Table of Content

01 February 2026, Volume 38 Issue 1

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An Innovative Grey Intelligent Prediction Algorithm with Fractional Differentiation and Fractional Accumulating Operator

Xiaozeng Xu, Yikun Wu, Bo Zeng, Wenjing Li

2026, 38(1):  1-14. 

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The limitation of the traditional grey prediction model in handling the derivative order value may lead to unstable or even abnormal prediction results, which suggests that this issue could be a significant influencing factor. In this paper, we unveil a groundbreaking approach to grey prediction by integrating fractional differentiation through a foundation rooted in fractional differential definitions and accumulation principles. The developed model not only allows for dynamic expansion and optimization of both derivative order and accumulation operator within the grey model but also effectively addresses inherent challenges in traditional models. The proposed model exhibits an average minimum MRPE of 2.293%, representing a reduction of 1.936% compared to the average minimum MRPE achieved by the latest comparative model. The result in a more robust and adaptable prediction framework, simultaneously amplifying the model's structural variability and self-accommodating capabilities. This study provides empirical evidence that substantiates the reliability and effectiveness of our innovative prediction algorithm, thereby making a significant contribution to the advancement of the theoretical foundations of grey prediction. Finally, the paper compares the GWO and PSO optimization algorithms. The results show that GWO performs better across all indicators. Therefore, the GWO algorithm is selected to optimize the parameters of the proposed grey prediction model and is applied to three different fields. The prediction results indicate that the average annual growth rates over the next five years will be 1.26%, -1.63%, and -0.23%, respectively.

China's Nuclear Power Generation Forecasting Based on A Novel Prediction Framework Containing A Proposed Grey Prediction Model and LSTM-RF

Junjie Wang, Ertai Cao, Ying Cai, Yaoguo Dang

2026, 38(1):  15-27. 

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Accurately predicting the remaining useful life (RUL) of lithium-ion batteries is crucial for safe operation. Grey prediction models, with advantages in handling small samples and uncertain information, offer a promising approach for RUL prediction. However, most of the existing grey prediction models focus on the degradation trend while neglecting the capacity regeneration phenomenon. To address this limitation, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is first applied to separate the capacity degradation trend from local regeneration trends, addressing their differences in magnitude and characteristics. A hybrid prediction method, which combines an improved grey multivariate model and Bayesian-optimized Gaussian process regression, is then proposed. For the capacity degradation trend, which exhibits information heterogeneity and an exponential nonlinear trend, a variable new information priority fractional discrete grey multivariate model is proposed for prediction. The model is not only based on the ideas of variable-order accumulation and discrete grey models, but also introduces an additional nonlinear correction term. For the local regeneration trend, which is nonstationary, nonlinear, and noisy, a denoising autoencoder is employed for noise reduction and feature enrichment, followed by the Bayesian-optimized Gaussian process regression model for prediction. Finally, the predictions of each component are reconstructed to obtain the complete capacity sequence. Multidimensional evaluations on multiple NASA battery datasets, including comparisons with common baseline models and ablation studies to verify the effectiveness of each module, demonstrate that the proposed method achieves superior accuracy, stability, and generalization in capacity degradation prediction.

Lithium-Ion Battery Remaining Useful Life Prediction Based on a Hybrid Method of Improved GM(1,N) Model and Gaussian Process Regression

Yang Cao, Min Sun, Qinqin Shen, Quan Shi

2026, 38(1):  28-40. 

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Accurately predicting the remaining useful life (RUL) of lithium-ion batteries is crucial for safe operation. Grey prediction models, with advantages in handling small samples and uncertain information, offer a promising approach for RUL prediction. However, most of the existing grey prediction models focus on the degradation trend while neglecting the capacity regeneration phenomenon. To address this limitation, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is first applied to separate the capacity degradation trend from local regeneration trends, addressing their differences in magnitude and characteristics. A hybrid prediction method, which combines an improved grey multivariate model and Bayesian-optimized Gaussian process regression, is then proposed. For the capacity degradation trend, which exhibits information heterogeneity and an exponential nonlinear trend, a variable new information priority fractional discrete grey multivariate model is proposed for prediction. The model is not only based on the ideas of variable-order accumulation and discrete grey models, but also introduces an additional nonlinear correction term. For the local regeneration trend, which is nonstationary, nonlinear, and noisy, a denoising autoencoder is employed for noise reduction and feature enrichment, followed by the Bayesian-optimized Gaussian process regression model for prediction. Finally, the predictions of each component are reconstructed to obtain the complete capacity sequence. Multidimensional evaluations on multiple NASA battery datasets, including comparisons with common baseline models and ablation studies to verify the effectiveness of each module, demonstrate that the proposed method achieves superior accuracy, stability, and generalization in capacity degradation prediction.

SGD Algorithm Based Parameter Optimization of the Improved GM(1,1) Power Model

Tong Huo, Yibo Sun, Wenjie Dong

2026, 38(1):  41-49. 

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The GM(1, 1) power model is frequently employed for forecasting nonlinear data, yet its parameter estimation faces challenges of slow convergence and susceptibility to local optima in existing methods. This study proposes an enhanced GM(1, 1) power model optimized via stochastic gradient descent (SGD), incorporating two innovations: dynamic background-value weights replace fixed ones to adaptively capture data trends, and full-information optimized initial conditions mitigate errors from insufficient information utilization. The SGD algorithm simultaneously optimizes the power exponent, background-value weights, full-information initial condition weights, and model parameters. Comparative experiments with particle swarm optimization (PSO) applied to the same model reveal that SGD, despite marginally slower convergence, achieves significantly higher precision and exhibits superior capability to escape local optima.

Conflict Resolution in Family Doctor Contract Services: A Grey OPA–GMCR Evolutionary Analysis of Four Stakeholders

Change Zhu, Aijun Xu, Xuebin Qiao, Baoxiang Song

2026, 38(1):  50-62. 

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The family doctor contract service system represents a multi-actor governance arrangement involving the joint participation of the government, hospitals, family doctors, and patients. this paper develops an integrated analytical framework combining the grey ordinal priority approach (OPA-G) and the graph model for conflict resolution (GMCR) to analyze their strategic conflicts and identify stable equilibrium outcomes. The OPA-G method is first employed to determine the strategic preferences of each stakeholder under uncertain and incomplete information. These preference vectors are then incorporated into the GMCR framework to construct state transitions and assess equilibrium stability under Nash, GMR, SMR, and SEQ criteria. Using the GMCR Plus v0.4 software, the analysis identifies state S1 (YNYYNNYNYNN) as the only strongly stable equilibrium across all four stability definitions. In this state, the government implements strong incentives with strict performance assessments, hospitals enforce internal regulation and evaluation, family doctors actively fulfill contracts and deliver high-quality services, and patients cooperate with service delivery. The proposed Grey OPA–GMCR approach offers methodological guidance for enhancing coordination and optimizing the implementation of family doctor contract services, and more broadly provides a systematic mixed-method framework for modeling multi-actor conflicts in primary healthcare governance.

A Grey Multivariable Model for Dynamic Prediction of Performance Degradation in Aircraft Hydraulic System Under Variable Working Conditions

Rui Shi, Zhigeng Fang, Liqing Xu, Guowei Lang, Youpeng Liu

2026, 38(1):  63-77. 

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During the operation of an aircraft, the performance of certain systems or components gradually degrades, and both environmental factors and working conditions influence this degradation. Accurately predicting performance degradation by comprehensively accounting for these influences is a critical and complex task aimed at reducing failure-related losses. Moreover, as time progresses, the degradation trend may vary, which imposes higher requirements on the flexibility and adaptability of prediction models. First, this paper introduces the concept of shock effects and models the impact of working condition change using a Beta function while incorporating environmental variables to characterize environmental influences, thereby constructing a grey multivariate prediction model. Second, an optimization model is established to minimize the mean absolute percentage error (MAPE), and the Whale Optimization Algorithm is employed to estimate the unknown parameters related to the shock effects. Subsequently, a recursive least squares-based iterative mechanism is proposed to dynamically and flexibly update the model's structural parameters. Finally, empirical analysis is conducted using hydraulic pressure data from an aircraft hydraulic system under varying working conditions. The results demonstrate that the proposed model exhibits excellent predictive performance and practical value.

A Hybrid Improved Grey Generalized Verhulst Model and Its Application

Xunqian Xu, Shuyong Pan, Cheng Zhou, Wenxuan Ge, Pengxiang Qian, Boshun Liu, Zijun Zhou, Xu Wu

2026, 38(1):  78-91. 

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Accurate prediction of fiber-asphalt bond-slip behavior is crucial for evaluating asphalt concrete performance. Although the generalized grey Verhulst model (GGVM) handles saturated growth sequences well, it exhibits limitations in modeling highly nonlinear interfacial curves due to insufficient "new information priority" in its accumulation operator and crude background value construction that causeserror accumulation. To address these issues, this paper proposes a hybrid improved grey generalized Verhulst model (Mar-NIP�FO_GGVM). First, the new information priority fractional-order (NIP-FO) accumulation operator and background value adjustment coefficient are introduced to structurally enhance the model's capability to capture recent information and nonlinear dynamics. Subsequently, particle swarm optimization is employed for global parameter optimization, while the Markov chain is utilized for stochastic correction of prediction residuals. Finally, the model is applied to predict interfacial bond-slip curves of basalt, glass, and polyester fibers. Comparisons with the Popovics model and three other grey models validate the proposed model's effectiveness, applicability, and robustness. Results show that Mar-NIP-FO_GGVM outperforms all benchmarks across three scenarios, with simulation and prediction MAPEs below 7%, while accurately capturing different fibers' stage-wise characteristics. Statistical tests further confirm its significant advantages, establishing a high-precision, data-driven tool for fiber-asphalt micromechanical analysis.

A Gauss-Legendre Integral and GOOSE-Optimized GM(1,1) Model for Forecasting Emergency Material Demand in Hebei Province, China

Lanxiang Yi, Wenguang Yang, Meiling Chen

2026, 38(1):  92-102. 

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Major sudden events have emergency material demands characterized by strong uncertainty, small sample size and temporal complexity. To achieve accurate prediction and scientific scheduling of such material demand, this paper proposes an improved GM(1,1) model. This model is optimized by using Gauss-Legendre integral and GOOSE optimization algorithm, named GLG-GM(1,1). Gauss-Legendre integral is employed to reconstruct both the background value and the time response equation. Meanwhile, to further enhance the model’s adaptability, a global adjustment factor is introduced in the background value calculation. With the mean relative error as the optimization objective, the coefficient is globally optimized using the GOOSE optimization algorithm. To verify the model performance, two actual cases are selected to conduct comparative experiments with genetic algorithm GM(1,1) (GAGM(1,1)), discrete fractional�order GM(1,1) (DFOGM(1,1)), Simpson GM(1,1) (SPGM(1,1)), and GM(1,1) models. The performances of these models areobjectively evaluated by the mean relative error (MRE), mean square error (MSE), mean absolute error (MAE) and mean absolute percentage error (MAPE). The results show that the prediction accuracy of the GLG-GM(1,1) is significantly better than that of the comparison models, and its adaptability in scenarios with small samples and strong fluctuations is more prominent. The research has confirmed that this model can provide reliable quantitative basis for the planning and dynamic allocation of emergency materials in Hebei Province, China, and provide effective methodological support for improving the regional disaster emergency response capability.

A Novel Seasonal Weighted Fractional Nonlinear Grey Bernoulli Model and Its Application in Forecasting China's Natural Gas Production

Fangli He, Lihua Ning, Zhenxiu Cao, Xiangyan Zeng

2026, 38(1):  103-119. 

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Accurate forecasting of natural gas production is essential for ensuring energy supply, optimizing the energy structure, and promoting high-quality economic development. This paper proposes a novel Seasonal Weighted Fractional Nonlinear Grey Bernoulli Model (SWFNGBM(1,1|sinx)) to predict China's natural gas production. First, a seasonal weighted fractional accumulation generation operator is introduced and combined with a sine function to enhance the traditional NGBM(1,1) model, aiming to eliminate seasonal fluctuations in the data and capture nonlinear dynamic characteristics. Second, to address the limitations of the traditional Grey Wolf Optimizer (GWO) algorithm in terms of search efficiency and global optimization capability, an improved algorithm incorporating Levy flight is employed to optimize the hyperparameters of the proposed model. Finally, comparative experiments with other models validate the superior predictive accuracy of the proposed model. Based on the newly developed model, China's seasonal natural gas production is forecasted, and the results demonstrate the strong practicality of the model.

Model Selection of a Class of Discrete Grey Models Based on Connotation Research

Zhicheng Jiang, Yunlian Li, Wenjing Li

2026, 38(1):  120-132. 

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This paper first rewrites the first-order accumulated sequence into two distinct sequences, thereby introducing a novel research perspective. Secondly, it reveals the intrinsic relationships and geometric interpretations among the DGM(1,1), NDGM(1,1), QPDGM(1,1), and DGMP(1,1,N) models, and proposes a new criterion for determining the degree N in the DGMP(1,1,N) model. Finally, numerical examples and practical applications demonstrate the validity of the proposed theory and the operational feasibility of the method. The approach improves efficiency for applied researchers in model selection and is straightforward to implement.