Nonlinear Grey Bernoulli Model Research Articles

Huizhou GDP forecast based on fractional opposite-direction accumulating nonlinear grey bernoulli markov model

<abstract><p>In this paper, a fractional opposite-direction accumulating nonlinear grey Bernoulli Markov model (FOANGBMKM) is established to forecast the annual GDP of Huizhou city from 2017 to 2021. The optimal fractional order number and nonlinear parameters of the model are determined by particle swarm optimization (PSO) algorithm. An experiment is provided to validate the high fitting accuracy of this model, and the effect of prediction is better than that of the other four competitive models such as autoregressive integrated moving average model (ARIMA), grey model (GM (1, 1)), fractional accumulating nonlinear grey Bernoulli model (FANGBM (1, 1)) and fractional opposite-direction accumulating nonlinear grey Bernoulli model (FOANGBM (1, 1)), which proves the robustness of the opposite-direction accumulating nonlinear Bernoulli Markov model. This research will provide a scientific basis and technical references for the economic planning industries.</p></abstract>

A novel matrixed nonlinear grey Bernoulli model for interval prediction of power generation

Development trends of power generation sequences have nonlinearity and uncertainty. The nonlinearity can be reflected by the nonlinear grey Bernoulli model. The uncertainty can be reflected by the interval sequence. A matrixed nonlinear grey Bernoulli model based on interval sequences is proposed in this paper. In order to make the model directly applicable to interval sequence, the traditional grey Bernoulli model is linearized by power function transformation, then the parameters are matrixized, and finally the recursive prediction formula is obtained by Cramer's law. Particle swarm optimization algorithm is used to estimate the nonlinear parameter. By adjusting the parameter, the new model can be used to predict exponential, saturated and parabolic interval sequences. The new model is used to predict the intervals of China's thermal power, hydro power, nuclear power, wind power, national power generation and power generation in east China. The prediction results show that the new model has higher prediction accuracy than the competing models. Finally, the power generation of four modes and national power generation from 2022 to 2025 are predicted and analyzed.

An Optimized Damping Grey Population Prediction Model and Its Application on China's Population Structure Analysis.

Population, resources and environment constitute an interacting and interdependent whole. Only by scientifically forecasting and accurately grasping future population trends can we use limited resources to promote the sustainable development of society. Because the population system is affected by many complex factors and the structural relations among these factors are complex, it can be regarded as a typical dynamic grey system. This paper introduces the damping accumulated operator to construct the grey population prediction model based on the nonlinear grey Bernoulli model in order to describe the evolution law of the population system more accurately. The new operator can give full play to the principle of new information first and further enhance the ability of the model to capture the dynamic changes of the original data. A whale optimization algorithm was used to optimize the model parameters and build a smooth prediction curve. Through three practical cases related to the size and structure of the Chinese population, the comparison with other grey prediction models shows that the fitting and prediction accuracy of the damping accumulated–nonlinear grey Bernoulli model is higher than that of the traditional grey prediction model. At the same time, the damping accumulated operator can weaken the randomness of the original data sequence, reduce the influence of external interference factors, and enhance the robustness of the model. This paper proves that the new method is simple and effective for population prediction, which can not only grasp the future population change trend more accurately but also further expand the application range of the grey prediction model.

Comparing forecasting accuracy of selected grey and time series models based on energy consumption in Brazil and India

The study compares the forecasting performance of the grey type models, represented by an optimized nonlinear grey Bernoulli model (ONGBM), a nonlinear grey Bernoulli model with particle swarm optimization (NGBM-PSO), and a standard GM with a classic time series model, represented by Auto-Regressive Integrated Moving Average (ARIMA). The models are compared based on simulations and energy consumption forecasting in Brazil and India at aggregate and disaggregate levels from 1992 to 2019. The study illustrates the picture of the Brazil and India energy consumption nexus at aggregate and disaggregated levels. The forecasting accuracy is compared using standard measures such as MAPE, MSE, RMSE, and normalized RMSE. The Diebold-Mariano test findings validates the ARIMA (1,1,1), GM (1,1), ONGBM (1,1) and NGBM (1,1)-PSO models' equal predictive performance. The models are used to compute forecast combinations, ensuring smaller forecasting errors than single models. Optimizing NGBM (1,1) using two algorithms ensures the highest forecasting efficiency for short time series. The results allow a recommendation to use ONGBM (1,1) and NGBM (1,1)-PSO models in the short-term forecasting of energy consumption and combine these forecasts with forecasts from GM (1,1) and ARIMA (1,1,1) models in practical applications.

Forecasting share of renewables in primary energy consumption and CO2 emissions of China and the United States under Covid-19 pandemic using a novel fractional nonlinear grey model

China and the United States (U.S.) are in the first two places among the countries that consume the most primary energy and emit CO2 emissions in the world. Considering the Sustainable Development Goal 7.2 and Paris Agreement’s goals, forecasting of CO2 emissions and the share of renewables in primary energy consumption for China and the United States in the coming years will be a guide for energy policies. In this context, this study aims to forecast the share of renewables in primary energy consumption and CO2 emissions of China and the U.S. using a novel optimized fractional nonlinear grey Bernoulli model with rolling mechanism, briefly as ROFANGBM(1,1), under pandemic and non-pandemic scenarios. This study also analyzed the gap in the energy consumption and CO2 emissions for the year 2020 caused by the Covid-19 pandemic. The results showed that ROFANGBM(1,1) gave the highest prediction performance with having the lowest mean absolute percentage error (MAPE) value for all cases and the share of renewables in primary energy consumption in 2025 is forecasted as 12.3% for the U.S. and 16.6% for China by using ROFANGBM(1,1). Additionally, CO2 emissions of China and the U.S. are forecasted by using ROFANGBM(1,1) as 10112 Mt and 4583 Mt in 2025, respectively. It is believed that this study will provide new avenues for researchers to make more accurate predictions by addressing the fluctuations due to the Covid-19 pandemic thanks to the proposed novel grey prediction model.

A novel conformable fractional non-homogeneous grey Bernoulli model based on Salp Swarm Algorithm and its application

The nonlinear grey Bernoulli model uses the first-order accumulated generating operation (1-AGO) to accumulate the sequence. However, 1-AGO violates the principle of new information priority, and the prediction accuracy needs to be improved. For this purpose, this paper proposed a conformable fractional non-homogeneous grey Bernoulli model by combing the conformable fractional accumulation operator and the non-homogeneous grey Bernoulli model to forecast biofuels production. The article discussed the properties of the proposed model and proved that the novel model is a more general extension of other grey models. The Salp Swarm Algorithm was used to optimize the nonlinear multi-objective parameters and improve prediction accuracy. Furthermore, three examples were used to verify the proposed model’s performance capability, which verified that the competitiveness of the new model is a better grey model. Then the novel model could be used to predict biofuels production in the US and China. Biofuels production from 2009 to 2016 was used to construct models, and the data from 2017 to 2020 was used to verify the model’s accuracy. The results indicated that the proposed model has higher accuracy in terms of prediction. Finally, the novel model could predict biofuels production of the above countries from 2021 to 2025. The forecast results suggested that biofuels production in the US will change moderately. And in China, biofuels production will show an increasing trend in the next few years.

An MPA-based optimized grey Bernoulli model for China’s petroleum consumption forecasting

The remarkable prediction of petroleum consumption is of significance for energy scheduling and economic development. Considering the uncertainty and volatility of petroleum system, this paperpresents a nonlinear grey Bernoulli model with combined fractional accumulated generation operator to forecast China’s petroleum consumption and terminal consumption. The newly designed model introduces a combined fractional accumulated generation operator by incorporating the traditional fractional accumulation and conformable fractional accumulation; compared to the old accumulation, the newly optimized accumulation can enhance flexible ability to excavate the development patterns of time-series. In addition, to further improve the prediction performance of the new model, marine predation algorithm is applied to determine the optimal emerging coefficients such as fractional accumulation order. Furthermore, the proposed model is verified by a numerical example of coal consumption; and this newly established model is applied to predict China’s petroleum consumption and terminal consumption. Our tests suggest that the designed ONGBM(1,1,k,c) model outperforms the other benchmark models. Finally, we predict China’s petroleum consumption in the following years with the aid of the optimized model. According to the forecasts of this paper, some suggestions are provided for policy-makers in the relevant sectors.

Predicting the production and consumption of natural gas in China by using a new grey forecasting method

Reasonable prediction of natural gas consumption and production in China is helpful for decision-makers formulate policies to solve the increasingly prominent contradiction between supply and demand of natural gas in China. Taking into account the data environment of China’s small sample, this paper considers using the grey prediction model with characteristic of small sample modeling to study the consumption and production of natural gas in China. To achieve accurate prediction, this paper developed a novel discrete fractional nonlinear grey Bernoulli model with power term (DFNGBM(1,1, α)), and used the latest equilibrium optimizer to solve the hyper-parameters of the model. The proposed model has the advantages of most existing grey prediction models, such as fractional-order accumulation operation and time power term. In addition, the proposed model is a unified expression form of many other grey forecasting models. The consumption and production of natural gas in China from 2003 to 2020 are used as examples to verify the feasibility and validity of the proposed model compared with other competitive models that have been used to study natural gas. It turns out that the prediction ability of the proposed model is better than other models. Therefore, the feasibility and effectiveness of the proposed model have been confirmed. Finally, the DFNGBM (1,1, α) model is used to predict China’s natural gas consumption and production in the next few years, and some corresponding recommendations are given based on the prediction results.

Prediction of WEEE Recycling in China Based on an Improved Grey Prediction Model

Accurate waste electrical and electronic equipment (WEEE) recycling forecast is an essential reference for optimizing e-waste industry layout and division of labor policies, conducive to better guiding enterprises’ recycling activities and improving the efficiency of WEEE recycling in China. The nonlinear grey Bernoulli model (NGBM (1,1)) was constructed by analyzing the recycling data characteristics of WEEE from 2012 to 2020, and a particle swarm optimization (PSO) algorithm was introduced to solve the model parameters and optimize the background value coefficients. The prediction results were compared with other grey prediction models to verify the effectiveness of the improved NGBM (1,1) model for WEEE recycling prediction in China and the applicability of the PSO algorithm for improving the prediction accuracy of each grey model. Statistical data were used to forecast the WEEE recycling volume in China from 2021 to 2023, and the results show that the value of WEEE recycling will continue to grow at 9%. The value of recycling will reach 16 billion yuan by 2023, while the quantity of WEEE recycling will see a slight decline. Based on the calculation results, the WEEE recycling industry development trend is predicted to guide the promotion of the WEEE industry recycling program and the national circular economy program.

Predictive analysis of the industrial water-waste-energy system using an optimised grey approach: A case study in China

To estimate the dynamic trend of industrial water-waste-energy (hereinafter referred to as WWE) system, this paper proposes a new method for forecasting specific indicators in such a system. First, the fractional accumulated generation operator, fractional derivative and classic nonlinear grey Bernoulli model are simultaneously coupled to develop an optimised nonlinear grey Bernoulli model that identifies the nonlinear trends in industrial WWE systems. Second, the particle swarm optimization algorithm is employed to determine the optimal model parameters in the newly-designed model. Based on this, simulation studies are conducted to examine the stability of the proposed model. Finally, the model is applied in the industrial WWE system. The results demonstrate that (1) the proposed model outperforms other competitive models in terms of error-value metrics and (2) industrial water use and industrial energy consumption will increase, whereas industrial wastewater discharge will decline. Furthermore, the rationality of the predicted results redis analyzed from a policy perspective.

Prediction of CO2 emissions in Saudi Arabia using Nonlinear Grey Bernoulli Model NGBM (1,1) compared with GM (1,1) model

Abstract One of the most critical solution for tackling the challenges of global warming and climate change is to study and know the accurate prediction of carbon dioxide (CO2) emissions. Thus, aid to develop appropriate strategic plans that will reduce future damages caused by these emissions into the atmosphere. This study utilizes annual time series data on CO2 emissions in Saudi Arabia from 1970 to 2016. The goal of this study is to predict CO2 emissions using the Nonlinear Grey Bernoulli model NGBM (1,1), and compared with the GM (1,1) model based on MAPE metrics to achieve a high-accuracy prediction. The NGBM (1,1) is a newly created grey model with wide ranging applications in diverse fields due to its precision in handling small time-series datasets with nonlinear variations. The NGBM (1,1) with power γ is a nonlinear differential equation that can control the predicted result and adjust the solution to fit the 1-AGO of previous raw data. Thus, the findings show that at sample sizes of N=10 and N=5, the Nonlinear Grey Bernoulli Model (NGBM) is more precise than the Grey Model GM (1, 1). The findings could help the government develop future economic policies.

UFNGBM (1,1): A novel unbiased fractional grey Bernoulli model with Whale Optimization Algorithm and its application to electricity consumption forecasting in China

The electricity distribution in a planned way has been a difficult issue that the power supply bureau wishes to solve, and it is the lifeblood of economic development. Forecasting annual electricity consumption is very crucial for the planning of the power supply bureau and for booming economic development. We propose a novel unbiased fractional nonlinear grey Bernoulli model [i.e, UFNGBM (1,1)] to forecast China’s annual electricity consumption based on the nonlinear grey Bernoulli model [i.e, NGBM (1,1)]. First, UFNGBM (1,1) approach is designed to derive calculation formula of the novel model, and validity of the model is proved by the matrix perturbation theory. Second, a novel optimization algorithm is introduced based on the whale algorithm to find the optimal parameters (i.e., order and power) of the proposed model. Third, the accuracy, stability, and effectiveness of our method are verified through three real-world cases in China. Finally, we collect the electricity consumption data of three provinces in China and successfully apply the proposed algorithm to predict the electricity consumption from 2019 to 2024. The experimental results demonstrate that our proposed model is significantly superior to nine alternative models on the electricity consumption data of Jilin and Jiangsu. The performance of our novel method is close to the state-of-the-art deep learning method on the electricity consumption data of Shandong. It is noticed that our method [as an extended version of NGBM(1,1)] is significantly better than NGBM(1,1) on these three real-world datasets, which further shows the effectiveness of the our proposed algorithm. Meanwhile, the electricity consumption of these three provinces in the next six years (2019–2024) is forecasted, which has a very good guiding significance and provides a more reliable reference for the economic and power bureau.

An integrated forecasting model for the coffee bean supply chain

ABSTRACT Coffee is the most traded commodity after petroleum. The Vietnamese coffee bean industry has raised concerns lately over an inefficient coffee value chain; bets on coffee price uncertainty are increasing worldwide in the current. Accurate optimization of coffee bean prices helps manufacturers to control an unpredictable market and upgrade cooperativeness in sustainable agriculture. The authors proposed a forecasting method to deal with demand volatility and uncertainty in volumes and coffee bean prices. In this paper, we applied the forecasting nonlinear grey Bernoulli model (NGBM) (1,1). NGBM (1,1), which is based on the parameter optimization algorithm, can increase the precision of predictions. NGBM (1,1) was integrated with Fourier residual modification model to forecast coffee bean price, which was a crucial factor in the Vietnamese coffee bean supply chain. The price of coffee beans was calculated using a differential equation in an uncertain system, along with actual data collected over the past six years. The results of this study demonstrate that an integrated forecasting model is an effective forecasting method. This research can help companies to control risks that come with uncertain coffee prices and reduce risks in the sustainable agriculture supply chain.

A novel composite forecasting framework by adaptive data preprocessing and optimized nonlinear grey Bernoulli model for new energy vehicles sales

To accurately predict the time series having limited data with stochastic disturbances and nonlinearity, this paper proposed a composite forecasting model by adaptive data preprocessing and optimized nonlinear grey Bernoulli model. Specifically, improvements in the proposed model lie in the following aspects: firstly, the new-information-based buffer operator is utilized to eliminate stochastic disturbance effects and thereby enhance the smoothness of the system series, which can effectively extract the potential patterns of recent development. Secondly, based on the preprocessed data, the nonlinear grey Bernoulli model provided with the newly-designed initial condition that abides by the principle of new information priority without data lapses enables the forecasting precision and applicability enhancement. Thirdly, due to the nonlinear relationships between the prediction errors and the parameters introduced by the above optimization paths, the Particle Swarm Optimization algorithm is employed to ascertain the optimal parameters simultaneously, whose efficacy and robustness have been validated by experimental comparison and sensitivity analysis. On the foundation of the above functional improvements, the novel composite forecasting framework successfully overcomes the limitations of conventional grey models to obtain more accurate and robust forecasts, which are substantiated by the applications for predicting the sales of new energy vehicles in China and Norway. Experimental results and discussions demonstrate that the new-information-based buffer operator can be regarded as an excellent alternative tool for data preprocessing. Besides, the grey models with buffered preprocessing can deliver much more accurate forecasts than their corresponding versions without buffered preprocessing. Furthermore, the proposed model with buffered preprocessing outperforms any of the other competitors and can be considered as the most promising technique for prognosticating NEVs sales.

A novel fractional discrete nonlinear grey Bernoulli model for forecasting the wind turbine capacity of China

PurposeA new method for forecasting wind turbine capacity of China is proposed through grey modelling technique.Design/methodology/approachFirst of all, the concepts of discrete grey model are introduced into the NGBM(1,1) model to reduce the discretization error from the differential equation to its discrete forms. Then incorporating the conformable fractional accumulation into the discrete NGBM(1,1) model is carried out to further improve the predictive performance. Finally, in order to effectively seek the emerging coefficients, namely, fractional order and nonlinear coefficient, the whale optimization algorithm (WOA) is employed to determine the emerging coefficients.FindingsThe empirical results show that the newly proposed model has a better prediction performance compared to benchmark models; the wind turbine capacity from 2019 to 2021 is expected to reach 275954.42 Megawatts in 2021. According to the forecasts, policy suggestions are provided for policy-makers.Originality/valueBy combing the fractional accumulation and the concepts of discrete grey model, a new method to improve the prediction performance of the NGBM(1,1) model is proposed. The newly proposed model is firstly applied to predict wind turbine capacity of China.

Nonlinear Grey Bernoulli model NGBM (1, 1)'s parameter optimisation method and model application

<p style='text-indent:20px;'>In the grey prediction, the nonlinear Grey Bernoulli model NGBM (1, 1) is an important type. The NGBM (1, 1) has good adaptability to data fitting and then small prediction errors, and thus has been applied widely. However, if we improve the modelling method, the prediction precision shall be improved to some extent. The important factors of prediction error are the approximation of background value and the approximation of power exponent. Therefore, the paper tries to combine the optimisation of background value with the optimisation of the power exponent of NGBM (1, 1) model and then improves the model from parameter estimation. The paper gives three methods for the following three cases respectively: the background value in the form of exponential curve, the background value in the form of the polynomial curve and the background value in the form of interpolation function, to combine background value optimisation with power exponent optimisation for parameter optimisation. The final section of the paper builds the NGBM (1, 1) models of China's GDP and energy consumption with three improvement methods. The simulation and prediction results show the three improvement methods all have high precision. The methods given offer good approaches for the in-depth study on nonlinear grey Bernoulli model, enrich the method system of grey modelling and can be applied to the studies on other grey models to promote the study and wide application of the grey model.</p>

Forecasting fuel combustion-related CO2 emissions by a novel continuous fractional nonlinear grey Bernoulli model with grey wolf optimizer.

Foresight of CO2 emissions from fuel combustion is essential for policy-makers to identify ready targets for effective reduction plans and to further improve energy policies and plans. A new method for forecasting the future development of China's CO2 emissions from fuel combustion is proposed in this paper by using grey forecasting theory. Although the existing fractional nonlinear grey Bernoulli model (denoted as FNGBM(1,1)) has been theoretically proven to enhance the adaptability to diverse sequences, its fixed integer-order differential derivative still impairs the performance to some extent. To this end, a varying-order differential derivative is introduced into the existing differential equation to enable a more flexible structure, thus improving the prediction ability of FNGBM(1,1). Specifically, because of the advantages of conformable fractional accumulation, the traditional differential derivative is first replaced by the conformable fractional differential derivative. As a consequence, the continuous conformable fractional nonlinear grey Bernoulli model (hereinafter referred to as CCFNGBM(1,1)) is proposed. To further increase the validity of the model, a metaheuristic algorithm, namely Grey Wolf Optimizer (GWO), is then applied to search for the optimal emerging coefficients for the proposed model. Two real examples and China's CO2 emissions from fuel combustion are considered to verify the effectiveness of the newly proposed model, the experimental results show that the newly proposed model outperforms other benchmark models in terms of forecasting accuracy. The proposed model is finally employed to forecast the future China's CO2 emissions from fuel combustion by 2023, accounting for 10,039.80 million tons. Based on the forecasts, several policy suggestions are provided to curb CO2 emissions.

An Improved Nonlinear Grey Bernoulli Model Based on the Whale Optimization Algorithm and Its Application

In order to improve the prediction performance of the existing nonlinear grey Bernoulli model and extend its applicable range, an improved nonlinear grey Bernoulli model is presented by using a grey modeling technique and optimization methods. First, the traditional whitening equation of nonlinear grey Bernoulli model is transformed into its linear formulae. Second, improved structural parameters of the model are proposed to eliminate the inherent error caused by the leap jumping from the differential equation to the difference one. As a result, an improved nonlinear grey Bernoulli model is obtained. Finally, the structural parameters of the model are calculated by the whale optimization algorithm. The numerical results of several examples show that the presented model’s prediction accuracy is higher than that of the existing models, and the proposed model is more suitable for these practical cases.

A non-linear systematic grey model for forecasting the industrial economy-energy-environment system

To estimate the internal structure and forecast dynamic trends in the industrial economy-energy-environment (3E) system, a non-linear systematic grey model (NSGM(1, m)) is proposed. This model is derived from a non-linear grey Bernoulli model (NGBM(1, 1)) and consisted of multiple non-linear equations in an embedded manner. To minimise the mean absolute percentage error (MAPE), a particle swarm optimisation (PSO) algorithm is used to optimise the parameters of the model. The empirical analysis based on a dataset of Zhejiang Province, China from 2000 to 2018 shows that the NSGM(1, m) model can identify a non-linear relationship among industrial economic growth, energy consumption, and pollutant emissions. The MAPEs of training sets of three industrial 3E systems are 2.95%, 4.90%, and 3.75%, while those of test sets are 1.36%, 6.94%, and 1.72%, respectively. From 2019 to 2023, it is predicted that, with constant industrial and economic growth, industrial energy consumption and discharge of exhaust and solid waste will continue to increase while industrial wastewater discharge will decline. Moreover, efficiency indices indicate that the economic efficiency of the industrial 3E systems in Zhejiang Province will increase.

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

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