A Stock Prediction Method Based on Multidimensional and Multilevel Feature Dynamic Fusion

Abstract

Stock price prediction has long been a topic of interest in academia and the financial industry. Numerous factors influence stock prices, such as a company’s performance, industry development, national policies, and other macroeconomic factors. These factors are challenging to quantify, making predicting stock price movements difficult. This paper presents a novel deep neural network framework that leverages the dynamic fusion of multi-dimensional and multi-level features for stock price prediction, which means we utilize fundamental trading data and technical indicators as multi-dimensional data and local and global multi-level information. Firstly, the model dynamically assigns weights to multi-dimensional features of stocks to capture the impact of each feature on stock prices. Next, it applies the Fourier transform to the global features to capture the long-term trends of the global environment and dynamically fuses these with local and global features of the stocks to capture the overall market environment’s impact on individual stocks. Finally, temporal features are captured using an attention layer and an RNN-based model, which incorporates historical price data to forecast future prices. Experiments on stocks from various industries within the Chinese CSI 300 index reveal that the proposed model outperforms traditional methods and other deep learning approaches in terms of stock price prediction. This paper proposes a method that facilitates the dynamic integration of multi-dimensional and multi-level features in an efficient manner and experimental results show that it improves the accuracy of stock price predictions.

1. Introduction

Stock price prediction is one of the core applications of financial data mining, and the stock market reflects the overall state and expectations of the economy. Studying the stock market gives valuable insights regarding economic trends, industry development, and company performance. By forecasting stock market movements, investors and financial institutions can develop appropriate risk management strategies in advance, reducing potential risks, obtaining more options, and achieving higher efficiency, thus making more informed investment decisions [1]. Consequently, there has been significant interest in stock prediction from both academia and the industry in recent years [2,3,4]. However, the inherent complexity and randomness of the stock market present significant challenges. Historical stock price data are fraught with noise and irregularities, complicating the decision-making process for investors. Moreover, stock prices are influenced by a multitude of factors—including macroeconomic trends, regulatory policies, and investor sentiment—that are often difficult to quantify [5,6]. This vast and intricate dataset poses a formidable barrier to achieving accurate predictions of stock market movements.

The Efficient Market Hypothesis (EMH) [7] suggests that stock prices reflect all available information, and thus, using price information alone can efficiently accomplish prediction tasks [8,9]. In the past, researchers have attempted to use traditional time series analysis methods for stock prediction, such as Vector Autoregression (VAR) [10] and Autoregressive Integrated Moving Average (ARIMA) [11]. However, these methods are predicated on the assumptions of linearity and stationarity, which do not hold true for the inherently nonlinear and non-stationary nature of stock price data. Consequently, these traditional approaches often falter in the face of the noise embedded in historical stock data [12]. Another common approach, fundamental analysis, entails evaluating a company’s financial statements and broader industry conditions. However, this method is resource-intensive and requires a depth of expertise, making it challenging to apply consistently across diverse market conditions. As a result, fundamental analysis can yield suboptimal predictive performance.

In recent years, deep learning technology has rapidly developed and has been widely applied in fields such as medicine, transportation, and finance. Due to the limitations of traditional methods, deep learning techniques have gradually become mainstream in stock prediction [13]. Deep learning can automatically extract useful features from large amounts of historical data without requiring manual feature definition, reducing the workload of feature engineering and improving prediction accuracy. Moreover, deep learning models can adapt to dynamic changes in market conditions through continuous training and updates [14], which significantly enhances their robustness and predictive performance. However, despite these advantages, most existing methods primarily focus on the impact of different features on stock prices in a separate way. This narrow approach often overlooks the complex interactions between multiple features and fails to consider the broader global environment’s influence on stock market dynamics. Such limitations can lead to suboptimal predictions, as stock prices are affected by a multitude of interrelated factors.

Recognizing these challenges, our research aims to develop a stock prediction method that employs the dynamic fusion of multi-dimensional and multi-level features. By integrating data smoothing strategies, we can mitigate the noise often present in global historical stock data, enhancing the reliability of our predictions. Additionally, incorporating attention mechanisms allows us to effectively capture both global and local environmental impacts on stock prices, thereby providing a richer understanding of the market.

To validate the effectiveness of our MDML model, we utilized stock data from the China Securities Index 300 (CSI 300), covering the period from January 2020 to April 2024, as a representation of local market features. Simultaneously, data from the CSI 300 index for the same period were employed to capture global market dynamics. The key contributions of this study are listed below.

• We provide a stock prediction method based on the dynamic fusion of multidimensional and multilevel data that successfully captures the impact of both global and local factors on stock prices. By combining variables from many dimensions and levels, the model provides a thorough knowledge of the factors that influence stock price movements. This novel approach analyzes not only the influence of individual features but also the interactions between features, hence increasing the model’s expressive power.
• A dynamic weight allocation method is presented that enables the model to dynamically modify the weights of various characteristics according to their relative importance. This guarantees an accurate representation of each feature’s effect on stock prices. The model can represent the changing importance of information across several time points and market conditions by dynamically allocating weights, which enhances the accuracy and dependability of prediction outcomes.
• We introduce a Fourier transform method for global features, applying Fourier transform to global features to capture long-term trends in the global environment. This technique helps in understanding the impact of macroeconomic and other broad factors on stock prices over an extended period, providing the model with support from long-term information.
• We conducted extensive experiments on stocks from different industries within the CSI 300 index in the Chinese market. The results indicate that the proposed model performs exceptionally well in stock price prediction, significantly outperforming traditional methods and other deep learning approaches, thereby demonstrating its substantial potential in practical applications.

The remainder of the paper will be organized as follows: Section 2 will review related work on stock prediction, including both traditional methods and deep learning approaches. Section 3 will provide a detailed description of the proposed stock prediction method based on dynamic fusion of multi-dimensional and multi-level features. Section 4 will present the design and results of the experiments. Section 5 discusses the limitations of this work and future research directions, while Section 6 summarizes the main findings and contributions of the study.

2. Related Work

There are various methods available for stock price prediction, including traditional time series forecasting methods, classic machine learning algorithms, and deep learning approaches.

Table 1. Information of literature review.

Model	Inputs	Strengths	Weaknesses
ARCH/GARCH/…	Single feature	Provides relatively accurate predictions	Requires manual feature selection
SVM/RF/KNN/…	Multiple features	Effectively captures nonlinear relationships between features	Struggles with highly complex nonlinear data
LSTM/CNN-LSTM/ALSTM/…	Multiple features	Captures temporal dependencies well	Cannot utilize multi-level features
ATFN/SFM/…	Multiple features	Captures periodicity, trends, and multi-scale patterns	Fails to capture interrelationships between features
CMoEs/Att-MoE/…	Multiple features	Combines results from multiple expert networks	Not widely applied in the finance sector

provides a summary of the literature review below.

2.1. Traditional Method

The stock prediction problem is often modeled as a time series forecasting problem. Classic models include the Autoregressive Conditional Heteroskedasticity (ARCH) [15] model, the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) [16] model, and the AutoRegressive Integrated Moving Average (ARIMA) [11] model, and so on, which are based on statistical theory and can provide relatively accurate predictions when market trends are clear or volatility is predictable. These models are widely used in the field of time series forecasting. However, they often require manual feature selection and have stringent requirements for data stationarity, which limits their applicability in the domain of stock prediction.

Machine learning methods have also found numerous applications in the field of stock prediction, such as Support Vector Machines (SVM) [17] and Random Forests (RF) [18]. These methods often do not require data to be stationary and can effectively capture the nonlinear relationships between features, making them widely used in stock prediction. For instance, in the paper [19], various machine learning algorithms, including support vector machines, decision trees, and neural networks, are explored to predict stock market trends. Additionally, in [20], the authors compare the performance of different machine learning algorithms, such as linear regression, SVM, and k-nearest neighbors (KNN) [21], in predicting stock prices. However, classic machine learning algorithms typically perform poorly when dealing with high-dimensional and complex nonlinear data. In contrast, deep learning models possess powerful nonlinear modeling capabilities, allowing them to capture intricate patterns and relationships within the data. Furthermore, classic machine learning algorithms often require additional feature extraction and transformation, such as sliding windows and time lags, when handling time series data.

Numerous deep learning-based stock prediction models have been developed in the past, such as Long Short-Term Memory (LSTM) networks [8,22], CNN-LSTM [23,24], ALSTM [25] and so on. They are specifically designed to process time series data, enabling them to better capture temporal dependencies and dynamic changes. There have been numerous applications in this area. For instance, the study [26] examines the use of LSTM networks in this scenario, predicting future stock price trends based on historical prices and technical analysis indicators. Hence, in this paper, we employ LSTM networks to capture temporal features from historical data.

2.2. Predict with Frequency Transformation

In [25], it was found that separating various features before capturing temporal features and then assigning weights based on the cell’s hidden states can enhance prediction accuracy. Therefore, in this paper, we also adopt this two-stage approach to perform multidimensional feature fusion and temporal information capture.

To better utilize global features, it is essential to process these features initially. Frequency transformation has been widely applied in the field of time series forecasting, often combining frequency transformation with deep learning models to enhance prediction accuracy. The significance of frequency transformation lies in its ability to capture the periodicity, trends, and multi-scale patterns of time series by converting them to the frequency domain, thereby providing a more global perspective, as illustrated in the Figure 1. Common frequency transformations include the Discrete Fourier Transform (DFT), Discrete Cosine Transform (DCT), and Discrete Wavelet Transform (DWT) [27]. In the study [28], a frequency domain block is proposed in the model named ATFN to capture the dynamic and complex periodic patterns of time series data, thereby improving prediction accuracy. Additionally, in [29,30], new modules are designed using Fourier transformation to capture patterns in time series. Moreover, ref. [31] combines Fourier analysis with the Transformer architecture, enhancing the Transformer’s ability to capture the global properties of time series.

In the field of stock prediction, many researchers have applied frequency transformation techniques. For instance, in [8], trading patterns are decomposed into various frequency components, with each component modeling the specific frequency of underlying trading patterns that influence stock price fluctuations. In this study, we employ Fourier transformation to extract the top K frequencies from the global features, aiming to capture long-term trends in the overall market to guide the prediction of individual stock prices.

2.3. Predict with Dynamic Fusion

In previous work, there have been few studies that utilize multidimensional features for prediction, while the use of attention mechanisms can effectively facilitate their integration. For example, in [32], the attention patterns in the encoder and decoder are modified to reduce the space and time cost of training while improving the accuracy of long-term predictions. In this paper, we introduce cross-attention and self-attention mechanisms [33] to capture future information contained in both global and local features. There have been precedents for using global and local information for stock prediction. For instance, in [34], a context aggregation module is employed to fuse global and local information, though this process introduces hyperparameters. To better integrate global and local information, we introduce the Mixture-of-Experts (MoE) [35] mechanism, which dynamically adjusts the weights of global and local information. The MoE mechanism typically comprises two parts: expert networks and a gating network, which enhances the scalability of information fusion without adding hyperparameters. This approach has been widely adopted in various fields. Jinguo Zhu et al. (2022) [36] utilized the Mixture-of-Experts (MoE) mechanism to alleviate task interference in general models, thereby enhancing performance across multiple tasks. They proposed Conditional Mixture-of-Experts (CMoEs), which activate sub-networks based on specific conditions to mitigate interference while maintaining versatility. Similarly, Jinhua Liu et al. (2020) [37] introduced Att-MoE, which employs a single gating network in combination with multiple expert networks. These experts collaborate on different sub-tasks, with the gating network guiding their adaptive and cooperative utilization to achieve efficient cell segmentation.

Upon surveying the literature mentioned above, we have identified that there remains significant room for improvement in the accuracy of stock prediction. Moreover, previous research focusing on the fusion of multi-dimensional and multi-level features has been relatively limited. Consequently, we propose a novel model that synergistically combines the capabilities of DFT to capture long-term trends, along with the feature-capturing and fusion prowess of MOE and attention mechanisms.

3. Methodology

3.1. Problem Formulation

Our goal is to predict the future price of a stock given historical data $X=\{x_1,x_2,\dots ,x_t\}$, where $x_i$ is the feature vector of the stock on the i-th day. The observation window is defined as the time interval of t days, and our objective is to predict the stock price on the $t+1$-th day, denoted as $y_{t+1}$. We divide the stock features into global features $X_g$ and local features $X_l$, where the global features represent the overall market trend and are represented by the CSI 300 index data, while the local features are specific to an individual stock. The input to our model consists of the global features $X_g$ and local features $X_l$, and the output is the close stock price on the $t+1$-th day $y_{t+1}$.

3.2. Data Process

We utilize historical stock data to train our model, encompassing a variety of data points such as the stock’s opening price, closing price, highest price, lowest price, trading volume, and so forth. To enhance the model training process, we have normalized the data, scaling the input data for the model to a range between 0 and 1. The expression is as follows:

$$x_{norm}={\displaystyle \frac{x-\min \left(x\right)}{\max \left(x\right)-\min \left(x\right)}}$$

(1)

3.3. Model Architecture

The model proposed in this study is built upon a dynamic fusion mechanism that integrates both multi-dimensional and multi-level features. The term “multi-dimensional” refers to a diverse set of features that encapsulate various aspects of financial data, including fundamental attributes such as opening and closing prices, trading volumes, and technical indicators like the Relative Strength Index (RSI). These features collectively provide a comprehensive representation of the financial market’s micro-level dynamics. In contrast, “multi-level” features pertain to the aggregation of data across different scales, capturing localized characteristics at the level of individual stocks, as well as global attributes that reflect macro-level trends and conditions in the broader stock market. This dual-level structure enables a more holistic analysis by accounting for both granular stock-specific details and overarching market-wide patterns. Taking into account the distinctive characteristics of our model, which synergistically combines these Multi-Dimensional and Multi-Level features, we have designated it as the MDML model.

The MDML model is divided into four components: the Global Feature Encoder, the Local Feature Encoder, the Dynamic Feature Fusion Module, and the Temporal Feature Capture Module. The overall architecture of the model is depicted in Figure 2:

3.3.1. Global Feature Encoder

The Global Feature Encoder aims to extract long-term trends from the global features. We employ the Fourier Transform to identify the top K frequencies within the global features, thereby capturing the long-term trends inherent in these features. The expression for the Fourier Transform is as follows:

$$X_k=\sum _{n=0}^{N-1}{x}_n{e}^{-j2\pi kn/N}$$

(2)

We input the global features $X_g$ into the Fourier Transform module to obtain the frequency components of the global features. Subsequently, we select the top K frequencies and perform an inverse transform on the global features. The expression for the inverse transform is as follows:

$$x_n={\displaystyle \frac{1}{K}}\sum _{k=0}^{K-1}{X}_k{e}^{j2\pi kn/K}$$

(3)

After obtaining the result of the inverse transform, we feed this output into the feature mapping module to derive the latent feature representation of the global long-term trends (a fully connected network followed by a ReLU activation function) as mentioned in [38]:

$$h_g=\mathrm{ReLU}(x_n{W}_g+b_g)$$

(4)

Here, the global features post-Fourier Transform are denoted as $x_n\in {\mathbb{R}}^{t\times d}$, with $W_g\in {\mathbb{R}}^{d\times d_p}$ and $b_g\in {\mathbb{R}}^{d_p}$ being trainable parameters. The latent feature representation of the global features, $h_g$, will then be inputted alongside the local features into the dynamic feature fusion module.

3.3.2. Local Feature Encoder

The purpose of the local feature encoder is to extract the prospective trends from the features of individual stocks. We utilize an LSTM network in conjunction with an attention mechanism to integrate and encode the local features. Specifically, we input the local features $X_l$ into the LSTM network. Unlike the method for capturing temporal features, we treat the local information as a sequence of features, where for the n-th feature, our input is ${\mathbf{x}}^n=(x_1^n,x_2^n,\dots ,x_t^n)\in {\mathbb{R}}^t$. Based on the hidden state $h_t$ and cell state $s_t$ of the LSTM cell, we calculate the weights for various features and then perform a weighted summation of these weights and features, which can be expressed as follows:

$$e_t^n=v_e^{\top }tanh(W_e[h_{t-1};s_{t-1}]+U_e{x}^n)$$

(5)

$${\alpha }_t^n={\displaystyle \frac{exp\left(e_t^n\right)}{{\sum }_{i=1}^{d}exp\left(e_t^i\right)}}$$

(6)

$$x_l={\alpha }_t^1{x}_1+{\alpha }_t^2{x}_2+\dots +{\alpha }_t^d{x}_d$$

(7)

After this, the output passes through a feature mapping module to obtain the latent feature representation of the local features $h_l\in {\mathbb{R}}^{t\times d_p}$, which will then be inputted together with the global features into the dynamic feature fusion module.

3.3.3. Dynamic Feature Fusion Module

After obtaining the latent feature representations of both global and local features, we dynamically integrate these features to capture the impact of the global and local environments on stock prices. The self-attention mechanism is capable of capturing dependencies between different time points, aiding the model in comprehending how price fluctuations within a specific past period influence future price trends. On the other hand, the cross-attention mechanism can be utilized to manage the interactions between multiple time series, namely the correlation between stock prices and macroeconomic indicators. Therefore, We employ self-attention mechanisms to capture the historical patterns of the global information $h_g$, which can be specifically represented as:

$$Q=h_g{W}^Q,K=h_g{W}^K,V=h_g{W}^V$$

(8)

$$\alpha =\mathrm{softmax}\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)$$

(9)

$${\tilde{h}}_g=\alpha \ast V$$

(10)

Here, $W^Q,W^K,W^V\in {\mathbb{R}}^{d_p\times d_k}$ are trainable parameters, and $d_k$ is the dimension of the attention mechanism.

To capture the relationship between global and local information, we introduce a cross-attention mechanism that performs cross-attention between global and local information. Additionally, to preserve the features of individual stocks, we incorporate residual connections, which can be represented as:

$$Q=h_g{W}^Q,K=h_l{W}^K,V=h_l{W}^V$$

(11)

$$\alpha =\mathrm{softmax}\left({\displaystyle \frac{Q{K}^T}{\sqrt{d_k}}}\right)V$$

(12)

$${\tilde{h}}_l=\alpha \ast V+h_l$$

(13)

After obtaining ${\tilde{h}}_g$ and ${\tilde{h}}_l$, we need to fuse these two types of information. We introduce the MOE (Mixture of Experts) mechanism, which dynamically adjusts the weights of global and local information without introducing hyperparameters:

$$h_{emb}=\mathrm{Concat}(h_l,h_g)$$

(14)

$${\alpha }_i=\mathrm{softmax}(W{h}_{emb}+b)$$

(15)

$$\widehat{h_{emb}}={\alpha }_1{h}_g+{\alpha }_2{h}_l$$

(16)

In this representation, ${\alpha }_1$ and ${\alpha }_2$ are the weights of the global and local information, respectively, which are dynamically adjusted during the training process. $\widehat{h_{emb}}$ is the fused result of global and local information, which will then be input into the temporal feature capture module.

3.3.4. Temporal Feature Capture Module

In the temporal feature module, we employ an LSTM network combined with a temporal attention mechanism to capture the temporal features within the historical data. We input ${\widehat{h}}_{\mathrm{emb}}$ into the LSTM network to obtain the hidden state of the LSTM cell, $h_t$. Subsequently, we feed $h_t$ into the temporal attention mechanism to acquire the weights for each time step and then perform a weighted summation of these weights with $h_t$ to obtain the final prediction result. This can be specifically represented as:

$$e_t={\mathbf{v}}_e^{\top }tanh({\mathbf{W}}_e[{\mathbf{h}}_{t-1};{\mathbf{s}}_{t-1}]+{\mathbf{U}}_e{\mathbf{h}}_t)$$

(17)

$${\alpha }_t={\displaystyle \frac{exp\left(e_t\right)}{{\sum }_{i=1}^{t}exp\left(e_i\right)}},$$

(18)

$${\widehat{h}}_{t+1}={\alpha }_t{h}_t$$

(19)

Finally, $\widehat{h_{t+1}}$ is input into a fully connected network to yield the ultimate prediction result $\widehat{y_{t+1}}$.

4. Experiments

4.1. Dataset

During the data collection process, taking into account that stocks in the same or related industries may exhibit similar trend variations, we combine the classification results based on the enterprise types from the East Money website with the clustering results based on the close price as shown in

Table 2. Dataset information.

Industry	Stocks	Training Set	Validation Set	Test Set
Finance	38	31,426	3838	3838
Lifestyle Services	36	29,772	3636	3636
Pharmaceuticals and Biotechnology	34	28,118	3434	3434
Electronic Technology	32	26,464	3232	3232
Chemical Industry	30	24,810	3030	3030
Manufacturing and Transportation	29	23,983	2929	2929
Information Technology	22	18,194	2222	2222
Construction	19	15,713	1919	1919

. This approach allows us to segment the stocks into different datasets for separate training.

4.2. Data Process and Model Training

The dataset is divided temporally into training, testing, and validation sets with a ratio of approximately 8:1:1, followed by min–max normalization of the data.

An Adam optimizer is utilized with a learning rate of 0.001 and a batch size of 64, employing the Mean Squared Error (MSE) as the loss function. To investigate the impact of the observation window on the predictive outcome, experiments were conducted with observation windows set to 5, 10, 15, 20, and 25. The results indicated that the model performed optimally with an observation window of 15; thus, this setting was adopted. During the Fourier transformation of global features, the top K frequencies were selected, and it was discovered through experimentation that the model’s predictive performance was best when the top K was set to 7. To mitigate overfitting, an early stopping strategy was employed, halting the training process if the loss value on the validation set did not decrease for five consecutive epochs.

The historical data indicators obtained from the website encompass a total of 20 types, as shown in

Table 3. Features used in the proposed model.

Type	Features
Fundamental trading attributes	Opening Price, Closing Price, Highest Price, Lowest Price, Trading Volume, Transaction Value, Turnover Rate, Price Change Percentage, Price Change Amount, Amplitude and Pivot
Technical indicators	MA5, MA30, EMA5, EMA30, RSI14, MOM14, OBV, WILLR, CMO14

, which include: Opening Price, Closing Price, Highest Price, Lowest Price, Trading Volume, Transaction Value, Turnover Rate, Price Change Percentage, Price Change Amount, Amplitude, Pivot and some technical indicators. Among the historical data indicators obtained from the website, the term “Pivot” refers to the average value of the Highest Price, Lowest Price, and Closing Price. In the global information capture module, we utilize the CSI 300 Index as a representative of global information. The CSI 300 is a significant index in China’s securities market, encompassing the stocks of 300 companies and representing the overall trend of China’s securities market.

4.3. Evaluation Parameters

In our quest to forecast stock prices with precision, we evaluated the reliability of the experimental results by incorporating three widely used metrics: Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE). These metrics serve as a multidimensional framework for assessing the predictive efficacy of our model. The Mean Absolute Error (MAE) offers a direct measure of the average prediction error magnitude, while the Mean Absolute Percentage Error (MAPE) translates these errors into a percentage of actual values, highlighting relative inaccuracies especially pertinent in finance. The Root Mean Square Error (RMSE) amplifies the impact of larger errors through squaring, making it an effective tool for gauging average error severity and identifying outliers [25]. The formulas for the calculation of these metrics are as follows:

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-{\widehat{y}}_i|$$

(20)

$$MAPE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{|y_i-{\widehat{y}}_i|}{y_i}}$$

(21)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(22)

where n is the number of samples, $y_i$ is the actual value, ${\widehat{y}}_i$ is the predicted value, and $\overline{y}$ is the average value.

4.4. Baseline

Our experiments are benchmarked against a diverse array of analytical techniques, including traditional models like ARIMA and Moving Average (MA) [39,40], which are cornerstones in time series forecasting. We also leverage machine learning algorithms such as SVM and RF for their efficacy with complex data patterns. Additionally, we engage deep learning models: CNN-LSTM, DTML, and LSTM-BN to harness advanced pattern recognition capabilities. This comprehensive comparison with our approach aims to affirm its effectiveness and superiority in predictive performance.

Here is a brief introduction to the baselines:

• ARIMA [41]: The ARIMA model is a time series forecasting model composed of the Autoregressive (AR) and Moving Average (MA) components. In this approach, we utilize historical closing prices as variables to predict future closing prices. To identify the optimal hyperparameters, we utilize the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) for parameter specification.
• MA [42]: The MA model is a time series forecasting model that consists solely of the Moving Average component. The predictions of the MA model are influenced by historical data and are suitable for stationary time series data.
• ES [43]: Exponential Smoothing (ES) effectively captures trends and seasonal variations in data, demonstrating strong adaptability to rapidly respond to market changes and adjust forecasts in a timely manner. This makes it suitable for dynamic financial markets. The relevant code for this part of the experiment can be found at https://github.com/DONGTangYuan/multi-dimension-data (accessed on 1 October 2024) for reference.
• lightGBM [44]: LightGBM is an efficient gradient-boosting tree algorithm that excels in handling nonlinear relationships and high-dimensional data, enabling it to capture complex market patterns.
• SVM [45]: Support Vector Machine is a supervised learning algorithm that finds an optimal hyperplane to separate data into two categories. It applies to both classification and regression problems. The SVM model can capture complex nonlinear relationships in historical data and is therefore widely used in stock price forecasting.
• RF [46]: Random Forest is an ensemble learning algorithm that makes predictions through multiple decision trees. In this stock prediction model, in addition to utilizing fundamental features such as opening price, highest price, lowest price, and trading volume, we have also incorporated technical indicators such as moving averages, Relative Strength Index (RSI), Bollinger Bands, and Moving Average Convergence Divergence (MACD).
• CNN-LSTM [23]: The CNN-LSTM model is a model that combines Convolutional Neural Networks and Recurrent Neural Networks. It extracts features from the input sequence using convolutional networks and then makes predictions using recurrent networks.
• DTML [34]: The DTML model is a transformer-based model. It learns the correlations between stocks in an end-to-end manner. DTML captures asymmetric and dynamic correlations by learning the temporal correlations within each stock and generates multi-level context based on the global market context.
• LSTM-BN [47]: The LSTM-BN model is a model based on Recurrent Neural Networks and Batch Normalization. It accelerates the training process and improves prediction accuracy through batch normalization.

4.5. Experiment Result

To substantiate the efficacy of our model, we conducted experiments on both the baseline models and our proposed model. During the validation process of the model’s performance, we set different random seeds and performed the experiments five times, taking the average results. The following are the outcomes from testing our model and the baseline models on a test set composed of stocks from eight different industries. The results are shown in

Table 4. Experiment results of chemical industry and electronic technology.

Model	Chemical Industry			Electronic Technology
	MAE	RMSE	MAPE	MAE	RMSE	MAPE
MA	0.4079	0.7308	1.8467	1.2687	2.4755	2.3378
ARIMA	0.4157	0.7410	1.8725	1.3042	2.5116	2.4076
ES	0.3928	0.7040	1.7760	1.2197	2.3518	2.2518
lightGBM	0.7817	1.8609	2.8179	1.0674	2.6782	2.8231
SVM	0.5006	1.3710	2.5500	0.7711	2.1557	2.8213
RF	0.5928	1.4952	2.7548	0.9590	2.1307	2.7948
CNN-LSTM	0.4880	0.8755	2.8209	2.5269	3.6165	2.7770
DTML	0.5578	0.9844	1.7309	1.0100	1.8743	2.5719
LSTM-BN	0.2088	0.4517	1.7028	1.2174	2.1548	2.5328
MDML	0.1027	0.1459	1.6872	0.8234	1.8006	2.2485

The bold text indicates the model with the best performance on a specific metric, while the underlined text indicates the second-best model.

,

Table 5. Experimental results of information technology and lifestyle services.

Model	Information Technology			Lifestyle Services
	MAE	RMSE	MAPE	MAE	RMSE	MAPE
MA	1.0878	2.9288	2.8132	1.5186	4.8575	1.8150
ARIMA	1.1200	2.9803	2.8194	1.5012	4.9519	1.8251
ES	1.0517	2.7599	2.6305	1.4041	4.4568	1.7240
lightGBM	0.9310	2.2063	3.0421	1.3021	3.7700	4.8085
SVM	0.7052	1.8305	3.8833	0.9629	3.1642	3.6098
RF	0.7244	0.9616	2.5948	0.6543	2.1395	2.6010
CNN-LSTM	0.4621	0.8327	3.6609	0.7024	1.3908	3.4719
DTML	0.5823	0.6854	3.2309	0.6704	1.9733	2.7539
LSTM-BN	0.5165	0.9517	2.9328	0.8442	1.9148	2.9648
MDML	0.4689	0.8164	2.1534	0.6461	1.0178	2.6474

The bold text indicates the model with the best performance on a specific metric, while the underlined text indicates the second-best model.

,

Table 6. Experimental results of manufacturing and transportation and pharmaceuticals andbiotechnology.

Model	Mfg. & Trans			Pharm. & Biotech.
	MAE	RMSE	MAPE	MAE	RMSE	MAPE
MA	0.6380	1.441	1.7689	1.0708	1.9695	2.0742
ARIMA	0.6146	1.3578	1.7551	1.0698	1.9688	2.0773
ES	0.5991	1.3498	1.6670	1.0141	1.9027	1.9762
lightGBM	0.4944	1.2618	2.6941	2.6998	6.0171	5.9236
SVM	0.4531	0.6033	1.5601	1.5048	2.4574	4.7238
RF	0.3901	1.0698	2.0490	2.1983	5.2635	4.7586
CNN-LSTM	0.3674	1.0010	2.0234	2.2238	5.3072	4.8086
DTML	0.3810	0.9845	1.2346	1.9456	3.6883	3.3224
LSTM-BN	0.5087	1.8596	2.0328	1.8465	3.6549	4.5415
MDML	0.3014	0.5712	1.7907	0.5565	1.2545	1.7354

The bold text indicates the model with the best performance on a specific metric, while the underlined text indicates the second-best model.

and

Table 7. Experimental results of finance and construction.

Model	Finance			Construction
	MAE	RMSE	MAPE	MAE	RMSE	MAPE
MA	0.1623	0.2936	1.3287	0.1989	0.3465	1.8554
ARIMA	0.1657	0.2995	1.3450	0.2008	0.3492	1.8639
ES	0.1526	0.2790	1.2419	0.1846	0.3259	1.7004
lightGBM	0.1057	0.2802	1.0162	0.7542	1.6749	7.6478
SVM	0.0534	0.1758	0.6086	0.6085	1.4318	6.1272
RF	0.0540	0.1766	0.6061	0.4059	0.5443	2.1129
CNN-LSTM	0.1202	0.2265	1.6542	0.3874	0.5425	1.5445
DTML	0.1823	0.2814	3.2455	0.5361	0.5733	2.3559
LSTM-BN	0.1344	0.3999	1.2357	0.5878	0.8109	2.4365
MDML	0.0679	0.0938	1.1058	0.1093	0.2093	1.1401

The bold text indicates the model with the best performance on a specific metric, while the underlined text indicates the second-best model.

. We have bolded the best metrics for all models and underscored the second-best metrics.

Table 6 and Table 7 demonstrate that the ES model and SVM model exhibit notably strong performance in the Pharm. and Biotech. and Finance sectors, respectively. This indicates that traditional time series forecasting methods and conventional machine learning algorithms can perform well in certain cases. On the other hand, the LSTM-BN and CNN-LSTM algorithms show superior performance in the Chemical Industry and Information Technology datasets, respectively, suggesting that employing alternative deep learning methods for stock price prediction also holds significant promise. Overall, across all eight datasets, our proposed models consistently achieve the lowest or second-lowest values for MAE, RMSE, and MAPE metrics. This signifies that the absolute and relative errors between predicted and actual results are minimal, indicating that our models possess a degree of stability and accuracy. This further validates the effectiveness of our proposed method for dynamic fusion of multi-level and multi-dimensional features.

4.6. Case Study

To substantiate the efficacy of our predictive model, an in-depth case study was conducted focusing on a select group of the most representative stocks within the CSI 300 index. This study was designed to meticulously assess the model’s forecasting capabilities by comparing its predicted values against the actual market outcomes. The selected stocks were chosen based on their market capitalization, liquidity, and influence on the overall index, ensuring a comprehensive reflection of the model’s performance across various sectors and market conditions. The testing period for this case study spanned from 1 April 2024, to 1 June 2024, a timeframe that encapsulates a variety of market conditions, including seasonal fluctuations and potential macroeconomic events that could influence stock prices.

The results of the case study were compelling, with the model demonstrating a high degree of accuracy in its predictions. The predicted values were found to be near the actual values as shown in Figure 3a,b, Figure 4a,b, Figure 5a,b and Figure 6a,b, indicating the model’s robustness and reliability in forecasting stock prices.

4.7. Ablation Study

To verify the effectiveness of our model, we conducted ablation studies. The setup of the ablation studies is as follows:

• my-model-1: The Discrete Fourier Transform (DFT) is removed, while dynamic weights and feature dynamic fusion are retained, to verify the impact of the DFT on the model.
• my-model-2: Dynamic weights are removed, while the DFT and feature dynamic fusion are retained, to verify the impact of dynamic weights on the model.
• my-model-3: Feature dynamic fusion is removed, while the DFT and dynamic weights are retained, to verify the impact of feature dynamic fusion on the model.
• my-model: The complete model, including the DFT, dynamic weights, and feature dynamic fusion.

In this experiment, we conducted experiments on two datasets from the finance and construction industries within the CSI 300. The results of the experiment are as follows:

The results of the ablation experiments in

Table 8. Ablation analysis of MDML model.

Model	Finance			Construction
	MAE	RMSE	MAPE	MAE	RMSE	MAPE
my-model-1	0.0786	0.1385	1.1326	0.1964	0.2737	1.2735
my-model-2	0.0906	0.2498	1.4358	0.2056	0.4051	1.8960
my-model-3	0.0979	0.2032	1.3254	0.3031	0.4396	1.9256
MDML	0.0679	0.0938	1.1058	0.1093	0.2093	1.1401

indicate that the removal of the DFT module, dynamic weights module, and feature dynamic fusion all lead to an increase in MAE, RMSE, and MAPE metrics, implying a decline in the model’s predictive accuracy. Thus, it can be concluded that the DFT module, MOE module, and feature fusion module all contribute to the enhancement of the model’s performance. Specifically, the impact of removing the DFT module on the experimental results is minimal, while the removal of the feature fusion module causes a significant increase in the MAE metric. Conversely, the removal of the MOE module results in a more substantial rise in RMSE and MAPE metrics. These experiments demonstrate that adding fusion modules is beneficial for reducing both the relative and absolute discrepancies in predictions, thus validating the effectiveness of multi-dimensional and multi-feature fusion. The minimal impact of the DFT module suggests that there is still room for improvement in capturing the global characteristics of stocks, which could be a focus for future work.

5. Limitations and Future Research

The model discussed in this text has demonstrated impressive performance in stock price forecasting, yet it is not without its limitations. We will examine these limitations and suggest possible avenues for future research. The limitations can be summarized as follows:

• The model captures long-term trends through Fourier transformation, but balancing the influence of long-term trends and short-term fluctuations in practical applications remains a challenge. Over-reliance on long-term trends may overlook short-term market changes, while an excessive focus on short-term fluctuations could lead to instability in the forecast results.
• Financial markets are significantly influenced by external factors such as policy changes, economic fluctuations, and unexpected events. These factors are often nonlinear and sudden, making them difficult to predict through historical data. The model may not adapt to these changes promptly, especially in the face of black swan events or economic crises, which could significantly reduce the model’s forecasting power.

Based on the aforementioned limitations, we propose the following directions for future research:

• Incorporate a dynamic weight adjustment mechanism based on market conditions to flexibly adjust the influence of long-term trends and short-term fluctuations under different market states. For example, using market volatility indicators to dynamically adjust the model’s attention to long-term trends and short-term fluctuations to cope with various market conditions.
• Develop an event-driven model capable of capturing the impact of unexpected events. Introduce exogenous variables based on events (such as policy changes, news events, etc.) into the time series model to help the model identify and respond to these sudden changes. For instance, using an attention mechanism-based event detection model to automatically identify and quantify the impact of unexpected events on the market.

6. Conclusions

In conclusion, this paper presents a novel deep neural network model for stock price prediction that effectively integrates multi-dimensional and multi-level features. By dynamically assigning weights to various stock features and applying the Fourier transform to capture long-term trends, the model successfully combines global and local information to reflect the overall market environment’s impact on individual stocks. The incorporation of an attention mechanism and RNN-based structure further enhances the model’s ability to capture temporal dynamics, leveraging historical price data to improve prediction accuracy. Experimental results on stocks from different industries within the CSI 300 index demonstrate the model’s superior performance compared to traditional methods and other deep learning approaches, highlighting its potential for more accurate and robust stock price prediction. However, the research presented in this paper still has several limitations. For instance, the experiments were conducted exclusively within the context of the Chinese stock market and relied solely on numerical data, neglecting textual information, which may limit the model’s predictive performance and expressive capability. Therefore, in future work, it would be beneficial to incorporate a broader range of information into the model, including textual data, and to validate the model’s effectiveness across stock markets in other countries.