A Hybrid Secondary-Decomposition and Intelligent- Optimization Framework for Agricultural Product Price Forecasting

Abstract

With the rapid development of big data and artificial intelligence, agricultural product price forecasting is evolving toward more intelligent and accurate approaches. However, such prices are affected by complex factors including natural conditions, market dynamics, and policy changes, resulting in strong nonlinearity and noise. To address the above challenges and achieve accurate agricultural price forecasts, this study proposes a hybrid framework that integrates a secondary decomposition algorithm with an improved Human Evolutionary Optimization Algorithm specifically tailored for the agricultural domain. The original price series is first decomposed using complete ensemble empirical mode decomposition with adaptive noise, and the high-frequency component is further processed using variational mode decomposition to enhance feature extraction. The improved optimization algorithm introduces Gaussian mutation and adaptive weights to optimize neural network parameters. Experiments on wheat, Chinese cabbage, and broiler chicken demonstrate that the proposed model significantly improves prediction accuracy, with determination coefficients increasing by 6.69, 8.87, and 6.43 percentage points, respectively. The results confirm the model’s effectiveness in reducing noise, capturing multi-scale features, and improving forecasting performance.

1. Introduction

Agriculture serves as a cornerstone of the national economy, with fluctuations in agricultural product prices exerting a profound impact on market stability and the assurance of basic livelihoods. The pricing of agricultural goods is shaped by a multitude of factors, such as natural events like climate disasters and pest outbreaks, shifts in market supply and demand, and policy measures including fiscal subsidies and trade regulation adjustments. These influences together lead to pronounced nonlinearity, high volatility, and non-stationary characteristics in price trends. The prices of key agricultural commodities—such as broiler chicken, wheat, and Chinese cabbage—are particularly vulnerable to external shocks due to lengthy production cycles and intricate supply chains. Accurately forecasting price movements is essential for informing government policy decisions, optimizing enterprise inventory management, and aiding farmers in mitigating risks.

With the swift progress of big data and artificial intelligence in recent years, neural network-driven forecasting techniques have emerged as a leading focus in contemporary research. Models such as Long Short-Term Memory (LSTM) [1] networks and Gated Recurrent Units (GRU) [2] employ gating mechanisms to capture long-term dependencies in sequential data and have demonstrated strong performance in agricultural price forecasting. However, although neural networks possess powerful nonlinear modeling capabilities, they still face significant challenges when addressing the high nonlinearity and structural complexity caused by the interaction of multiple influencing factors in agricultural price fluctuations. These challenges hinder model fitting and limit generalization. To address this, a secondary decomposition strategy is proposed, breaking the original time series into multiple subsequences with distinct fluctuation patterns. Because these subsequences differ in frequency, amplitude, and trend, using fixed or manual hyperparameters cannot accommodate all types, reducing prediction accuracy. To overcome this, an optimization algorithm is introduced to adaptively tune parameters, enhancing modeling effectiveness and forecast precision.

To address the aforementioned challenges, this study proposes a price forecasting model that integrates secondary decomposition with an Improved Human Evolutionary Optimization Algorithm (HEOA) [3]. The model first applies Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN). The CEEMDAN method was first introduced by Torres et al. [4] to the original price series to suppress noise and extract multi-scale intrinsic mode functions (IMFs). Subsequently, it conducts a secondary decomposition of the high-frequency IMF0 component using Variational Mode Decomposition (VMD) [5], thereby enhancing the ability to capture local mutation features. In the second stage, the study develops an improved version of the Human Evolutionary Optimization Algorithm by introducing a Gaussian mutation operator and an adaptive weighting mechanism. This design dynamically balances global exploration and local exploitation, optimizes the hyperparameter configuration of the neural network, and mitigates the issue of premature convergence commonly observed in traditional heuristic algorithms. To validate the model, the study selects price data from three representative agricultural products—wheat, Chinese cabbage, and broiler chicken—and evaluates the model’s effectiveness from multiple dimensions.

The main contributions of this study are as follows:

• First, a CEEMDAN–VMD hierarchical decomposition is introduced. CEEMDAN reduces global non-stationarity, and VMD further refines high-frequency components. This approach provides a more detailed representation of price dynamics than conventional single-level methods.
• Second, an IHEOA is developed, featuring Gaussian mutation and adaptive weighting. IHEOA enhances search diversity and convergence stability in GRU hyperparameter tuning, reducing premature convergence. Its effectiveness is validated through ablation studies and significance tests, confirming that performance gains stem from the optimizer’s design.
• Third, the framework is validated on multiple agricultural products and benchmarked against various machine learning and deep learning models. Results show that integrating hierarchical decomposition with adaptive hyperparameter optimization consistently improves forecasting accuracy.

2. Materials and Methods

2.1. Traditional Agricultural Product Price Prediction

Agricultural product price forecasting has become a key focus within agricultural economics research and has attracted widespread attention. However, due to the combined influence of multiple factors—such as natural conditions, market supply and demand, and policy adjustments—agricultural prices exhibit strong volatility and high forecasting complexity. Traditional autoregressive (AR) [6] models assume a linear relationship between current values and several past observations, which limits their applicability to data with low volatility and smooth trends. Moving average (MA) [7] models, by contrast, use weighted averages of historical forecast errors to smooth fluctuations, making them more suitable for data affected by short-term noise. The ARMA [8] model combines the advantages of AR and MA and performs well on stationary time series. Nevertheless, its forecasting accuracy often declines when confronted with the common trend variations observed in agricultural price data. To address this issue, ARIMA introduces differencing operations to handle non-stationarity and improve its fitting ability for real-world data. However, the model still struggles to capture seasonal characteristics and multi-period fluctuations. In response, the SARIMA [9] model extends ARIMA by incorporating seasonal differencing, thereby effectively compensating for ARIMA’s limitations when dealing with data exhibiting seasonal patterns.

2.2. Signal Decomposition and Intelligent Optimization-Driven Agricultural Price Forecasting

Although traditional methods for agricultural price forecasting possess a certain level of predictive capability, they remain inadequate for capturing the underlying trends in data characterized by pronounced nonlinearity, seasonal fluctuations, and multiple influencing factors. These limitations hinder their ability to produce accurate forecasts for agricultural prices. To address the challenges of insufficient trend extraction and limited prediction accuracy arising from the strongly nonlinear dynamics and complex factor interactions in agricultural price data, time series forecasting models that integrate signal decomposition algorithms have emerged as a promising solution.

Jayaparvathy et al. [10] combined CEEMDAN and LSTM networks to forecast soil moisture content at varying depths in Coimbatore, India. By decomposing complex nonlinear time series, their approach achieved a maximum improvement of 11.99% in the $R^2$ metric. Li et al. [11] investigated crude oil price forecasting based on VMD, employing a multi-decomposition framework to identify both global and local patterns across multiple scales in the data. This method preserved heterogeneous information to the greatest extent possible, thereby enhancing the predictive accuracy of the neural network model. Rezaei et al. [12] utilized Complete Ensemble Empirical Mode Decomposition (CEEMD) [13] in stock price prediction. By reconstructing denoised time series through decomposition, the approach improved forecasting accuracy and yielded lower error rates compared to the traditional Empirical Mode Decomposition (EMD) [14]. Özger et al. [15] assessed the impact of time series decomposition using EMD and wavelet-based methods based on the Wasserstein Distance (WD) [16] on drought prediction. Their wavelet analysis, leveraging WD’s time localization properties, identified transient features in both high- and low-frequency components. In a study conducted in Antalya and Adana, located in the southern Mediterranean region of Turkey, the WD-based model demonstrated higher accuracy and outperformed the EMD approach.

After effectively extracting multi-scale features from time series through signal decomposition algorithms, enhancing the predictive performance of neural networks becomes a key challenge. Metaheuristic algorithms provide an effective solution by intelligently optimizing network parameters. Kumar et al. [17] proposed an improved sailfish optimizer for crop yield prediction, which maintains a balance between exploration and exploitation phases and reduces the risk of local optima. They applied this algorithm to optimize the parameters of LSTM-GRU networks, including activation functions, number of epochs, optimizers, number of hidden neurons, and batch size, thereby improving both accuracy and precision. Given the strong correlation between rice growth and soil moisture balance, Majumdar et al. [18] developed a soil moisture prediction model that integrates Support Vector Machines (SVM) [19] with a Grey Wolf Optimizer (GWO) [20] enhanced by Chebyshev chaotic mapping and opposition-based learning. By leveraging GWO’s social hierarchy and hunting strategies, the model achieved higher predictive accuracy and faster global convergence. Samantaray and Sahoo [21] combined Particle Swarm Optimization (PSO) [22] and GWO to construct a hybrid PSO-GWO algorithm. Although PSO often produces good results during optimization, it tends to become trapped in local minima due to its convergence behavior. PSO alone may guide solutions toward arbitrary points that do not escape local optima, which can significantly deviate from the global minimum. The integration of GWO’s scanning ability effectively reduces the risk of stagnation at shallow local minima and enhances global search capability.

Although metaheuristic algorithms have shown significant advantages in optimizing neural network hyperparameters, neural networks themselves—due to their strong nonlinear modeling capabilities—have become one of the core methods for agricultural product price forecasting. Recurrent Neural Networks (RNNs) [23], in particular, enable time-aligned modeling of sequential data. A forecasting system based on RNNs for agricultural price time series has achieved approximately 27% lower mean squared error (MSE) compared to traditional ARIMA models. However, RNNs still suffer from issues such as gradient vanishing and explosion during training. Therefore, Ray [24] introduced the LSTM model to forecast legume price indices and proposed an ARIMA-LSTM hybrid model based on random forest techniques. This model extracted and modeled both linear trends and nonlinear patterns, achieving a 29% improvement in the Mean Absolute Scaled Error (MASE) compared to traditional methods.

With further advancements in technology, the GRU is a simplified variant of the LSTM that reduces model complexity and computational cost by employing fewer gating mechanisms. Montano [25] proposed a Holt-Winters-GRU model, in which the GRU component was employed to capture nonlinear trends in the time series. The model achieved a root mean square error (RMSE) of 27.7532 on the test set, with a gap of only 0.0695 from the training error, indicating strong generalization performance. Kaur [26] employed Artificial Neural Network (ANN) [27] techniques to model and forecast energy consumption in wheat production in India. The experimental results demonstrated high predictive accuracy, with the ANN model achieving a coefficient of determination ($R^2$) of 0.99 during training and 0.973 during validation. In comparison to the standalone ANN model, Shastry [28] proposed a hybrid forecasting model (H-ANN) that integrates ANN with modified Particle Swarm Optimization (m-PSO), incorporating weighted Principal Component Analysis (w-PCA) for feature extraction. This hybrid model improved the RMSE by 30% over the conventional ANN, demonstrating superior performance. Liu et al. [29] proposed an integrated model of bee colony optimization [30], Multivariate Empirical Mode Decomposition [31] and eXtreme Gradient Boosting [32] to improve the prediction accuracy of stock market time series. In industrial applications, Ma et al. [33] designed a dual-channel attention mechanism that combines local convolutional features and global temporal dependencies to optimize the dynamic weight allocation of key features, thereby improving the detection accuracy of pipelines under complex working conditions.

In recent years, the combination of signal decomposition techniques and machine learning algorithms has become a mainstream strategy for improving time series forecasting performance.Integrating decomposition with metaheuristic optimization algorithms allows the automatic fine-tuning of model hyperparameters, thereby enhancing convergence efficiency and generalization performance. Hou et al. [34] first adaptively decomposed the original water-level series by CEEMDAN; the components that still exhibited strong fluctuations were further split into smoother sub-series via VMD. These sub-series were then fed as multi-channel inputs into a BiLSTM, and the final prediction was reconstructed from the network outputs.Targeting industrial flexible loads, Li et al. [35] proposed a “CEEMDAN-VMD-GRU” framework to evolve and simulate load profiles, whereby the two-stage decomposition preserves the nonlinear, multi-scale nature of flexible demand and the GRU captures its temporal dynamics. However, this pipeline overlooks the fact that the GRU hyper-parameter space is high-dimensional, non-convex and computationally expensive to search; manual or grid tuning is thus inefficient and prone to local optima. To overcome these limitations, the present study incorporates the HEOA to automatically optimize the GRU architecture and training hyper-parameters, ensuring both convergence efficiency and global optimality while retaining the original CEEMDAN-VMD decomposition benefits.

2.3. Proposed Models

This study proposes a time series forecasting model that integrates CEEMDAN-VMD decomposition, GRU neural network modeling, and optimization via the Improved Human Evolutionary Optimization Algorithm (IHEOA), and the specific structure diagram is shown in Figure 1. First, the CEEMDAN method is applied to perform an initial decomposition of the original price series, extracting components at different frequencies to reduce data complexity. Then, VMD is employed for secondary decomposition of the intrinsic mode functions (IMFs) generated by CEEMDAN, enabling further separation of non-stationary features embedded in the data and facilitating finer-grained temporal feature extraction. Each decomposed sub-series is then individually modeled using GRU to capture its temporal dependencies. To enhance the forecasting performance of the GRU network, IHEOA is introduced to optimize key model parameters, improving convergence speed while maintaining global search capability. This approach effectively combines the precision of signal decomposition, the temporal modeling strength of deep learning, and the efficiency of intelligent optimization, offering a novel solution for forecasting complex nonlinear time series.

2.3.1. Secondary Decomposition

(1)

CEEMDAN

To address the challenges posed by nonlinearity and non-stationarity in the data, we first apply the CEEMDAN to preprocess the original series. CEEMDAN is an enhanced version of the EMD, incorporating the Gaussian noise addition strategy and ensemble averaging mechanism from the Ensemble Empirical Mode Decomposition (EEMD) [36]. Compared with previous decomposition methods, CEEMDAN yields lower residual noise, effectively reduces reconstruction error, and improves decomposition efficiency.

Through CEEMDAN, the original price series is decomposed into a finite set of IMFs and a residual component, capturing oscillatory patterns at different frequency scales. The detailed mathematical derivation follows the standard CEEMDAN formulation and is therefore omitted for brevity. The final decomposition can be expressed as:

$$h\left(t\right)=\sum _{k=1}^K{\mathrm{IMF}}_k\left(t\right)+R\left(t\right)$$

(1)

where K denotes the total number of intrinsic mode functions, and $R\left(t\right)$ represents the residual component.

(2)

VMD

This study reveals that the preliminary decomposition of raw data using the CEEMDAN method generates high-frequency oscillatory components with insufficient prediction accuracy, which may fail to comprehensively characterize the multidimensional characteristics and latent patterns in the time-series data. To address this limitation, a secondary decomposition strategy is implemented to perform in-depth processing of the initial decomposition results, significantly enhancing the granularity of feature resolution and effectively overcoming the inherent drawbacks of conventional single-level decomposition approaches. By synergistically integrating multiple signal decomposition techniques, the proposed system demonstrates improved capability in accurately identifying core feature elements from composite signals. This innovative solution exhibits particular efficacy in analyzing time-series data with pronounced nonlinear and time-varying characteristics. The VMD algorithm separates a non-stationary signal into several IMFs. If we denote the signal as $f\left(t\right)$ and recognize it is broken down into k IMF components, the underlying optimization principle governing VMD is expressed in Equation (8):

$$\underset{\left\{u_k\right\},\left\{w_k\right\}}{min}\left\{\sum _k{\left|{\partial }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k\left(t\right)\right]e^{-j{\omega }_{k}t}\right|}_2^2\right\}\mathrm{such}\mathrm{that}\sum _k{u}_k\left(t\right)=f\left(t\right)$$

(2)

In the equation, ${\partial }_t$ denotes the partial derivative with respect to t, and $\delta \left(t\right)$ refers to the Dirac delta function. The terms $u_k\left(t\right)$ represent the individual IMF components, while ${\omega }_k\left(t\right)$ corresponds to the central frequency of each IMF. The symbol j is the imaginary unit. To obtain the optimal solution of the variational formulation in Equation (8), a penalty factor $\alpha$ and a Lagrange multiplier $\lambda \left(t\right)$ are introduced. The resulting augmented expressions are provided in Equation (9):

$$\begin{array}{cc}\hfill L=\left(\left\{u_k\right\},\left\{{\omega }_k\right\},\lambda \right)=& \alpha \sum _k{\left|{\partial }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast u_k\left(t\right)\right]{\mathrm{e}}^{-j{\omega }_{k}t}\right|}_2^2+{\left|f\left(t\right)-\sum _k{u}_k\left(t\right)\right|}_2^2\hfill \\ \hfill & +〈\lambda \left(t\right),f\left(t\right)-\sum _k{u}_k\left(t\right)〉\hfill \end{array}$$

(3)

The VMD optimization problem is addressed by employing the Alternating Direction Method of Multipliers (ADMM) [37]. In each iteration, we update $u_k^{n+1}\left(\omega \right)$, ${\omega }_k^{n+1}$ and ${\lambda }^{n+1}$ to gradually approach the optimal solution. The corresponding iteration formulas are presented as follows, with n denoting the current iteration number.

$$u_k^{n+1}\left(\omega \right)={\displaystyle \frac{f\left(\omega \right)-{\sum }_{i<k}{u}_i\left(\omega \right)+\frac{\lambda \left(\omega \right)}{2}}{1+2\alpha {\left(\omega -{\omega }_k\right)}^2}}$$

(4)

$${\omega }_k^{n+1}={\displaystyle \frac{{\int }_0^{+\infty }\omega {\left|u_k^{n+1}\left(\omega \right)\right|}^2\mathrm{d}\omega }{{\int }_0^{+\infty }{\left|u_k^{n+1}\left(\omega \right)\right|}^2\mathrm{d}\omega }}$$

(5)

$${\lambda }^{n+1}\left(\omega \right)={\lambda }^n\left(\omega \right)+\tau \left(f\left(\omega \right)-\sum _k{u}_k^{n+1}\left(\omega \right)\right)$$

(6)

In the equation, $u_k^{n+1}\left(\omega \right)$, ${\omega }_k^{n+1}\left(\omega \right)$, ${\lambda }^{n+1}\left(\omega \right)$ represent the modal component function, center frequency, and Lagrange multiplier at iteration n + 1, respectively. Here, $\tau$ indicates the bandwidth constraint, while $f\left(\omega \right)$ and $\lambda \left(\omega \right)$ denote the spectral density functions corresponding to $f\left(t\right)$ and $\lambda \left(t\right)$, respectively.

The iterative procedure is stopped when the following convergence conditions are satisfied.

$$\sum _k{\displaystyle \frac{{∥u_k^{n+1}-u_k^n∥}_2^2}{{∥u_k^n∥}_2^2}}<\epsilon$$

(7)

Here, $\epsilon$ denotes the predefined convergence threshold.

The main parameters of the CEEMDAN and VMD algorithm used in this study are summarized in

Table 1. Parameter settings of the CEEMDAN and VMD algorithms.

Algorithm	Parameter	Implication	Value
CEEMDAN	Nstd	Noise Standard Deviation	0.2
	NR	Number of Iterations	50
	MaxIter	Maximum Sifting Iterations	1000
	SNRFlag	Noise Mode	1
VMD	$\alpha$	Bandwidth Parameter	200
	$\tau$	Noise Tolerance	0
	K	Number of Modes	3
	DC	Direct Current Component	0
	init	Center Frequency Initialization	1
	tol	Convergence Tolerance	$1.0\times {10}^{-7}$

.

2.3.2. GRU Networks

The GRU adopted in this study demonstrates remarkable advantages in time-series forecasting applications. As an enhanced variant of recurrent neural networks, this architecture innovatively incorporates dual gating mechanisms (update gate and reset gate), effectively addressing the gradient anomaly issues inherent in conventional RNNs during temporal information propagation. Compared with LSTM networks, GRU exhibits a more streamlined topological design characterized by: (1) approximately 33% reduction in total parameters, significantly decreasing model complexity; (2) 20–40% improvement in computational efficiency, facilitating practical deployment; and (3) over 25% reduction in training time while maintaining comparable prediction accuracy. Figure 2 illustrates the detailed network architecture and information flow mechanisms of the proposed model:

The formulas are:

$$z_t=\sigma (W_z{h}_{t-1}+U_z{x}_t+b_z)$$

(8)

$$r_t=\sigma (W_r{h}_{t-1}+U_r{x}_t+b_r)$$

(9)

$${\tilde{h}}_t=tanh\left(W_{\mathrm{h}}{x}_t+U_{\mathrm{h}}\left(r_t\otimes h_{t-1}\right)+b_{\mathrm{h}}\right)$$

(10)

$$h_t=(1-z_t)\otimes h_{t-1}+z_t\otimes {\tilde{h}}_t$$

(11)

where $W_z$, $W_r$, $W_h$ and $U_z$, $U_r$, $U_h$ are the corresponding weight matrices, while $b_z$, $b_r$, and $b_h$ denote the associated bias vectors. The reset gate $r_t$ controls the extent to which the previous hidden state $h_{t-1}$ is combined with the current input $x_t$. The update gate $z_t$ determines how much information from $h_{t-1}$ is preserved. In addition, $b_t$ represents the candidate hidden state computed from $h_{t-1}$ and $x_t$. ⊗ denotes element-wise multiplication (Hadamard product).

Table 2. GRU1D Model Framework.

Layer	Parameter
Input	24
GRU	64
Dense	64
Dropout	0.2
Output	1

summarizes the structure of the GRU1D model. The network begins with an input layer comprising 24 neurons. This is followed by a GRU layer with 64 units to capture temporal dependencies. Next, a fully connected Dense layer with 64 neurons is employed to further extract features. A Dropout layer is then introduced with a rate of 0.2 to mitigate overfitting. Finally, the model ends with an output layer containing a single neuron for prediction.

Table 3. GRU2D Model Framework.

Layer	Parameter
Input	24
GRU	64
GRU	64
Dense	64
Dropout	0.2
Output	1

illustrates the configuration of the GRU2D model. The network starts with an input layer composed of 24 neurons. Subsequently, two consecutive GRU layers are employed, each containing 64 units to learn sequential features. This is followed by a Dense layer with 64 neurons for further representation learning. A Dropout layer with a rate of 0.2 is then introduced to reduce the risk of overfitting. Finally, the architecture concludes with an output layer consisting of a single neuron.

2.3.3. Improved Human Evolutionary Optimization Algorithm

(1)

Human Evolutionary Optimization Algorithm

The Human Evolution Optimization Algorithm draws inspiration from the adaptive strategies developed by humans in complex environments. By introducing chaotic universe theory for population initialization, the algorithm enhances diversity and global search capabilities. HEOA consist of two main stages: Human Exploration Stage, simulating early human trial-and-error and adaptation through experience; Societal Development Stage, where individuals are classified into four roles—leaders, who guide the search based on current best solutions; explorers, who expand the search space; followers, who refine solutions locally; and eliminated individuals, who are removed due to poor performance. To boost convergence and maintain diversity, new individuals are introduced in promising regions, and distinct update strategies are applied to each role. The corresponding mathematical representation of the algorithm can be expressed as follows:

<1> Population Initialization

HEOA models the chaotic stage of early human evolution by employing a logistic chaotic map to initialize the population. The corresponding mathematical formulation is given in Equation (18).

$$x_i=\alpha ·x_{i-1}·(1-x_{i-1}),0\le x_0\le 1,i=1,2,\dots ,N,\alpha =4$$

(12)

Here, $x_i$ denotes the value at the ith iteration, whereas $x_{i-1}$ corresponds to the value obtained in the preceding iteration. The mapping function that projects the chaotic variable $x_i$ onto the search domain is defined in Equation (19) as follows:

$$X_i^0=lb+(ub-lb)·x_i$$

(13)

<2> Human Exploration Phase

During the experiments, the exploration stage is set to one quarter of the total maximum iterations. In the human development phase, when individuals encounter unfamiliar regions with insufficient prior information, the population generally follows a collective search strategy. The corresponding mathematical expression is given as follows:

$$\begin{array}{cc}\hfill X_i^{t+1}& =\beta ·\left(1-{\displaystyle \frac{t}{Ma{x}_{iter}}}\right)·(X_i^t-X_{best})·Levy\left(dim\right)\hfill \\ \hfill & +X_{best}·\left(1-{\displaystyle \frac{t}{Ma{x}_{iter}}}\right)\hfill \\ \hfill & +(X_{mean}^t-X_{best})·floor\left({\displaystyle \frac{rand}{f_{jump}}}\right)·f_{jump}\hfill \end{array}$$

(14)

$$X_{mean}^t={\displaystyle \frac{1}{N}}\sum _{k=1}^N{X}_k^t$$

(15)

$$\beta =0.2\left(1-{\displaystyle \frac{t}{Ma{x}_{iter}}}\right)·(X_i^t-X_{mean}^t)$$

(16)

In this formulation, $\beta$ represents the adaptive function, while t denotes the current iteration and $Ma{x}_{iter}$ is the predefined maximum number of iterations. The parameter $dim$ indicates the problem dimensionality. $X_i^t$ refers to the position of the ith individual at iteration t, whereas $X_i^{t+1}$ denotes its updated position in the next iteration. $X_{best}$ corresponds to the best solution obtained so far, and $X_{mean}^t$ stands for the mean position of the population at the current iteration. The operator $floor(·)$ denotes the floor function, and $Levy$ indicates a random variable following the Lévy distribution.

<3> Human Development Phase

During the human development stage, HEOA divides the population into four different social roles: leaders, explorers, followers, and losers. Each group adopts a specific search mechanism to enhance the ability to locate the global optimum.

Leaders: Leaders are characterized by richer experience and greater knowledge, enabling them to recognize promising regions in the search space. As a result, they are usually located near optimal areas. In the experiments, the top 40% of individuals are selected as leaders. Their search strategy can be formulated as follows:

$$X_i^{t+1}=\left\{\begin{array}{cc}\omega ·X_i^t·exp\left({\displaystyle \frac{-t}{rand·Ma{x}_{iter}}}\right),\hfill & \mathrm{if}R<A\hfill \\ \omega ·X_i^t+R_n·\mathrm{ones}(1,\dim ),\hfill & \mathrm{if}R\ge A\hfill \end{array}\right.$$

(17)

$$\omega =0.2cos\left({\displaystyle \frac{\pi }{2}}\left(1-{\displaystyle \frac{t}{Ma{x}_{iter}}}\right)\right)$$

(18)

where $R_n$ is a random variable following a normal distribution. The function $\mathrm{ones}(1,\dim )$ produces a $1\times dim$ vector whose entries are all equal to 1. The parameter R denotes a uniformly distributed random number within the interval $[0,1]$. The term A refers to the contextual evaluation factor, which is set to $A=0.6$. Moreover, the knowledge acquisition difficulty coefficient $\omega$ gradually decreases as the development stage advances.

Explorers: Explorers aim to identify promising solutions by investigating unexplored areas of the search space. In the experiments, individuals whose fitness ranks between the top 40% and 80% are assigned the role of explorers. Their corresponding search strategy is described by the following mathematical formulation.

$$X_i^{t+1}=X_i^t+\omega ·Rd·(X_{best}^t-X_i^t)$$

(19)

where $X_{best}^t$ denotes the location of the best-performing individual in the population at iteration t, and $Rd$ is a randomly selected integer within the interval $[1,\dim ]$.

Failures: Failures correspond to individuals that are unable to effectively adapt during the evolutionary process. In the experiments, the lowest 10% of the population are classified as failures. To maintain population diversity, these individuals are replaced through a reproduction mechanism that relocates them into more promising regions of the search space. The reproduction operation is formulated as follows:

$$X_i^{t+1}=X_{best}+(X_{best}-X_i^t)·R_n$$

(20)

(2)

Improved Human Evolutionary Optimization Algorithm

The original HEOA exhibits limited effectiveness when applied to multimodal optimization problems and uncertain scenarios, mainly because of its weak exploration diversity. To address this issue, an adaptive weighting framework combined with Gaussian mutation is incorporated into the algorithm. This hybrid mechanism enhances global search capability and significantly alleviates premature convergence in locally optimal regions.

<1> Introducing Gaussian Mutation to Improve Optimization Accuracy

The precision of local optimization is significantly improved when Gaussian mutation is employed to investigate narrow neighborhoods around candidate solutions.

$$Gauss={\displaystyle \frac{1}{\sqrt{2\pi \sigma }}}{e}^{-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}}$$

(21)

$$x_{new}=x(1+Gauss\left(x\right))$$

(22)

<2> Introduce Adaptive Weights to Enhance Global Search Capability

The algorithm updates the weights according to the fitness gap between each individual and the corresponding best and worst solutions. Adaptive weighting functions as a flexible mechanism widely used in different optimization and learning tasks. Its fundamental principle lies in dynamically modifying weights throughout the optimization process in response to performance variations or specific criteria. Such an approach enhances the achievement of global optimality and improves the algorithm’s adaptability. With this strategy, HEOA is able to generate more diverse candidate solutions across the entire search space at the end of each iteration, thereby reducing population diversity loss caused by solution aggregation and reinforcing the global exploration ability of the algorithm.

$$w={\displaystyle \frac{fitnes{s}_{best}-fitnes{s}_i}{fitnes{s}_{best}-fitnes{s}_{worst}}}$$

(23)

The detailed pseudocode of the this part is presented in Algorithm 1 as follows:

Algorithm 1 Improved Human Evolutionary Optimization Algorithm

Input: Population size N, maximum iterations T, bounds $[lb,ub]$, objective function $f\left(x\right)$ Output: Global best solution $GBest$  1: Initialize population X using logistic chaotic mapping  2: Evaluate fitness $f_i=f\left(X_i\right)$ for all individuals  3: for $t=1$ to T do  4:     Compute adaptive weights: $$w_i={\displaystyle \frac{f_{best}-f_i}{f_{best}-f_{worst}}}$$  5:     for each individual $i=1,2,\dots ,N$ do  6:         Apply Gaussian mutation: $$X_i\leftarrow X_i\times (1+\mathcal{N}(0,{\sigma }^2))$$  7:         Update $X_i$ according to exploration or exploitation strategy  8:         Perform boundary control to ensure $X_i\in [lb,ub]$  9:     end for 10:     Evaluate new fitness $f_i=f\left(X_i\right)$ and update $GBest$ 11:     Record current best fitness in convergence curve 12: end for 13: return $GBest$

For clarity and readability, the main abbreviations used in this paper are summarized in

Table 4. List of abbreviations used in this study.

Abbreviation	Full Form
PD	Primary Decomposition
SD	Secondary Decomposition
WOA	Whale Optimization Algorithm
GRU1D	One-layer GRU
GRU2D	Two-layer GRU

.

2.4. Parameter Configuration of Representative Hybrid Models

The hyperparameter settings of the simpler baseline approaches (such as ANN and GRU) are provided in their respective experimental discussions, whereas this subsection concentrates on representative hybrid frameworks involving more sophisticated decomposition and optimization mechanisms.

To ensure the reproducibility and fairness of the comparative experiments, this subsection reports the parameter configurations of two representative hybrid forecasting models, namely the CEEMDAN–TDNN model [38] and the GA–VMD–LSTM [39] model. These models were selected because they integrate signal decomposition techniques with neural network-based predictors and involve relatively complex parameter structures.

Table 5. Parameter settings of representative hybrid forecasting models.

Model	Parameter	Setting
CEEMDAN–TDNN	Noise type	Gaussian white noise
	Noise variance	Unit variance (adaptive noise addition)
	Ensemble strategy	Complete ensemble decomposition
	Stopping criterion	Residual with at most two extrema
	TDNN architecture	Single hidden-layer TDNN
	Activation function	Logistic
	Training algorithm	Levenberg–Marquardt backpropagation
GA–VMD–LSTM	Number of modes K	2–25 (GA optimized)
	Bandwidth parameter $\alpha$	$\{10,{10}^2,{10}^3,{10}^4,{10}^5\}$
	Noise tolerance $\tau$	$\{0,1\}$
	Convergence tolerance $\epsilon$	$\{{10}^{-4},{10}^{-5},{10}^{-6},{10}^{-7}\}$
	GA encoding	Binary encoding
	Population size	100
	Number of generations	25
	Crossover probability	0.5
	Mutation probability	0.1
	Fitness function	Residual energy ratio

lists the main parameter settings of the CEEMDAN–TDNN and GA–VMD–LSTM models, enabling clear comparison and supporting the reproducibility of the experimental results.

3. Results

3.1. Datasets

To further verify the robustness of the proposed model for agricultural commodity price forecasting, extensive experiments are performed on three representative categories: grains, vegetables, and livestock. The dataset covers the period from 2011 to 2024 and includes over 10,000 records. For grains, the data spans from 21 May 2013, to 30 November 2024, with a total of 4211 records. For vegetables and livestock, the data ranges from 1 January 2011, to 30 November 2024, with 5082 records. These datasets contain average daily wholesale prices in the Chinese market. The target products include wheat, Chinese cabbage, and broiler chicken, while the influencing factors include pork, beef, eggs, mutton, the agricultural product price index, soybean meal, corn, wheat bran, and others. To improve data quality for subsequent analysis and modeling, the study implemented data preprocessing comprising handling missing values, data transformation, and normalization. And dataset is split into 70% for training and 30% for testing. The data are split chronologically into 70% for training and 30% for testing to reflect real-world forecasting scenarios, where future prices are predicted based on historical observations. This time-ordered split avoids information leakage and preserves the temporal structure of the series.

In this study, agricultural price forecasting is formulated as a multivariate time-series prediction problem. In addition to the target price series, several auxiliary market-related variables are included as exogenous inputs. These variables are intended to provide complementary information on market conditions and are not treated as causal drivers. For model construction, the exogenous variables are used jointly with the price data rather than being modeled separately. After the CEEMDAN–VMD decomposition of the target price series, the resulting IMFs and the exogenous variables are concatenated along the feature dimension to form the input matrix of the GRU model. All variables are normalized using the same scaling strategy prior to model training to ensure numerical stability and consistency.

3.2. Evaluation Metrics

MSE measures the differences between predicted values and actual observed values, helping to evaluate model accuracy. By calculating the average of the squared differences, MSE highlights larger errors, making it a valuable tool for assessing how well a model fits the data. The formula is as follows:

$$MSE={\displaystyle \frac{1}{N}}\sum _{t=1}^N{(y_t-{\widehat{y}}_t)}^2$$

(24)

MAE is a way to measure how well a model predicts results. It calculates the average difference between the model’s predictions and the actual results, focusing only on the absolute values. This means it looks at how far off each prediction is, regardless of whether it overestimated or underestimated. A lower MAE shows that the model is more accurate in its predictions. The formula is as follows:

$$MAE={\displaystyle \frac{1}{N}}\sum _{t=1}^N|y_t-{\widehat{y}}_t|$$

(25)

MAPE is a statistical metric that evaluates the accuracy of forecasting models. It measures the difference between predicted values and actual observed values, expressing this discrepancy as a percentage. This provides a clear assessment of prediction accuracy, helping analysts make informed decisions based on the reliability of their forecasts.

$$MAPE={\displaystyle \frac{100\%}{n}}\sum _{t=1}^n\left|{\displaystyle \frac{y_t-{\widehat{y}}_t}{y_t}}\right|$$

(26)

$R^2$ evaluates the goodness of fit of a model by measuring how well the predicted values match the actual values. Its value ranges between 0 and 1, with values closer to 1 indicating stronger predictive performance.

$$R^2={\displaystyle \frac{\sum {({\widehat{y}}_t-{\overline{y}}_t)}^2}{\sum {(y_t-{\overline{y}}_t)}^2}}$$

(27)

3.3. Experimental Results

3.3.1. Results of the Secondary Decomposition

(1)

CEEMDAN’s Results

By applying the CEEMDAN algorithm to the agricultural product price data series, the decomposition yielded modal components with different frequencies, as shown in Figure 3. These results demonstrate that the CEEMDAN technique decomposes agricultural price time series into a set of IMFs with different frequency characteristics. By separating the original signal into multiple IMFs, the underlying trends and periodic patterns within price variations can be more effectively revealed.

(2)

VMD’s Results

For the most intricate component, IMF0, produced by the initial CEEMDAN decomposition, a further decomposition is carried out using the VMD technique. This secondary decomposition acts as an additional refinement step, which contributes to improving the forecasting performance of high-frequency fluctuations. The corresponding decomposition outcomes for the three categories of agricultural products are illustrated in Figure 4.

(3)

Prediction Results of the Secondary Decomposition Model

To verify the effectiveness of the secondary decomposition algorithm in improving accuracy, this study compares the results of ANN, GRU1D, and GRU2D models. Figure 5 shows the prediction results of these three neural network models on the test sets of three agricultural product price series.

3.3.2. Results Based on Decomposition Reconstruction and IHEOA

In IHEOA, the search activity is primarily concentrated around the best or near-best solutions obtained in earlier iterations. Each individual adjusts its position by incorporating collective knowledge gathered from the current population. The baseline HEOA framework contains five distinct update strategies, consisting of one exploration scheme and four development-oriented mechanisms. These strategies are designed to coordinate global exploration with local exploitation, ensuring an efficient optimization process. Furthermore, IHEOA enhances the original algorithm by integrating Gaussian mutation along with an adaptive weighting approach, and the iterative search is terminated once the specified convergence criteria are met. The table below summarizes the optimized hyperparameter settings used for forecasting each decomposed subsequence with different metaheuristic optimization techniques. In this context, IMFn represents the subsequences extracted from the original series through CEEMDAN decomposition, while IMF0(SD) denotes the IMF0 component that is further refined using the VMD approach.

Table 6. Parameter selection of neural network model for wheat price prediction by IHEOA.

Series	Neuron Count(X0)	Neuron Count(X1)	Neuron Count(X2)	Dropout Rate(X3)	Neural Network Layers(X4)
IMF0	111	111	74	0	1
IMF0(SD)	114	124	126	0.2	1
IMF1	94	82	12	0.5	1
IMF2	124	104	8	0.4	1
IMF3	124	111	86	0.2	0
IMF4	99	107	73	0	1
IMF5	40	40	38	0.5	1

,

Table 7. Parameter selection of IHEOA’s neural network model for cabbage price prediction.

Series	Neuron Count(X0)	Neuron Count(X1)	Neuron Count(X2)	Dropout Rate(X3)	Neural Network Layers(X4)
IMF0	105	109	74	0.1	1
IMF0(SD)	124	111	86	0.2	0
IMF1	94	82	12	0.3	0
IMF2	124	104	8	0.4	1
IMF3	120	104	8	0.1	0
IMF4	111	111	74	0	1
IMF5	123	104	8	0.1	1
IMF6	117	108	81	0.5	1

and

Table 8. Parameter selection of IHEOA’s neural network model for broiler chicken price prediction.

Series	Neuron Count(X0)	Neuron Count(X1)	Neuron Count(X2)	Dropout Rate(X3)	Neural Network Layers(X4)
IMF0	13	98	104	0.3	0
IMF0(SD)	111	111	74	0	1
IMF1	114	124	126	0.2	1
IMF2	45	75	36	0.5	1
IMF3	48	80	116	0.5	1
IMF4	124	104	8	0.4	1
IMF5	111	111	74	0	1
IMF6	50	125	27	0.2	1

illustrate the optimized parameters of crops by IHEOA.

3.3.3. Model Prediction Results

To validate the effectiveness of the model, the WOA (Whale Optimization Algorithm) algorithm and ANN (Artificial Neural Network) model are now introduced for verification. The fitting curves of prediction results and the true values in the test set are shown in Figure 6, Figure 7 and Figure 8. The curves shown in orange, gray, yellow, blue, light green, pink, and dark green correspond to SD-IHEOA, PD-IHEOA, SD-HEOA, SD-WOA, SD-GRU2D, SD-GRU1D, and SD-ANN, respectively. Overall, the fitting curves indicate a high similarity between the predicted results and true values across all three categories of agricultural products, with consistent fluctuation patterns. From Figure 7, Figure 8 and Figure 9, the SD-IHEOA model performs best in capturing the overall trend and periodic fluctuations of agricultural product prices, showing the highest agreement with true values. By comparison, the SD-ANN approach exhibits more pronounced prediction errors, particularly in periods characterized by abrupt price changes. The SD-HEOA and SD-IHEOA models deliver moderate results, successfully tracking the overall long-term tendency of price movements, yet showing limitations in accurately reflecting short-term fluctuations. In general, the SD-IHEOA optimized framework achieves the best forecasting performance among all considered models, highlighting its stronger fitting capability for agricultural price prediction tasks.

The study systematically evaluated the applicability of seven prediction models—SD-IHEOA, PD-IHEOA, SD-HEOA, SD-WOA, SD-ANN, SD-GRU1D, and SD-GRU2D—across five major agricultural product price series. The models’ performances underwent comprehensive assessment using statistical metrics including $R^2$, MAE, MAPE, and MSE. The evaluation results are presented in Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19 and Table 20.

• Prediction Results for Wheat

Table 9. $R^2$ indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	58.72%	–	46.91%	48.05%	35.96%	44.80%	45.86%
IMF1	86.34%	86.34%	71.96%	68.73%	65.16%	67.76%	67.10%
IMF2	93.92%	93.92%	91.74%	92.99%	89.68%	89.79%	90.64%
IMF3	99.60%	99.60%	96.37%	95.62%	93.63%	93.93%	95.59%
IMF4	99.73%	99.73%	97.18%	96.77%	94.10%	95.85%	96.75%
IMF5	98.78%	98.78%	98.24%	98.03%	97.47%	97.27%	96.54%
Refactored results	98.62%	97.43%	95.96%	95.18%	89.62%	91.54%	93.69%

Table 9. $R^2$ indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	58.72%	–	46.91%	48.05%	35.96%	44.80%	45.86%
IMF1	86.34%	86.34%	71.96%	68.73%	65.16%	67.76%	67.10%
IMF2	93.92%	93.92%	91.74%	92.99%	89.68%	89.79%	90.64%
IMF3	99.60%	99.60%	96.37%	95.62%	93.63%	93.93%	95.59%
IMF4	99.73%	99.73%	97.18%	96.77%	94.10%	95.85%	96.75%
IMF5	98.78%	98.78%	98.24%	98.03%	97.47%	97.27%	96.54%
Refactored results	98.62%	97.43%	95.96%	95.18%	89.62%	91.54%	93.69%

Table 10. MAE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	19.5334	19.5334	–	–	–	–	–
IMF0(SD)	10.6028	–	9.5467	9.6087	10.2272	9.9980	9.9477
IMF1	4.6316	4.6316	5.0182	8.8323	9.1573	8.3432	8.3060
IMF2	12.6959	12.6959	14.8706	12.9002	20.1110	19.1955	19.1154
IMF3	4.9518	4.9518	13.0014	15.2234	14.9750	17.1651	16.1645
IMF4	9.8062	9.8062	27.4300	29.7472	39.3960	33.4206	31.1152
IMF5	5.3782	5.3782	6.2790	6.9638	7.6677	8.3387	9.3368
Refactored results	21.7252	27.3391	36.2480	38.5634	58.5630	51.1372	46.4079

Table 10. MAE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	19.5334	19.5334	–	–	–	–	–
IMF0(SD)	10.6028	–	9.5467	9.6087	10.2272	9.9980	9.9477
IMF1	4.6316	4.6316	5.0182	8.8323	9.1573	8.3432	8.3060
IMF2	12.6959	12.6959	14.8706	12.9002	20.1110	19.1955	19.1154
IMF3	4.9518	4.9518	13.0014	15.2234	14.9750	17.1651	16.1645
IMF4	9.8062	9.8062	27.4300	29.7472	39.3960	33.4206	31.1152
IMF5	5.3782	5.3782	6.2790	6.9638	7.6677	8.3387	9.3368
Refactored results	21.7252	27.3391	36.2480	38.5634	58.5630	51.1372	46.4079

Table 11. MAPE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	3.4457	3.4457	–	–	–	–	–
IMF0(SD)	15.6058	–	5.0591	11.5135	25.4509	9.9646	20.3458
IMF1	0.9337	0.9337	0.5807	1.9676	1.5215	4.0458	10.1391
IMF2	0.3433	0.3433	0.5784	0.3514	1.4144	0.9870	1.7404
IMF3	0.1411	0.1411	0.2119	0.7540	0.8738	0.9061	0.8386
IMF4	0.2848	0.2848	0.3139	0.6219	0.7744	22.5905	0.5519
IMF5	0.0019	0.0019	0.0022	0.0024	0.0027	0.0029	0.0033
Refactored results	0.0074	0.0094	0.0126	0.0137	0.0206	0.0177	0.0162

Table 11. MAPE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	3.4457	3.4457	–	–	–	–	–
IMF0(SD)	15.6058	–	5.0591	11.5135	25.4509	9.9646	20.3458
IMF1	0.9337	0.9337	0.5807	1.9676	1.5215	4.0458	10.1391
IMF2	0.3433	0.3433	0.5784	0.3514	1.4144	0.9870	1.7404
IMF3	0.1411	0.1411	0.2119	0.7540	0.8738	0.9061	0.8386
IMF4	0.2848	0.2848	0.3139	0.6219	0.7744	22.5905	0.5519
IMF5	0.0019	0.0019	0.0022	0.0024	0.0027	0.0029	0.0033
Refactored results	0.0074	0.0094	0.0126	0.0137	0.0206	0.0177	0.0162

Table 12. MSE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	1020.66	1020.66	–	–	–	–	–
IMF0(SD)	298.88	–	277.41	266.57	276.63	283.59	302.07
IMF1	38.62	38.62	74.15	130.26	138.48	118.52	120.06
IMF2	273.08	273.08	362.50	323.17	652.31	523.15	511.14
IMF3	31.27	31.27	215.66	284.10	419.92	365.23	331.51
IMF4	115.64	115.64	935.93	1062.81	1908.93	1344.39	1118.22
IMF5	29.75	29.75	41.73	54.89	84.15	69.75	87.31
Refactored results	801.28	1487.41	1875.88	2318.59	5030.76	3786.58	3156.13

Table 12. MSE indicators for individual models of wheat prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	1020.66	1020.66	–	–	–	–	–
IMF0(SD)	298.88	–	277.41	266.57	276.63	283.59	302.07
IMF1	38.62	38.62	74.15	130.26	138.48	118.52	120.06
IMF2	273.08	273.08	362.50	323.17	652.31	523.15	511.14
IMF3	31.27	31.27	215.66	284.10	419.92	365.23	331.51
IMF4	115.64	115.64	935.93	1062.81	1908.93	1344.39	1118.22
IMF5	29.75	29.75	41.73	54.89	84.15	69.75	87.31
Refactored results	801.28	1487.41	1875.88	2318.59	5030.76	3786.58	3156.13

Numerical comparisons show that in the reconstruction results, the proposed SD-IHEOA model achieves the best performance across all four metrics, with an $R^2$ of 98.62%, MAE of 21.73, MAPE of 0.01, and MSE of 801.28, confirming the model’s effectiveness. For component-level predictions, the SD-IHEOA model performs exceptionally well on multiple components (IMF1 to IMF5). Its $R^2$ values for IMF1 to IMF5 reach 86.34%, 93.92%, 99.60%, 99.73%, 98.73%, and 98.78%, significantly outperforming SD-HEOA, SD-WOA, SD-ANN, SD-GRU1D, and SD-GRU2D models. Additionally, the SD-HEOA model shows good performance in predicting the high-frequency component IMF0(SD), with an $R^2$ of 46.91%. Although lower than SD-IHEOA, it achieves MAE of 9.55, MAPE of 5.06, and MSE of 277.41, all better than other models. While SD-WOA’s metrics fall below those of the IHEOA and HEOA combined models, it still surpasses models without metaheuristic optimization. Comparing IMF0 and IMF0(SD), IMF0(SD) shows better $R^2$, MAE, and MSE, demonstrating the effectiveness of the secondary decomposition algorithm in predicting high-frequency components. For overall data reconstruction, the SD-IHEOA model also performs best, with an $R^2$ of 98.62%, MAE of 21.73, MAPE of 0.0074, and MSE of 801.28, all significantly lower than those of other models.

• Prediction Results for Cabbage

Table 13. $R^2$ indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	74.89%	–	73.65%	72.18%	62.33%	69.99%	71.57%
IMF1	97.57%	97.57%	93.78%	93.65%	87.38%	91.59%	91.46%
IMF2	95.42%	95.42%	94.45%	94.12%	89.86%	91.40%	92.40%
IMF3	99.50%	99.50%	97.87%	97.61%	94.69%	96.16%	96.98%
IMF4	99.98%	99.98%	97.07%	96.92%	92.91%	95.93%	94.90%
IMF5	99.92%	99.92%	98.88%	98.50%	96.47%	98.19%	98.33%
IMF6	99.17%	99.17%	98.36%	98.09%	96.59%	96.08%	97.49%
Refactored results	97.94%	96.03%	95.32%	94.52%	91.06%	90.87%	92.35%

Table 13. $R^2$ indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	74.89%	–	73.65%	72.18%	62.33%	69.99%	71.57%
IMF1	97.57%	97.57%	93.78%	93.65%	87.38%	91.59%	91.46%
IMF2	95.42%	95.42%	94.45%	94.12%	89.86%	91.40%	92.40%
IMF3	99.50%	99.50%	97.87%	97.61%	94.69%	96.16%	96.98%
IMF4	99.98%	99.98%	97.07%	96.92%	92.91%	95.93%	94.90%
IMF5	99.92%	99.92%	98.88%	98.50%	96.47%	98.19%	98.33%
IMF6	99.17%	99.17%	98.36%	98.09%	96.59%	96.08%	97.49%
Refactored results	97.94%	96.03%	95.32%	94.52%	91.06%	90.87%	92.35%

Table 14. MAE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0465	0.0465	–	–	–	–	–
IMF0(SD)	0.0239	–	0.0237	0.0223	0.0257	0.0238	0.0240
IMF1	0.0218	0.0218	0.0336	0.0346	0.0516	0.0433	0.0483
IMF2	0.0294	0.0294	0.0343	0.0348	0.0445	0.0397	0.0383
IMF3	0.0110	0.0110	0.0210	0.0243	0.0292	0.0291	0.0278
IMF4	0.0027	0.0027	0.0283	0.0324	0.0467	0.0369	0.0397
IMF5	0.0007	0.0007	0.0022	0.0027	0.0040	0.0030	0.0025
IMF6	0.0027	0.0027	0.0040	0.0040	0.0055	0.0062	0.0048
Refactored results	0.0440	0.0588	0.0620	0.0660	0.0848	0.0816	0.0841

Table 14. MAE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0465	0.0465	–	–	–	–	–
IMF0(SD)	0.0239	–	0.0237	0.0223	0.0257	0.0238	0.0240
IMF1	0.0218	0.0218	0.0336	0.0346	0.0516	0.0433	0.0483
IMF2	0.0294	0.0294	0.0343	0.0348	0.0445	0.0397	0.0383
IMF3	0.0110	0.0110	0.0210	0.0243	0.0292	0.0291	0.0278
IMF4	0.0027	0.0027	0.0283	0.0324	0.0467	0.0369	0.0397
IMF5	0.0007	0.0007	0.0022	0.0027	0.0040	0.0030	0.0025
IMF6	0.0027	0.0027	0.0040	0.0040	0.0055	0.0062	0.0048
Refactored results	0.0440	0.0588	0.0620	0.0660	0.0848	0.0816	0.0841

Table 15. MAPE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	8.3498	8.3498	–	–	–	–	–
IMF0(SD)	1.9644	–	1.4644	3.0263	4.3006	4.5634	3.7591
IMF1	0.7222	0.7222	1.0128	0.7105	1.1518	1.4980	3.0130
IMF2	0.3850	0.3850	0.5601	0.9469	0.8718	1.4484	0.6897
IMF3	0.1507	0.1507	0.3469	2.5336	0.6239	0.9264	0.6242
IMF4	0.0126	0.0126	0.1307	0.2698	0.3913	1.0185	0.3105
IMF5	0.0834	0.0834	0.1547	0.1558	0.5927	0.3421	0.0924
IMF6	0.0013	0.0013	0.0019	0.0019	0.0026	0.0030	0.0023
Refactored results	0.0216	0.0288	0.0300	0.0323	0.0409	0.0395	0.0412

Table 15. MAPE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	8.3498	8.3498	–	–	–	–	–
IMF0(SD)	1.9644	–	1.4644	3.0263	4.3006	4.5634	3.7591
IMF1	0.7222	0.7222	1.0128	0.7105	1.1518	1.4980	3.0130
IMF2	0.3850	0.3850	0.5601	0.9469	0.8718	1.4484	0.6897
IMF3	0.1507	0.1507	0.3469	2.5336	0.6239	0.9264	0.6242
IMF4	0.0126	0.0126	0.1307	0.2698	0.3913	1.0185	0.3105
IMF5	0.0834	0.0834	0.1547	0.1558	0.5927	0.3421	0.0924
IMF6	0.0013	0.0013	0.0019	0.0019	0.0026	0.0030	0.0023
Refactored results	0.0216	0.0288	0.0300	0.0323	0.0409	0.0395	0.0412

Table 16. MSE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0037	0.0037	–	–	–	–	–
IMF0(SD)	0.0009	–	0.0009	0.0008	0.0010	0.0009	0.0009
IMF1	0.0009	0.0009	0.0020	0.0024	0.0044	0.0036	0.0036
IMF2	0.0013	0.0013	0.0021	0.0020	0.0033	0.0029	0.0026
IMF3	0.0002	0.0002	0.0006	0.0007	0.0014	0.0011	0.0010
IMF4	0.00001	0.00001	0.0013	0.0014	0.0028	0.0017	0.0021
IMF5	0.000001	0.000001	0.00001	0.00001	0.00002	0.00001	0.00001
IMF6	0.00001	0.00001	0.00002	0.00002	0.00004	0.00004	0.00003
Refactored results	0.0031	0.0060	0.0066	0.0075	0.0127	0.0122	0.0109

Table 16. MSE indicators for individual models of cabbage prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0037	0.0037	–	–	–	–	–
IMF0(SD)	0.0009	–	0.0009	0.0008	0.0010	0.0009	0.0009
IMF1	0.0009	0.0009	0.0020	0.0024	0.0044	0.0036	0.0036
IMF2	0.0013	0.0013	0.0021	0.0020	0.0033	0.0029	0.0026
IMF3	0.0002	0.0002	0.0006	0.0007	0.0014	0.0011	0.0010
IMF4	0.00001	0.00001	0.0013	0.0014	0.0028	0.0017	0.0021
IMF5	0.000001	0.000001	0.00001	0.00001	0.00002	0.00001	0.00001
IMF6	0.00001	0.00001	0.00002	0.00002	0.00004	0.00004	0.00003
Refactored results	0.0031	0.0060	0.0066	0.0075	0.0127	0.0122	0.0109

The experimental results for Chinese cabbage show that the SD-IHEOA model achieves the highest $R^2$ values in most IMF components, especially reaching 99.98% and 99.92% in IMF4 and IMF5 respectively, significantly outperforming other models. In terms of MAE, the SD-IHEOA model performs best across IMF1 to IMF6, with values of 0.0218, 0.0294, 0.0110, 0.0027, 0.0007, and 0.0027, all markedly lower than those of competing models. MAPE and MSE metrics further confirm the superiority of the SD-IHEOA model, showing substantially lower errors than others. Comparing IMF0 and IMF0(SD), IMF0(SD) outperforms in all four metrics, indicating the effectiveness of the secondary decomposition algorithm in predicting high-frequency components. The four evaluation metrics for reconstruction results also demonstrate that the SD-IHEOA model delivers the highest prediction accuracy and stability, with values of 97.94%, 0.0440, 0.0216, and 0.0031, all surpassing other models. These findings highlight the advantage of the SD-IHEOA model in capturing the trends and periodic variations of mid- to low-frequency components when processing Chinese cabbage data.

• Prediction Results for Broiler Chicken

Table 17. $R^2$ indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	58.43%	–	55.18%	55.88%	39.47%	48.43%	51.38%
IMF1	97.24%	97.24%	96.63%	96.32%	90.96%	94.58%	95.17%
IMF2	99.97%	99.97%	98.23%	97.97%	93.90%	97.60%	97.01%
IMF3	99.99%	99.99%	98.73%	98.57%	94.61%	98.14%	97.72%
IMF4	99.96%	99.96%	99.92%	99.93%	98.87%	99.89%	99.08%
IMF5	93.59%	93.59%	90.96%	89.71%	72.78%	86.78%	88.70%
Refactored results	98.42%	93.92%	98.21%	97.76%	93.56%	95.84%	96.06%

Table 17. $R^2$ indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	\	\	–	–	–	–	–
IMF0(SD)	58.43%	–	55.18%	55.88%	39.47%	48.43%	51.38%
IMF1	97.24%	97.24%	96.63%	96.32%	90.96%	94.58%	95.17%
IMF2	99.97%	99.97%	98.23%	97.97%	93.90%	97.60%	97.01%
IMF3	99.99%	99.99%	98.73%	98.57%	94.61%	98.14%	97.72%
IMF4	99.96%	99.96%	99.92%	99.93%	98.87%	99.89%	99.08%
IMF5	93.59%	93.59%	90.96%	89.71%	72.78%	86.78%	88.70%
Refactored results	98.42%	93.92%	98.21%	97.76%	93.56%	95.84%	96.06%

Table 18. MAE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0844	0.0844	–	–	–	–	–
IMF0(SD)	0.0398	–	0.0388	0.0394	0.0340	0.0410	0.0347
IMF1	0.0196	0.0196	0.0211	0.0220	0.0348	0.0256	0.0242
IMF2	0.0025	0.0025	0.0179	0.0206	0.0380	0.0216	0.0240
IMF3	0.0026	0.0026	0.0278	0.0318	0.0644	0.0346	0.0402
IMF4	0.0109	0.0109	0.0138	0.0108	0.0498	0.0119	0.0456
IMF5	0.0363	0.0363	0.0523	0.0420	0.1047	0.0637	0.0578
Refactored results	0.0636	0.1146	0.0706	0.0777	0.1620	0.1151	0.1039

Table 18. MAE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0844	0.0844	–	–	–	–	–
IMF0(SD)	0.0398	–	0.0388	0.0394	0.0340	0.0410	0.0347
IMF1	0.0196	0.0196	0.0211	0.0220	0.0348	0.0256	0.0242
IMF2	0.0025	0.0025	0.0179	0.0206	0.0380	0.0216	0.0240
IMF3	0.0026	0.0026	0.0278	0.0318	0.0644	0.0346	0.0402
IMF4	0.0109	0.0109	0.0138	0.0108	0.0498	0.0119	0.0456
IMF5	0.0363	0.0363	0.0523	0.0420	0.1047	0.0637	0.0578
Refactored results	0.0636	0.1146	0.0706	0.0777	0.1620	0.1151	0.1039

Table 19. MAPE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	5.1324	5.1324	–	–	–	–	–
IMF0(SD)	1.7937	–	4.5604	4.6696	0.0748	2.8745	0.0758
IMF1	0.7616	0.7616	2.0537	0.8460	0.7438	1.6236	1.7099
IMF2	0.0810	0.0810	1.2755	0.6173	1.9414	0.3112	0.4157
IMF3	0.3752	0.3752	0.1411	0.5605	1.0493	0.5411	0.5735
IMF4	0.2136	0.2136	0.2812	0.3532	0.5206	0.0636	0.3980
IMF5	0.0020	0.0020	0.0030	0.0024	0.0059	0.0036	0.0032
Refactored results	0.0036	0.0065	0.0040	0.0044	0.0092	0.0066	0.0059

Table 19. MAPE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	5.1324	5.1324	–	–	–	–	–
IMF0(SD)	1.7937	–	4.5604	4.6696	0.0748	2.8745	0.0758
IMF1	0.7616	0.7616	2.0537	0.8460	0.7438	1.6236	1.7099
IMF2	0.0810	0.0810	1.2755	0.6173	1.9414	0.3112	0.4157
IMF3	0.3752	0.3752	0.1411	0.5605	1.0493	0.5411	0.5735
IMF4	0.2136	0.2136	0.2812	0.3532	0.5206	0.0636	0.3980
IMF5	0.0020	0.0020	0.0030	0.0024	0.0059	0.0036	0.0032
Refactored results	0.0036	0.0065	0.0040	0.0044	0.0092	0.0066	0.0059

Table 20. MSE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0229	0.0229	–	–	–	–	–
IMF0(SD)	0.0035	–	0.0035	0.0034	0.0028	0.0039	0.0026
IMF1	0.0008	0.0008	0.0008	0.0010	0.0022	0.0013	0.0012
IMF2	0.000009	0.000009	0.0006	0.0007	0.0023	0.0008	0.0010
IMF3	0.00001	0.00001	0.0014	0.0018	0.0062	0.0022	0.0029
IMF4	0.0001	0.0001	0.0003	0.0003	0.0040	0.0004	0.0032
IMF5	0.0019	0.0019	0.0029	0.0024	0.0131	0.0042	0.0034
Refactored results	0.0074	0.0280	0.0082	0.0101	0.0370	0.0198	0.0183

Table 20. MSE indicators for individual models of broiler chicken prices.

	SD-IHEOA	PD-IHEOA	SD-HEOA	SD-WOA	SD-ANN	SD-GRU1D	SD-GRU2D
IMF0	0.0229	0.0229	–	–	–	–	–
IMF0(SD)	0.0035	–	0.0035	0.0034	0.0028	0.0039	0.0026
IMF1	0.0008	0.0008	0.0008	0.0010	0.0022	0.0013	0.0012
IMF2	0.000009	0.000009	0.0006	0.0007	0.0023	0.0008	0.0010
IMF3	0.00001	0.00001	0.0014	0.0018	0.0062	0.0022	0.0029
IMF4	0.0001	0.0001	0.0003	0.0003	0.0040	0.0004	0.0032
IMF5	0.0019	0.0019	0.0029	0.0024	0.0131	0.0042	0.0034
Refactored results	0.0074	0.0280	0.0082	0.0101	0.0370	0.0198	0.0183

In the performance evaluation of the broiler chicken experiment, the model’s fitting results across different IMFs underwent quantitative analysis. For the $R^2$ metric, the SD-IHEOA model excelled in IMF2, IMF3, and IMF4, achieving values of 99.97%, 99.99%, and 99.96%, respectively, clearly outperforming other models. The SD-HEOA model’s $R^2$ value on IMF0(SD) reached 0.5518, slightly lower than SD-IHEOA’s 0.5843. Regarding MAE, the SD-IHEOA model performed best on IMF1, IMF3, IMF4, and IMF5, with values of 0.0196, 0.0026, 0.0109, and 0.0363, significantly lower than competing models. The MAPE metric further confirmed SD-IHEOA’s superiority on IMF2, IMF3, and IMF5, with values of 0.0810, 0.3752, and 0.0020, respectively, all below other models. On MSE, SD-IHEOA showed the best performance from IMF1 to IMF5, with values markedly lower than the rest.Comparing IMF0 and IMF0(SD), all four metrics favored IMF0(SD), demonstrating the effectiveness of the secondary decomposition algorithm in predicting high-frequency components. The reconstruction metrics also indicate that SD-IHEOA delivers the highest prediction accuracy and stability, with R², MAE, MAPE, and MSE values of 98.42%, 0.0636, 0.0036, and 0.0074, respectively, outperforming other models. These results confirm that SD-IHEOA holds a significant advantage in processing broiler chicken data, particularly in capturing trends and periodic variations of mid- to low-frequency components.

To verify whether the improvement of the proposed SD-IHEOA model is statistically significant, the Diebold–Mariano (DM) test was performed. The results in

Table 21. DM Test Results of SD-IHEOA vs. Other Models Across Datasets.

Dataset	Model Comparison	DM t-Statistic	p-Value
wheat	SD-IHEOA vs. PD-IHEOA	−3.1536	0.00190000
	SD-IHEOA vs. SD-HEOA	−6.7055	0.00000000
	SD-IHEOA vs. SD-WOA	−6.6333	0.00000000
	SD-IHEOA vs. SD-GRU2D	−6.6333	0.00000000
	SD-IHEOA vs. SD-GRU1D	−8.1559	0.00000000
	SD-IHEOA vs. SD-ANN	−8.4698	0.00000000
cabbage	SD-IHEOA vs. PD-IHEOA	−4.5964	0.00000700
	SD-IHEOA vs. SD-HEOA	−5.3735	0.00000020
	SD-IHEOA vs. SD-WOA	−6.0010	0.00000001
	SD-IHEOA vs. SD-GRU2D	−7.9425	0.00000000
	SD-IHEOA vs. SD-GRU1D	−6.3609	0.00000000
	SD-IHEOA vs. SD-ANN	−7.2815	0.00000000
chicken	SD-IHEOA vs. PD-IHEOA	−3.0675	0.00240000
	SD-IHEOA vs. SD-HEOA	−0.9405	0.34800000
	SD-IHEOA vs. SD-WOA	−3.3380	0.00100000
	SD-IHEOA vs. SD-GRU2D	−7.4528	0.00000000
	SD-IHEOA vs. SD-GRU1D	−6.9500	0.00000000
	SD-IHEOA vs. SD-ANN	−11.3772	0.00000000

show that the differences in forecasting accuracy between the proposed and benchmark models are statistically significant (p < 0.05), confirming the robustness of the proposed method.

3.4. Comparative Experiments

To validate the predictive performance advantage of the proposed model (SD-GRU2D), this study selected several baseline models for comparison experiments. These include traditional machine learning models such as Random Forest (RF), Support Vector Regression (SVR), and Light Gradient Boosting Machine (LGB), as well as ANN and LSTM. The experiments used the same dataset and evaluation metrics: $R^2$, MAE, MAPE, and MSE. The comparison results are shown in Figure 9, Figure 10 and Figure 11, and the evaluation metrics are presented in

Table 22. Evaluation indicators of wheat in different models.

	ANN	GRU1D	GRU2D	LSTM	SVR	RF	LGB	CEEMDAN–TDNN	GA–VMD–LSTM
$R^2$	79.99%	80.06%	87.00%	81.05%	\	\	\	83.47%	81.37%
MAE	74.49	74.90	60.12	73.90	213.64	304.32	310.73	75.49	65.04
MAPE	0.026	0.026	0.021	0.026	0.078	0.117	0.120	0.042	0.078
MSE	9026.87	8484.38	5950.29	8460.43	62921.45	141500.63	146712.72	12579.34	9568.26

,

Table 23. Evaluation indicators of cabbage in different models.

	ANN	GRU1D	GRU2D	LSTM	SVR	RF	LGB	CEEMDAN–TDNN	GA–VMD–LSTM
$R^2$	79.17%	83.98%	83.48%	81.15%	59.78%	69.19%	72.42%	78.97%	80.32%
MAE	0.125	0.101	0.108	0.113	0.152	0.150	0.117	0.097	0.167
MAPE	0.061	0.049	0.054	0.056	0.073	0.073	0.056	0.059	0.087
MSE	0.029	0.021	0.023	0.025	0.043	0.043	0.032	0.027	0.030

and

Table 24. Evaluation indicators of broiler chicken in different models.

	ANN	GRU1D	GRU2D	LSTM	SVR	RF	LGB	CEEMDAN–TDNN	GA–VMD–LSTM
$R^2$	82.56%	89.51%	89.63%	84.13%	31.64%	74.37%	79.20%	81.58%	79.24%
MAE	0.221	0.160	0.155	0.204	0.365	0.291	0.296	0.285	0.247
MAPE	0.013	0.009	0.009	0.012	0.020	0.017	0.017	0.015	0.031
MSE	0.082	0.050	0.048	0.075	0.192	0.138	0.129	0.067	0.058

.

3.5. Ablation Experiment

To clarify the individual contributions of the two improvements—Gaussian mutation and adaptive weights—this study conducts an ablation analysis by testing the effects of adding only Gaussian mutation and only adaptive weights separately. This approach clearly demonstrates the independent effectiveness of each improvement and their combined impact.

The performance analysis in

Table 25. Comparison of evaluation indicators of different models for wheat prices.

Model	${\mathit{R}}^2$	MAE	MAPE	MSE
SD-GRU-IHEOA	98.62%	21.7252	0.0074	801.2814
SD-GRU-HEOA (Gaussian Mutation)	97.19%	31.4357	0.0109	1478.0427
SD-GRU-HEOA (Adaptive Weights)	96.75%	33.3784	0.0115	1708.6671
SD-GRU-HEOA	95.96%	36.2480	0.0126	1875.8800

,

Table 26. Comparison of evaluation indicators of different models for cabbage prices.

Model	${\mathit{R}}^2$	MAE	MAPE	MSE
SD-GRU-IHEOA	97.94%	0.0440	0.0216	0.0031
SD-GRU-HEOA (Gaussian Mutation)	96.00%	0.0623	0.0304	0.0062
SD-GRU-HEOA (Adaptive Weights)	96.12%	0.0604	0.0293	0.0060
SD-GRU-HEOA	95.32%	0.0620	0.0300	0.0066

and

Table 27. Comparison of evaluation indicators of different models for broiler chicken prices.

Model	${\mathit{R}}^2$	MAE	MAPE	MSE
SD-GRU-IHEOA	98.42%	0.0636	0.0036	0.0074
SD-GRU-HEOA (Gaussian Mutation)	98.32%	0.0694	0.0039	0.0078
SD-GRU-HEOA (Adaptive Weights)	98.19%	0.0697	0.0040	0.0083
SD-GRU-HEOA	98.21%	0.0706	0.0040	0.0082

shows that the HEOA improved with Gaussian mutation and adaptive weights effectively enhances accuracy and reduces errors. Among the ten model groups, nine demonstrate significantly better R² values, and error metrics also improve. Comparing with the SD-GRU-HEOA model confirms that these two improvements enhance the model’s precision by strengthening global search capabilities and fine-tuning local search.

4. Conclusions

This study proposed a hybrid forecasting framework integrating CEEMDAN–VMD decomposition, GRU neural networks, and IHEOA to enhance the accuracy and stability of agricultural product price prediction. Beyond the numerical superiority observed in $R^2$, MAE, and MSE metrics, improved short-term forecasting stability is particularly valuable in volatile agricultural markets, as it helps reduce uncertainty in inventory planning and price risk management. More accurate predictions of price fluctuations can indirectly support economic decision-making by improving the timing of market interventions and supply-chain adjustments.

The comparative analysis indicates that the primary advantage of the proposed model lies in its hierarchical signal decomposition strategy. Agricultural price series often exhibit strong non-stationarity and multi-scale volatility, which can limit the effectiveness of models based on raw inputs or single-stage decomposition. By combining CEEMDAN for global non-stationarity reduction with VMD for high-frequency refinement, the framework transforms complex price signals into more interpretable components, facilitating more effective learning by the forecasting model. Moreover, the comparison between GRU models optimized using conventional metaheuristic algorithms and those optimized with the proposed IHEOA highlights the importance of adaptive hyperparameter optimization. The incorporation of Gaussian mutation and adaptive weighting enhances search diversity and convergence stability, resulting in more robust and consistent forecasting performance across different agricultural products. Ablation results further confirm that these improvements stem from the optimizer design rather than incidental parameter tuning.

From an applied perspective, the improved forecasting stability under both normal and volatile market conditions suggests that the proposed framework can serve as a useful decision-support tool for early warning systems and price stabilization mechanisms. Reliable short-term forecasts can assist farmers, enterprises, and policymakers in making timely and informed decisions.

Nevertheless, this study has several limitations. The empirical analysis focuses on specific products within a particular market, which may restrict the generalizability of the results. Additionally, the multi-stage decomposition and optimization framework requires greater computational resources compared to simpler benchmark models. Although this work emphasizes forecasting accuracy and stability, future research could integrate explainable artificial intelligence to improve interpretability and highlight the influence of exogenous variables. Potential extensions include expanding to multi-regional contexts, enhancing computational efficiency, and exploring multi-horizon forecasting.