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Fractal Fract
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Applications of Fractional-Order Grey Models
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Applications of Fractional-Order Grey Models

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: 14 March 2025 | Viewed by 10753

Special Issue Editors

Dr. Shuli Yan

E-Mail Website
Guest Editor
School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
Interests: fractional grey models; prediction; decision analysis
Prof. Dr. Lifeng Wu

E-Mail Website
Guest Editor
College of Management Engineering and Business, Hebei University of Engineering, Handan 056038, China
Interests: fractional models; grey system model
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional-order grey systems refers to a class of emerging theories that incorporate concepts from fractional-order calculus into the establishment of grey models. Most of the systems in real life are fractional-order, and the fractional-order grey model can essentially reflect the characteristics and behavior of objects. Compared with traditional grey models, the fractional-order grey models have stronger stability, higher flexibility and wider application prospects. They can be further divided into fractional-order accumulating grey models and fractional-order derivative grey models. The main difference is whether the modeling is implemented directly using the raw data or accumulating the raw data firstly. In addition, fractional grey models have been widely used in various fields such as agriculture, industry, society, economy, management, transportation, and energy, and relevant achievements are constantly emerging.

The focus of this Special Issue is to advance research on topics relating to the application of fractional-order grey models. The submitted papers must demonstrate sufficient novelty in the solution of practical problems by using fractional-order grey models. Topics that are invited for submission include (but are not limited to):

  • Applications of fractional-order grey models for energy consumption;
  • Fractional-order grey modelling algorithm;
  • Fractional-order grey model theory improvements;
  • Applications of fractional-order grey models for air quality;
  • Applications of fractional-order grey models for traffic flow;
  • Applications of fractional-order grey models in agriculture;
  • Applications of fractional-order grey models in settlement prediction;
  • Applications of fractional-order grey models in supply chains;
  • Applications of fractional-order grey models for natural disasters.

Dr. Shuli Yan
Prof. Dr. Lifeng Wu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional-order
  • grey model
  • prediction
  • algorithm
  • application

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Published Papers (9 papers)

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Research

18 pages, 10964 KiB  
Article
Forecasting Carbon Dioxide Emission Regional Difference in China by Damping Fractional Grey Model
by Haolei Gu
Fractal Fract. 2024, 8(10), 597; https://doi.org/10.3390/fractalfract8100597 - 11 Oct 2024
Viewed by 581
Abstract
The emission of carbon dioxide is the main reason for many global warming problems. Although China has made tremendous efforts to reduce carbon emission, the space–time dynamics of the carbon emission trend is still imbalanced. To forecast CDED in China, the Dagum Gini [...] Read more.
The emission of carbon dioxide is the main reason for many global warming problems. Although China has made tremendous efforts to reduce carbon emission, the space–time dynamics of the carbon emission trend is still imbalanced. To forecast CDED in China, the Dagum Gini coefficient was applied to measure regional CDED. Then, a grey correlation model was used to select potential influence factors and a wrapping method for selecting the optimal subset. DGMC is proposed to forecast CDED. The research results showed that the DGMC generalization performance is significantly superior to other models. The MAPE of DGMC in six cases are 1.18%, 1.11%, 0.66%, 1.13%, 1.27% and 0.51%, respectively. The RMSPEPR of DGMC in six cases are 1.08%, 1.21%, 0.97%, 1.36%, 1.41% and 0.57%, respectively. The RMSPEPO of DGMC in six cases are 1.29%, 0.69%, 0.02%, 0.58%, 0.78% and 0.32%, respectively. In future trends, the eastern carbon dioxide emission intraregional differences will decrease. Additionally, the intraregional differences in western and middle-region carbon dioxide emissions will expand. Interregional carbon emission difference will display a narrowing trend. Compared with the traditional grey model and ANN model, integrating the influence factor information significantly improved forecasting accuracy. The proposed model will present better balanced historical information and accurately forecast future trends. Finally, policy recommendations are proposed based on the research results. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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18 pages, 2859 KiB  
Article
Forecasting Carbon Sequestration Potential in China’s Grasslands by a Grey Model with Fractional-Order Accumulation
by Lei Wu, Chun Wang, Chuanhui Wang and Weifeng Gong
Fractal Fract. 2024, 8(9), 536; https://doi.org/10.3390/fractalfract8090536 - 14 Sep 2024
Viewed by 772
Abstract
This study aims to predict the carbon sequestration capacity of Chinese grasslands to address climate change and achieve carbon neutrality goals. Grassland carbon sequestration is a crucial part of the global carbon cycle. However, its capacity is significantly impacted by climate change and [...] Read more.
This study aims to predict the carbon sequestration capacity of Chinese grasslands to address climate change and achieve carbon neutrality goals. Grassland carbon sequestration is a crucial part of the global carbon cycle. However, its capacity is significantly impacted by climate change and human activities, making its dynamic changes complex and challenging to predict. This study adopts a fractional-order accumulation grey model, using 11 provinces in China as samples, to analyze and forecast grassland carbon sequestration. The study finds significant differences in grassland carbon sequestration trends across the sample regions. The carbon sequestration capacity of the grasslands in Xizang (Tibet) and Heilongjiang province is increasing, while it is decreasing in other provinces. The varying prediction results are influenced not only by regional climatic and natural conditions, but also by human interventions such as overgrazing, irrational reclamation, excessive mineral resource exploitation, and increased tourism development. Therefore, more region-specific grassland management and protection strategies should be formulated to enhance the carbon sequestration capacity of grasslands and promote the sustainable development of ecosystems. The significance of this study lies not only in providing scientific guidance for the protection and sustainable management of Chinese grasslands, but also in contributing theoretical and practical insights into global carbon sequestration strategies. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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19 pages, 6182 KiB  
Article
Multi-Point Seawall Settlement Prediction with Limited Data Volume Using an Improved Fractional-Order Grey Model
by Peng Qin, Chunmei Cheng, Zhenzhu Meng, Chunmei Ding, Sen Zheng and Huaizhi Su
Fractal Fract. 2024, 8(7), 423; https://doi.org/10.3390/fractalfract8070423 - 19 Jul 2024
Viewed by 543
Abstract
Settlement prediction based on monitoring data holds significant importance for engineering maintenance of seawalls. In practical engineering, the volume of the collected monitoring data is often limited due to the restrictions of devices and engineering budgets. Previous studies have applied the fractional-order grey [...] Read more.
Settlement prediction based on monitoring data holds significant importance for engineering maintenance of seawalls. In practical engineering, the volume of the collected monitoring data is often limited due to the restrictions of devices and engineering budgets. Previous studies have applied the fractional-order grey model to time series prediction under the situation of limited data volume. However, the performance of the fractional-order grey model is easily affected by the inappropriate settings of fractional order. Also, the model cannot make dynamic predictions due to the characteristic of fixed step size. To solve the above problems, in this paper, the genetic algorithm with enhanced search capabilities was employed to solve the premature convergence problem. Additionally, to solve the problem of the fractional-order grey model associated with fixed step size, the real-time tracing algorithm was introduced to conduct equal-dimensionally recursive calculation. The proposed model was validated using monitoring data of four monitoring points at Haiyan seawall in Zhejiang province, China. The prediction performance of the proposed model was then compared with those of the fractional-order GM(1,1), integer-order GM(1,1), and fractal theory model. Results indicate that the proposed model significantly improves the prediction performance compared to other models. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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15 pages, 384 KiB  
Article
A New Fractional-Order Grey Prediction Model without a Parameter Estimation Process
by Yadong Wang and Chong Liu
Fractal Fract. 2024, 8(7), 396; https://doi.org/10.3390/fractalfract8070396 - 2 Jul 2024
Cited by 1 | Viewed by 805
Abstract
The fractional-order grey prediction model is widely recognized for its performance in time series prediction tasks with small sample characteristics. However, its parameter-estimation method, namely the least squares method, limits the predictive performance of the model and requires time to address the ill-conditioning [...] Read more.
The fractional-order grey prediction model is widely recognized for its performance in time series prediction tasks with small sample characteristics. However, its parameter-estimation method, namely the least squares method, limits the predictive performance of the model and requires time to address the ill-conditioning of the system. To address these issues, this paper proposes a novel parameter-acquisition method treating structural parameters as hyperparameters, obtained through the marine predators optimization algorithm. The experimental analysis on three datasets validate the effectiveness of the method proposed in this paper. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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17 pages, 1678 KiB  
Article
Dynamic Fractional-Order Grey Prediction Model with GWO and MLP for Forecasting Overseas Talent Mobility in China
by Geng Wu, Haiwei Fu, Peng Jiang, Rui Chi and Rongjiang Cai
Fractal Fract. 2024, 8(4), 217; https://doi.org/10.3390/fractalfract8040217 - 8 Apr 2024
Cited by 1 | Viewed by 1211
Abstract
International students play a crucial role in China’s talent development strategy. Thus, predicting overseas talent mobility is essential for formulating scientifically reasonable talent introduction policies, optimizing talent cultivation systems, and fostering international talent cooperation. In this study, we proposed a novel fractional-order grey [...] Read more.
International students play a crucial role in China’s talent development strategy. Thus, predicting overseas talent mobility is essential for formulating scientifically reasonable talent introduction policies, optimizing talent cultivation systems, and fostering international talent cooperation. In this study, we proposed a novel fractional-order grey model based on the Multi-Layer Perceptron (MLP) and Grey Wolf Optimizer (GWO) algorithm to forecast the movement of overseas talent, namely MGDFGM(1,1). Compared to the traditional grey model FGM(1,1), which utilizes the same fractional order at all time points, the proposed MGDFGM(1,1) model dynamically adjusts the fractional-order values based on the time point. This dynamic adjustment enables our model to better capture the changing trends in the data, thereby enhancing the model’s fitting capability. To validate the effectiveness of the MGDFGM(1,1) model, we primarily utilize Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) as the evaluation criteria for the prediction accuracy, as well as standard deviation (STD) as an indicator of the model stability. Furthermore, we perform experimental analysis to evaluate the predictive performance of the MGDFGM(1,1) model in comparison to NAÏVE, ARIMA, GM(1,1), FGM(1,1), LSSVR, MLP, and LSTM. The research findings demonstrate that the MGDFGM(1,1) model achieves a remarkably high level of prediction accuracy and stability for forecasting overseas talent mobility in China. The implications of this study offer valuable insights and assistance to government departments involved in overseas talent management. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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21 pages, 4049 KiB  
Article
Identification of the Dynamic Trade Relationship between China and the United States Using the Quantile Grey Lotka–Volterra Model
by Zheng-Xin Wang, Yue-Ting Li and Ling-Fei Gao
Fractal Fract. 2024, 8(3), 171; https://doi.org/10.3390/fractalfract8030171 - 15 Mar 2024
Cited by 1 | Viewed by 1679
Abstract
The quantile regression technique is introduced into the Lotka–Volterra ecosystem analysis framework. The quantile grey Lotka–Volterra model is established to reveal the dynamic trade relationship between China and the United States. An optimisation model is constructed to solve optimum quantile parameters. The empirical [...] Read more.
The quantile regression technique is introduced into the Lotka–Volterra ecosystem analysis framework. The quantile grey Lotka–Volterra model is established to reveal the dynamic trade relationship between China and the United States. An optimisation model is constructed to solve optimum quantile parameters. The empirical results show that the quantile grey Lotka–Volterra model shows higher fitting accuracy and reveals the trade relationships at different quantiles based on quarterly data on China–US trade from 1999 to 2019. The long-term China–US trade relationship presents a prominent predator–prey relationship because exports from China to the US inhibited China’s imports from the United States. Moreover, we divide samples into five stages according to four key events, China’s accession to the WTO, the 2008 global financial crisis, the weak global economic recovery in 2015, and the 2018 China–US trade war, recognising various characteristics at different stages. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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10 pages, 406 KiB  
Article
Utilizing a Fractional-Order Grey Model to Predict the Development Trends of China’s Electronic Commerce Service Industry
by Jianhong Guo, Che-Jung Chang and Yingyi Huang
Fractal Fract. 2024, 8(3), 169; https://doi.org/10.3390/fractalfract8030169 - 14 Mar 2024
Viewed by 1257
Abstract
Electronic commerce plays a vital role in the digital age, and the creation of a good electronic commerce ecosystem is crucial to maintaining economic growth. The electronic commerce service industry is a leading indicator of electronic commerce development, and its possible changes imply [...] Read more.
Electronic commerce plays a vital role in the digital age, and the creation of a good electronic commerce ecosystem is crucial to maintaining economic growth. The electronic commerce service industry is a leading indicator of electronic commerce development, and its possible changes imply the future trends and innovation directions of the electronic commerce industry. An accurate grasp of the possible future revenue scale of the electronic commerce service industry can provide decision-making information for government policy formulation. Electronic commerce companies must formulate operational plans based on the latest information to determine strategic directions that are reasonable and consistent with the actual situation. Although there exist many prediction methods, they often fail to produce ideal results when the number of observations is insufficient. The fractional-order grey model is a common method used to deal with small data set prediction problems. This study therefore proposes a new modeling procedure for the fractional-order grey model to predict the revenue scale of China’s electronic commerce service industry. The results of experiments demonstrate that the proposed procedure can yield robust outputs under the condition of small data sets to reduce decision-making risks. Therefore, it can be regarded as a practical small data set analysis tool for managers. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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26 pages, 2121 KiB  
Article
Applications of Fractional Order Logistic Grey Models for Carbon Emission Forecasting
by Xiaoqiang He, Yuxin Song, Fengmin Yu and Huiming Duan
Fractal Fract. 2024, 8(3), 145; https://doi.org/10.3390/fractalfract8030145 - 29 Feb 2024
Cited by 1 | Viewed by 1430
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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24 pages, 2519 KiB  
Article
Research on the Corporate Innovation Resilience of China Based on FGM(1,1) and Fuzzy-Set Qualitative Comparative Analysis Model
by Houxue Xia, Jingyao Jiao, Pengcheng Wang, Xiaowei Tang, Chunyan Xiong and Liusan Wu
Fractal Fract. 2024, 8(1), 2; https://doi.org/10.3390/fractalfract8010002 - 19 Dec 2023
Viewed by 1316
Abstract
Over the past few years, the uncertain business environment has shaped the resilient development thinking of firms. Measuring and predicting innovation resilience plays a crucial role in fostering the sustainable development of enterprises. This paper used the entropy-weight TOPSIS model and FGM(1,1) model [...] Read more.
Over the past few years, the uncertain business environment has shaped the resilient development thinking of firms. Measuring and predicting innovation resilience plays a crucial role in fostering the sustainable development of enterprises. This paper used the entropy-weight TOPSIS model and FGM(1,1) model to measure the innovation resilience of companies based on an indicator system, covering aspects such as tolerance for factor scarcity, R&D safety, core technology self-sufficiency, and organizational change capacity. The results show that the MAPE of the FGM(1,1) model is 0.0136, which is lower than that of the GM(1,1) model, with the predicted annual growth rate of the resilience being −0.95% from 2020 to 2025. Consequently, the study investigated what policy configuration may improve innovation resilience using the fuzzy-set qualitative comparative analysis (fsQCA) model. It identified four policy configuration paths, of which the combination of a tax policy for an additional deduction of enterprise R&D expenses and an income tax reduction policy is an effective policy configuration. This research expands the application of the FGM(1,1) model and inspires managers to develop innovative policies to enhance corporate resilience. Full article
(This article belongs to the Special Issue Applications of Fractional-Order Grey Models)
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