Fractional Order Accumulation Research Articles

A novel seasonal grey prediction model with fractional order accumulation for energy forecasting

To accurately predict sequence data with seasonal characteristics, we combine data restart technology and fractional order accumulation into a novel seasonal grey model (FSGM (1,1, α)). The particle swarm optimization (PSO) algorithm is used to solve the fractional order and background value coefficients of the model, and the effectiveness of FSGM (1,1, α) is verified using three cases. Finally, we use FSGM (1,1, α) to predict quarterly electricity generation in Beijing and Henan Province and quarterly petroleum coke production in China from 2023 to 2027. The research results indicate that, first, FSGM (1,1, α) is reasonable and effective and has the ability to accurately capture the dynamic trend of seasonal data. Second, compared with the grey model (GM (1,1)), seasonal grey model (SGM (1,1)), data grouping grey model (DGGM (1,1)), data grouping seasonal model (DGSM (1,1)), and data grouping seasonal time model (DGSTM (1,1)), which have seasonal characteristics, FSGM (1,1, α) can better fit the original data, achieve higher prediction accuracy, and perform better. Third, from 2023 to 2027, it is predicted that there will be no significant change in Beijing's electricity generation, and the current stable trend will be maintained. Both the power generation in Henan Province and the petroleum coke production in China will steadily increase to a certain extent, with obvious seasonal cyclical fluctuations. Notably, the power generation and petroleum coke production in Henan Province in the fourth quarter of 2027 will increase by 11.50 % and 10.93 %, respectively, compared to those in the fourth quarter of 2023.

A novel grey fractional model based on model averaging for forecasting time series

The introduction of fractional order accumulation has played a crucial role in the development of grey forecasting methods. However, accurately identifying a single fractional order accumulation for modeling diverse sequences is challenging due to the dependence of different fractional order accumulations on data structure over time. To address this issue, we propose a novel fractional grey model abbreviated as FGMMA, incorporating a model averaging method. The new model combines existing fractional grey models by using four judgment criteria, including Akaike information criteria, Bayesian information criteria, Mallows criteria, and Jackknife criteria. Meanwhile, the cutting-edge algorithm named breed particle swarm optimization is employed to search the optimal fractional order for each candidate model to enhance the effectiveness of the designed model. Subsequently, we conduct a Monte Carlo simulation for verification and validation purposes. Finally, empirical analysis based on energy consumption in three countries is conducted to verify the applicability of the proposed model. Compared with other benchmark models, we can conclude that the proposed model outperforms the other competitive models.

Applications of Fractional Order Logistic Grey Models for Carbon Emission Forecasting

In recent years, global attention to carbon emissions has increased, becoming one of the main drivers of global climate change. Accurate prediction of carbon emission trends in small and medium-sized countries and scientific regulation of carbon emissions can provide theoretical support and policy references for the effective and rational use of energy and the promotion of the coordinated development of energy, environment, and economy. This paper establishes a grey prediction model using the classical Logistic mathematical model in a determined environment to investigate the carbon emission system. At the same time, we use the basic principle of fractional-order accumulation to establish a grey prediction model with fractional-order Logistic and obtain the parameter estimation and time-response equation of the new model by solving the model through the theory related to fractional-order operators. The particle swarm optimization algorithm is used to complete the optimization process of the order of the fractional order grey prediction model and obtain the optimal model order. Then, the new model is applied to predict carbon emissions in five medium-emission countries: Ethiopia, Djibouti, Ghana, Belgium, and Austria. The new model shows better advantages in the validity analysis process, and the simulation results indicate that the new model proposed in this paper has stronger stability and better simulation and prediction accuracy than other comparative models, proving the model’s validity. Finally, the model is used to forecast the carbon emissions of these five countries for the five years of 2021–2025, and the results are analyzed, and relevant policy recommendations are made.

Seasonal electricity consumption forecasting: an approach with novel weakening buffer operator and fractional order accumulation grey model

PurposeAccurate prediction of seasonal power consumption trends with impact disturbances provides a scientific basis for the flexible balance of the long timescale power system. Consequently, it fosters reasonable scheduling plans, ensuring the safety of the system and improving the economic dispatching efficiency of the power system.Design/methodology/approachFirst, a new seasonal grey buffer operator in the longitudinal and transverse dimensional perspectives is designed. Then, a new seasonal grey modeling approach that integrates the new operator, full real domain fractional order accumulation generation technique, grey prediction modeling tool and fruit fly optimization algorithm is proposed. Moreover, the rationality, scientificity and superiority of the new approach are verified by designing 24 seasonal electricity consumption forecasting approaches, incorporating case study and amalgamating qualitative and quantitative research.FindingsCompared with other comparative models, the new approach has superior mean absolute percentage error and mean absolute error. Furthermore, the research results show that the new method provides a scientific and effective mathematical method for solving the seasonal trend power consumption forecasting modeling with impact disturbance.Originality/valueConsidering the development trend of longitudinal and transverse dimensions of seasonal data with impact disturbance and the differences in each stage, a new grey buffer operator is constructed, and a new seasonal grey modeling approach with multi-method fusion is proposed to solve the seasonal power consumption forecasting problem.HighlightsThe highlights of the paper are as follows:A new seasonal grey buffer operator is constructed.The impact of shock perturbations on seasonal data trends is effectively mitigated.A novel seasonal grey forecasting approach with multi-method fusion is proposed.Seasonal electricity consumption is successfully predicted by the novel approach.The way to adjust China's power system flexibility in the future is analyzed.

Predicting China’s elderly population by using a fractional grey prediction model with time power term

Accurate prediction of the aging population can provide valuable reference and corresponding theoretical support for the adjustment of national population development policy and economic development strategy. To explore the future development trend of China’s aging population, this paper establishes a novel fractional grey prediction model with the time power term (abbreviated as FGM (1, 1, t α) model) to study China’s aging population. FGM (1, 1, t α) has the properties of fractional order accumulation operation and GM (1, 1, t α) model, which makes it good at capturing nonlinear features in time series. Furthermore, the quantum genetic algorithm is used to search for unknown parameters in the model to facilitate the solving task of the model. Data on China’s aging population from 2000 to 2009 are used to train the prediction models, and data from 2010 to 2019 are used to evaluate the models’ prediction performance. The results show that the FGM (1, 1, t α) model outperforms the other competing models, which means that it has good generalization. Finally, the FGM (1, 1, t α) model is used to forecast China’s aging population from 2020 to 2029.

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.

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

A novel grey multivariate convolution model based on the improved marine predators algorithm for predicting fossil CO2 emissions in China

The effective prediction of fossil carbon dioxide (CO2) emissions provides a foundation for environmental protection authorities to make relevant management and decisions. CO2 emissions are influenced by various factors and exhibit a nonlinear and complex trend. First, the weighted conformable fractional accumulation (WCFA) method is proposed. WCFA is a generalized form of the existing weighted accumulation and conformable fractional order accumulation. A nonlinear discrete grey Bernoulli multivariate convolution model is introduced to model the sequences processed by WCFA. Then, an improved marine predators algorithm (IMPA) is obtained by improving the marine predators algorithm by the tent map and cosine control parameter. The performance of IMPA is verified by algorithm comparison experiments and Friedman test. The feasibility and predictive effectiveness of the proposed model is validated through three case studies. In prediction of fossil CO2 emissions in China, the fitting and prediction errors of the proposed model reach 1.1773% and 0.5402%, respectively. Finally, the total fossil CO2 emissions in China from 2022 to 2026 are predicted and analyzed. The results indicate a gradual decrease in fossil CO2 emissions over the next five years. Some reasonable suggestions are made based on the prediction results and sources of fossil CO2 emissions.

Memory-dependent derivative grey Bernoulli model and its application in electricity generation forecast

PurposeThis paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.Design/methodology/approachThis paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.FindingsThe selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.Research limitations/implicationsThe model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.Practical implicationsIn practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.Originality/valueBased on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

Prediction of demand for staple food and feed grain by a novel hybrid fractional discrete multivariate grey model

Increasing geopolitical conflicts, extreme weather and other emergencies are continuously affecting global agriculture and food systems, threatening national and regional food security. Therefore, it is particularly important to accurately predict national or regional grain demand. In this paper, a novel fractional discrete dynamic multivariate grey model is proposed to predict grain demand. Based on the discrete modeling idea, the model considers the impacts of the change trend of the related factor sequences, adds the linear correction term and the random disturbance term, and applies the fractional order accumulation strategy, which improves the accuracy and robustness. An algorithm based on sailfish optimizer is introduced to optimize the fractional order accumulation parameter of the model. Taking the Yangtze River Economic Belt as an example, the fitting and prediction results of the novel and the existing three models are compared. The novel model can better fit and predict the demand for staple and feed grain, which is superior compared to other models. The predictions show that demand for staple food and urban feed grain in the Yangtze River Economic Belt will increase to varying degrees, while rural feed grain demand will remain stable overall. This paper will help the government to better grasp the changes in the structure and quantity of staple food and feed grain demand in the Yangtze River Economic Belt and formulate efforts to ensure food security.

Predicting of elderly population structure and density by a novel grey fractional-order model with theta residual optimization: a case study of Shanghai City, China

Background Accurately predicting the future development trend of population aging is conducive to accelerating the development of the elderly care industry. This study constructed a combined optimization grey prediction model to predict the structure and density of elderly population.MethodsIn this paper, a GT-FGM model is proposed, which combines Theta residual optimization with fractional-order accumulation operator. Fractional-order accumulation can effectively weaken the randomness of the original data sequence. Meanwhile, Theta residual optimization can adjust parameter by minimizing the mean absolute error. And the population statistics of Shanghai city from 2006 to 2020 were selected for prediction analysis. By comparing with the other traditional grey prediction methods, three representative error indexes (MAE, MAPE, RMSE) were conducting for error analysis.ResultsCompared with the FGM model, GM (1,1) model, Verhulst model, Logistic model, SES and other classical prediction methods, the GT-FGM model shows significant forecasting advantages, and its multi-step rolling prediction accuracy is superior to other prediction methods. The results show that the elderly population density in nine districts in Shanghai will exceed 0.5 by 2030, among which Huangpu District has the highest elderly population density, reaching 0.6825. There has been a steady increase in the elderly population over the age of 60.ConclusionsThe GT-FGM model can improve the prediction accuracy effectively. The elderly population in Shanghai shows a steady growth trend on the whole, and the differences between districts are obvious. The government should build a modern pension industry system according to the aging degree of the population in each region, and promote the balanced development of each region.

A novel Weighted Average Weakening Buffer Operator based Fractional order accumulation Seasonal Grouping Grey Model for predicting the hydropower generation

A novel Weighted Average Weakening Buffer Operator based Fractional order accumulation Seasonal Grouping Grey Model for predicting the hydropower generation

Chinese Provincial Air Pollutant Concentration Prediction over the Long Term

Although China’s urban air quality has improved, there are still many cities that do not meet China’s ambient air quality standards and experience serious air pollution problems, causing tremendous damage to people’s health and sustainable social development. For the sake of obtaining the specific time when China’s ambient air quality will reach the standard, the annual mean air pollutant concentrations of 27 Chinese provinces are predicted and analyzed. Based on original data from air pollutant concentrations in 27 Chinese provinces from 2017 to 2021, a gray prediction model with fractional order accumulation is established to analyze and predict the concentration of pollutants in 27 provinces. The applicability of the model is then validated by mean absolute percentage error values. According to the forecast results, by 2026, the concentrations of six pollutants, PM2.5, PM10, SO2, NO2, CO, and O3, will all meet Class II air quality standards in 25 Chinese provinces, namely Beijing, Chongqing, Shanghai, Hebei, Jiangsu, Shanxi, Zhejiang, Liaoning, Anhui, Jilin, Fujian, Heilongjiang, Jiangxi, Hubei, Qinghai, Hunan, Guangdong, Hainan, Guangxi, Guizhou, Shaanxi, Gansu, Yunnan, Inner Mongolia, and Xinjiang (corrected for the effect of sandstorms). Tianjin, Sichuan, and Xinjiang (not corrected for the effect of sandstorms) still exceed the standard in the annual mean concentration of PM2.5, NO2, and PM10, respectively. Sichuan and Tianjin are, respectively, expected to meet Class II air quality standards in 2027 and 2030, and Xinjiang (not corrected for the effect of sandstorms) is expected to fail to meet Class II standards in the next 15 years. Finally, the current situation with respect to China’s ambient air quality in 27 Chinese provinces is analyzed, and corresponding suggestions are put forward to offer an explicit direction for relevant departments.

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.

Application of a novel hybrid accumulation grey model to forecast total energy consumption of Southwest Provinces in China

PurposeConsidering the small sample size and non-linear characteristics of historical energy consumption data from certain provinces in Southwest China, the authors propose a hybrid accumulation operator and a hybrid accumulation grey univariate model as a more accurate and reliable methodology for forecasting energy consumption. This method can provide valuable decision-making support for policy makers involved in energy management and planning.Design/methodology/approachThe hybrid accumulation operator is proposed by linearly combining the fractional-order accumulation operator and the new information priority accumulation. The new operator is then used to build a new grey system model, named the hybrid accumulation grey model (HAGM). An optimization algorithm based on the JAYA optimizer is then designed to solve the non-linear parameters θ, r, and γ of the proposed model. Four different types of curves are used to verify the prediction performance of the model for data series with completely different trends. Finally, the prediction performance of the model is applied to forecast the total energy consumption of Southwest Provinces in China using the real world data sets from 2010 to 2020.FindingsThe proposed HAGM is a general formulation of existing grey system models, including the fractional-order accumulation and new information priority accumulation. Results from the validation cases and real-world cases on forecasting the total energy consumption of Southwest Provinces in China illustrate that the proposed model outperforms the other seven models based on different modelling methods.Research limitations/implicationsThe HAGM is used to forecast the total energy consumption of the Southwest Provinces of China from 2010 to 2020. The results indicate that the HAGM with HA has higher prediction accuracy and broader applicability than the seven comparative models, demonstrating its potential for use in the energy field.Practical implicationsThe HAGM(1,1) is used to predict energy consumption of Southwest Provinces in China with the raw data from 2010 to 2020. The HAGM(1,1) with HA has higher prediction accuracy and wider applicability compared with some existing models, implying its high potential to be used in energy field.Originality/valueTheoretically, this paper presents, for the first time, a hybrid accumulation grey univariate model based on a new hybrid accumulation operator. In terms of application, this work provides a new method for accurate forecasting of the total energy consumption for southwest provinces in China.

Multivariate Grey Prediction Model Application in Civil Aviation Carbon Emission Based on Fractional Order Accumulation and Background Value Optimization

The GM(1,N) model, as a classical multivariate grey prediction model, can make a holistic and dynamic analysis of multiple factors and reflect the dynamic change relationship between the variable series and the related factor series. However, numerous works in the literature show that the GM(1,N) model has mechanistic defects, parametric defects, and structural defects. Therefore, the thesis establishes the OGM(1,N) model based on the GM(1,N) model by adding the linear correction term and the amount of grey action. According to the principle of dynamic optimization, the PSO algorithm is used to determine the background value. On this basis, the fractional order idea is introduced to push the model order from the integer field to the real field, and the FOBGM(1,N) model is established to systematically reduce the model error. Second, the literature in the ScienceDirect database for the last ten years is reviewed, and the carbon emission impact factors of civil aviation are selected. The calculated carbon emission values are taken as sample data based on Method 2 of Civil Aviation in Volume 2 of the 2006 IPCC Guide to National Greenhouse Gas Inventories. The results show that the prediction accuracy of the model has an increasing trend after multi-layer and multi-angle optimization. Among them, the MAPE of the OGM model and FOBGM model decreased by 24.40% and 31.86% compared with the GM(1,N) model. The 5-year average prediction accuracy of the FOBGM model reaches 99.996%, which verifies the effectiveness and practicality of the model improvement and has certain practical significance and application prospects.

A novel conformable fractional nonlinear grey multivariable prediction model with marine predator algorithm for time series prediction

To promote the development of multivariate prediction modelling in small sample environments, this paper constructs a new multivariate prediction model named CFDNGBM(r,N) by integrating the marine predator algorithm (MPA), the modified conformable fractional-order accumulation operation (MCFAO) and the grey prediction model. In CFDNGBM(r,N), the MCFAO and time-delay polynomial are used to enhance the model prediction performance, the backward difference operation is used to activate the unbiasedness, and the unbiased regularization algorithm is used to estimate the model parameters. In addition, the MPA is used to optimize the hyperparameters in the model. The experimental results show that the CFDNGBM(r,N) outperforms the 12 benchmark algorithms and all the optimization measures are effective, both of which confirm the effectiveness of the proposed methods.

A novel fractional-order discrete grey Gompertz model for analyzing the aging population in Jiangsu Province, China

PurposeChina's population aging is gradually deepening and needs to be actively addressed. The purpose of this paper is to construct a novel model for analyzing the population aging.Design/methodology/approachTo analyze the aging status of a region, this study has considered three major indicators: total population, aged population and the proportion of the aged population. Additionally, the authors have developed a novel grey population prediction model that incorporates the fractional-order accumulation operator and Gompertz model (GM). By combining these techniques, the authors' model provides a comprehensive and accurate prediction of population aging trends in Jiangsu Province. This research methodology has the potential to contribute to the development of effective policy solutions to address the challenges posed by the population aging.FindingsThe fractional-order discrete grey GM is suitable for predicting the aging population and has good performance. The population aging of Jiangsu Province will continue to deepen in the next few years.Practical implicationsThe proposed model can be used to predict and analyze aging differences in Jiangsu Province. Based on the prediction and analysis results, identified some corresponding countermeasures are suggested to address the challenges of Jiangsu's future aging problem.Originality/valueThe fractional-order discrete grey GM is firstly proposed in this paper and this model is a novel grey population prediction model with good performance.

A New Conformable Fractional-Order Time-Delay Grey Bernoulli Model with the Arithmetic Optimization Algorithm and Its Application in Rural Regional Economy

To further promote the development of the grey system theory, this paper develops a novel conformable fractional-order grey Bernoulli model with a time-delay effect, namely, the CFTDNGBM (1, 1) model. In addition, the arithmetic optimization algorithm (AOA) is incorporated into the system of the model to solve the hyperparameters existing in the model. Compared with the previous grey prediction models, the CFTDNGBM (1, 1) model with a conformable fractional-order accumulation operation (CFAO), time-delay factor, and Bernoulli parameter has stronger compatibility in structure. The proposed model and its nine competitive models with excellent performance are used to predict and analyze the consumption level and per capita consumption expenditure of rural residents in China to verify the feasibility of the proposed method. The case results show that in both cases, the seven descriptive indicators of the CFTDNGBM (1, 1) model are higher than those of its competing models. Therefore, the CFTDNGBM (1, 1) model has a certain application value.

Machine learning-based prediction for time series damage evolution of Ni-based superalloy microstructures

The Directionally Solidified (DS) superalloy DZ125 developed time-series related microstructure damage with serve time. The conventional grey prediction model (GM (1,1)), the fractional order accumulation grey prediction model (FAGM (1,1)), and the long short-term memory (LSTM) neural network prediction model were used to forecast the values of the time-related damage variables. The damage variables of the superalloy microstructure evolution were obtained through experimental observations of the evolution characteristics of the γ' precipitates and γ matrix. The microstructure damage variables were predicted by three machine learning models, and the fatigue damage evolution mechanism of the superalloy was analyzed. It was found that the root mean square error of the three prediction models was 0.072, 0.001, and 0.008, respectively. Finally, a model to predict fatigue life was established based on the Chaboche fatigue damage theory, and the fatigue life was obtained using the damage variables predicted by machine learning. The results revealed that the fractional order cumulative grey model and the long short-term memory neural network prediction model were both more accurate, with the conventional grey model being better for exponentially small sample time series data. • Damage variable was predicted with time-series small sample using grey model. • The prediction accuracy of the GM (1,1), FAGM (1,1), and LSTM were compared. • Fatigue life prediction model of DZ125 superalloy containing damage was developed.