GCN-based stock relations analysis for stock market prediction

Self-generating relation algorithm

We consider determining whether relations exist between two stocks and how to design rules to capture them. Here, we refer to realistic stock relations as entity relations, such as industrial relations and competitive relations. A stock may have entity relations with many other stocks. The combined relations ultimately act on stock pairs’ trading behavior, especially prices in different periods. So that their prices show a specific pattern, such as two stocks’ closing prices rising or falling together or one stock’s five-day average price rising while another falls. Therefore, the correlation between two stock prices reflects a combination of multiple relations between them. There is a mapping: (1)${\displaystyle \left(R_{i,j}^1,R_{i,j}^2,\dots ,R_{i,j}^N\right)\underset{}{\overset{f}{\to }}{r}_{i,j}^k}$ where R_i,j denotes an entity relation between stock i and stock j, $r_{i,j}^k$ denotes the correlation between stock i and stock j in certain trading data. Both are Bool data, represented by 0 and 1.

 
 
 Data: stocks S, trading days Tr, trading features F, minimum support 
              min_support 
    Result:  Stock relation adjacency matrix Asgr[len(S)][len(S)][len(F)] 
  1  Function gen_Stocks_Feature_Event(feature): 
     2   Event_Bit_Set[len(S)][len(Tr)] = {0} 
3   for t in Tr do 
    4   for s in S do 
5   if trading feature of stock s in day t meet feature threshold 
   then 
6   Event_Bit_Set[s][t] = 1 
7   end 
8   end 
9   end 
10   return Event_Bit_Set 
11 
12  for f in F do 
     13   E = gen_Stocks_Feature_Event(S, Tr, f) 
14   for (s1,s2) pair from S do 
    15   dis = Hamming Distance between E[s1] and E[s2] 
16   support = (len(Tr) - dis) / len(Tr) 
17   if support >min_support then 
18   Asgr[s1][s2][f] = 1 
19   else 
20   Asgr[s1][s2][f] = 0 
21   end 
22   end 
23  end 
 
              Algorithm 1: Self-generating relation algorithm    

Therefore, we use the correlation of stock prices, namely self-generating relations (SG-relations), as an alternative to entity relations. The SGR algorithm mines stocks’ SG-relations. Its input is multiple trading features F, and the output is a 3D adjacency matrix A_sgr. The flow of the algorithm is represented in pseudocode in Algorithm 1. In short, two stocks have a SG-relation on a trading feature, which means they meet the same threshold on the feature for enough days. The threshold here can be similar to that the closing price is greater than 0, the five-day average price is less than 0, and so on.

Weighted graph convolutional neural network

GCN requires two inputs: an adjacency matrix A representing the relationship, and the other is a matrix H representing stocks’ time-series features. We use a simplified GCN proposed by Kipf & Welling (2017) which is the state-of-the-art formulation (Chen, Wei & Huang, 2018; Feng et al., 2019). It consists of two convolutional layers, one for input-to-hidden and the other for hidden-to-output: (2)${\displaystyle \mathit{A}=\mathit{D}-1/2\left(\mathit{A}+\mathit{I}\right)\mathit{D}-1/2\mathit{Y}=f\left(\mathit{X},\mathit{A}\right)=softmax\left(\mathit{A}\left(ReLU\left(\mathit{A}\mathit{X}\mathit{W}0\right)\right)\mathit{W}1\right)}$

Here, A + I is the adjacency matrix of stock relations undirected graph G with added self-connections, D ∈ ℝ^N×N is the degree matrix of G.W0 ∈ ℝ^C₁×C₃ is an input-to-hidden weight matrix, and W1 ∈ ℝ^C₃×C₄ is the hidden-to-output weight matrix.

Note that A is a 2D adjacency matrix, which means only a single relation between stock nodes. However, two stocks may have multiple SG-relations as defined in the previous section. Therefore, we designed the WGCN to utilize relational information. It adds a weight layer to the traditional GCN to map various SG-relations into a synthetic relation: (3)${\displaystyle \mathit{A}{sgr}^{\left(N,N,n_r\right)}\to \mathit{W}{r}^{\left(n_r,1\right)}\mathit{A}\left(N,N\right)\mathit{A}=\mathit{W}r\mathit{A}sgr}$ where A_sgr denotes the adjacency matrix of multiple SG-relations, and W_r denotes the weight vector.

Figure 2 illustrates mapping the entity relations to a synthetic relation. Suppose that A is a traditional auto company actively turning to new energy, and B is a power battery company. They used to have mutually exclusive competition and are gradually increasing cooperation. These complex entity relationships are reflected in market transactions, which may be the reverse trend of the 60-day average price because it reflects longer periods. It may also show the same trend of the daily trading price because it reflects the latest situation. These SG-relations are combined by weights to form a synthetic relation that is fed into the GCN.

Framework in whole

The TRMF model comprises feature selection, information extraction, and target prediction. Figure 3 shows the framework in whole.

Experimental data

Experiments data is from the New York Stock Exchange (NYSE) and the Nasdaq Stock Exchange (NASDAQ). Because they are the top two stock markets in terms of global market capitalization in 2021 (https://data.worldbank.org/indicator/CM.MKT.LCAP.CD), and the most developed markets in the financial industry in the world. Then the constituent stocks of the S&P 500 are chosen for stock selection. This method, on the one hand, reduces the amount of calculation. On the other hand, these stocks contain 80% of the US stock market value (https://en.wikipedia.org/wiki/S%26P_500), so they can represent the entire market.

The experimental period was 01/02/2018–06/28/2019 and was divided into three intervals: the training (01/02/2018–12/28/2018), validation (01/02/2019–03/29/2019), and test (04/01/2019–06/28/2019) sets. We made this choice because S&P 500 Index shows periods of volatility, plummeting, and soaring in the test set, which is representative. Table 1 lists the number of trading days in the NYSE and NASDAQ.

We screened 134 stocks on the NASDAQ and 325 on the NYSE during the experimental period based on the two filters mentioned above. Table 2 shows the number of stocks used in the experiment and Appendix lists all stocks.

We collected the opening price, closing price, highest price, lowest price, trading volume, and turnover rate as stock features. Because they are typical and common, and any stock exchange will provide them. Furthermore, all data were normalized to [0,1] using Min-Max normalization.

We selected four trading features: closing price, 5-day moving average, 20-day moving average, and 60-day moving average as input of SGR algorithm to generate SG-relations. They represent the stock’s trading behavior on the day, week, month, and quarter, respectively. Each trading feature has two different forms of correlation: the same trend and the reverse trend. So, there is eight SG-relations in total. Table 3 list all SG-relations.

Trading strategy

We used a daily cycle “buy-hold-sell” trading strategy to simulate market investments, which has been commonly used in many articles (Feng et al., 2019; Chen & Ge, 2019; Fischer & Krauss, 2018). On each t + 1 day of the test time set, at the opening moment, we simulated a trader selling out all stocks held at the opening price (buy on t day) and then buying the stock with the highest expected revenue.

Transaction rates are ignored (Feng et al., 2019).

Evaluation metrics

Our purpose was to predict stock prices accurately and balance the return and risk of stock investment. So we employed the following four metrics to evaluate and compare the models: mean square error (MSE), cumulative investment return ratio (IRR), maximum drawdown (MDD), and Sharpe ratio(SR) (Wang, Zhuang & Feng, 2022; Feng et al., 2019; Deng et al., 2016). The detailed formulas are presented in the Table B1 in the Appendix B.

MSE has been a standard evaluation metric for regression tasks in machine learning. Since directly reflecting the effect of stock investment, IRR is our primary metric. It is calculated by summing the return ratios of the selected stock on each testing day. MDD describes the worst possible situation in the investment process, which significantly affects investors’ pessimism. It is a significant risk metric. Finally, SR comprehensively evaluates the performance of an investment behavior from the two dimensions of return and risk and is familiar to investors. It should be noted that the r_f and δ_p in the SR formula are very sensitive to the calculation frequency. This study performs the calculation daily since back-testing uses the daily “buy-hold-sell” strategy. Smaller MSE and MDD values and larger IRR and SR values indicate better performance.

Experimental models & parameter settings

Table 4 shows all compared models in back-testing experiments. It is worth emphasizing that TGC is the main one because it incorporates industry relations from third-party platform into stock predictions and is the state-of-the-art stock relation-based solution. Moreover, its source code and data are open access (Feng et al., 2019).

We draw on previous literature to select the hyperparameters for existing models (Dai, Shao & Lu, 2013; Chen & Ge, 2019; Feng et al., 2019; Liu, Liu & Zhang, 2022). In particular, for SVR, Attn-LSTM and TGC, we follow the original setting in Dai, Shao & Lu (2013); Chen & Ge (2019) and Feng et al. (2019), respectively. For all deep learning methods, we apply the Adam optimizer with a learning rate of 0.001. Furthermore, We employ grid search to select the optimal hyperparameters regarding IRR for other models. We tune two hyperparameters for Attn-LSTM layer in TGC, the length of sequential input T_seq and hidden units C₁ within {5, 10, 20, 60} and {32, 64, 128}, respectively. Besides T_seq and C₁, we further tune the length of trading days T_r in SGR layer. Specifically, we tune T_r within {20, 60, 120, 240}. We futher tune C₃ and C₄ in WGCN layer within {64, 128, 256} and {64, 128, 256}, respectively.

Finally, the specified parameters of our trained TRMF model are as follows:

• Input layer with T_seq = 20, which corresponds to a trading month.

• LSTM layer with C₁ = 64 hidden neurons.

• SGR layer with T_r = 240, which corresponds to a trading year.

• WGCN layer with C₃ = 128 input-to-hidden neurons and C₄ = 128 hidden-to-output neurons.

• Output layer with one neuron and ReLU activation function.

Study of SG-relations

We use the proposed SGR algorithm to mine SG-relations in the NASDAQ and NYSE markets. Then, an experiment will compare SG-relations with the industry relations. Industry relations are extracted from sector-industry data maintained by NASDAQ Inc (https://www.nasdaq.com/screening/industries.aspx). Feng et al. (2019) used them in their TGC model and presented in its appendix.

Figure 4 qualitatively presents the coverage of SG-relations and industry relations via the distribution of intersection colors. Firstly, we can see that the number of red points representing coverage far exceeds the number of blue points representing non-coverage, so SG-relations can largely cover industry relations. Furthermore, a large number of black points are scattered across the graph, which indicates that the SGR algorithm is able to expand the industry relations.

Figure 5 shows examples of SG-relations. Among them, we can see that SG-relations can cover many industry relations (https://en.wikipedia.org/wiki/List_of_S%26P_500_companies), such as FB.O & GOOG.O, AEP.O & LNT.O, and APA.O & FANG.O. Furthermore, there are several non-industry relations in SG-relations. For example, NVDA.O is the supplier of AAPL.O. Their 60-day average prices show the same trend, yet AAPL.O slightly lags behind NADA.O. In addition, NXPI.O has an exclusive relation with XEL.O because the former represents the new energy vehicle industry while the latter represents the traditional fossil energy industry. Their 20-day average prices fluctuate reversely. Finally, EBAY.O’s 60-day average price shows an apparent reverse trend with EXC.O’s for a long time while we cannot figure it out by industry analysis. There may be some internal connection between them. This information can help predict their price more accurately.

Table 5 quantitatively shows coverages of SG-relations with the non-zero number to industry relations. We can see that the SG-relation can largely cover the industry relations provided by the third-party platform. Among them, SG-relation can cover more than 60% of industry relations in the NYSE market, and even more than 80% in the NASDAQ market. On the other hand, the SGR algorithm adds many stock relations that the third-party platforms fail to provide. In both the NYSE and NASDAQ markets, industry relations contain no more than 5% of SG-relations. Although these stock pairs with SG-relation are not in the same industry, there is a strong correlation between their prices in different periods. These relations can be used as important information for stock price prediction. SG-relations generated by different trading data have different coverage rates for industry relations. Of these, the SG-relation corresponding to the daily price has the highest coverage, and the coverage of other trading data decreases with the growth of the period. Figure 6 shows the rule more intuitively. This phenomenon reveals that the short-term prices of companies in the same industry are correlated, while the long-term price correlation can reflect other relationships.

Study on back-testing return of TRMF

Figure 7 presents the cumulative rate of return of all models. Their performance follows the order of TRMF > TGC > Attn-LSTM > WGCN > SVR > Baseline > Five-Factors. And most of the time, TRMF is on top. Therefore, TRMF can bring maximum benefits to investors in the experimental environment.

Table 6 gives the results of each experimental model under four evaluation metrics. We can see that TRMF model outperforms comparison models on most metrics. First, we found both TRMF and TGC outperforms traditional Five-Factors model, SVR, and Attn-LSTM on two markets with great performance w.r.t. IRR, MDD, and SR. It varifies that incorporating stock relation information into the model can significantly improve the stock price prediction. In terms of MDD, the prediction of both TRMF and TGC are significantly smaller than those of other comparable models and benchmark indices. Both them integrate stock relations into the prediction. Therefore, we hypothesized that the relation category factor could effectively reduce the drawdown because the pricing of related stocks gives investors confidence and reduces the overselling situation because of short-term market sentiment. Furthermore, TRMF’s performance was better than the TGC using industry relations, which may be related to the higher number of SG-relations. This indicates that our proposed SG-relations can capture more relation information to enhance the robustness of model predictions. However, TRMF failed to reach the best performance w.r.t. MSE, though it was still the second of all models. The reason could be because TRMF has the most parameters. The WGCN, which only uses relation information, is inferior to the Attn-LSTM which only uses time information in terms of IRR and SR, and it cannot achieve high prediction accuracy with respect to MSE. Therefore, the time series information of stocks is more important, and the stock prices of different stocks retain a large degree of autonomy.