Empirical mode decomposition using deep learning model for financial market forecasting

Deep learning model based on EMD

EMD is a spectrum analysis method proposed by NASA’s signal processing expert (Huang et al., 1998), which analyzes nonlinear and non-stationary data series. EMD is also known as Hilbert-Huang Transform (HHT). It includes two processes: EMD and HHT. Any complex signal can be decomposed into several IMFs through EMD, and the number of IMFs is often limited. These IMF series can well describe each local oscillation of the original data series, with well-performed HHT. Therefore, the Hilbert spectrum obtained has excellent energy time-frequency features (Fu, 2018; Nan et al., 2018).

The EMD method considers that any signal is composed of several eigenmode functions. A signal can contain thousands of eigenmode functions at any time. If the eigenmode functions overlap each other, a composite signal is formed. EMD decomposition aims to obtain the eigenmode function and then perform HHT on each eigenmode function, thereby obtaining the Hilbert spectrum. In this case, the original signal is mentioned below. (1)${\displaystyle S\left(t\right)=\sum _{k=1}^K{\mathrm{IMF}}_k\left(t\right)+r_K\left(t\right).}$

In Eq. (1), S(t) express the original signal which iteratively decomposes a time series, ${\mathrm{IMF}}_k\left(t\right)$ is are called intrinsic mode functions,plus a nonoscillatory trend called the residual term expressed by $r_K\left(t\right)$ and K-level IMF is obtained after EMD decomposition (Zhu et al., 2019), where k = 1, 2, …, K. (2)${\displaystyle r_K\left(t\right)=S\left(t\right)-\sum _{k=1}^K{\mathrm{IMF}}_k\left(t\right).}$

By rearranging Eq. (1), the residual term can be calculated by Eq. (2). Individually, IMFs at all levels are not acquired by explicit convolution calculations; instead, they are obtained through an algorithm. After IMFs at all levels are obtained through the algorithm, each IMF can be a decomposition series and substituted into the BPNN of separation and integration. Therefore, a complete EMD-based separation and integration BPNN model is built. Here, the concept of level is generated by multiple EMD iterations of the data. Therefore, it does not correspond to a strict level of time scale. It is a scale-space representation that reflects the features of local oscillations (Fang et al., 2018; Luo et al., 2019; Ullah et al., 2018; Wen, Gao & Li, 2019). In general, the choice of the objective function is determined by the specific problem. If it is a backpack problem, fitness is the total price of the object in the package (Zhang, Han & Deng, 2018).

FTSEMD

FTS contains information that is different from the general time series in contents and formats. Therefore, financial time series empirical mode decomposition (FTSEMD) is also different from the general time series. Generally, an FTS can be represented by five-time series as Eq. (9). (3)${\displaystyle X\left(t\right)=\left(X.O\left(t\right),X.H\left(t\right),X.L\left(t\right),X.C\left(t\right),X.V\left(t\right)\right).}$

In Eq. (3), $X.O\left(t\right)$ is the time series of opening market price, $X.H\left(t\right)$ is the time series of the highest price, $X.L\left(t\right)$ is the time series of the lowest price, $X.C\left(t\right)$ is the time series of closing market price, and $X.V\left(t\right)$ is the time series of transaction volume. The FTSEMD can utilize various combinations of the above times series to perform EMD. Precisely, the FTS exhibits nonlinear, non-stationary, multiscale, and interval features, its EMD processing is also different from that of the general time series. The daily price of FTS fluctuates between the highest price and the lowest price. Therefore, when building a mathematical model to predict its fluctuation trend, all the information on transaction prices must be thoroughly considered. The conclusions drawn by modelling only with the closing market price will be biased because it ignores other transaction prices. The aim is to build a unique structure of FTS. Hence, an interval EMD algorithm is proposed, which combines the time series of the highest price, the lowest price, and the forecast signal.

Dimensionality reduction after FTSEMD

During processing, multiple regression models often contain more explanatory variables; besides, these variables are correlated, and the information they contained overlaps, making the analysis more complicated. Therefore, for the solutions to the above problems and demonstration of the essential features of the original data, variables that are connected are often indicated by several indicators. This process is dimensionality reduction. Afterwards, these indicators, which will be unconnected, contain most of the information in the original data, thereby benefiting the mathematical modelling.

The second crucial step of the FEPA model is reducing the dimensionality of the IMF components after EMD. The FTS extract data through the forward-scrolling window, and many IMF components are obtained through EMD. Due to the scrolling feature, most of the data entered into the scrolling window are the same each time, except that the last batch of data is deleted, and the latest batch of data is added. After EMD, the data, which are extracted by the forward-scrolling window, contains much redundant information; hence, dimensionality reduction is necessary. Here, the PCA algorithm is adopted to reduce the dimensions of the decomposed IMF components. Afterwards, several principal components, containing most of the information of the original signals, are obtained. The cumulative variance contribution rate of these components must meet particular conditions. The PCA dimensionality reduction after FTSEMD is a significant innovation, and PCA is an essential step in FEPA modelling.

PSO

The PSO algorithm was first presented in Kennedy & Eberhart (1995). The PSO is a random search strategy based on a population of particles. The principle concept of PSO reaches from the social behaviour of flocks birds. In this algorithm, each particle drives in a D-dimension based on its own experience and other particles as well. The PSO algorithm is easy to understand, simple to code, and easy to implement. However, the setting of parameters has a great influence on the performance of the algorithm, such as control convergence, avoiding premature and so on Li et al. (2018); Liang et al. (2018); Wang & Li (2017); Wei-Chang et al. (2010). In PSO, the position of particle i can be represented by the D dimension vector in Eq. (4). (4)${\displaystyle X_i\left(t\right)=\left\{x_{i1},x_{i2},x_{i3},\dots ,x_{iD}\right\}.}$

The velocity at the time is expressed by $V_i\left(t\right)$ which is calculated using Eq. (5). (5)${\displaystyle V_i\left(t\right)=\left\{v_{i1},v_{i2},v_{i3},\dots ,v_{iD}\right\}.}$

The best position of the particle itself is expressed by $P_i\left(t\right)$ which is calculated using Eq. (6). (6)${\displaystyle P_i\left(t\right)=\left\{P_{i1},P_{i2},P_{i3},\dots ,P_{iD}\right\}.}$

The current optimal position of the entire particle swarm is expressed by $P_g\left(t\right)$ which is calculated using Eq. (7). (7)${\displaystyle P_g\left(t\right)=\left\{P_{g1},P_{g2},P_{g3},\dots ,P_{gD}\right\}.}$

The tth generation particle updates velocity and position expressed by $V_i\left(t\right)$ and $X_i\left(t\right)$which are calculated using Eqs. (8) and (9) (Liu et al., 2021).

(8)${\displaystyle V_i\left(t\right)=w{V}_i\left(t-1\right)+c_1{r}_1\left(P_i-X_i\left(t-1\right)\right)+c_2{r}_2\left(P_g-X_i\left(t-1\right)\right)}$ (9)${\displaystyle X_i\left(t\right)=X_i\left(t-1\right)+V_i\left(t\right).}$

The EMD algorithm and PSO algorithm are used to extract the original FTS and obtain the dataset. Then, the dataset is decomposed into eigenmode functions with different scales by the EMD method; meanwhile, the PSO algorithm is used for parameter optimization and prediction. Combining the EMD and PSO algorithms can understand the data features from multiple dimensions, which will effectively improve the control over financial market transactions and accurately predict future financial market transactions.

Empirical analysis of the prediction effect of interval EMD model

EMD decomposes IMFs successively through multiple screening processes, during which the local average of signals is calculated from their upper and lower envelopes of them. The upper and lower envelopes are the local maxima and minima of the signal given by the spline interpolation algorithm. Since both ends of the signal cannot be at the maximum and minimum values at the same time, the upper and lower envelopes will inevitably appear divergently at both ends of the data series. Errors are introduced into the screening process (Cai et al., 2017). As the screening process continues, the result of such divergence will gradually “contaminate” inward the entire data series, causing severe distortions in the results. For long data series, data at both ends can be discarded according to the extreme point, thereby ensuring that the resulting envelope distortion is minimized. However, for short data series, discarding data at both ends becomes completely infeasible.

In general, fluctuations in data series of trading prices in the financial market are random, nonlinear, and non-stationary. The current prediction model is difficult to fully understand the features of various types of data and obtain good prediction results. If a model has an excellent predictive ability for trading prices in the financial market, its value is self-evident.

As a new method to process nonlinear and non-stationary signals, EMD time-frequency analysis is fundamentally different from traditional signal time-frequency analysis methods and has achieved excellent results in practical applications. The EMD decomposition algorithm obtains the IMF components of the signal feature scales at different time points through layer-by-layer screening (Nait Aicha et al., 2018). The primary goal of EMD decomposition is to smooth the signal, perform HHT on the IMF component, and finally, obtain the instantaneous frequency component corresponding to the IMF component. The instantaneous frequency obtained has a reasonable physical meaning. The Hilbert spectrogram obtained is a two-variable function of time and frequency, from which the frequency information at any time can be obtained (Zhang & Zeng, 2017). For example, the magnitude and amplitude of the frequency, as well as the corresponding moments appearing, can be obtained, which can describe the time-frequency features of the non-stationary and nonlinear signal in detail. (10)${\displaystyle \mathrm{MAE}=\frac{1}{n}\times \sum _{i=1}^n\left|\begin{array}{ccc}\hfill T_i\hfill & \hfill -\hfill & \hfill A_i\hfill \end{array}\right|}$ (11)${\displaystyle \mathrm{MAPE}=\frac{1}{n}\times \sum _{i=1}^n\left|\frac{T_i-A_i}{T_i}\right|\times 100\text{\%}}$

(12)${\displaystyle \mathrm{RSME}=\sqrt{\frac{1}{n}\times \sum _{i=1}^n{\left(T_i-A_i\right)}^2}}$ (13)${\displaystyle \mathrm{Hit}\mathrm{Rate}\left(\text{\%}\right)=\frac{\text{Correct Predictions}}{\text{Number of Test Data}}\ast 100}$ (14)${\displaystyle \mathrm{DS}=\frac{100}{n}\times \sum _{u=1}^n{d}_i}$ (15)${\displaystyle d_i=\left\{\begin{array}{cc}1\hfill & \left(T_i-T_{i-1}\right)\left(A_i-A_{i-1}\right)\ge 0\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.}$

(16)${\displaystyle \mathrm{MAD}=\frac{\sum |T_i-A_i|}{n}}$ (17)${\displaystyle \mathrm{TS}=\frac{\sum \left(T_i-A_i\right)}{\mathrm{MAD}}.}$

Equations (10), (11), (12), (13), (16) and (17) are used to calculate the mean absolute error (MAE), mean absolute percentage error (MAPE), root mean squared error (RMSE), hit rate percentage, mean absolute deviation (MAD), and the tracking signal (TS), respectively where T_i and A_i express the actual and forecast value. Although obtaining a precise prediction of the stock index is challenging, a rough prediction of the price trend will help in investment decisions. The EMD algorithm decomposes the time series of the stock index and produces a stationary IMF series, which improves the predictive ability of the model. The time series can be mastered from different scales to reveal the intrinsic laws of data.

Empirical analysis of major financial markets

The financial market is undoubtedly complex, uncertain, and dynamic. In the financial markets, people cannot use a single strategy or model simultaneously; otherwise, they may suffer huge losses. The same model may behave differently in different financial environments. Has the model been applied since many years ago? Is the mature experience of other countries also applicable to the Chinese market? Answers to these questions are unknown. Nevertheless, Shanghai, Shenzhen, Hang Seng, and Dow Jones stock market index data can describe the problems in applications, and the above questions can be answered by empirical analysis.

At present, artificial intelligence (AI) has been widely used in the Internet and manufacturing industries (Hansen & Bøgh, 2021; Qiu, Suganthan & Amaratunga, 2017; Rizvi et al., 2021; Zeba et al., 2021). Whether for the continuous expansion of application fields or the continuous optimization of deep learning algorithms, AI has dramatically improved the traditional working and thinking modes. According to the information collected from news and reports, AI can comprehensively consider whether the content in the collected information is positive or negative for the fundamentals of the financial market; then, it rates the information as very bad, bad, moderate, good, and very good. The deep learning model is trained to determine whether an article conveys positive or negative information through fundamental logic (Zhu et al., 2018). AI algorithms show the excellent ability of market background interpretation in various stages of back-tests. Also, the risk preference index has individual rationality, which shows better early-warning capability when the market upswing terminates. If the market index increases but the risk preference index decreases, the market will be prompted to withdraw the risks, and the effect is noticeable. As the model learns information, the sentiment index and risk preference measures will become accurate.

In Table 2, the statistics of the Shenzhen Index Yield indicate that the kurtosis is 2.286, with a thick tail phenomenon. The thick tail characteristic of the negative deviation direction is more evident than the index. Therefore, the market of the Shenzhen Stock Exchange is more volatile and dynamic than that of the Shanghai Stock Exchange. However, Shanghai Stock Exchange is also more volatile and dynamic than Hang Seng and Dow Jones respectively based on kurtosis.

The EMD-with BPNN has a higher hit rate than other single reference models, and its prediction error is also small. Hence, the EMD algorithm can improve the prediction accuracy of neural networks. The above results indicate that principal component analysis (PCA) can reduce data dimensionality, compress redundant data, improve prediction accuracy, and shorten the data training time of neural networks.

Table 3 provided the evaluation of the forecast model based on RMSE, MAPE, MAE and TS. In all aspects of statistical comparison, our proposed forecasting method performed better than the benchmark RW method.

Figure 2 expresses the IMF1, IMF2, IMF3, and IMF4 component map on the provided data for the US dollar against the CNY exchange rate. The comparison of USD to CNY exchange rate’s actual data and forecast value as the output system is expressed in Fig. 3. The upper part of Fig. 3 represents the graphical view of actual data and forecast data for visualization of the differences. Moreover, the bottom part of Fig. 3 represents the curve fitting plots of USD to CNY exchange rate for the forecast versus actual data. In the case of USD to CNY exchange rate forecasting proposed method’s accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.011061, 0.001423, 0.009247, and 10.36, respectively. Similarly, benchmark method RW accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.037337, 0.004162, 0.027068, and −66.88, respectively. From this comparison, it is clear that the proposed method performs better than the benchmark method for USD to CNY exchange rate forecasting.

Figure 4 expresses the IMF1, IMF2, IMF3, and IMF4 component map on the provided data for the EURO against the CNY exchange rate. The comparison of EURO to CNY exchange rate’s actual data and forecast value as the output system is expressed in Fig. 5. The upper part of Fig. 5 represents the graphical view of actual data and forecast data for visualization of the differences. Moreover, the bottom part of Fig. 5 represents the curve fitting plots of EURO to CNY exchange rate for the forecast versus actual data. For EURO to CNY exchange rate forecasting proposed method’s accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.018999, 0.002051, 0.016009, and −18.44, respectively. Similarly, benchmark method RW accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.039198, 0.004184, 0.032679, and −47.80, respectively. From this comparison, it is clear that the proposed method performs better than the benchmark method for EURO to CNY exchange rate forecasting. Figure 6 expresses the IMF1, IMF2, IMF3, and IMF4 component map on the provided data for the JPY against the CNY exchange rate. The comparison of JPY to CNY exchange rate’s actual data and forecast value as the output system is expressed in Fig. 7. The upper part of Fig. 7 represents the graphical view of actual data and forecast data for visualization of the differences. Moreover, the bottom part of Fig. 7 represents the curve fitting plots of JPY to CNY exchange rate for the forecast versus actual data. In the case of JPY to CNY exchange rate forecasting proposed method’s accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.000206, 0.002450, 0.000149, and −0.32, respectively. Similarly, benchmark method RW accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.004327, 0.049092, 0.002951, and −72.49, respectively. From this comparison, it is clear that the proposed method performs better than the benchmark method for JPY to CNY exchange rate forecasting.

Figure 8 expresses the IMF1, IMF2, IMF3, and IMF4 component map on the provided data for the CHF against the CNY exchange rate. The comparison of CHF to CNY exchange rate’s actual data and forecast value as the output system is expressed in Fig. 9. The upper part of Fig. 9 represents the graphical view of actual data and forecast data for visualization of the differences. Moreover, the bottom part of Fig. 9 represents the curve fitting plots of CHF to CNY exchange rate for the forecast versus actual data. For CHF to CNY exchange rate forecasting proposed method’s accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.019752, 0.002249, 0.016032, and −18.09, respectively. Similarly, benchmark method RW accuracy in terms of RMSE, MAPE, MAE, and TS are reported as 0.165387, 0.0139802, 0.100183, and −50.31, respectively. From this comparison, it is clear that the proposed method performs relatively better than the benchmark method for CHF to CNY exchange rate forecasting.

The EMD-BPNN model has a higher hit rate than other single reference models, while the prediction error is smaller. This shows that the EMD decomposition algorithm can improve the prediction accuracy of the neural network. This indicates that principal component analysis can reduce dimensionality and compress redundant data, improve prediction accuracy to a certain extent, and shorten the time of neural network training data.Notably, while predicting, one or more components of the highest frequency may be discarded. Therefore, the influence of high-frequency noise on prediction can be suppressed. Therefore, to eliminate the trend, except for the last component or components, all the extracted IMFs are added as decomposition results. Such a process can be easily combined with the smoothing of the results obtained if the highest frequency components have been discarded from the process of adding up the components.

Combination of deep learning and financial transaction

Deep learning can be used in various frequency trading, from low-frequency stock-picking models to high-frequency algorithmic trading models. Deep learning has been a thriving industry case at both levels of investment decision-making and transaction execution. For example, the hedge fund Cerebellum, which was established in 2009, manages assets of $90 billion, uses AI for adjunct forecasting and has been profitable every year since 2009. Man Group, one of the world’s most significant hedge funds, adopted AI to implement passive investment five years ago. Currently, the assets managed by AI have stable profits. Wall Street investment banks, such as Goldman Sachs and JPMorgan Chase, have also invested in AI stock-picking models. It is believed that machines can predict the results accurately through “deep learning” and reduce unnecessary transaction risks.

Deep learning is a method of learning the laws in massive data through DNN models. Deep learning ANNs are widely interconnected by numerous neurons, which are imitations of biological neural networks (brains). It is a nonlinear, distributed parallel processing, as well as a self-confirming algorithm model. Neurons are the fundamental units constituting a neural network. A neuron receives input signals sent from other neurons and produces inputs. In mathematics, a neuron is equivalent to a nonlinear transformation (excitation function). When a group of neurons is combined and has a hierarchical structure, a neural network model is formed. As deep learning develops, AI has made technical breakthroughs in many fields, such as image, speech, and natural speech processing. Currently, practical applications of AI are various and those in the financial field are also flourishing. In the meantime, deep learning is very suitable for financial prediction analysis in the context of big data. If deep learning is used, supplemented by a technique similar to knowledge maps, various events that have a significant influence on finance can be expressed in the form of knowledge maps. Then, features are automatically selected through deep networks for parameter and weight adjustments. The results can be more accurate and objective, and even those that have not been anticipated can be achieved.

Due to congregational psychology, humans are easily influenced by the surrounding environment during investments. The circular neural network has been widely applied in the field of natural language processing and has achieved great success. Such technologies make it possible to comprehend public opinion more accurately, thereby extracting the events that may affect the financial market. Combined with the above methods, various market states can be understood, providing users with better services.

Deep learning is used in areas such as financial risk control and big data credit. Hopefully, more new applications will emerge in the future. The use of deep learning in the financial field will lead to more intelligent management and consumption methods.

Empirical results and discussion

Here, an interval EMD algorithm is proposed based on the FEPA model. The research sample is the return rate of each index, and the forecast performance of the interval EMD model is tested empirically. The major conclusions include:

(1) The interval EMD model improves the prediction performance based on the FEPA model. Compared to the FEPA model, the prediction error of the interval EMD model is reduced. Compared to other reference models, the prediction error of the interval EMD model is much smaller. Compared to the FEPA model, the hit rate of the interval EMD model in predicting the closing market price increases by only 2%. However, the hit rate of the interval EMD model in predicting the highest and lowest prices increases by about 6% to 8% more than the FEPA model. Such increases show that the interval EMD model is useful in predicting FTS, especially the short-term fluctuation trends of the highest and lowest prices.

(2) Comprehensive and efficient utilization of transaction price information help improve forecast accuracy. In actual transactions, analysts will utilize comprehensive price information to predict the price trends of the future market. The empirical results suggest that if only the closing market price is considered, almost all data series are very similar to random walks. However, the interval EMD model that utilizes comprehensive transaction price information can improve the predictive ability of fluctuation trends in the stock index.