Nonlinear Grey Bernoulli Model Research Articles

Water–carbon–economy multivariate spatial–temporal collaborative relationships and nonlinear projections in urban agglomerations

A complex dynamic coupling process exists among water resource consumption, carbon emissions, and economic growth; however, their intricate collaborative mechanisms remain unclear. This study proposes a new barycenter formula-based multivariate spatial–temporal collaborative relationship model for identifying the complex relationships and interactions among the water footprint (WF), carbon footprint (CF), and gross domestic product (GDP) in the three largest urban agglomerations in the Yangtze River Economic Belt (YREB) of China. A nonlinear grey Bernoulli model is then advanced for illustrating the future evolution in WF–CF–GDP and their multivariate collaborative relationship. Results reveal that WF and CF of all urban agglomerations increase continuously, with annual growth rates of 0.59% and 5.63%, respectively. WF, CF, and GDP are significantly positively correlated, and they exhibit similar spatial clustering characteristics and centroid migration directions along the main course of the Yangtze River. Moreover, their multivariate temporal collaborative relationship (0.79) significantly surpasses their spatial (0.63), while the spatial–temporal (0.77) is intermediate. WF has the greatest impact on the binary spatial–temporal collaborative relationship between CF and GDP. In contrast, GDP has the least impact on the relationship between CF and WF. The projected WF, CF, and GDP indicate that the YREB will face more severe water and carbon pressures as its economy grows. Their multivariate spatial–temporal collaborative relationships are projected to decrease by 10.26% in 2050. These findings contribute to promoting collaborative and sustainable development of inter-regional socio-economic and ecological environment.

Forecasting medical waste in Istanbul using a novel nonlinear grey Bernoulli model optimized by firefly algorithm.

Waste management has gained global importance, aligning with the escalating impact of the COVID-19 pandemic and the associated concerns regarding medical waste, which poses threats to public health and environmental sustainability. In Istanbul, medical waste is considered a significant concern due to the rising volume of this waste, along with challenges in collection, incineration and storage. At this juncture, precise estimation of the waste volume is crucial for resource planning and allocation. This study, thus, aims to estimate the volume of medical waste in Istanbul using the nonlinear grey Bernoulli model (NGBM(1,1)) and the firefly algorithm (FA). In other words, this study introduces a novel hybrid model, termed as FA-NGBM(1,1), for predicting waste amount in Istanbul. Within this model, prediction accuracy is enhanced through a rolling mechanism and parameter optimization. The effectiveness of this model is compared with the classical GM(1,1) model, the GM(1,1) model optimized with the FA (FA-GM(1,1)), the fractional grey model optimized with the FA (FA-FGM(1,1)) and linear regression. Numerical results indicate that the proposed FA-NGBM(1,1) hybrid model yields lower prediction error with a mean absolute percentage error value 3.47% and 2.57%, respectively, for both testing and validation data compared to other prediction algorithms. The uniqueness of this study is rooted in the process of initially optimizing the parameters for the NGBM(1,1) algorithm using the FA for medical waste estimation in Istanbul. This study also forecasts the amount of medical waste in Istanbul for the next 3 years, indicating a dramatic increase. This suggests that new policies should be promptly considered by decision-makers and practitioners.

Forecasting natural gas consumption in Turkey using fractional non-linear grey Bernoulli model optimized by grey wolf optimization (GWO) algorithm

Forecasting natural gas consumption in Turkey using fractional non-linear grey Bernoulli model optimized by grey wolf optimization (GWO) algorithm

Data-driven multi-step solar photovoltaic predictions with limited and uncertain information: Insights from a collaboratively-optimized nonlinear grey Bernoulli model

As solar energy grows as a crucial substitute for fossil energy worldwide, it is worth investigating solar photovoltaic predictions characterized by complex patterns and a lack of prior knowledge. Accordingly, an optimized nonlinear grey Bernoulli model is proposed, encompassing three essential advancements. Firstly, the damping accumulated generating operation is initially integrated with the grey Bernoulli model to reduce random disturbances and improve the smoothness of the original collections, extracting more valuable information via preprocessing the raw data. Secondly, the background value is modified to remove the jumping errors arise from converting differential to difference functions based on the Simpson rule. Thirdly, the heuristic intelligent algorithm is employed to collaboratively explore the optimal damping coefficient and power index in the proposed model. For illustration purposes, experiments on predicting the global photovoltaic installed capacity (case one) and solar electricity net generation (case two) are carried out to examine the effectiveness and generality of this new approach. Experimental results indicate that the proposed model achieves significant improvements over the prevalent models in one-to-three-step ahead forecasting, obtaining the average 323.94% and 211.86% improvement rates over the seven benchmarks measured by MAPE and RMSE. Furthermore, robustness tests in terms of Monte-Carlo simulation, algorithm comparison, and hyper-parameter sensitivity analysis are performed to explore the proposed model’s forecasting stability against the intelligent algorithm randomness, choices, and hyper-parameters. Generally, the proposed model is proven to be a reliable method and is deployed for future global solar photovoltaic predictions.

Conformable fractional accumulation in triangular fuzzy sequences grey nonlinear model for tertiary industry gross output forecast

The prediction of the tertiary industry's output value is essential for enhancing the market economic system and providing guidance for the government in formulating relevant economic policies. The tertiary industry encompasses various sectors such as finance and transportation, and its total output value exhibits uncertain and nonlinear trends. Triangular fuzzy number sequences contain more comprehensive information than exact number sequences. Therefore, a nonlinear grey Bernoulli model with conformable fractional accumulation for triangular fuzzy sequences is proposed. Firstly, the conformable fractional accumulation generating operation is utilized to mitigate the volatility tendency of the original sequences. Additionally, a time power function term is introduced to capture the power exponential trend. Furthermore, an improved whale optimization algorithm incorporating opposition-based learning strategy and adaptive control parameters is employed to optimize the model parameters, which has been verified through algorithm comparison experiments. Three existing grey models are used as competing models to validate the prediction accuracy of our proposed model. Finally, we utilize this proposed model to predict the tertiary industry's output values in both Yangtze and Pearl River Deltas for 2024–2026.

A novel multi-scale fractional-order Bernoulli grey model and its application on primary school scale prediction

Primary education plays an important role in the whole educational structure. The prediction of the development scale of primary schools is of great significance to the optimal allocation of educational resources by education departments. The paper introduces a novel multi-scale fractional-order Bernoulli grey model by combining a local cumulative operator with a global cumulative operator, abbreviated as MFGM. First, a nonlinear grey Bernoulli model based on conformable fractional derivative is proposed, where a hybrid accumulation is presented. Then, the MFGM parameters are obtained by an PSO optimizer, which are adaptive to data sets. Finally, the proposed model is validated and used to predict the scale of primary school scale. Experimental results show that the MFGM is superior to the other grey models and machine learning models in the fitting and testing primary school scale. The prediction results contribute to the decision-making of education sectors.

An IDE-based nonlinear grey Bernoulli model and applications to daily traffic flow pattern identification

The nonlinear grey Bernoulli model is a powerful tool for modelling and forecasting time series exhibiting a rough inverted U-shape behaviour. Traditionally, it is considered an indirect dynamical model, informed by empirical knowledge that the accumulation of the inverted U-shape series leads to a growth behaviour, which is subsequently characterized by the Bernoulli ordinary differential equation (ODE). However, its mechanism remains incomplete, particularly concerning the modelling paradigm and parameter estimation. To this end, we propose an equivalent and simplified variant in the form of an integro-differential equation (IDE), enabling direct modelling of the inverted U-shaped series and thereby enhancing the interpretability. By taking the IDE as a state equation in the state space framework, separable nonlinear least squares (SLS) and nonlinear least squares (NLS) are respectively formulated to simultaneously estimate structural parameters and the initial condition. Then, by combining the state equation (ODE/IDE) and parameter estimation approach (SLS/NLS), the generated ODE-based indirect paradigms are comprehensively compared with the IDE-based direct ones. Numerical simulations show that (i) IDE-based models outperform ODE-based ones and (ii) NLS outperforms SLS. Real-world applications indicate that the nonlinear grey Bernoulli model combining IDE and NLS can accurately discover the underlying dynamics of daily traffic flow, outperforming the competitive models.

A novel Seasonal Fractional Incomplete Gamma Grey Bernoulli Model and its application in forecasting hydroelectric generation

With the arrival of the first truly global energy crisis, how to precisely forecast the hydroelectric generation becomes a hot spot for allowing governments to obtain more valuable information. A novel forecasting model, Seasonal Fractional Incomplete Gamma Nonlinear Grey Bernoulli Model (SFIGNGBM(1, 1)), is proposed in this paper to precisely forecast the hydroelectric generation in some countries. First, the seasonal raw data are classified into four seasonal groups based on their significant seasonal fluctuations. Second, a novel SFIGNGBM(1, 1) model is established by combining the Bernoulli equation, the fractional-order accumulation operator, and the incomplete gamma function to further optimize partial parameters in the forecasting model and improve the forecasting performance. Third, the Whale Optimized Algorithm (WOA) is employed to optimize the Bernoulli power exponent η, the fractional order parameter r, and the incomplete coefficient h for minimizing the MAPE values and enhancing the fitting precision. Finally, our results present that our proposed model outperforms a set of baseline forecasting models with the smallest three error measure values in all fitting results, and its MAPE values converge before 10 iterations. This indicates that our proposed model has a favorable forecasting performance with fast-convergence for hydroelectric generation in the elected countries.

Prediction and assessment of marine fisheries carbon sink in China based on a novel nonlinear grey Bernoulli model with multiple optimizations

The vigorous development of marine fisheries carbon sinks (MFCS) has become a momentous pathway to mitigate global warming and effectively cope with the climate crisis. Deservedly, based on clarifying mechanism of carbon sequestration, this paper designs a research paradigm for predicting and evaluating the potential of MFCS. Specifically, a novel nonlinear grey Bernoulli model, namely MFCSNGBM(1,1), is proposed by innovatively mining the original data law through adaptive cumulative series and introducing the compound Simpson formula to optimize background values. More precisely, we utilize a heuristic Grey Wolf Optimization algorithm to find the best power index, which enhances the adaptability. To prove usefulness and robustness of MFCSNGBM(1,1) model, yields of seven common shellfishes (oyster, clam, mussel, scallop, razor clam, bloody clam, and snail) and three main algae (kelp, pinnatifid undaria, and laver) are predicted and compared with six competing models. Based on prediction results, new model has the most accurate predictions, with all prediction errors being <10 %, and thus can achieve effective prediction of shellfish and algae production from 2022 to 2025. Further, the capacity and potential of MFCS in China are scientifically evaluated using a removable carbon sink model, considering various yield levels and biological parameters of shellfish and algae. The assessment results show that during the sample period, China's marine fisheries carbon sinks steadily increased with an annual growth rate of 57,000 tons. From 2022 to 2025, with support of policy of MFCS and improvement of disaster prevention and mitigation capacity, the potential of MFCS will be further released. The growth rate of MFCS will be increased to 94,000 tons per year, and its overall scale is expected to reach 2,198,245 tons by 2025, equivalent to fixing 8.06 million tons of CO2. The carbon sink's economic value is significantly estimated to be over 400 billion yuan.

A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area

A Nonlinear Grey Bernoulli Model with Conformable Fractional-Order Accumulation and Its Application to the Gross Regional Product in the Cheng-Yu Area

Nonlinear grey Bernoulli model with physics-preserving Cusum operator

The Cusum (cumulative sum) operator is a fundamental prerequisite for the nonlinear grey Bernoulli model. Traditionally, it is believed to visually identify the underlying dynamic pattern of the original time series. This paper presents the misconceptions concerning the Cusum operation and the over-optimization of the initial condition in the classical nonlinear grey Bernoulli model, both of which inspire the proposal of a physics-preserving Cusum operator. Under a state space framework, separable nonlinear least squares and nonlinear least squares are formulated to generate simultaneous estimates of structural parameters and initial condition. By combining the Cusum operators and parameter estimation methods, four modeling paradigms are generated and comprehensively compared. The simulation results show that (i) the physics-preserving Cusum outperforms the traditional Cusum and (ii) nonlinear least squares outperforms separable nonlinear least squares, especially in irregular sampling settings with large time intervals and high noise levels. Finally, the proposed approach is used to identify the underlying dynamics from short-term traffic flow data, and the results validate its effectiveness.

Assessing the current and future effects of Covid-19 on energy related-CO2 emissions in the United States using seasonal fractional grey model

Accurate CO2 forecasting plays an important role in energy planning. However, in the annual forecasting studies on CO2 emissions, the seasonal effects cannot be predicted. To overcome this problem, this study proposed a novel prediction model based on the seasonally optimised fractional nonlinear grey Bernoulli model (SOFANGBM(1,1)), combining the seasonal fluctuation technique with optimisation of the background, power index, and fractional order values. The proposed novel model offers two important improvements in prediction performance: (1) This model combined optimised fractional nonlinear grey Bernoulli model (OFANGBM(1,1)) with the seasonal fluctuation technique to enable monthly and quarterly predictions (2) The seasonally optimised fractional nonlinear grey Bernoulli model (SFANGBM(1,1)) was improved by optimising the background value. CO2 emissions had the largest share in global GHG emissions, and the United States was the second largest CO2 emission emitter worldwide after China in 2019. However, cases and deaths from Covid-19 continue in the United States, and important questions arise: How has Covid-19 affected CO2 emissions by fossil fuel type in the past, and how will it reshape them in the future? This study aimed to analyse how Covid-19 affects CO2 emissions from fossil fuels in the U.S., how it will reshape its future, and also contribute to Sustainable Development Goals (SDGs). Quarterly CO2 emissions from coal, natural gas, petroleum, and total CO2 emissions in the U.S. were forecasted using a novel grey prediction model under pandemic and pandemic-free scenarios. The pandemic-free scenario determined the CO2 emissions gap due to Covid-19, and the pandemic scenario presented forecasted results of quarterly and annual CO2 emissions by 2025. The prediction performance was tested from 2022-Q1 to 2022-Q4 by simulated from 2015-Q1 to 2021-Q4. Using the SOFANGBM(1,1), Covid-19 caused 2 %, 2 %, 16 %, and 12 % reductions in CO2 emissions from coal, natural gas, petroleum, and total CO2 emissions, respectively, in 2020. SOFANGBM(1,1) also forecasts that total CO2 emissions will reach 4520.6 Mt by 2025.

Forecast of clean energy generation in China based on new information priority nonlinear grey Bernoulli model.

The forecast of clean energy power generation is of major prominence to energy structure adjustment and the realization of sustainable economic development in China. In order to scientifically predict clean energy power generation data, a structure-adaptive nonlinear grey Bernoulli model submitted to the new information priority criterion (abbreviated as IANGBM) is established. Firstly, an improved conformable fractional accumulation operator that conforms to the priority of new information is proposed, which can effectively extract the information from small samples. Then, IANGBM is derived from the Bernoulli differential equation, and the perturbation bound theory proves that this model is suitable for the analysis of small sample data. In addition, the grey wolf optimization algorithm is utilized to optimize the model parameters to make the model more adaptable and generalized. To verify the superiority of the model, two cases consisting of wind and nuclear power generation prediction are implemented by comparing eight benchmark models involving IANGBM, GM, FGM, FANGBM, LR, SVM, BPNN, and LSTM. The experiment results demonstrate that the proposed model achieves higher prediction accuracy compared to the other seven competing models. Finally, the future nuclear and wind power generation from 2023 to 2030 are predicted by adopting the IANGBM(1,1) model. For the next 8 years, nuclear power generation will maintain stable development, while wind energy power generation will grow rapidly.

CO2 emission forecasting based on nonlinear grey Bernoulli and BP neural network combined model

CO2 emission forecasting based on nonlinear grey Bernoulli and BP neural network combined model

ESG Modeling and Prediction Uncertainty of Electronic Waste

Driven by a variety of factors, including the advent of digitalization, increasing population and urbanization, and rapid technological advancements, electronic waste (e-waste) has emerged as the fastest growing waste stream globally. Effective management of e-waste is inherently aligned with environmental, social, and governance (ESG) frameworks and is typically examined within this context. Accurate quantification of the current and future accumulation of e-waste is a key step towards ensuring its proper management. Numerous methodologies have been developed to predict e-waste generation, with the grey modeling approach receiving considerable attention due to its ability to yield meaningful results using relatively small datasets. This study aims to introduce a novel forecasting technique for predicting e-waste, particularly when limited historical data are available. The proposed approach, the non-linear grey Bernoulli model with fractional order accumulation NBGMFO(1,1) enhanced by Particle Swarm Optimization, demonstrates superior accuracy compared to alternative forecasting models. Additionally, the Fourier residual modification method is applied to enhance the precision of the forecast. To provide a practical illustration, a case study utilizing waste mobile phone data from Turkey is presented.

A novel nonlinear grey Bernoulli model NGBM(1,1,t^p,α) and its application in forecasting the express delivery volume per capita in China.

The grey prediction is a common method in the prediction. Studies show that general grey models have high modeling precision when the time sequence varies slowly, but some grey models show low modeling precision for the high-growth sequence. The paper researches the grey modeling for the high-growth sequence using the extended nonlinear grey Bernoulli model NGBM(1,1,t⌃p,α). To improve the nonlinear grey Bernoulli model NGBM(1,1,t⌃p,α)'s prediction precision and make data have better adaptability to the model, the paper makes improvements in the following three aspects: (1) the paper improves the accumulated generating sequence of original time sequence, i.e. making a new transformation of traditional accumulated generating sequence; (2) the paper improves the model's structure, extends the grey action and builds an extended nonlinear grey Bernoulli model NGBM(1,1,t⌃p,α); (3) the paper improves the model's background value and uses the value of cubic spline function to approximate the background value. Because the parameters of the new accumulated generating sequence transformed, the nonlinear grey Bernoulli model's time response equation and the background value are optimized simultaneously, the prediction precision increases greatly. The paper builds an extended nonlinear grey Bernoulli model NGBM(1,1,t⌃2,α) using the method proposed and seven comparison models for China's express delivery volume per capita. Comparison results show that the extended nonlinear grey Bernoulli model built with the method proposed has high simulation and prediction precision and shows the precision superior to that of seven comparison models.

Forecasting China's electricity generation using a novel structural adaptive discrete grey Bernoulli model

The generation forecast helps to develop the scheduling plan of power sources, build a power balance system and ensure the adequacy of power supply to the power system at the macro level. Therefore, a novel structurally adaptive discrete grey Bernoulli model, called DHGBM(1,1), is proposed in this paper for the prediction of power generation. Firstly, the Hausdorff fractional order cumulative generation operation makes it possible to achieve new information prioritisation. As for the structure of the model, we have extended the nonlinear grey Bernoulli model with the introduction of a nonlinear dynamic structure term, which allows the model to simulate variable generation data and improves the adaptability of the model. In a comparative test of the algorithms, the well-performing differential evolution algorithm was used for hyperparameter optimisation of the model to obtain better predictive performance. Monte Carlo simulations and probability density analysis verified the excellent robustness of the model. In addition, through the simulation tests, we obtained a reasonable range of parameters for the structural terms, thus limiting the complexity of the model, and the phenomenon of model overfitting was effectively prevented. Finally, five actual cases of China's hydro, China's thermal, China's nuclear, China's wind power generation and China's electricity exports are predicted.

Modelling and forecasting non-renewable energy consumption and carbon dioxide emissions in China using a PSO algorithm-based fractional non-linear grey Bernoulli model

In China, the consumption of non-renewable energy increases not only in general economic growth but also in large amounts of carbon dioxide (CO2) emissions which cause disasters and catastrophic damages to the environment. To alleviate environmental pressure, it is neccessary to forecast and model the relationship between energy consumption and CO2 emissions. In this study, a fractional non-linear grey Bernoulli (FANGBM(1,1)) model based on particle swarm optimization is proposed to forecast and model non-renewable energy consumption and CO2 emissions in China. Firstly, based on the FANGBM(1,1) model, non-renewable energy consumption in China is predicted. The comparison results of several competitive models show that the FANGBM(1,1) model has the best predictive performance. Then, the relationship between non-renewable energy consumption and CO2 emissions is modeled. On this basis, China's future CO2 emissions are effectively predicted based on the established model. The forecast results show that the growth trend of China's CO2 emissions will continue to grow to 2035, while the prediction results in different scenarios also show that that the different growth rates of renewable energy share lead to different times to peak CO2 emissions. In the end, relevant suggestions are proposed to support China's dual carbon goals.

Natural gas consumption forecasting using an optimized Grey Bernoulli model: The case of the world’s top three natural gas consumers

The accurate and reasonable prediction of natural gas consumption is important for enhancing national energy security and alleviating the environmental problems caused by energy consumption. Starting with the traditional nonlinear grey Bernoulli model, in this paper, we expand the development coefficient a and grey action quantity b, propose a new self-adaptive time-varying grey Bernoulli prediction model, deduce the time response formula of the model, and explore the relationship between model parameters and model accuracy. Then, the nonlinear parameter and power parameter coefficient of the new model are optimized by a genetic algorithm, and the effectiveness of the model is analyzed by the natural gas consumption of China, the United States, and Russia. The results show that, compared with other traditional grey prediction models, the new model has a better simulation effect, higher prediction accuracy, and stronger applicability. Meanwhile, the model was used to forecast the natural gas consumption of China, the United States, and Russia in the period from 2022 to 2026. The prediction results show that China’s natural gas consumption will continue to steadily increase in the future, which is consistent with the global goal of adhering to the orderly development of natural gas. The prediction results can provide a theoretical basis for relevant departments to formulate relevant natural gas policies.

Prediction of Vehicle Form Feature for Cross-model Evolution Based on Improved Nonlinear Grey Bernoulli Model

Prediction of Vehicle Form Feature for Cross-model Evolution Based on Improved Nonlinear Grey Bernoulli Model