Nonlinear Grey Model Research Articles

Forecasting the output of high-tech industry in China: A novel nonlinear grey time-delay multivariable model with variable lag parameters

Under the rapidly developing economy in China, accurate forecasting holds vital significance for policymaking and operational planning within the high-tech industry. However, the influencing factors affecting the output, accompanied by the time-delay effect, could be nonlinear, and uncertain. Thereby, this paper proposes a new nonlinear grey multivariable model with time-varying lag parameters. To be specific, the newly designed time delay function and power exponent are introduced, which can significantly enhance the adaptability and flexibility of the proposed method. The Grey Wolf Optimization algorithm is utilized to calculate the dynamic time lag parameters and power exponent to improve the prediction reliability. Furthermore, this new approach is applied to predict the high-tech industry’s output in China, Shanghai Municipality, and the Eastern Region, with due consideration given to the time-delay effect between input factors and outputs. To assess its efficacy, some leading models are selected for comparison to the proposed model. Furthermore, the utilization of Monte-Carlo simulation, the Probability Density Analysis, and the simulations are used to demonstrate the robustness and stability of this new method. The findings show that the proposed model is a feasible and applicable approach for prediction, exhibiting outstanding accuracy.

Enriched nonlinear grey compositional model for analyzing multi-trend mixed data and practical applications

The compositional data are interrelated, and analyzing the evolution of each component is crucial for understanding population dynamics. However, the complex structure and tedious process of modeling pose challenges to the reasonable construction of grey compositional models for analyzing multi-trend mixed data. To address this, a novel enriched nonlinear grey compositional model with global multi-parameter combinatorial optimization is firstly proposed. Secondly, two types of Monte Carlo simulations are designed to validate the performances, modeling characteristics and noise levels of our model. Finally, using the bioenergy power generation structure of China as a case study, the practicability of our approach is verified. The results demonstrate that our model significantly outperforms traditional mainstream models in multi-trend mixed sequences, and the interrelationships among components are effectively verified. Our model not only enriches the methodological base but also broadens the application scope of grey compositional model.

A non-linear grey Fourier model based on kernel method for seasonal traffic speed forecasting

This article introduces a non-linear grey Fourier model utilizing the kernel method to better characterize seasonal traffic speed with fluctuating trends. The linear trend term in the grey Fourier model is replaced with feature mapping, resulting in a non-linear form. Utilizing Lagrange multipliers and a kernel function, the parameter estimation is transformed into a linear equation-solving problem. Order selection is carried out via power spectrum analysis, and hyper-parameters are determined using Bayesian optimization. Two numerical examples from real problems are presented to validate the model’s performance. The results indicate that the developed model outperforms nine models across four categories. Specifically, the model is applied to predict traffic speed in Guangzhou, and comparative results with four neural networks demonstrate its commendable performance.

Onshore and offshore wind power generation forecasting using a novel flexible time-varying fractional nonlinear grey model

Onshore and offshore wind power generation forecasting using a novel flexible time-varying fractional nonlinear grey model

A CRITIC-TOPSIS and optimized nonlinear grey prediction model: A comparative convergence analysis of marine economic resilience

The strength of marine economic resilience (MER) determines whether a region can maintain its original development path or fall into stagnation in the face of economic shock, which is an important starting point for promoting marine economic sustainable development. We discuss the connotation and propose a logical framework for MER, which is estimated by CRITIC-TOPSIS method from 2006 to 2019 based on three dimensions (country, region, province). Furthermore, the temporal evolution trend and regional disparities are accessed by Dagum Gini coefficient. Results reveal that MER has shown a significant upward trend, and it is undergoing gradient change in China, and unfortunately, the specific performance is in the imbalance, and lack of coordination of MER in the southern marine economic circle is the most prominent. Foreseeably, the novel fractional nonlinear grey model with double optimization (DOFNGBM(1,1)) combining Grey wolf optimization (GWO) algorithm is proposed and its performance is examined. Ulteriorly, as predicted by DOFNGBM(1,1) and presented by σ convergence and β convergence models, based on the cross-comparison between in-sample and out-of-sample, significant findings indicate that weak convergence of China’s MER indicates that coastal areas with a low level have faster improvement rates, and MER in China will present different ways of growing delightedly. Comparatively, considering the driving factors of marine economic resilience, all of them are proven to be conducive to raising the steady-state value of MER to a higher level, despite the regional heterogeneity of their effects. Purposefully, it implies that diversified development and synergistic cooperation can improve the endogenous power of marine economic resilience.

Forecasting education expenditure with a generalized conformable fractional-order nonlinear grey system model

As an important human capital investment, education is an effective means to improve the comprehensive quality of people. Education expenditure is an important material guarantee for the development of educational undertakings. Education expenditure data is highly susceptible to numerous economic and social factors that complicate its nonlinear structure. In order to model the complex nonlinear problems of the system, this paper proposes a generalized conformable fractional-order nonlinear grey prediction model for the first time by analyzing the traditional time series-based modeling method in a nonlinear grey domain. The proposed model expands on the classical grey Bernoulli model by introducing the generalized conformable fractional accumulation as a new accumulation generator and utilizes error minimization principles in the modeling process. By altering the optimal order of the model and the cumulative generation operator, this model can adapt to various time series and reduce errors. Finally, the model is applied to education expenditure forecasting, and it is proved that the proposed model achieved good results and has higher accuracy than other models.

Estimating China's CO2 emissions under the influence of COVID-19 epidemic using a novel fractional multivariate nonlinear grey model.

The COVID-19 pandemic has posed a severe challenge to the global economic recovery and China's economic growth. Although China has achieved stage victory against the epidemic, the impact and influence of the epidemic on China's economy will linger. On the positive side, the epidemic has led to a dramatic reduction in air pollution levels and a sharp slowdown in greenhouse gas emissions such as CO2. Forecasting China's CO2 emissions under the impact of the COVID-19 epidemic is crucial for formulating policies that contribute to smooth economic development and rational energy consumption. To this end, this paper improves on the traditional grey model, GM(1, N), by developing a novel fractional multivariate nonlinear grey model, FMNGM(q, N). These improvements include two key points. First, different power exponents are assigned to each relevant factor variable to explain their nonlinear, uncertain, and complicated relationships with the system characteristic variables. Second, the integer accumulation is changed to fractional accumulation to preprocess the data, and the Caputo-type fractional derivative is used to characterize the endogenous relationships among the system characteristic data. In addition, the quantum particle swarm optimization (QPSO) algorithm is used to optimize the above parameters with the goal of minimizing MAPE. The collected data on China's CO2 emission from 2001 to 2018 were divided into two parts according to different stages of economic development: 2001-2009 and 2010-2018. Both sets of data were modelled and analysed separately and compared with other models. Results showed that the proposed model had better prediction accuracy than other models. Finally, the model built using the 2010-2018 data was used to forecast China's CO2 emissions. Results show that China's CO2 emissions in 2020 under the impact of COVID-19 will decrease by approximately 3.15% from 2019 to 9893 million tons. The results bear important policy implications for planners in investment in clean energy and infrastructure to achieving the goal of a low-carbon transition.

Two-Outputs Nonlinear Grey Box Model for Lithium-Ion Batteries

Storing energy efficiently is one of the main factors of a more sustainable world. The battey management system in energy storage plays an extremely important role in ensuring these systems’ efficiency, safety, and performance. This battery management system is capable of estimating the battery states, which are used to give better efficiency, a long life cycle, and safety. However, these states cannot be measured directly and must be estimated indirectly using battery models. Therefore, accurate battery models are essential for battery management systems implementation. One of these models is the nonlinear grey box model, which is easy to implement in embedded systems and has good accuracy when used with a good parameter identification method. Regarding the parameter identification methods, the nonlinear least square optimization is the most used method. However, to have accurate results, it is necessary to define the system’s initial states, which is not an easy task. This paper presents a two-outputs nonlinear grey box battery model. The first output is the battery voltage, and the second output is the battery state of charge. The second output was added to improve the system’s initial states identification and consequently improve the identified parameter accuracy. The model was estimated with the best experiment design, which was defined considering a comparison between seven different experiment designs regarding the fit to validation data, the parameter standard deviation, and the output variance. This paper also presents a method for defining a weight between the outputs, considering a greater weight in the output with greater model confidence. With this approach, it was possible to reach a value 1000 times smaller in the parameter standard deviation with a non-biased and little model prediction error when compared to the commonly used one-output nonlinear grey box model.

An Optimized Fractional Nonlinear Grey System Model and Its Application in the Prediction of the Development Scale of Junior Secondary Schools in China

As part of China’s compulsory nine-year education system, junior secondary education (JSSE) plays a vital role in supporting students’ physical and mental development. The accurate prediction of the development scale trend of JSSE is helpful for the government to estimate the scale of educational development within a chosen time frame so as to aid decision making.Nevertheless, China’s education system is complex, highly dimensional, and largely influenced by policy and other factors, which results in difficulty in modeling the education sample. Based on gray system theory, this paper proposes an improved fractional-order grey prediction model, OCFNGBM(1,1), to predict the development scale of JSSE. We describe the basic expressions of the model, the parameter estimation method, and the optimization method for hyperparameters and construct a scheme for optimizing the background value coefficients. Data collected from official websites from 2011 to 2021 are used to build the forecasting model, and data from 2011 to 2017 are used to evaluate the model’s accuracy. Our experimental results indicate that the OCFNGBM(1,1) model has higher accuracy than the classical nonlinear gray prediction model. The OCFNGBM(1,1) model was employed to forecast the development scale of JSSE in China from 2022 to 2024, which provided useful information. This research provides a resource to help the national education department to develop a comprehensive and long-term plan for the development goals, scale, speed, steps, and measures of relevant education.

Comparison of Data-Driven Modeling and Identification Approaches for a Self-Balancing Vehicle

This paper gives a systematic comparison of different state–of–the–art modeling approaches and the corresponding parameter identification processes for a self–balancing vehicle. In detail, a nonlinear grey box model, its extension to consider friction effects, a parametric black box model based on regression neural networks, and a hybrid approach are presented. The parameters of the models are identified by solving a nonlinear least squares problem. The training, validation, and test datasets are collected in full–scale experiments using a self–balancing vehicle. The performance of the different models used for ego–motion prediction are compared in full–scale scenarios, as well. The investigated model architectures can be used to improve both, simulation environments and model–based controller design. This paper shows the upsides and downsides arising from using the different modeling approaches. Videos showing the self–balancing vehicle in action are available at: https://tinyurl.com/mvn8j7vf

Multi-objective optimization of recycling and remanufacturing supply chain logistics network with scalable facility under uncertainty

ABSTRACT Recycling and remanufacturing logistics network affects the scale and efficiency of sustainable development of the manufacturing industry. This paper designs a multi-level closed-loop supply chain network with supplier, manufacturer, recycling centers, preprocessing centers and processing centers. An improved nonlinear grey Bernoulli-Markov model is developed to predict the recycled quantity. The capacity of recycling center and preprocessing center, the demand of manufacturer and the inventory of preprocessing center are formulated as constraints. A dynamic multi-objective model, which is based on scalable logistics facilities, takes into account the minimization of system operating costs and minimization of time costs related to the out-of-stock and inventory in each operating cycle. This model realizes the dynamic selection of the scale of facilities. Objective weighted genetic algorithm is adopted to transform multi-objective optimization problem into a single-objective. A scrap automobile products calculations are analyzed to verify the effectiveness and practicability of this model.

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.

Developing a Grey Forecasting Model for the Air Flowing across the Parallel Plate Duct

The focus of this work is on developing a nonlinear grey model for the laminar flow regime of carrier fluid across the flat plate collector. The trust region reflective algorithm was used to solve the nonlinear laminar flow problem and handle the sparse matrix. The stream function, dimensionless velocity, gradients of velocity components, carrier fluid temperature, and the absorber plate temperature were estimated for a flat plate collector operated with and without a circular chimney. Similarly, the same technique was adopted to determine a model for heat transfer across the absorber plate. The numerically obtained solution was also compared with the parameters obtained through the instrumental measurement. The relative tolerance was kept at 10−5 for the white and grey box solutions. The developed model was noticed to have a smaller deviation than the conventional analytical model to predict the experimental values. The slip condition was observed for the proposed scheme. The uncertainty in the temperature measurement through the proposed model varies from ±0.53 to ±1.47 K. The minimum absolute percentage error (MAPE) lies in the domain of 0.52–1.67% for the nonlinear grey model, whereas the linear grey model for measuring absorber plate temperature has a percentage error of 0.0011–0.02%.

The research on modeling and application of dynamic grey forecasting model based on energy price-energy consumption-economic growth

The unbalanced development among energy price, energy consumption, and economic growth will destroy the stability of the energy market. The nonlinear differential equations among several factors are established based on the actual situation of energy market changes and the causal relationship between energy price and economic growth in an economic stage and discuss the relationship among the factors. The nonlinear grey prediction model group of energy price-energy consumption-economic growth is established by using grey difference information. The least-square method is utilized to give the parameter estimation formula of the prediction model group, and the approximate time response formula of the model is obtained by recursive relation transformation. Finally, the new models are applied to the prediction of coal price and coal consumption in China, and the effectiveness of the two models analyzes the same data set. It shows that the prediction results of coal price and coal consumption are better than the other three classical grey prediction models, and it can effectively predict the coal price and consumption in China from 2021 to 2025. The prediction results can provide adequate information for China's energy system and have theoretical and practical significance for the stability of the energy system.

A Minimum Variance Hedging Ratio Model Based on Nonlinear Grey Classification Model

The risk transfer function of futures market is mainly realized by hedging strategy. Futures price yield and spot price yield tend to show different fluctuations before and during hedging, which leads to the distortion of hedging ratio, that is, the calculated hedging effect is weaker than the traditional hedging effect. On the basis of MV (minimum variance) hedging model, this paper introduces NGCM (nonlinear grey classification model) to solve the nonlinear correlation between futures and spot returns, which can improve the hedging effect. The results show that, due to the existence of basis, the price change model violates the linear assumption of OLS (ordinary least squares) parameters, and there is a problem of model missetting. It is estimated that HR (hedging ratio) should choose the price model, which can better depict the linkage between futures price and spot price. The effectiveness of HR in this study is higher than that of existing models. Applying this model to hedge can effectively avoid the spot price risk. Investors can reasonably choose the hedging model according to their own needs.

Application of a novel grey Bernoulli model to predict the global consumption of renewable energy

Accurate prediction of renewable energy can provide an important basis for national energy security and the government to formulate policies. Therefore, according to the non-linearity of energy prediction system, this paper proposes a new nonlinear grey Bernoulli optimization model by the grey Bernoulli extended model, which is a nonlinear grey prediction model, and studies the properties of the new model. The order and nonlinear coefficient of the new model are optimized by Particle Swarm Optimization. Then, through the consumption of global renewable energy, such as solar, wind and hydropower as empirical analyses, the results of the four evaluation indicators show that the new model works better than the original model, which has higher prediction accuracy than before and makes the prediction model more applicable. At the same time, the model results were compared with the weighted grey model, Verhulst and the discrete grey model, and the new model has the highest accuracy. Finally, the new model is used to forecast the global consumption of wind, solar and hydropower energy in 2019–2023. The results will provide important forecasting information for global energy conservation and emission reduction policies.

Battery Model Identification Approach for Electric Forklift Application

Electric forklifts are extremely important for the world’s logistics and industry. Lead acid batteries are the most common energy storage system for electric forklifts; however, to ensure more energy efficiency and less environmental pollution, they are starting to use lithium batteries. All lithium batteries need a battery management system (BMS) for safety, long life cycle and better efficiency. This system is capable to estimate the battery state of charge, state of health and state of function, but those cannot be measured directly and must be estimated indirectly using battery models. Consequently, accurate battery models are essential for implementation of advance BMS and enhance its accuracy. This work presents a comparison between four different models, four different types of optimizers algorithms and seven different experiment designs. The purpose is defining the best model, with the best optimizer, and the best experiment design for battery parameter estimation. This best model is intended for a state of charge estimation on a battery applied on an electric forklift. The nonlinear grey box model with the nonlinear least square method presented a better result for this purpose. This model was estimated with the best experiment design which was defined considering the fit to validation data, the parameter standard deviation and the output variance. With this approach, it was possible to reach more than 80% of fit in different validation data, a non-biased and little prediction error and a good one-step ahead result.

Predictions and mitigation strategies of PM2.5 concentration in the Yangtze River Delta of China based on a novel nonlinear seasonal grey model.

High delicate particulate matter (PM2.5) concentration can seriously reduce air quality, destroy the environment, and even jeopardize human health. Accordingly, accurate prediction for PM2.5 plays a vital role in taking precautions against upcoming air ambient pollution incidents. However, due to the disturbance of seasonal and nonlinear characteristics in the raw series, pronounced forecasts are confronted with tremendous handicaps, even though for seasonal grey prediction models in the preceding researches. A novel seasonal nonlinear grey model is initially designed to address such issues by integrating the seasonal adjustment factor, the conventional Weibull Bernoulli grey model, and the cultural algorithm, simultaneously depicting the seasonality and nonlinearity of the original data. Experimental results from PM2.5 forecasting of four major cities (Shanghai, Nanjing, Hangzhou, and Hefei) in the YRD validate that the proposed model can obtain more accurate predictive results and stronger robustness, in comparison with grey prediction models (SNGBM(1,1) and SGM(1,1)), conventional econometric technology (SARIMA), and machine learning methods (LSSVM and BPNN) by employing accuracy levels. Finally, the future PM2.5 concentration is forecasted from 2020 to 2022 using the proposed model, which provides early warning information for policy-makers to develop PM2.5 alleviation strategies.

A new grey prediction model and its application to predicting landslide displacement

Landslide displacement prediction is an important part of reducing landslide hazard losses. The existing methods to predict landslide displacement are too complicated to be applied to engineering practice. The landslide displacement evolution and the grey model prediction mechanism show a good consistency. However, the existing grey prediction model also has some shortcomings including neglecting time term. A new grey prediction model called the background value optimization nonlinear grey prediction model (BNGM(1,1, t2)) is proposed to overcome the deficiencies. BNGM(1,1, t2) is a univariate prediction model that incorporates the influence of the time term. The integration method is used to determine the background value, and the minimum value method is employed to obtain a constant term of time response functions. Five accuracy test methods for BNGM(1,1, t2) are examined. BNGM(1,1, t2) can show better performances than other multivariate prediction models including the recursive discrete multivariate grey prediction model. BNGM(1,1, t2) is applied to four typical landslide case studies. The results indicate that the BNGM(1,1, t2) has the best prediction accuracy. The complexity of BNGM(1,1, t2) is lower than the nonlinear grey Bernoulli model, the Weibull–Bernoulli grey model, and the fractional accumulation nonlinear grey Bernoulli model. The comparison of comprehensive results demonstrates that the BNGM(1,1, t2)-based method has a wide application potential to predict landslide displacement.

A Predictive Analysis of China’s CO2 Emissions and OFDI with a Nonlinear Fractional-Order Grey Multivariable Model

Since the implementation of the Belt and Road Initiative, China’s total amount of outward foreign direct investment (OFDI) has increased each year and this has caused its relationship with carbon emissions (CO2e) to receive great attention recently. Forecasting China’s CO2e accurately by considering the impact of OFDI is important since the government can use it to formulate an appropriate emissions plan to fulfill its carbon reduction commitments. Because the relationship between OFDI and CO2e has nonlinear characteristics, a nonlinear grey multivariable model with fractional-order accumulation (NFGM(1,N)) was proposed in this study. To enhance the prediction accuracy, an optimization process was used to determine the parameters. The outcomes of the variable fractional order showed that fractional-order accumulation can better extract the grey information hidden in the original data, which confirmed the principle of new information priority. The result of the power coefficient indicated a nonlinear relationship between the CO2e and OFDI. Based on the prediction performance, the prediction accuracy of the NFGM(1,N) model was proven to be superior to those of the ARMA model, linear regression model, the GM(1,1), GM(1,N), and FGM(1,N) models. The empirical results also revealed that OFDI increased the CO2e in China. The relationship between the CO2e and OFDI exhibits a U-shaped development based on further predictions for the 2018–2030 period. Finally, some suggestions for long-term CO2e reduction plans were provided in this paper.