Application of grey system-based benchmark models in marine industry forecasting

The marine economic system is complex, dynamic, and open, facing significant uncertainties driven by multiple factors such as climate change, shifts in the international geopolitical landscape, and events such as the discharge of treated nuclear wastewater by Japan. This situation necessitates scientific decision-making and proactive, strategic management to ensure stable development. In 2024, China's marine fisheries achieved a value added of RMB 488 billion, representing a 4.0% year-on-year increase. The marine transportation industry reached 819.4 billion yuan, up 7.4%, with cargo throughput growing by 9.5%, reflecting robust demand for container shipping and improved port service capacity. Meanwhile, the seawater desalination and comprehensive utilization sector achieved RMB 33.7 billion, representing a 3.9% growth. Core industries such as port shipping and fisheries have become key drivers of national wealth growth in China and beyond. Scientific forecasting of these industries’ development trends facilitates optimized resource allocation, enhanced decision-making effectiveness, and strengthened industrial stability and resilience, thereby promoting the high-quality and sustainable development of the marine economy. Against this backdrop, grey system theory, with its unique advantages in handling uncertainty and incomplete information, has emerged as a vital tool for modeling and forecasting complex marine economic systems. It provides an appropriate theoretical foundation and technical framework.

Since Professor Deng first proposed grey system theory in 1982 (Ju-Long, 1982), the theory has gradually evolved into a set of more advanced models due to its unique advantages in modeling small samples and systems with uncertain information. The GM(1,1) model has been widely applied in environmental monitoring and energy consumption forecasting, but its assumption of exponential growth limits its ability to capture nonlinear fluctuations. To improve the modeling of complex dynamic data, researchers proposed the Non-Homogeneous Discrete Grey Model (NDGM), which introduces non-homogeneous terms to enhance fitting accuracy (Xie and Liu, 2015). In recent years, fractional-order grey modeling has become a major focus. He et al. (2024) applied fractional-order logistic grey models to forecast carbon emissions in multiple countries, demonstrating improved stability and accuracy compared to classic models (He et al., 2024). Additionally, Wu et al. (2023) presented an optimized conformable fractional grey approach for industrial water–energy–waste systems, showing superior predictive performance in complex environmental data (Wu et al., 2023). The Bernoulli grey model family has also gained prominence in nonlinear modeling. Wang et al. (2025a) proposed a continuous fractional nonlinear grey Bernoulli model (FBernoulliGM), optimized via grey wolf optimizer for CO₂ emissions forecasting, yielding substantially higher accuracy than traditional models (Xie et al., 2021). As model complexity increases, intelligent optimization algorithms have been widely applied for parameter tuning and structure optimization. Recently developed models – such as the tunable fractional nonhomogeneous Bernoulli grey model (Qu, 2024), the continuous fractional Bernoulli model, and the matrix-weighted conformable fractional nonlinear grey model (MWCFNGBM) (1,1, $tα$⁠) (Zeng et al., 2025) have significantly enhanced short-term forecasting performance through structural innovation and algorithmic integration.

With the rapid growth of the marine economy, there is a pressing need for forecasting tools that can handle nonlinear and incomplete data in areas like fisheries, port logistics, marine energy, and ecological protection. Grey system theory, known for its effectiveness in small-sample and uncertain environments, has shown strong adaptability and predictive power across these fields. For example, using data from the China Marine Economic Statistical Bulletin, a study compared three grey models in the Bohai Rim region and found that the standard GM(1,1) offered the most stable medium- and long-term forecasts, serving as a valuable reference for regional marine economic trend analysis (Liu and Yan, 2020). In another study, integrated grey relational analysis with the GM model to forecast port cargo throughput resulted in enhancing the model's ability to accommodate multiple input factors (Wang and Ma, 2019). To capture nonlinear growth patterns, the grey Verhulst model has been applied to model the S-shaped growth curve of marine tourism revenues, making it suitable for forecasting during saturation or decline phases (Tian et al., 2020). Grey models have also been employed in assessing development trends and investment risks in the marine energy sector, particularly under conditions of data scarcity or high system complexity (Wang and Jv, 2021). In recent years, the integration of grey models with other approaches has become a major research trend. For example, combining grey forecasting with machine learning model, and particle swarm optimization (PSO) has further improved generalization capability and predictive accuracy (Wang and Li, 2019). Improved models, such as the grey Verhulst model, GM models with buffer operators, and non-equidistant GM models with optimized background values, have also shown strong adaptability and scalability in analyzing marine economic data (Wang and Ma, 2019). Additionally, researchers have begun applying grey system theory to regional economic disparity analysis and marine ecological compensation evaluation, expanding the scope of grey model applications (Zhang, 2024; Qirong et al., 2023). Recently, Wang et al. (2025a) developed an innovative grey forecasting framework integrating optimal data preprocessing to predict China’s coastal marine economy, demonstrating enhanced robustness under highly uncertain environments (Wang et al, 2025a). Similarly, Wang et al. (2025b) demonstrates that grey system theory effectively predicts and analyzes sturgeon aquaculture production dynamics, offering insights for sustainable management (Wang et al, 2025b).

Forecasting marine economic data is of great importance for supporting policy-making, resource allocation, and sustainable development. However, existing studies on short-term marine economic forecasting remain relatively limited. Current approaches often struggle to capture highly nonlinear and dynamic time series, while also exhibiting sensitivity to outliers and lacking sufficient theoretical support for parameter selection. These shortcomings have constrained the broader application of models in practical contexts. Compared with traditional ARIMA and machine learning models, grey models demonstrate significant advantages in forecasting marine economic sectors such as fisheries, seawater utilization, and marine transportation. First, grey models require relatively small and incomplete datasets, enabling high-accuracy modeling under conditions of data scarcity and strong uncertainty, whereas ARIMA relies on the stationarity of time series and machine learning models typically require large historical datasets for stable training. Second, grey models feature fewer parameters, simple structures, and high computational efficiency, making them well suited to marine industries where data collection is limited and conditions are highly dynamic. In addition, grey models exhibit strong robustness and interpretability, allowing them to reveal the intrinsic evolutionary patterns of complex systems and provide scientific support for analyzing fluctuations in fishery output, assessing growth trends in seawater desalination, and performing short-term forecasts of port transportation volumes. Therefore, grey models offer unique applicability and theoretical value in marine economic forecasting tasks characterized by small samples, high uncertainty, and pronounced nonlinear features.

Moreover, although grey models have been applied in prior research, systematic comparisons and comprehensive evaluations across different types of grey models are still scarce, and empirical analyses remain insufficient, particularly within specific marine economic sub-sectors such as fisheries, seawater utilization, and marine transportation. These limitations underscore the necessity of developing a unified research framework that not only encompasses diverse grey model structures but also evaluates their forecasting performance and applicability in a comparative manner, thereby providing both theoretical and practical guidance for model selection. To address these gaps, this study conducts a comprehensive and systematic evaluation of fourteen representative grey models, including basic, fractional-order, new information priority, and nonlinear Bernoulli types, using annual output value data from 2006 to 2023 across three key marine industries. Furthermore, particle swarm optimization (PSO) is incorporated into the parameter estimation of nonlinear grey models to enhance adaptability and robustness. Collectively, these efforts not only fill an important gap in the existing literature but also provide new insights into the strengths and limitations of different grey models, while extending their applicability to the complex and dynamic context of marine economic forecasting. Section 2 introduces the grey models and the PSO algorithm; Section 3 presents the application cases, while Sections 4 and 5 provide the discussion and conclusions, respectively.

Given its strong ability to capture underlying nonlinear patterns in systems with limited or uncertain data, the grey system model has proven particularly well suited for forecasting tasks. Its robustness and adaptability allow it to provide reliable trend predictions even when information is incomplete. As shown in Figure 1, this paper employs fourteen grey prediction models. Based on their structural characteristics, these benchmark models include the basic first-order accumulation structure, the fractional-order accumulation structure, the new information priority accumulation structure, and the nonlinear grey Bernoulli structure.

Specifically, the first-order accumulated generating operator (1-AGO), which constitutes the core operation of the classical GM(1,1) model, effectively reduces random fluctuations in the raw data and highlights cumulative trends. The fractional-order accumulated generating operator (r-FAGO) extends the accumulation process from integer to real order, providing a flexible balance between data smoothing and the retention of raw fluctuations, which makes it particularly suitable for dynamic time series. The new information priority accumulated generating operator (NIPAGO) incorporates an exponentially decaying weight factor $λ$⁠, which assigns greater importance to recent observations under the principle of “new information priority,” making it well-suited for rapidly changing environments. And the nonlinear grey Bernoulli structure introduces a nonlinear term $Xm$ into the grey differential equation, thereby overcoming the linear limitations of the traditional grey model and improving its capacity to characterize nonlinear dynamics. The mathematical formulations and structural characteristics of these models are summarized in Table 1.

Grey models based on fractional-order accumulation, new information priority accumulation, and the grey Bernoulli structure all include nonlinear parameters. To optimize these parameters and enhance both the fitting accuracy and forecasting stability of the grey models, this study employs the particle swarm optimization (PSO) algorithm. The PSO algorithm is a widely used evolutionary technique for global optimization, inspired by the collective foraging behavior of bird flocks.

In this study, the PSO algorithm is implemented using the pyswarm library in Python (https://github.com/esa/pygmo2), with a maximum of 500 iterations and a swarm size of 30. The inertia weight $w$ is set within the range $[0.4,0.9]$⁠, while the cognitive and social learning factors, $c1$ and $c2$⁠, are both selected from the range $[0,1]$⁠. The search ranges for the grey model parameters are defined as $r∈(−3,3)$⁠, and $m∈(−10,1)$⁠. To ensure the stability of the optimization process, the nonlinear parameters of each grey model are optimized through five independent runs. The pseudocode for the PSO algorithm is presented in Algorithm 1.

Furthermore, Section 2.2 defines $Xˆ(0)$ as the restored values corresponding to the original series $X(0)$⁠. The performance of the GM model is evaluated by the mean absolute percentage error (MAPE), which also serves as the fitness function during the optimization process. The time series datasets are divided into training, validation, and testing subsets. The training and validation sets (in-sample) are used for model fitting and parameter selection, whereas the testing set (out-of-sample) is reserved for final performance evaluation. Given training and validation sizes $s1$ and $s2$⁠, the MAPE calculation is computed as

where $X(0)$ and $Xˆ(0)$ represent the original values and the predicted values, respectively. Considering the stochastic nature of intelligent optimization algorithms, each nonlinear grey model was subjected to multiple independent optimization experiments. The results showed only negligible variations across runs, thereby confirming the robustness and reliability of the obtained solutions.

This study applies grey models to the field of marine economic forecasting, aiming to systematically compare the predictive accuracy and generalization capabilities of different grey modeling approaches. According to the China Marine Statistical Yearbook, this paper focuses on three representative primary industries, including marine fisheries, seawater utilization, and marine transportation. Annual gross output data for these industries are used as the original dataset, covering the period from 2006 to 2024 for marine fisheries and marine transportation, and from 2006 to 2021 for seawater utilization. Given the inherent temporal dependencies and sequential structure of time series data, and to reduce the risks of information leakage and model overfitting, this study adopts a data partitioning strategy inspired by the leave-one-out approach. The dataset is sequentially divided into training, validation, and test sets in a ratio of 7:1:2. The training set is used to fit and estimate model parameters, the validation set is employed to support hyperparameter tuning through intelligent optimization algorithms, and the test set is reserved for assessing the model's out-of-sample predictive performance.

To comprehensively evaluate the predictive performance of the models, four widely used statistical metrics are employed: mean absolute percentage error (MAPE), mean absolute error (MAE), root mean squared error (RMSE), and mean arctangent absolute percentage error (MAAPE). These metrics capture different dimensions of prediction accuracy, including relative error, absolute deviation, and sensitivity to outliers. Specifically, MAPE and MAAPE measure relative accuracy by normalizing errors with respect to actual values, whereas MAE and RMSE reflect the absolute magnitude of prediction errors, with RMSE assigning a greater penalty to larger deviations. The definitions and corresponding mathematical formulations of these evaluation metrics are summarized in Table 2.

1. Case 1: marine fisheries

Figure 2 illustrates the complete forecasting results for the marine fisheries sector, while Figure 3 displays the corresponding residual plots. The data preceding the red dashed line represent the in-sample period, encompassing both the training and validation sets, whereas the data following the red line correspond to the test set. Most grey models effectively capture the historical growth patterns during the in-sample phase. The traditional GM model and its improved variants, including fractional grey model (FGM), fractional non-homogeneous discrete grey model (FNDGM), and NDGM, generate prediction curves that closely match the observed values. However, during the test phase, several models – particularly NIPNDGM and NIPNGM – demonstrate substantial deviations, indicating limited generalization capability. As depicted in the residual plots, fractional-order models maintain relatively small and stable residuals, whereas the basic models exhibit larger fluctuations, especially near turning points. These results suggest that the dataset requires precise trend modeling, and excessively complex nonlinear models are prone to overfitting.

1. Case 2: seawater utilization industry

Figure 4 presents the complete forecasting results for the seawater utilization industry, while Figure 5 displays the corresponding residual plots. The data preceding the red dashed line represent the in-sample set, whereas the data following the red line correspond to the test set. The original series in this sector exhibits a steady upward trend with minimal fluctuations, allowing most grey models to achieve accurate fitting and forecasting across the entire period. Models such as GM, DGM, FGM, and NPDGM demonstrate strong agreement between predicted and observed values, indicating high robustness when modeling stable time series. The residual plots further corroborate these findings: both fractional-order and conventional models produce small, evenly distributed residuals, reflecting stable performance. In contrast, nonlinear models (e.g. NIPNDGM and NIPNGM) exhibit larger residuals and pronounced oscillations during the test phase, suggesting potential overfitting and reduced predictive reliability in practical applications.

1. Case 3: marine transportation industry

Figure 6 presents the complete forecasting results for the marine transportation industry, while Figure 7 illustrates the corresponding residual plots. The data preceding the red dashed line represent the in-sample set, whereas the data following the red line correspond to the test set. The series in this sector shows a generally upward trend during the training phase but exhibits pronounced fluctuations in the test phase, posing greater challenges for model generalization. As shown in Figure 6, models such as FDGM, FGM, and GM perform consistently well, capturing the overall trend with high stability. In contrast, nonlinear models – particularly NIPNDGM and NIPNGM – demonstrate substantial deviations and even extreme prediction outliers during testing. The residual plots in Figure 7 further corroborate this observation: fractional-order models produce smaller and more uniformly distributed residuals, whereas nonlinear and new-information-based models display wider residual ranges and sharp oscillations. These results suggest that fractional-order models provide more reliable forecasts for datasets with moderate volatility, while overly complex nonlinear structures tend to compromise stability and increase prediction errors.

Table 3 presents a comparison of the performance of fourteen grey models across three datasets: marine fisheries, seawater utilization, and marine transportation. The FNDGM model achieves the highest overall accuracy, with MAPE values of 5.78%, 1.85%, and 5.08% respectively, and consistently records the lowest MAE and RMSE among all models. NDGM and FBernoulliGM also perform well, with MAPE values generally below 10%, indicating strong adaptability. In contrast, basic models such as GM and DGM exhibit higher errors, with MAPE values exceeding 13% for marine fisheries and approximately 7% for seawater utilization, suggesting a limited capacity to capture complex relationships. Models incorporating NIP structures demonstrate less stable performance; for example, NIPNDGM shows extremely poor results on the marine transportation dataset, with a MAPE of 616.29%, RMSE of 51,911.32, and MAAPE of 1.3845, reflecting severe overfitting and low robustness. Overall, fractional-order models – particularly FNDGM – display superior accuracy and stability across all datasets. The performance gap between the best and worst models exceeds 600% points in MAPE, highlighting substantial differences in predictive capability among the grey model variants.

As shown in Figures 8–10, different grey models produce distinct forecasting outcomes on the same dataset. This indicates clear variability in their predictive capabilities.

In Case 1, as shown in Figure 8, all fourteen grey models successfully reproduce the overall upward trend of marine fisheries, demonstrating consistent short-term fitting capability. However, clear differences emerge in the extrapolated segment. Traditional models such as GM, FGM, and FDGM produce prediction curves that closely follow the observed data and maintain smooth continuity into the test phase. In contrast, several nonlinear and new-information models – particularly NIPNDGM and NIPNGM – exhibit substantial deviations or unrealistic downturns in later years, indicating overfitting and limited generalization ability. Fractional-order models, including FDGM and FNDGM, perform relatively better by incorporating system memory and gradual adjustment mechanisms, enabling more stable trend representation. Overall, while basic GM-type models may oversimplify dynamic variations, overly complex nonlinear structures tend to distort long-term forecasts under volatile conditions.

In Case 2, as illustrated in Figure 9, all models capture the stable and continuous upward trend of seawater utilization, reflecting the regular growth of desalination and related industries. Differences among models are minor, with GM, FGM, NPDGM, and FDGM showing nearly perfect alignment with observed values throughout the series. New-information models (NIPGM, NIPNGM) perform comparably during the fitting period but display slight underestimation in later years, likely due to their adaptive weighting mechanisms responding to limited recent variations. Given the smooth and monotonic nature of the dataset, simpler fractional or polynomial grey models provide the most reliable performance, whereas highly nonlinear models contribute little additional benefit and may introduce unnecessary oscillations. These findings suggest that for stable, gradually evolving systems, model simplicity and robustness outweigh excessive adaptivity.

In Case 3, as shown in Figure 10, the marine transportation dataset exhibits greater volatility and variability across models. Most models, including GM, FGM, and FDGM, capture the general growth pattern during the training period. However, several nonlinear models – particularly NIPNGM and NIPNDGM – produce extreme downward forecasts in the test phase, even generating unrealistic negative values, indicating severe instability and overfitting. Fractional-order and traditional grey models demonstrate more stable and reasonable trends, better reflecting the sector's gradual expansion. These results indicate that while nonlinear extensions can enhance flexibility, they may also amplify numerical errors when data exhibit abrupt shifts. Consequently, moderate-complexity grey models with fractional or polynomial structures achieve a superior balance between accuracy and stability in marine transportation forecasting.

By comparing the prediction results across the three datasets, clear performance differences among model types can be observed. Fractional-order models, such as FDGM and FNDGM, perform better when the data show nonlinear characteristics, as they can capture dynamic changes more effectively. New information models, including NIPGM and NIPNGM, perform well in Case 2, suggesting their advantage in using newly available information to enhance prediction accuracy. Nonlinear Bernoulli models, such as NIPBernoulliGM, also show accurate predictions in Case 3, especially under high data volatility, due to their strong ability to model nonlinear behavior. In contrast, basic models like GM and DGM perform poorly in all three cases, mainly because they cannot effectively capture nonlinear features or adapt to complex data patterns.

Basic grey models, such as GM and DGM, perform poorly across all datasets. Their limited modeling capability prevents them from effectively capturing nonlinear features and complex data variations. In contrast, fractional-order grey models show strong overall performance. FNDGM maintains high prediction accuracy and stability across different datasets, demonstrating good generalization and adaptability to complex data structures. New information models, including NIPGM and NIPNGM, perform well in Case 2, but their accuracy declines sharply when facing highly volatile or structurally complex data, such as in Case 3, indicating limited model stability. Bernoulli-type models perform well in some situations but show larger errors in others. For example, NIPBernoulliGM produces significant errors in Case 2, suggesting weaker adaptability to varying data characteristics and greater sensitivity to fluctuations. Therefore, careful evaluation of data features is necessary when applying these models.

This study provides a comprehensive evaluation of the predictive performance of fourteen grey system models in the context of the marine economy. Three key ocean sectors, marine fisheries, seawater utilization, and maritime transportation, are analyzed to examine model applicability under small-sample and nonlinear data conditions. The evaluated models encompass four structural types: basic first-order accumulation, fractional-order accumulation, new-information priority accumulation, and nonlinear Bernoulli structures. Four statistical metrics are used to assess forecasting accuracy and stability. Empirical results show that the fractional-order nonlinear discrete grey model (FNDGM) consistently achieves the highest forecasting accuracy across all examined sectors. Specifically, its MAPE values are 5.775%, 1.852%, and 5.077% across the three marine-economy datasets. These findings confirm the robustness of the FNDGM model in capturing nonlinear and uncertain patterns in marine economic data. Overall, the results highlight that the FNDGM model offers distinct advantages in forecasting marine economic indicators, providing high predictive accuracy, strong stability, and substantial practical value for data-driven forecasting and policy analysis. Future research should aim to further enhance the adaptability and generalization capability of grey system models through integration with advanced techniques.

The superior performance of the FNDGM model demonstrates that grey forecasting methods can effectively support decision-making in the marine economy under conditions of data scarcity and uncertainty. Policymakers can utilize these models to monitor and predict the development trends of key marine sectors, such as fisheries, seawater utilization, and maritime transportation, enabling timely and data-driven policy adjustments. Moreover, accurate forecasts of marine industry output can assist governments and regional authorities in allocating resources more efficiently, optimizing industrial structures, and designing targeted support measures for emerging marine sectors. The robustness of grey models further indicates their potential integration into early-warning systems for marine economic risks, contributing to long-term strategic planning and the sustainable use of ocean resources. In addition, by combining grey system models with policy analysis tools, decision-makers can quantitatively assess the potential impacts of policy interventions, thereby enhancing the scientific basis, transparency, and effectiveness of marine governance. Future research can further strengthen the linkage between forecasting outcomes and marine policy formulation. Integrating grey system forecasting methods with spatial and environmental policy models can provide multidimensional decision support and strategic insights to promote balanced regional development and sustainable utilization of marine resources.