A model based LSTM and graph convolutional network for stock trend prediction

Traditional machine learning techniques

Traditional machine learning methods are widely used for stock trend prediction, such as K-nearest neighbor (KNN), Random Forest (RF), and support vector machine (SVM). Specifically, SVM, RF, and combined approaches are highly favored.

Impact of many factors on stock price makes the stock price prediction a difficult and highly complicated task. In order to overcome such difficulties, Kumar et al. (2018) applied machine learning techniques for the stock price prediction i.e., SVM, RF, KNN, Naive Bayes, and Softmax, and applied several technical indicators to stock prices data. Menon et al. (2016) opted for a basic ARMA-type predictor for their bulk price prediction. Kara, Boyacioglu & Baykan (2011) used artificial neural network (ANN) and SVM to predict the movement direction of Istanbul Stock Exchange National 100 Index. ANN predicts the index significantly better than SVM (Kara, Boyacioglu & Baykan, 2011). In Zhang, Li & Pan (2016), a status box method is proposed. Then an ensemble method integrated with AdaBoost algorithm and probabilistic multi-class SVM is constructed to perform the status boxes classification for stock trend prediction. A weighted kernel least squares SVM method was used to predict stock trend in Markovic et al. (2017). The feature ranking and selection were finished through the analytic hierarchy process providing feature weights. In Özorhan, Toroslu & Şehitoğlu (2019), the authors proposed a novel approach based on a modified version of the Zigzag technical indicator, expectation maximization, and SVM for predicting short-term trends in the financial time series of foreign exchange market. In Parray et al. (2020), SVM, perceptron, and logistic regression were used to predict the trend of stocks at next day. In Picasso et al. (2019), the authors combine the technical and fundamental analysts approaches to market trend forecasting through the use of machine learning techniques applied to time series prediction and sentiment analysis in the NASDAQ100 index.

Deep learning methods

With increasing computational intelligence capacity, deep learning methods have been widely used to predict stock trend in recent years. Deep learning methods perform well and outperform shallow counterparts, due to their ability to effectively extract high-level features from stock data. LSTM was introduced by Hochreiter & Schmidhuber (1997) aiming to achieve a better performance by tackling the vanishing gradient issue that recurrent networks would suffer when dealing with long data sequences. LSTM can effectively extract the time series features and achieves better success.

In Fischer & Krauss (2017), the authors made an earlier attempt to apply LSTM network to predict out-of-sample directional movements for the constituent stocks on the S&P 500. The experimental results demonstrate that LSTM network outperforms memory free classification methods, i.e., RF, deep neural net, and logistic regression classifier. In Wu et al. (2020), a novel data annotation method was proposed to extract continuous trend features from financial time series data. The extracted trend features were used for four supervised machine learning methods and two deep learning models for financial time series forecasting. In Wu et al. (2020), a new approach to data annotation was employed to capture the continuous trend characteristic of financial time series data. These features were then utilized in four different supervised machine learning techniques and two deep learning models for predicting financial time series. In Nelson, Pereira & de Oliveira (2017), the authors studied the use of LSTM network to predict future trends of stock prices based on stock market price history and technical analysis indicators. In Qin et al. (2017), the authors propose a dual-stage attention-based recurrent neural network. The proposed method consists of an encoder with an input attention mechanism for adaptively selecting the most relevant input features and a decoder with a temporal attention for appropriately capturing the long-term temporal dependencies of the time series. In Zhao et al. (2021), three prediction models were developed to predict stock trends using the attention mechanism on RNN, LSTM, and gated recurrent unit. In Kim & Won (2018), the authors proposed a new hybrid LSTM network to predict stock price volatility that combines the LSTM network with various generalized autoregressive conditional heteroscedasticity-type models. In Chen & Ge (2019), the authors explore the attention mechanism in LSTM network based stock price movement prediction. The evaluation results show the effectiveness of the proposed method in the Hong Kong stock market.

The latest deep learning framework, Transformer, has recently been applied to stock prediction (Wang et al., 2022; Lai et al., 2023; Zhang et al., 2022; Zeng et al., 2023), because it can better characterize the underlying rules of stock market dynamics by using the encoder–decoder architecture and the multi-head attention mechanism. In Wang et al. (2022), a Transformer is utilized to predict major stock market indices worldwide. A moving window approach is utilized to construct features and labels from the observed time series. Unlike the original decoder in Vaswani et al. (2017), the Transformer do not use the mask attention mechanism. In Zeng et al. (2023), the authors propose a method to forecast whether the price will go up, down or remain the same in the future, employing CNNs to model short-term dependencies and Transformers to learn long-term dependencies within a time series.

Graph-based approach

In recent years, many studies have shown that considering the correlation among stocks can further improve the prediction performance of stock trend.

Kipf & Welling (2017) proposed an efficient variant of convolutional neural networks for semi-supervised classification on graph-structured data. In a number of experiments on citation networks and on a knowledge graph dataset, the proposed approach outperforms the related methods by a significant margin. In Chen et al. (2020), the authors propose a graph convolutional feature based convolutional neural network for stock trend prediction, in which an improved graph convolutional network was used to extract stock market features from the overall stock market information and a Dual-CNN was used to extract individual stock features from specific individual stock information. In Li et al. (2022a), to solve the quantification difficulty problem of the technical charts of stock movement, an arbitrary graph kernel and graph convolutional network are proposed to mine the information in the technical charts for stock movement prediction after extracting the key point sequence from the stock price series. In Zhao et al. (2022), the authors proposed a self-generating relations algorithm based on feature threshold and Hamming Distance to extract relational features automatically from related stocks data. The relational features are used as the adjacency matrix to an graph convolutional neural network. In Zhao et al. (2023), the authors utilized a time series relational model (TSRM) to automatically classify stocks through a K-means model and derives stock relationships, such as industry and upstream/downstream connections. The authors used an LSTM network to extract the time series information and a Graph convolutional network to extract relationship information. In Li et al. (2020), to leverage the connection among stocks, the authors adopted relational graph convolutional networks (RGCN) (Schlichtkrull et al., 2018) to encode the graph structure with their correlation matrix. In the correlation matrix, two kinds of relationship representing positive and negative correlation . The authors proposed to add an LSTM between RGCN layers so that the LSTM can dynamically select which part of the information should be transmitted to upper layers. In Li et al. (2022b), the authors construct graph data to achieve data fusion based on a comprehensive data set for stocks. LSTM and GRU are used to complete the aggregation process. An attention mechanism graph neural network based on nodes and edges is used to complete the classification and prediction.

LSTM network

An LSTM network is used for extracting features of individual stock information. The LSTM network comprises three kinds of gating units and one cell, namely input gate i_t, output gate o_t, forgetting gate f_t and memory cell state c_t.t is a time step. The memory cell is like a conveyor belt. These gate units adaptively keep or override information into the memory cell. The function of f_t is to control what information will be discarded from the memory cell. The function of i_t is to control what new information will be stored into the memory cell. The function of o_t is to select the information to be output from the memory cell (Hochreiter & Schmidhuber, 1997). The gates and the memory cell is calculated as in Eqs. (1)–(6). (1)i t = σ W i ⋅ h t − 1 ; x t + b i (2)f t = σ W f ⋅ h t − 1 ; x t + b f (3)g t = t a n h W c ⋅ h t − 1 ; x t + b c (4)c t = f t ⊙ c t − 1 + i t ⊙ g t (5)o t = σ W o ⋅ h t − 1 ; x t + b o (6)h t = o t ⊙ t a n h c t where W_i,  W_f,  W_o,  W_c ∈ R^d×2d are the weight matrices of input gate, forget gate, output gate and memory cell, and b_i,  b_f,  b_o,  b_c ∈ R^d are the bias term of input gate, forget gate, output gate and memory cell. These weight matrices and bias term will be updated when minimizing loss function during LSTM training. σ is an activation functions, ⊙ is the element-wise multiplication. h_t denotes a current hidden state. h_t−1 denotes a prior hidden state. x_t denotes an input sequence.

Graph node creation and its timing sequence feature

Stock market information includes individual stock data and the overall market topology structure. The topology structure can represented as a network graph where each graph node is a stock item s_i. The information of individual stock is transformed into an time sequence feature as the graph nodes of stock market network. The similarity of the feature of individual stock is transformed into the edge weight to construct the connections between graph nodes. Detail process of generating graph node embedding was presented in Fig. 2 and Eq. (7).

In Eq. (7), S^t denotes stocks information sampled at timestep t. H^t denotes all output of hidden states at timestep t. p 1 t + w , p 2 t + w , … , p n t + w are the target price of stocks at next timestep t + w. As such, at timestep t X i t, we feed the sampled historical stock i information to the LSTM network and take all hidden states H i t as the embedding e i t of stock i (notes that e i t = H i t). We have Eq. (8) (8)E t = L S T M X t where E t = e 1 t , … , e N t T r a n s p o s e ∈ R N × U denotes the sequential embeddings of all stocks. U is the embedding size and the number of hidden units of LSTM. N is the number of sampled stocks.

Adjacency matrix representing the relationship

There are inter-stocks entity relationship, such as industry and upstream/downstream connections, and these relationship interact and influence each other (Zhao et al., 2023). In our work, PCC is used to measure these entity relationship. In statistics, PCC is also the most commonly correlation coefficient in financial analysis (Chun-Xiao & Fu-Tie, 2018; Thakkar, Patel & Shah, 2021). In Chun-Xiao & Fu-Tie (2018), the authors used PCC to study the correlation of time series is to construct a correlation coefficient matrix. The authors found that the degree of a node is related to the average correlation coefficient between the node and other nodes. In Thakkar, Patel & Shah (2021), the authors apply PCC for weight initialization instead of random initialization for a feed-forward vanilla neural network to enhance the prediction performance. The results demonstrate that the proposed weight initialization techniques provided comparable results compared to random initialization.

In this paper, PCC is used to construct adjacency matrix to handle adjacency relationship that exist among stocks. The PCC for variable X and Y can be obtained by using Eq. (9) (Benesty et al., 2009). (9)r = N ∑ X Y − ∑ X ∑ Y N ∑ X 2 − ∑ X 2 N ∑ Y 2 − ∑ Y 2 where N denotes the number of stocks. ∑ is sum function. X and Y denote the sequential embeddings of all stocks. The explanation of X and Y can be referred to Fig. 2 and Eq. (7).

Graph convolutional network

GCN is a special kind of graph-based learning methods to model the graph structure (Kipf & Welling, 2017), which is the state-of-the-art formulation for stock price prediction (Chen, Wei & Huang, 2018; Feng et al., 2018). In this paper, we use a simplified GCN (Kipf & Welling, 2017) to learn the relationship among stocks that is shown in Eq. (10). The GCN consists of two convolutional layers, one for input-to-hidden and the other for hidden-to-output. The GCN needs to be provided with two inputs: a matrix of node features H representing the time-series features of stocks, a graph structure in the form of an adjacency matrix A representing the inter-stocks relationship: (10)H l + 1 = σ A ̂ H l W l where A ̂ = D ˇ − 1 2 A ˇ D ˇ − 1 2 is a normalized symmetric adjacency matrix. Here, A ˇ = A + I, A is the adjacency matrix of the undirected graph G without added self-connections, I is an identity matrix. D ˇ is the degree matrix of A ˇ and calculated by using the formula D ˇ i i = ∑ j A ˇ i i. H^(l) is a l layer-specific feature matrix and represents the hidden representations of all the nodes in the l-th layer. H^(l+1) is the aggregated neighbor information for the (l + 1)-th layer. W^(l) is a l layer-specific learnable weight matrix for the directed cross-correlation of stocks. σ is a nonlinear activation function. In our case, we use ReLU(…) as the activation function.

In this paper, we build a two-layer GCN model based LSTM network for stock price movements prediction. The graph convolutional of the GCN is described in Eq. (11): (11)Y = σ A ̂ σ A ̂ X W 0 W 1 where W⁽⁰⁾ ∈ R^C×H is an input-to-hidden weight matrix, the hidden layer has feature matrix H.W⁽¹⁾ ∈ R^H×F is a hidden-to-output weight matrix.

Target prediction

The GCN output was used as the input for the output layer in order to make predictions of the stock’s closing price on the following day. The output layer utilized in our model is a fully connected layer, as shown in Eq. (12): (12)z ̂ i = W y i + b where y_i is the output of stock i from the GCN Layer. z ̂ i is the predicted close price of stock i on the next day.

The proposed model was trained using smooth L1 loss (Girshick, 2015) over all stock data, because it is less sensitive to outliers than L2 loss. When the regression targets are unbounded, training with L2 loss may require careful tuning of learning rates to prevent exploding gradients (Girshick, 2015). For a batch of size: math N, the unreduced loss can be described as Eqs. (13) and (14): (13)L o s s = ∑ l 1 , … , l N (14)l i = 0 . 5 z i − z ̂ i 2 , if | z i − z ̂ i | < 1 | z i − z ̂ i | − 0 . 5 , otherwise where, z_i is the observed close price of stock i on the next day.

Dataset description

Experimental data in this paper is from the daily trading of Chinese stock market. The constituent stocks of the MSCI China A 50 Connect Index (MSCI, 2023) were used as stock pool because the Index aims to reflect the performance of the 50 largest securities. The securities represent each Global Industry Classification Standard sector and reflect the sector weight allocation of the MSCI China A Index.¹ We defined an screening criteria of stocks that the stocks within the stock pool have uninterrupted data from 2018-01-02 to 2023-03-10 were selected as the experimental data. According to the screening criteria, 20 stocks from China Shanghai stock market were selected and 15 stocks from China Shenzhen stock market were selected. In this paper, the index change over days of the screened stocks is showed in Fig. 3. The computer method of the index is showed in Eq. (15), where i denotes the ith stock, the index was computed at the t-th day. (15)s t o c k i n d e x t = ∑ i = 1 35 c o l s e p r i c e i t c o l s e p r i c e i 1 . We use a publicly available API TdSdk (TqSdk, 2023) to get the historical daily quote data (including opening prices, high prices, low prices, closing prices, volumes etc.) of the screened stocks between 2nd January 2018 and 10th March 2023.

Data preprocessing

We proposed a sliding window method (See Fig. 4), which was inspired by the sliding window method in Chen et al. (2020). The proposed sliding window method, with a fixed start training time followed by a fixed length test period, effectively addresses the data alignment problem. The size of the training dataset started to steadily increase by increments of 44 days from 2nd January 2018, followed by a 44 days testing period in our experiments.

The data of the selected stocks is continuous from January 2, 2018 to March 1, 2023 without any interruptions, ensuring no issues with data alignment. The data values that do not fall within the stock opening period are removed. The quality of the dataset is excellent, with no errors or missing values.

Normalization is a very good practice for the input data preprocessing to standardize the effect of price changes between different stocks. We rescaled the input sequences and target values to the range of 0 to 1 to avoid neuron saturation because the gates of the LSTM unit have an sigmoid activation function. (16)X i t = X i t m a x X i t ∗ 1 + 0 . 1 − m i n X i t ∗ 1 − 0 . 1 (17)Z i t + 1 = Z i t + 1 m a x X i t ∗ 1 + 0 . 1 − m i n X i t ∗ 1 − 0 . 1 .

In this paper, a new normalization method for input sequences and target values is proposed that is show in Eqs. (16)–(17). We increase the maximum value of each input data by 10% and decrease the minimum value by 10%, and then normalize the data to 0 to 1.

Comparison methods

To show the performance and returns of the proposed model, we compare our model with some baseline methods, including an linear regression (LR), a naive Bayes (NB), a RF, a Transformer based method, LSTM and GCN. The experiment was repeated five times for each method to minimize variability.

• Linear regression: this method takes the historical numerical information of stocks as the input data, applies linear logistic regression (https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression) to predict stock price trend.

• Naive Bayes: this method takes the historical numerical information of stocks as the input data, and applies Naive Bayes classifier (https://scikit-learn.org/stable/modules/linear_model.html#bayesian-regression) to predict stock price trend.

• Random Forest (Khaidem, Saha & Dey, 2016): this method is an ensemble of multiple decision trees that minimizes forecasting error by treating the forecasting problem as a classification problem (https://scikit-learn.org/stable/modules/ensemble.html#forest). Some of the technical indicators such as relative strength index, stochastic oscillator etc were used as the inputs features.

• Transformer based: In Zeng et al. (2023), the authors proposed a method for financial time series forecasting (CTTS) that based CNN and Transformer. Instead of the decoder of original Transformer (Vaswani et al., 2017), the proposed method used a multilayer perceptron with softmax activation.

• LSTM (Nelson, Pereira & de Oliveira, 2017): this method is a basic LSTM network. Based on the price history, the future trends of stock price were predicted alongside with technical analysis indicators.

• GCN (Chen, Wei & Huang, 2018): A three-layer GCN (Kipf & Welling, 2017) was used to incorporate information of related corporations of a target company for its stock price prediction.

• TSRM (Zhao et al., 2023): TSRM is an LSTM and GCN based model for stock price prediction, in which the stock relationships are measured by using a K-means model to cluster stock trading data, time series features are extracted by LSTM.

Setting

We obtained the hyper-parameters for the proposed model based on validation set with AdaGrad algorithm (Duchi, Hazan & Singer, 2011), as shown in Table 1. Where, we look back 12 days to construct historical information. In particular, we use historical dat of the selected stocks from day t − 12 to day t as an input X^t, and predict the observed close price Z^t+1 on day t + 1 . Z t + 1 ̂ is the output of the proposed model that represents the predicted stock close price on day t + 1.

The experimental environment was set as follows. The computer system was equipped with an Intel Core i5-12400 processor clocked at 2.50 GHz, 16 GB of RAM, a GeForce RTX 3070Ti graphics card, and Windows 11 operating system. The proposed model was implemented with Python 3.9.12, PyTorch 1.11.0 and CUDA 11.6.112. The DataParallel function of PyTorch 1.11.0 enabled parallel computation on the GPU. The integrated development environment utilized for the task was Visual Studio Code 1.73.1, which is provided by Microsoft.

Evaluation metrics

The following metrics are adopted for prediction performance evaluation in the experiment: mean absolute error (MAE), mean square error (MSE), accuracy (Acc), precision (Pre) and recall (Rec) F-measure(F1) (Chen et al., 2020; Zhao et al., 2022; Zhao et al., 2023). The corresponding formulas are listed in Eqs. (18)–(26). (18)M A E = 1 M N ∑ i = 1 M ∑ j = 1 N | z i j − z i j ˆ | (19)M S E = 1 M N ∑ i = 1 M ∑ j = 1 N z i j − z i j ˆ 2 where, M is the number of the screened stocks and N is the number of samples in a stock. z is the observed stock close price, and z ˆ is the output value of the proposed model. (20)A c c = T P + T N T P + F P (21)P r e p o s = T P T P + F P (22)P r e n e g = T N T N + F N (23)R e c p o s = T P T P + F N (24)R e c n e g = T N T N + F P (25)F 1 p o s = 2 P r e p o s R e c p o s P r e p o s + R e c p o s (26)F 1 n e g = 2 P r e n e g R e c n e g P r e n e g + R e c n e g where, true positive (TP) for correctly predicted event values, false positive (FP) for incorrectly predicted event values, true negative (TN) for correctly predicted no-event values, false negative (FN) for incorrectly predicted no-event values.

In stock trading backtest, the average annual return (AAR), sharp ratio (SR), maximum drawdown (MDD) and total returns money (TRMoney) metrics are used to focus on the trading risk and returns of the proposed model (Chen et al., 2020; Zhao et al., 2022; Wang, Zhuang & Feng, 2022). The corresponding formulas are showed in Eqs. (27)–(30). An annual risk free rate of 2.5% is adopted in the stock trading backtest. (27)T R M o n e y = A v a i l a b l e C a p i t a l − S t a r t M o n e y + C l o s i n g P r i c e × P o s i t i o n o f S t o c k s (28)A A R = T R M o n e y S t a r t M o n e y 1 / N u m b e r o f Y e a r × 100 % (29)S R = R e t u r n − R i s k f r e e r a t e S t a n d a r d D e v i a t i o n o f R e t u r n (30)M D D = M a x P x − P y P x where, P_x is the net value of stocks on day x, and P_y is the net value of stocks on day y, x is a certain day, y is a certain day after x, annual risk free rate is 2.5%.

Models training

The proposed model was trained end-to-end through error back-propagation algorithm. The training process was to minimize the loss function and adjust the model parameters step by step. AdaGrad algorithm was used to finished our training because its gradient-based adaptive optimization can regulate the learning rate of neural networks (Duchi, Hazan & Singer, 2011). AdaGrad algorithm is listed in Eqs. (31)–(33). (31)θ t + 1 , i = θ t , i − η G t , i i + ɛ ⋅ g t , i (32)g t , i = ∇ θ J θ i (33)G t , i i = ∑ t g t , i 2

At time step t, the process to optimize parameter θ_t+1,i is performed according to Eq. (23). Where, learning rate η is set to 0.01. G_t ∈ R^d×d is a diagonal matrix. Each element on the diagonal line G_t,ii is an square sum over history gradient value θ_i.ɛ is a smoothing term that avoids division by zero (usually on the order of 1e − 8).

Comparison results of different methods on stock 600031

The comparison results between prediction and output of different methods on stock 600031 are shown in Figs. 5–11. The prediction performance of the proposed model is more stable than baseline methods:

(1) As can be seen clearly from Figs. 5–11, LSTM-GCN (see Fig. 11) is more stable than other comparative methods.

(2) LSTM-GCN has no outliers unlike the GCN based method (see Fig. 9).

(3) LSTM-GCN has better continuity than LR (see Fig. 5), NB (see Fig. 6) and RF Regression based methods(see Fig. 7).

(4) The difference between two values of LSTM-GCN is smalle all of CTTS (see Fig. 8), GCN (see Fig. 9) and LSTM (see Fig. 10) based methods.

The evaluation metrics values of different methods on stock 600031 are list in Table 2. To better visualize the prediction performance of the proposed method, we have highlighted the best results in bold and created the bar charts based on these findings (See Fig. 12). The observation indicates that:

(1) LSTM-GCN outperforms all baseline methods in Acc, Pre_pos and Pre_neg metrics, remains medium values of Rec_pos, Rec_neg, F1_pos and F1_neg metrics, outperforms the deep learning based methods (CTTS, GCN and LSTM) and is the same order as the machine learning based methods (LR, NB and RF) in MAE and MSE metrics.

(2) LSTM-GCN performs well than GCN and LSTM in MAE, MSE, Acc and Pre metrics, and remains medium values in Rec and F1 metrics. This indicates that the prediction performance can be improved by combining LSTM with GCN to integrate time series information and relational features of stocks.

(3) LSTM-GCN outperforms the deep learning based methods (CTTS and LSTM) in MAE, MSE, Acc, Pre_pos, Pre_neg metrics due to its application on multiple stock input data.

(4) LSTM-GCN is the same order as the machine learning based methods (LR, NB and RF) in MAE and MSE metrics, and outperforms the machine learning based methods in Acc, Pre_pos, Pre_neg, Rec_neg, and F1_neg metrics. This indicates that deep network structure is more effective for stock trend prediction.

(5) LSTM-GCN outperforms most baseline methods, but does not always stand out in all the evaluation metrics. It is clear that when predicting future trends of stock 600031, none of the methods always remain high values of MAE, MSE, Accuracy, Precision, Recall or F1.

Comparison results of the proposed model and baseline methods on China A50 stocks

Table 3 shows the comparison results of the proposed model and baseline methods on China A50 stocks. We have highlighted the best results in bold. The conclusion derived in Table 3 is similar to those derived in Table 2. We drew bar charts based on Table 3 to clearly observe the prediction performance of LSTM-GCN (see Fig. 13).

Visualization of feature maps

The Figs. 14 and 15 display two maps that were obtained by registering a forward hook in the Implementation code of Grid-LSTM. The hook was called every time function ‘forward’ computed an output. Figure 14 visualizes a test input for LSTM-GCN, while Fig. 15 shows the corresponding feature extracted by LSTM-GCN. In Fig. 14, columns represent stock indices and rows represent time steps of stock data sampling. The visualization in Fig. 15 highlighting and distributing in a continuous manner compared to the input data in Fig. 14 demonstrates that information from the input data can be effectively extracted and centrally recorded. The experimental results based on stock market data from China further validate the effectiveness of these features.

Stock trading backtest

We firstly identify a set of trading strategies as part of the trading plan to make more money than they lose. The strategies is following:

(1) The initial capital of each stock for trading was 300,000 RMB.

(2) The data from 2018-01-02 to 2023-03-10 was used for trading that was divided into many trading intervals. A trading interval is set to 44 days. The data before trading intervals is used for training.

(3) The predicted price is at the next time step.

(4) Buy stocks when their forecast prices increase by 5% at next time step.

(5) Sell stocks when their profits exceed 5% at current time step to take profit from the curent stock trading.

(6) Sell stocks when holding them for more than 10 days at current time step. The predetermined trading intervals is employed instead of stop loss for the curent stock trading to prevent premature exit during stock liquidation. Figure 16 shows the choice of trading intervals and their impact on the results. The profits decrease when the trading intervals exceed 10 days or are less than 10 days. We can see that 10days is an appropriate trading intervals. The predetermined trading intervals are set to 10 days.

(7) Has no stock trading in other cases.

On financial evaluation, we train modes and simulated stock trading based on the screened stocks with using above strategies.

Table 4 shows the stock trading backtest results of LSTM-GCN and the baseline methods based on China A50 stocks. We have highlighted the best results in bold. We drew the profits changes over trading times (See Fig. 17) and the profits changes over trading days (See Fig. 18). The financial evaluation of LSTM-GCN is clearly observed. It is obvious that:

(1) LSTM-GCN has achieve more returns, has the best AAR and SR, medium level MDD than the comparative methods (see 4).

(2) The returns of all the methods are positive. AAR values of all the methods are more than 2%, SR values of five methods are more than 0 (see 4).

(3) LSTM-GCN can gain better returns regardless of the upward or downward range of the stock index (see Fig. 3), has good stability and robustness (see Figs. 17 and 18).