified internal parameter

estimation like development coefficient and grey input coefficient. Some scholars have

established GM (1,1) model with residuals modification like Hsu [15] and Wang et al. [24]. In

addition, many hybrid models based on GM (1,1) were proposed. These include the grey

econometric model [25], the grey Markov model [27, 28], and the grey fuzzy model [21], etc.

Despite its improvement in prediction accuracy, the prediction accuracy of the GM (1,1) model

is always monotonic. As a result, GM (1,1) model may not be always satisfactory. The recently

developed nonlinear grey Bernoulli model NGBM (1,1) is a new grey forecasting model [28]. It

has a power exponent n that can effectively manifest the nonlinear characteristics of real

systems and flexibility determines the shape of the model’ curve. Unlike GM (1,1) and the grey

Verhulst model which rely on a constant number such as 0 or 2, the NGBM (1,1) does not

require such a number (excluding 1).

Therefore, forecasting of the fluctuation sequence can be performed by the fluctuation features

as long as the power exponent and structural parameters in the model are known. The NGBM

(1,1) was successfully used to simulate and forecast the values of the annual unemployment

rates of ten selected countries and foreign exchange rates of Taiwan’s major trading partners

[29, 30]. This success indicates that the NGBM (1,1) significantly improves the accuracy of the

simulation and forecasting predictions of the original GM (1,1). Zhou et al. [30] selected the

value of n by using a particle swarm optimization algorithm and used the model to forecast the

power load of the Hubei electric power network. Hsu [16] used the genetic algorithm to

optimize parameters of the NGBM (1,1) and applied it to forecast the economic trends in the

integrated circuit industries in Taiwan. Chen et al. [31] proposed a Nash NGBM (1,1) in which

an interpolated coefficient in the background value is introduced into the NGBM (1,1) and the

parameters solved for based on the Nash equilibrium concept. This strengthens the adaptability

of the model towards the original data and eventually improves the accuracy of the model.

Later, Wang, et al. [32] proposed optimized NGBM (1,1) model to forecast the qualified

discharge rate of the industrial waste water in 31 administrative areas in China by improved

background interpolation value p and exponential value n is put forward in an NGBM (1,1). Pao

et al. [33] forecasted CO2 emissions, energy consumption, and economic growth in China based

on ARIMA and NGBM (1,1). Performance evaluation results showed that the NGBM (1,1) can be

used safely for future projection of these indicators in clean energy economy.

All these improvements focus on the model parameters and the background value. In fact, the

initial condition is also an important factor determining the grey modeling accuracy. This is

because the initial condition is a part of the predictive function. The current paper aims to

develop an approach to increase the predictive precision of the NGBM (1,1) by modifying the

residual error obtained from NGBM (1,1) with Fourier series. Numerical example and practical

application shows that the proposed Fourier-NGBM (1,1) model has higher performance on

models prediction. The remainder of this paper is organized as follows. A brief introduction to

the original NGBM (1,1) and the method of optimization of the modified residual error by

Fourier series are given in Section 2. Section 3 proves the effectiveness of using the proposed

Fourier-NGBM (1,1) by comparison with the Coupling model of Grey system and Multivariate