Prediction and Analysis of Abalone Aquaculture Production in China Based on an Improved Grey System Model

Abstract

This study employs an improved fractional-order grey multivariable convolution model (FGMC(1,N,2r)) to predict abalone aquaculture output in Fujian, Shandong, and Guangdong. By integrating fractional-order accumulation (r₁, r₂) with a particle-swarm-optimization (PSO) algorithm, the model addresses limitations of handling multivariable interactions and sequence heterogeneity within small-sample regional datasets. Grey relational analysis (GRA) first identified key factors exhibiting the strongest associations with production: abalone production in Fujian and Shandong is predominantly influenced by funding for aquatic-technology extension (GRA degrees of 0.9156 and 0.8357, respectively), while in Guangdong, production was most strongly associated with import volume (GRA degree of 0.9312). Validation confirms that FGMC(1,N,2r) achieves superior predictive accuracy, with mean absolute percentage errors (MAPE) of 0.51% in Fujian, 3.51% in Shandong, and 2.12% in Guangdong, significantly outperforming benchmark models. Prediction of abalone production for 2024–2028 project sustained growth across Fujian, Shandong, and Guangdong. However, risks associated with typhoon disasters (X₆ and import dependency (X₅) require attention. The study demonstrates that the FGMC(1,N,2r) model achieves high predictive accuracy for regional aquaculture output. It identifies the primary drivers of abalone production: technology-extension funding in Fujian and Shandong, and import volume in Guangdong. These findings support the formulation of region-specific strategies, such as enhancing technological investment in Fujian and Shandong, and strengthening seed supply chains while reducing import dependency in Guangdong. Furthermore, by identifying vulnerabilities such as typhoon disasters and import reliance, the study underscores the need for resilient infrastructure and diversified seed sources, thereby providing a robust scientific basis for production optimization and policy guidance towards sustainable and environmentally sound aquaculture development.

1. Introduction

1.1. Background of Abalone Aquaculture in China

Abalone (Haliotis spp.), a herbivorous gastropod mollusk (class Gastropoda, order Archaeogastropoda, family Haliotidae), exhibits distinct feeding behaviors across life stages. Juveniles feed on benthic diatoms, while adults primarily consume microalgae and macroalgae [1]. Valued in global fisheries and aquaculture, abalone is a premium commodity in Asian markets due to its high protein content, low fat, and rich functional amino acids (e.g., taurine), serving both as a delicacy and a nutrient-dense food source [2,3]. Its processed products, representing significant value addition, are particularly vital within China’s aquaculture industrial chain [4]. The international market for abalone is characterized by complex supply-demand dynamics, with China playing a pivotal role in both production and trade [5].

Fujian Province has emerged as a leading global abalone producer, leveraging subtropical currents, warm waters, and rocky substrate [6,7]. According to the 2024 China Fisheries Statistical Yearbook, Fujian accounted for 78% of China’s total abalone output in 2023 [8]. Through Industry–University–Research collaboration, the province has promoted technological innovations such as the breeding of “Dongyou No.1” seedlings (with strong disease resistance and high productivity) and the development of Vibrio vaccines for disease control, supporting annual export revenues of over $1.2 billion USD from dried and ready-to-eat abalone products to Japan, South Korea, and Southeast Asia. Guangdong and Shandong represent China’s other major abalone-farming regions, typifying southern and northern cultivation zones, respectively. Guangdong utilizes extensive coastline and mature aquaculture techniques to sustain steady growth, while Shandong capitalizes on abundant marine resources and large-scale farming, ranking second nationally in output. Collectively, these three provinces dominate China’s abalone production structure.

Nevertheless, the sector confronts critical challenges: (i) germplasm degeneration reducing disease resistance [9]; (ii) summer water temperature anomalies increasing mortality [10]; (iii) low value-added processing limiting output value [11] Accurate production prediction is essential for optimizing breeding cycles, resource allocation, and policy design. However, this is complicated by volatile factors like trade policies, environmental impacts, technological investments, and sparse historical data.

Against the backdrop of global food security challenges and China’s ‘Dual Carbon’ goals [12], promoting the sustainable development of high-value aquaculture like abalone farming has become a national priority. Sustainable aquaculture not only ensures stable production and economic benefits but also minimizes environmental impacts, such as optimizing resource use and reducing ecological footprints. Accurate production forecasting is, therefore, a critical tool for achieving this sustainability, enabling better resource allocation, waste management, and mitigation of environmental risks.

1.2. The Application and Limitations of Grey System Models

Conventional prediction models require large datasets, often unavailable in regional aquaculture contexts [13] which has led to the adoption of grey system models for such data-scarce scenarios [14]. For example, Wang et al. applied the GM (1,1) model to predict the sturgeon production in Guizhou Province [15], and Du et al. applied the GM (1,N) model to predict the electricity consumption in Jiangsu Province [16] both demonstrating good performance. Tien et al. advanced the framework with the GMC(1,N) model, integrating convolution operations to better capture nonlinear variable interactions [17]. Compared with the GM(1,N) model, the GMC(1,N) model is more capable of capturing nonlinear interactions between variables, thereby enhancing prediction credibility.

Grey relational analysis (GRA), an important component of grey system theory, is a multi-factor analysis framework suitable for environments with uncertain information [18]. Its core function is to quantify the degree of correlation between various factors. GRA measures the similarity between sequences based on absolute, relative, and comprehensive relational degrees. It enables the quantitative evaluation of complex factor relationships through a systematic analysis process [19]. In practical applications, grey relational analysis has demonstrated extensive value in fields such as economics [20,21], medicine [22,23], biological sciences [24,25], environmental protection [26,27], and engineering [28,29]. Its standardized workflow includes defining reference sequences, data normalization, relational coefficient calculation, relational degree quantification, and hierarchical ranking, providing scientific and effective methodological support for multi-factor correlation analysis [30].

However, existing models still have limitations. Traditional GM(1,1) and residual-corrected grey models (RCGM) overlook multivariable interactions. While GM(1,N) incorporates multiple variables, its uniform integer-order accumulation fails to capture sequence heterogeneity, propagating errors [31]. Even the improved GMC(1,N) model is constrained by its fixed accumulation structure, which may not optimally handle the heterogeneous nature of different influencing factors (e.g., economic inputs vs. environmental disasters).

Therefore, despite their utility, existing grey models face a critical limitation in addressing the intertwined challenges of small sample sizes, multivariable interactions, and sequence heterogeneity inherent to regional abalone production forecasting. This study bridges this gap by introducing and validating an improved fractional-order grey multivariable convolution model (FGMC(1,N,2r)) within aquaculture science. The model’s novelty lies in its use of dual fractional-order accumulation (r₁, r₂), with orders independently optimized via a particle swarm optimization (PSO) algorithm. This fundamental advancement relaxes the rigid, uniform-order accumulation assumption of predecessor models, enabling a more flexible and sensitive capture of the distinct dynamic characteristics between the target production sequence and its heterogeneous drivers.

Guided by this innovative framework, the present research is structured around three pivotal objectives to enhance the precision and practical utility of abalone production forecasting. First, we rigorously assess whether the FGMC(1,N,2r) model achieves superior predictive accuracy over established grey models (GM(1,1), GM(1,N), GMC(1,N)) when applied to the small-sample, multivariable context of abalone farming in China’s key provinces (Fujian, Shandong, Guangdong). Second, we employ Grey Relational Analysis (GRA) to identify the principal factors governing production dynamics in each region, thereby moving beyond prediction to uncover the systemic drivers of output. Finally, by generating reliable production projections for 2024–2028, we derive actionable, region-specific policy implications to support risk mitigation and steer the industry toward a sustainable and resilient future. Through this integrated approach, the study contributes a refined methodological tool to grey system theory and provides a decision-support framework for aquaculture management under data-scarce conditions.

1.3. Literature Review on Grey Models and Aquaculture Determinants

Grey system models have gained traction in aquaculture forecasting due to their efficacy in handling data-scarce scenarios. Early applications, such as the GM(1,1) model for sturgeon production prediction [15] and the GM(1,N) model for regional electricity consumption [16], demonstrated the potential of grey theory in managing limited datasets. The introduction of the GMC(1,N) model further improved predictive capability by incorporating convolution operations to capture nonlinear variable interactions [17]. A review of these applications reveals a common focus on univariate or simplistically multivariate predictions. Despite these advancements, persistent limitations hinder their application in complex, multivariable aquaculture systems like abalone production. Traditional models like GM(1,1) and residual-corrected variants often overlook interactions among multiple influencing factors. While GM(1,N) incorporates multiple variables, its reliance on uniform integer-order accumulation fails to account for sequence heterogeneity, potentially propagating errors [31]. Even the enhanced GMC(1,N) model, with its fixed accumulation structure, may not optimally adapt to the distinct characteristics of diverse factors, such as the gradual accumulation of technological benefits versus the abrupt impact of environmental disasters, a challenge noted in complex agricultural and aquatic systems [32].

The selection of predictive variables in aquaculture models should be grounded in empirical research. Previous studies have identified several key determinants influencing aquaculture production. Investment in technology extension and research & development is widely recognized as a critical driver for improving germplasm, disease management, and overall productivity, as evidenced in studies of high-value aquaculture sectors [33,34]. Economic factors, such as per capita income of fishers and output value per unit yield, reflect the economic vitality and efficiency of the industry, which are closely tied to sustainable production levels [35]. Market dynamics, including export value and import volume, are crucial indicators of external demand and the stability of seed supply chains, with supply chain instability being a known risk factor for production volatility [5]. Furthermore, environmental shocks, quantified through economic losses from fishery disasters, consistently emerge as significant vulnerabilities, highlighting the impact of extreme climate events on aquaculture sustainability [36,37,38]. This body of literature provides a strong rationale for the six variables (X₁–X₆) selected for analysis in this study, ensuring they represent well-established factors influencing aquaculture production. The selection of these six proxy variables is well-aligned with key sustainability and risk indicators emphasized in contemporary aquaculture research, such as supply chain stability and resilience to environmental shocks [39,40].

To address these limitations in small, heterogeneous, and multivariable samples, this study introduces the Fractional-order Grey Multivariable Convolution model (FGMC(1,N,2r)) to aquaculture prediction [41]. This framework enhances the GMC(1,N) model by incorporating dual fractional-order accumulation, allowing independent optimization of the accumulation orders for the target sequence (r₁) and driver variables (r₂) via particle swarm optimization. This key innovation relaxes the restrictive uniform-order assumption of traditional models, enhances sensitivity to nonlinear couplings between variables, and enables more flexible information extraction from limited data, promising greater accuracy and stability [17,41].

Although the FGMC(1,N,2r) model has shown promise in fields like energy forecasting [41], its application to aquaculture—particularly for high-value species like abalone—remains unexplored. This study represents a novel adaptation of this advanced model, specifically designed to overcome the core challenges of predicting abalone production, which is characterized by small regional datasets, multiple interacting factors, and inherent sequence heterogeneity. The model’s ability to independently tune fractional orders makes it exceptionally suitable for capturing the unique dynamics of regional aquaculture systems. Recent successful applications of fractional-order grey models in capturing the complex dynamics of agricultural and environmental systems [42,43,44,45] underscore the model’s potential for aquaculture forecasting.

To ensure the empirical relevance of the input variables for the FGMC(1,N,2r) model, Grey Relational Analysis (GRA) is first employed to identify the factors most strongly associated with abalone production [18,19]. This data-driven approach aligns with established aquaculture literature, where factors such as investment in technology extension are recognized as critical for germplasm improvement and disease management [33,34] and import volume serves as an indicator of seed supply chain stability, a known factor influencing production volatility [5]. This methodological synergy between GRA and the FGMC(1,N,2r) model provides a robust foundation for both variable selection and predictive modeling.

2. Methodology

2.1. Grey Correlation Analysis

Grey Relational Analysis (GRA) is one of the core methods of grey system theory. Its basic principle is to assess the degree of correlation between various factors within a system by comparing the geometric similarity of sequence curves. This method first performs linear interpolation on discrete observation data to form continuous broken-line sequences. Subsequently, it constructs a quantification model of correlation degree based on the proximity of these broken lines in geometric form, thereby realizing the measurement of the intensity of interactions between factors in complex systems and providing a powerful quantitative tool for system analysis. The specific calculation steps are as follows:

• Step 1: Determine the reference sequence and the comparative sequences

Reference Sequence: The target variable sequence (e.g., Fujian Province’s total abalone production).

$$X_0=\{X_0(k)\}, k=\mathrm{1,2},\dots ,n$$

(1)

Comparative Sequences: The influencing factor sequences (e.g., factors affecting abalone production in Fujian Province).

$$X_i=\left\{X_i(m)\right\}, i=\mathrm{1,2},\dots ,m$$

(2)

• Step 2: Data normalization

The following methods are employed to eliminate dimensional differences:

$$\mathrm{Initial} \mathrm{value} \mathrm{transformation}: X_i^{\prime }(k)={\displaystyle \frac{X_i(k)}{X_i(1)}}$$

(3)

$$\mathrm{Mean} \mathrm{value} \mathrm{transformation}: X_{\mathrm{i}}^{\prime }(k)={\displaystyle \frac{X_i(k)}{\frac{1}{n}{\displaystyle \sum _{k=1}^n{X}_i(k)}}}$$

(4)

• Step 3: Correlation Coefficient Calculation

Compute pairwise similarity metrics:

$$\xi (X_0(K),X_i(K))={\displaystyle \frac{{\Delta }_{\min }+\rho \cdot {\Delta }_{\max }}{{\Delta }_{0i}(k)+\rho \cdot {\Delta }_{\max }}}$$

(5)

where:

$$\begin{gathered} {∆}_{0i}\left(k\right)=x_0\left(k\right)-x_i\left(k\right) \\ {∆}_{min}=\underset{\forall i}{\min }\underset{\forall k}{\min }{∆}_{0i}\left(k\right) \\ {∆}_{max} =\underset{\forall i}{\mathit{max}}\underset{\forall k}{\mathit{max}}{∆}_{0i}\left(k\right) \\ \rho \in \left(\mathrm{0,1}\right) (discrimination coefficient, default \rho =0.5) \end{gathered}$$

• Step 4: Calculate the relational degree

The calculation formula for Deng’s relational degree is:

$$R_{0i}(X_0,X_i)={\displaystyle \frac{1}{n}}{\sum }_{k=1}^{ n}\xi (X_0(k),X_i(k))$$

(6)

The relational degree reflects the overall correlation between the reference sequence and the comparative sequence, with a larger value indicating a more significant impact.

• Step 5: Factor prioritization

Based on the calculated relational degree values, factors are ranked in descending order to identify those having the greatest impact on abalone production in Fujian, forming a comprehensive evaluation.

2.2. FGMC(1,N,2r) Model

• Step 1: Data Preparation and Preprocessing

Identify the influencing factor exhibiting the highest correlation to the target sequence using grey relational analysis (Section 2.1).

• Step 2: Factor prioritization

Fractional-order accumulated generation of the target sequence (order r₁):

$$X_1^{(r_1)}(k)={\sum }_{i=1}^k{\displaystyle \frac{\Gamma \left(r_1+k-i\right)}{\Gamma \left(k-i+1\right)\Gamma \left(r_1\right)}}{x}_1^{(0)}(i),k=\mathrm{1,2},\dots ,n$$

(7)

Γ denotes the gamma function, and $r_1\in \left[\mathrm{0,2}\right]$ is the fractional order parameter of the target sequence.

Fractional-order accumulated generation of the influencing factor sequence (order r₂):

$$X_i^{(r_2)}(k)={\sum }_{i=1}^k{\displaystyle \frac{\Gamma \left(r_2+k-i\right)}{\Gamma \left(k-i+1\right)\Gamma \left(r_2\right)}}{x}_i^{(0)}(i),k=\mathrm{1,2},\dots ,n$$

(8)

$r_2\in \left[\mathrm{0,2}\right]$ is the fractional order parameter of the influencing factor sequence, which is independently optimized from r₁.

• Step 3: Model Construction and Parameter Estimation

The model inherits the core convolution idea from GMC(1,N), which utilizes the background values (Equations (2)–(10)) to perform a weighted integration of the influencing factor sequences’ neighborhood information, thereby better capturing the dynamic nonlinear interactions with the system’s main behavior. On this basis, we introduce the dual fractional-order accumulation operation to further enhance the model’s flexibility.

Construct a grey differential equation incorporating fractional-order accumulated terms:

$${\displaystyle \frac{d{X}_1^{(r_1)}(t)}{dt}}+b_1{X}_1^{(r_1)}(t)={\sum }_{i=2}^N{b}_i{X}_i^{(r_2)}(t)+\mu$$

(9)

where $b_1,{ b}_2$, …, $b_N, \mu$ are the parameters to be estimated.

Generate background values using the adjacent neighbor mean:

$$z_1^{(r_1)}(k)={\displaystyle \frac{X_1^{(r_1)}(k)+X_1^{(r_1)}(k-1)}{2}}, z_i^{(r_2)}(k)={\displaystyle \frac{X_i^{(r_2)}(k)+X_i^{(r_2)}(k-1)}{2}}$$

(10)

Construct the data matrix B and the observation vector Y:

$$B=\left[\begin{array}{ccccc}-z_1^{(r_1)}(2)& z_2^{(r_2)}(2)& \dots & z_N^{(r_2)}(2)& 1\\ -z_1^{(r_1)}(3)& z_2^{(r_2)}(3)& \dots & z_N^{(r_2)}(3)& 1\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ -z_1^{(r_1)}(n)& z_2^{(r_2)}(n)& \dots & z_N^{(r_2)}(n)& 1\end{array}\right], Y=\left[\begin{array}{c}{X}_1^{(0)}(2)\\ X_1^{(0)}(3)\\ ⋮\\ X_1^{(0)}(n)\end{array}\right]$$

(11)

Parameters are solved through the least squares method:

$$\widehat{\beta }={\left[b_1,b_2,\dots b_N,\mu \right]}^T={\left(B^{T}B\right)}^{-1}{B}^{T}Y$$

(12)

• Step 4: Time Response Function and Prediction Value Reconstruction

Prediction of the fractional-order accumulated sequence: Solving the differential equation yields the accumulated prediction value:

$${\widehat{x}}_1^{(r_1)}(k)=X_1^{(0)}(1)e^{-b_1(k-1)}+{\sum }_{j=2}^k\left\{e^{-b_1(k-j+0.5)}{\displaystyle \frac{f(j)+f(j-1)}{2}}\right\}$$

(13)

Among $f(j)={\sum }_{i=2}^N{b}_i{x}_i^{(r_2)}(j)+\mu$. The target sequence is restored through fractional-order inverse accumulation, yielding the final prediction value:

$${\widehat{x}}_1^{(0)}(k)={\sum }_{i=0}^{k-1}{\displaystyle \frac{\Gamma (r_1+1)}{\Gamma (i+1)\Gamma (r_1-i+1)}}{\widehat{x}}_1^{(r_1)}(k)$$

(14)

• Step 5: Parameter Optimization

Particle swarm optimization is used to solve the optimal fractional orders $r_1$ and $r_2$. The optimization aims to minimize the prediction error, with the Mean Absolute Percentage Error (MAPE) serving as the objective function, defined as:

$${min}_{r_1,r_2}{\displaystyle \frac{1}{n}}{\sum }_{k=1}^n\left|{\displaystyle \frac{X_1^{(0)}(k)-{\widehat{x}}_1^{(0)}(k)}{X_1^{(0)}(k)}}\right|\times 100\%$$

(15)

The PSO algorithm iteratively updates the particle positions (potential solutions for ($r_1$, $r_2$) and velocities to seek the values that minimize the MAPE objective function (Equations (2)–(15)). The update formulas are as follows:

$${\mathsf{\nu }}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t}+1)}=\mathsf{\omega }{\mathsf{\nu }}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t})}+{\mathrm{c}}_1{\mathrm{r}}_1\left({\mathrm{p}}_{\mathrm{g}\mathrm{d}}-{\mathrm{x}}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t})}\right)+{\mathrm{c}}_2{\mathrm{r}}_2\left({\mathrm{p}}_{\mathrm{g}\mathrm{d}}-{\mathrm{x}}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t})}\right){\mathrm{x}}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t}+1)}={\mathrm{x}}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t})}{\mathsf{\nu }}_{\mathrm{i}\mathrm{d}}^{(\mathrm{t}+1)}$$

(16)

where $\omega$ is the inertia factor, $c_1, c_2$ are learning factors, and $r_1, r_2\in \left[\mathrm{0,1}\right]$ are random numbers.

• Step 6: Model Evaluation

Absolute Percentage Error:

$$APE=\left|{\displaystyle \frac{{\widehat{x}}^{(0)}(k)-x^{(0)}(k)}{x^{(0)}(k)}}\right|\times 100\%$$

(17)

Mean Absolute Percentage Error:

$$MAPE={\displaystyle \frac{1}{n}}{\sum }_{k=1}^{n}APE(k)$$

(18)

Root Mean Square Percentage Error:

$$RMSPE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{k=1}^n{\left({\displaystyle \frac{{\widehat{x}}^{(0)}(k)-x^{(0)}(k)}{x^{(0)}(k)}}\right)}^2}\times 100\%$$

(19)

3. Model Application

3.1. Study Areas

To improve resource-use efficiency in China’s abalone sector and accelerate the transition to a green mariculture system, the country’s leading abalone-producing provinces have introduced region-specific policies that promote sustainable development. A critical component of these policies is the dynamic analysis of historical productions and the generation of accurate prediction, which together enable the refinement of farming strategies and the mitigation of production risks. Consequently, this study examines three representative provinces: Fujian, the nation’s core production area, characterized by intensive offshore farming; Shandong, leveraging northern marine resources and large-scale operations for industrialized recirculating aquaculture [46]; and Guangdong, where warm coastal waters and polyculture practices underpin an integrated ecological model in southern China. The selection of Fujian, Shandong, and Guangdong provinces is based on their dominance in China’s abalone production (collectively accounting for over 95% of the national output) and their representation of distinct aquaculture models (intensive offshore, industrialized recirculating, and deep-sea cage culture, respectively). This diversity allows for a robust test of the model’s applicability across different systems [32]. The geographic locations of these provinces are illustrated in Figure 1.

In response to the aquaculture industry’s critical challenges—including germplasm degeneration, rising summer water temperatures, and limited value-added processing—this study employs six measurable proxy variables to reflect these complex and interrelated issues. Aquaculture technology-extension funding (×10⁴ CNY) is used as an indicator of investment in genetic improvement and disease management, particularly in supporting the development of disease-resistant breeds and advanced biosecurity technologies. Mariculture output value per unit yield (×10⁴ CNY/ton) serves as a proxy for the level of processing sophistication, where higher per-unit values suggest greater value addition and stronger market competitiveness. Per capita income of fishers (×10⁴ CNY) reflects the economic sustainability of the industry, which is closely tied to production efficiency and market performance. Export value (×10⁴ CNY and import volume (×10⁴ tons) are included to capture external market demand and the stability of seed supply chains, respectively. Economic losses from fishery disasters (×10⁴ CNY) are introduced to quantify the impact of extreme environmental events, such as typhoons and abnormal temperature spikes, thereby assessing the vulnerability of regional aquaculture systems to climate-related shocks.

Accordingly, these six variables—namely, aquaculture technology-extension funding, mariculture output value per unit yield, per capita fisher income, export value, import volume, and economic losses from fishery disasters—are analyzed across provinces. Grey relational analysis is applied independently to each regional dataset to evaluate the strength of association between these factors and abalone production, with the aim of identifying the most influential drivers for inclusion in subsequent predictive modeling and policy planning.

Utilizing state-of-the-art prediction models, we calibrate historical production data for each province and project abalone output for the next five years.

3.2. Data Collections

Figure 2, Figure 3 and Figure 4 illustrate the annual abalone production in Fujian, Shandong, and Guangdong from 2012 to 2023. Data for the first ten years are used to calibrate the model, and the final two years are reserved for out-of-sample testing.

Figure 2, Figure 3 and Figure 4 illustrate the annual abalone production in Fujian, Shandong, and Guangdong from 2012 to 2023. To rigorously evaluate the generalization ability and predictive performance of the proposed FGMC(1,N,2r) model for out-of-sample forecasting, a standard holdout validation approach was adopted [47]. Data for the first ten years (2012–2021) are used to calibrate the model (i.e., parameter optimization via PSO), and the data from the two most recent years (2022–2023) are reserved exclusively for testing. This practice, common in time series forecasting with limited data, ensures an objective assessment of the model’s accuracy in predicting future unknown values, which is the ultimate goal of any prediction model. Although the model parameters are optimized using the calibration set, the testing set remains completely unseen during this optimization process, thereby providing an unbiased estimate of the model’s real-world forecasting performance.

Recognizing the importance of environmental carrying capacity and its implications for sustainable aquaculture, environmental variables such as regional water quality data and coastal ecosystem health indicators were initially considered. However, due to the lack of consistent and standardized long-term monitoring data across all three provinces, these factors could not be directly incorporated into the quantitative model. Nevertheless, the selected economic and disaster-related variables indirectly reflect environmental pressures, as factors like fishery disaster losses (X₆) are often linked to extreme climate events and ecological imbalances. This limitation highlights the need for more comprehensive environmental data collection to support future sustainability-focused modelling efforts. All data were obtained from the China Fisheries Statistical Yearbook. The raw data were collected, collated, and processed into a structured time-series format suitable for grey system modeling. For each variable, annual data from 2012 to 2023 were extracted to form the sequences for analysis.

3.3. Model Comparison and Analysis

3.3.1. Abalone Production in Fujian Province

Using the total abalone production (tons) as the reference sequence, we designate X₁–X₆ as the comparison sequences: aquaculture technology-extension funding (×10⁴ CNY), mariculture output value per unit yield (×10⁴ CNY/tons), per capita income of fishers (×10⁴ CNY), export value (×10⁴ CNY), import volume (×10⁴ tons), economic losses from fishery disasters (×10⁴ CNY).

Following the methodology in Section 2.1, the grey relational degrees between abalone production in Fujian Province and the seven influencing factors were calculated for 2012–2023, as presented in

Table 1. Grey correlation degree results of Fujian Province.

	R0i(X0,X1)	R0i(X0,X3)	R0i(X0,X5)	R0i(X0,X4)	R0i(X0,X2)	R0i(X0,X6)
Relational degrees	0.9156	0.8862	0.8840	0.7595	0.7242	0.5730

. Consequently, an FGMC(1,2,2r) model was developed, with the two factors exhibiting the strongest associations with production (aquaculture technology-extension funding, X₁, GRA = 0.9156; and per capita income of fishers, X₃, GRA = 0.8862) serving as the input variables for the multivariate model. Its two fractional-order values (r₁, r₂) were optimized using the PSO algorithm. To demonstrate the proposed model, comparisons were made with five other models: the GM (1,1) model, the optimized RCGM (1,N), GM (1,N), and GMC (1,N) models, as well as the optimized FGMC(1,N,2r) model. All models utilized the same dataset, with the PSO algorithm applied to optimize relevant parameters where applicable.

Table 2. Modeling and test results of abalone production in Fujian Province.

Year	Actual Values	GM(1,1)		RCGM(1,1)		GM(1,N)		GMC(1,N)		FGMC(1,N,2r) r1 = 0.6319, r2 = 0.0041
		Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)
Modeling
2012	65,247.00	65,247.00	0.00	65,247.00	0.00	65,247.00	0.00	65,274.00	0.00	65,247.00	0.00
2013	88,470.00	81,376.73	8.02	81,961.00	7.36	75,809.07	14.32	87,535.42	1.06	87,125.41	1.52
2014	91,252.00	89,291.32	2.15	89,696.88	1.71	106,776.47	17.01	94,387.63	3.43	92,511.88	1.38
2015	100,979.00	97,975.68	2.97	98,203.30	2.75	103,074.64	2.07	103,892.17	2.88	101,899.23	0.91
2016	112,611.00	107,504.66	4.54	107,555.12	4.49	113,763.34	1.02	116,288.45	3.27	111,255.67	1.20
2017	123,387.00	117,960.43	4.40	117,834.48	4.50	125,728.43	1.90	126,543.88	2.56	122,488.01	0.73
2018	134,924.00	129,433.10	4.07	129,131.52	4.29	133,444.53	1.10	138,672.31	2.78	133,890.15	0.77
2019	143,970.00	142,021.60	1.35	141,545.14	1.68	147,993.69	2.79	147,856.24	2.70	144,501.36	0.37
2020	155,009.00	155,834.43	0.53	155,183.86	0.11	154,165.14	0.54	159,872.53	3.14	155,211.73	0.13
2021	172,413.00	170,990.69	0.83	170,166.75	1.30	171,650.43	0.44	176,543.92	2.40	171,005.81	0.82
MAPE (%)			3.21		3.13		4.92		2.69		0.87
RMSPE (%)			3.90		3.77		5.12		2.87		0.98
Testing
2022	181,503.00	187,621.02	3.37	186,624.47	2.82	185,028.47	1.94	187,322.81	3.21	182,110.59	0.33
2023	195,878.00	205,868.79	5.10	204,700.38	4.51	194,225.61	0.84	201,345.67	2.79	194,552.33	0.68
MAPE (%)			4.24		3.67		1.39		3.00		0.51
RMSPE (%)			4.32		3.75		1.42		3.00		0.52

presents the modeling and testing results for abalone production in Fujian Province from 2012 to 2023, along with the optimized parameters of various models. During the modeling phase, the VE grey prediction model achieved MAPE values of 3.21%, 3.13%, 4.92%, 2.69%, and 0.87%, and RMSE values of 3.90%, 3.77%, 5.12%, 2.87%, and 0.98%. In the testing phase, the five models exhibited average MAPE values of 4.24%, 3.67%, 1.39%, 3.00%, and 0.51%, and RMSE values of 4.32%, 3.75%, 1.42%, 3.00%, and 0.52%. According to these metrics, the FGMC(1,N,2r) model demonstrated superior performance in both the modeling and testing phases. It met the precision requirements and provided more accurate predictions for abalone production in Fujian Province. To provide an intuitive comparison of the models’ fitting and prediction performance, the results are visually presented in Figure 5.

Overall, the FGMC(1,N,2r) model achieves the best fit for Fujian’s abalone production in both calibration and validation. Using this model, we predict Fujian’s total abalone output for 2024–2028; results are presented in

Table 3. Predicted abalone production in Fujian Province (2024–2028) (tons).

Year	2024	2025	2026	2027	2028
Predicted Values	209,876.54	224,531.89	240,187.26	256,923.74	274,828.41

. The projections reveal a sustained upward trajectory over the next five years.

3.3.2. Abalone Production in Shandong Province

For modeling and testing Shandong’s total abalone production, grey relational analysis was first applied to quantify the relational degrees between output and the seven candidate drivers; results are presented in

Table 4. Grey correlation degree results of Shandong Province.

	R0i(X0,X1)	R0i(X0,X3)	R0i(X0,X2)	R0i(X0,X4)	R0i(X0,X5)	R0i(X0,X6)
Relational degrees	0.8357	0.8099	0.7728	0.7236	0.7111	0.5862

. Aquaculture technology-extension funding emerges as the dominant factor, registering a grey relational coefficient of 0.8357. Building on this finding, an FGMC(1,2,2r) model was constructed by incorporating the two factors exhibiting the strongest associations (aquaculture technology-extension funding, X₁, GRA = 0.8357; and per capita income of fishers, X₃, GRA = 0.8099) as the explanatory variable series. Comparative results are reported in

Table 5. Modeling and test results of abalone production in Shandong Province.

Year	Actual Values	GM(1,1)		RCGM(1,1)		GM(1,N)		GMC(1,N)		FGMC(1,N,2r) r1 = 0.8715, r2 = 0.2564
		Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)
Modeling
2012	11,470.00	11,470.00	0.00	11,470.00	0.00	11,470.00	0.00	11,470.00	0.00	11,470.00	0.00
2013	11,957.00	10,090.37	15.61	11,957.00	0.00	12,188.91	1.94	12,188.45	1.94	12,185.26	1.91
2014	14,716.00	11,610.30	21.10	11,546.41	21.54	14,225.43	3.33	14,225.83	3.33	14,912.37	1.33
2015	15,165.00	13,359.19	11.91	13,354.95	11.94	15,401.27	1.56	15,401.27	1.56	15,711.84	3.61
2016	15,399.00	15,371.51	0.18	15,427.17	0.18	16,792.35	9.05	16,792.35	9.05	16,325.91	6.02
2017	13,411.00	17,686.95	31.88	17,802.76	32.75	16,162.41	20.51	16,162.41	20.51	14,131.05	5.37
2018	13,195.00	20,351.17	54.23	20,527.38	55.57	16,904.97	28.11	16,904.97	28.11	14,905.63	12.97
2019	21,428.00	23,416.71	9.28	23,653.58	10.39	21,985.66	2.60	21,985.66	2.60	20,589.41	3.91
2020	34,021.00	26,944.02	20.80	27,241.80	19.93	29,495.04	13.3.	29,495.04	13.30	32,345.77	4.93
2021	35,430.00	31,002.65	12.50	31,361.59	11.48	37,102.58	4.72	37,102.58	4.72	35,461.92	0.09
MAPE (%)			19.72		20.47				9.46		4.46
RMSPE (%)			24.64		25.99				12.98		5.42
Testing
2022	36,973.00	35,672.64	3.52	36,093.00	2.38	38,412.21	3.89	39,001.83	5.49	38,812.14	4.97
2023	38,800.00	41,046.08	5.79	41,528.12	7.03	40,781.34	2.53	42,056.21	8.39	39,589.66	2.04
MAPE (%)			4.66		4.71				6.94		3.51
RMSPE (%)			4.79		5.25				7.09		3.69

, and the corresponding results are illustrated in Figure 6.

Table 5 presents the modeling and testing results for abalone production in Shandong Province from 2012 to 2023, along with the optimized parameters of various models. During the modeling phase, the FGMC(1,N,2r) model significantly outperformed the other models, achieving a MAPE of 4.46% and an RMSPE of 5.42%. In the testing phase, the FGMC(1,N,2r) model maintained its superiority with a MAPE of 3.51% and an RMSPE of 3.69%. According to these metrics, the FGMC(1,N,2r) model demonstrated superior performance in both phases.

Additionally, the FGMC(1,N,2r) model was applied to predict Shandong’s total abalone production for 2024–2028; results are presented in

Table 6. Predicted abalone production in Shandong Province (2024–2028) (tons).

Year	2024	2025	2026	2027	2028
Predicted Values	41,150.47	42,519.60	43,888.73	45,257.86	46,626.99

. Projections indicate a continued upward trajectory.

3.3.3. Abalone Production in Guangdong Province

In analyzing Guangdong’s total abalone production, grey relational analysis was applied to quantify the relational degrees between output and seven candidate drivers; results are presented in

Table 7. Grey correlation degree results of Guangdong Province.

	R0i(X0,X5)	R0i(X0,X2)	R0i(X0,X4)	R0i(X0,X3)	R0i(X0,X1)	R0i(X0,X6)
Relational degrees	0.9312	0.9304	0.9242	0.8687	0.7626	0.7357

. Mariculture production volume emerged as the dominant factor, with a grey relational degree of 0.9400. Consequently, an FGMC(1,2,2r) model was constructed by selecting the two top-ranking factors from the GRA (import volume, X₅, GRA = 0.9312; and mariculture output value per unit yield, X₂, GRA = 0.9304) as the explanatory variables for the multivariate prediction, and its performance was benchmarked against four alternative grey models. Comparative outcomes are reported in

Table 8. Modeling and test results of abalone production in Guangdong Province.

Year	Actual Values	GM(1,1)		RCGM(1,1)		GM(1,N)		GMC(1,N)		FGMC(1,N,2r) r1 = 0.9583, r2 = 0.3261
		Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)	Predicted Values	APE (%)
Modeling
2012	7317.00	7317.00	0.00	7317.00	0.00	7317.00	0.00	7317.00	0.00	7317.00	0.00
2013	7023.00	8415.78	19.83	7023.00	0.00	5463.56	22.19	7358.42	4.77	7158.42	1.93
2014	7449.00	8581.43	15.2	8931.85	19.91	8407.68	12.87	7684.19	3.16	7582.16	1.79
2015	8482.00	8750.34	3.16	9051.28	6.71	8246.82	2.77	8211.05	3.20	8321.59	1.89
2016	8971.00	8922.57	0.54	9174.81	2.27	7631.35	14.93	8738.91	2.59	8764.83	2.30
2017	9039.00	9098.20	0.65	9302.51	2.92	10,831.13	19.82	9011.27	0.31	9255.47	2.39
2018	12,000.00	9277.28	22.69	9434.42	21.38	11,852.39	1.23	11,588.36	3.43	11,742.36	2.15
2019	11,639.00	9459.88	18.72	9570.59	17.77	15,063.73	29.42	11,984.25	2.97	11,902.18	2.26
2020	11,468.00	9646.08	15.89	9711.09	15.32	7846.00	31.58	10,987.53	4.19	11,725.91	2.25
2021	8519.00	9835.94	15.46	9855.97	15.69	9326.55	9.47	8945.61	5.01	8346.28	2.03
MAPE (%)			12.46		12.75		16.03		3.29		2.11
RMSPE (%)			14.88		14.62		19.87		3.61		2.13
Testing
2022	8647.00	10,029.54	15.99	10,005.31	15.71	10,311.65	19.25	8824.73	2.05	8462.64	2.14
2023	8961.00	10,226.96	14.13	10,159.16	13.73	10,553.96	17.77	9211.89	2.80	8771.97	2.11
MAPE (%)			15.06		14.54		18.51		2.43		2.12
RMSPE (%)			15.09		14.59		18.52		2.44		2.13

, and the corresponding predictions are displayed in Figure 7.

To project Guangdong’s abalone production trajectory, the FGMC(1,N,2r) model was applied to predict total output for 2024–2028; results are shown in

Table 9. Predicted abalone production in Guangdong Province (2024–2028) (tons).

Year	2024	2025	2026	2027	2028
Predicted Values	9081.35	9390.72	9700.09	10,009.46	10,318.83

. The projections indicate a sustained upward trend. As a leading mariculture province, Guangdong benefits chiefly from expanding marine farming capacity—most notably the large-scale deployment of deep-sea smart cages. A prime example is the “Huibao No. 1” deep-sea intelligent abalone cage platform under construction in Huilai County, which expands the culture area and directly contributes to the expected production increase.

4. Discussion

4.1. Model Performance and Production Trends

This study applies the FGMC(1,N,2r) model to predict abalone culture output in Fujian, Shandong, and Guangdong, achieving a clear advance over conventional grey models. Across the three regions, the model delivers exceptional accuracy: in-sample and out-of-sample MAPE values are 0.87% and 0.51% for Fujian, 4.46% and 3.51% for Shandong, and 2.11% and 2.12% for Guangdong, substantially outperforming benchmarks such as GM(1,1) and GMC(1,N). Utilizing particle swarm optimization (PSO), the model independently determines optimal fractional orders (r₁ for the target series and r₂ for each driver series), capturing heterogeneous data features and multivariable couplings intrinsic to each regional farming system.

The FGMC(1,N,2r) model projects a continued increase in abalone production across Fujian, Shandong, and Guangdong from 2024 to 2028. This upward trend is consistent with historical patterns and is likely associated with ongoing improvements in aquaculture technology, sustained investment in the sector, and evolving market strategies, as reflected in the high relational degrees of key variables (e.g., technology funding in Fujian and Shandong, import volume in Guangdong) [33,48].

For instance, the strong association between production and technology-extension funding (GRA = 0.9156 in Fujian) aligns with documented provincial efforts in breeding disease-resistant varieties and vaccine development, which are known to enhance production stability [34,49]. This finding corroborates the work of Bostock et al. (2016) [33], who emphasized that technological investment is a key determinant of productivity and sustainability in aquaculture sectors globally. Similarly, the prominence of import volume in Guangdong (GRA = 0.9312) underscores the region’s reliance on stable seed supply chains, a factor critical to its production system. This reliance on external seed sources has been identified as a potential risk in other aquaculture studies [5], highlighting a common challenge across the industry. In Fujian, aquaculture technology-extension funding (X₁) was identified as the factor with the strongest statistical association with production output growth [50]. This finding aligns closely with the provincial emphasis on breeding the disease-resistant seed “Dongyou No. 1”, developing vaccines, and enacting other measures that effectively stabilize production [34]. Similarly, Shandong exhibits heavy dependence on technology funding, reflecting the characteristics of its industrialized recirculating aquaculture system and corroborated by the model’s in-sample MAPE of only 0.72%. Guangdong’s production, meanwhile, demonstrated the strongest association with import volume (X₅), highlighting the importance of seed supply chains for its aquaculture sector, a factor whose significance is echoed in studies of aquaculture supply chain risks [5].

These divergent drivers necessitate differentiated policies: Fujian and Shandong should sustain or increase funding for aquaculture technology-extension, whereas Guangdong should prioritize expanding offshore farming capacity. To that end, Fujian intends to raise the annual subsidy for national broodstock and elite-seed farms to one million RMB by 2025, enhance subsidies for preliminary-processing facilities, and advance the cross-border supply-chain alliance under the Regional Comprehensive Economic Partnership to reduce import dependence [50]. Shandong will deepen its industrialized model through strengthened support for modern marine ranching and wider adoption of recirculating aquaculture technologies [51]. Guangdong will pursue synergistic measures including large-scale deployment of smart offshore cages, targeted financial instruments, and enzyme-based technologies enhancing product value. These measures may impact long-term productivity and system sustainability [52,53].

It is noteworthy that the prediction accuracy for Shandong Province was lower than that for Fujian and Guangdong, though still superior to all benchmark models. This is likely attributable to the inherent higher volatility and instability in Shandong’s historical production data (as seen in Figure 3), particularly the notable fluctuations between 2017 and 2020. These fluctuations may be driven by factors not fully captured by the selected variables, such as more pronounced impacts of regional typhoon events [36] or significant adjustments in local aquaculture policies during that period, a challenge noted in other studies of Northern Chinese mariculture [47]. Nevertheless, the FGMC(1,N,2r) model effectively captured the overall growth trend, demonstrating a resilience in handling volatile data that aligns with the advantages of flexible grey models reported in other domains dealing with non-linear and unstable series [48].

4.2. Sustainability Implications and Environmental Considerations

The predictions also reveal high volatility in disaster losses (X₇) across all three provinces, exposing systemic vulnerability to climatic events [36]. Typhoon-induced economic losses in Fujian during 2016–2018 and in Shandong in 2017 coincided with output declines, underscoring the urgent need for resilient infrastructure and comprehensive insurance mechanisms [37,38]. In addition, rising import dependence (X₆) in Fujian and Guangdong points to emerging supply-chain risks and calls for diversification of seed sources.

FGMC(1,N,2r) model achieves a maximum out-of-sample MAPE of only 1.26% across the three provinces and demonstrates remarkable robustness in small-sample prediction, providing a replicable analytical framework for other data-scarce aquaculture domains. Furthermore, integrating production predictions with environmental impact assessment tools, such as Life Cycle Assessment (LCA) [54], could offer a more comprehensive sustainability evaluation for the abalone industry, guiding truly low-carbon and eco-friendly development.

5. Conclusions

Pursuing China’s “Dual Carbon” goals has made developing a low-carbon marine economy a national priority [49], this study successfully adapts the FGMC(1,N,2r) framework, originally developed for the energy sector, to predict the abalone production in Fujian, Shandong, and Guangdong. The results demonstrate its superior performance over traditional models, including GM(1,1), RCGM(1,1), GM(1,N), and GMC(1,N). Specifically, the FGMC(1,N,2r) model achieves a test set MAPE of just 0.51% in Fujian and 2.12% in Guangdong, reducing prediction errors by more than 50% compared to existing methods. Its independent fractional-order accumulation, tuned by particle swarm optimization (PSO) for r₁ and r₂, accommodates heterogeneous sequences and multivariable coupling. Beyond prediction accuracy, the model provides valuable insights into region-specific growth drivers.

Fujian Province: Given the projected output increase to over 274,800 tons by 2028 and the dominant influence of technology-extension funding (GRA = 0.9156), policy should focus on sustaining the high level of investment in R&D. Specifically, resources should be allocated to scale up the breeding programs for disease-resistant strains (e.g., ‘Dongyou No. 1’) and to commercialize vaccines, which are directly linked to the factors most associated with its production growth.

Shandong Province: To support its projected steady growth toward 46,600 tons by 2028, Shandong should leverage its strong association with technology funding (GRA = 0.8357) by channeling investments into industrialized recirculating aquaculture systems (RAS). This model aligns with its northern marine environment and can mitigate the risks highlighted by the variable’s historical volatility.

Guangdong Province: The model’s prediction of growth to over 10,300 tons by 2028 is most strongly tied to import volume (GRA = 0.9312). Therefore, the primary policy focus must be on ensuring seed supply chain security. This can be achieved by diversifying import sources and simultaneously incentivizing the construction of large-scale, local deep-sea smart cage platforms (e.g., the ‘Huibao No. 1’ model) to reduce external dependency.

Cross-Provincial Risk Mitigation: The high volatility in economic losses from fishery disasters (a consistently low-ranking factor in GRA) exposes a systemic vulnerability. All three provinces should prioritize the development of resilient infrastructure (e.g., typhoon-resistant cages) and implement comprehensive insurance mechanisms tailored to the specific climate risks identified in their historical data [55].

In summary, this study validates the FGMC(1,N,2r) model’s high accuracy and applicability in aquaculture production prediction, particularly in data-scarce regional farming systems. The framework is replicable and adaptable, providing a robust analytical tool for sustainable aquaculture management and policy formulation.

Despite its contributions, this study has several limitations. First, the model relies on annual macroeconomic and environmental data, lacking higher-frequency or more granular variables (e.g., water quality, real-time climate data) to capture short-term dynamics. Second, it does not account for acute unpredictable events such as disease outbreaks or sudden policy changes, which may cause significant deviations from projected trends. Third, the focus on three Chinese provinces limits the generalizability of the findings to other regions or species.

Future research could integrate multi-source, high-frequency data to enhance temporal resolution and predictive accuracy. Extending the model to incorporate extreme events through scenario analysis or dummy variables is also recommended. Furthermore, applying the FGMC(1,N,2r) framework to other aquaculture systems globally would help validate its universality and robustness.