# Financial Market Prediction and Simulation Based on the FEPA Model

Authors: Dehui Zhou
Journal: Journal of Mathematics
 (2021)

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## Abstract

Since the birth of the financial market, the industry and academia want to find a method to accurately predict the future trend of the financial market. The ultimate goal of this paper is to build a mathematical model that can effectively predict the short-term trend of the financial time series. This paper presents a new combined forecasting model: its name is Financial Time Series-Empirical Mode Decomposition-Principal Component Analysis-Artificial Neural Network (FEPA) model. This model is mainly composed of three components, which are based on financial time series special empirical mode decomposition (FTA-EMD), principal component analysis (PCA), and artificial neural network. This model is mainly used to model and predict the complex financial time series. At the same time, the model also predicts the stock market index and exchange rate and studies the hot fields of the financial market. The results show that the empirical mode decomposition back propagation neural network (EMD-BPNN) model has better prediction effect than the autoregressive comprehensive moving average model (ARIMA), which is mainly reflected in the accuracy of prediction. This shows that the prediction method of decomposing and recombining nonlinear and nonstationary financial time series can effectively improve the prediction accuracy. When predicting the closing price of Australian stock index, the hit rate (DS) of the FEPA model decomposition method is 72.22%, 10.86% higher than the EMD-BPNN model and 3.23% higher than the EMD-LPP-BPNN model. When the FEPA model predicts the Australian stock index, the hit rate is improved to a certain extent, and the effect is better than other models.


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## Body

## 1. Introduction

The financial market is a collection of very complex systems [1]. The complexity of various elements of the financial market makes the financial market difficult to predict. The study of the law of price fluctuations in financial markets has always attracted the attention of economists and has formed many traditional classic theories about market operations. Among these theories, the most notable are the technical analysis theory that guides people’s practice, the efficient market hypothesis that lays the foundation of the capital market theory building, and the bubble theory that explains huge fluctuations. However, these theories have failed to reveal the deep-seated complexity laws contained in the financial market, and therefore, it is difficult to help people avoid major financial market turmoil and serious financial crises. In short, the complexity of the financial market often causes turbulence and even severe recession in the domestic economy and even the world economy. Therefore, it is necessary and meaningful to carry out research on the complexity law of financial market price fluctuations [2]. The scientific theory of complexity has only become a theoretical hot spot in the study of the law of price fluctuations in financial markets in the past ten years. A large number of domestic and foreign studies have proved that the financial market is a chaotic dynamic system. This new discovery has opened up a new way for us to reunderstand the price fluctuation mechanism of financial markets. At present, international research on complexity science is in the ascendant. To scientifically explore the mechanism of financial market complexity, there are still many issues worthy of study. Among the theories that believe that the securities market is regular and can be mastered, there are mainly the technical school (also known as the chart school) and the basic school (also known as the basic solid school). The former believes that history is the basis of reality, and analysis and research based on historical data of previous market operations can predict future market price trends. The latter believes that value is the basis and basis of price, and securities have their intrinsic value. In different periods, the price of securities can be lower or higher than its intrinsic value, but in general, the price of securities will return to the level of its intrinsic value. Since the intrinsic value of securities can be obtained by estimating expected dividends and the discounted value of returns, people can make correct judgments on the trend of securities prices. Over the past two decades, high-frequency data have become more and more accessible, which has promoted the development of high-frequency algorithm trading, such as volume weighted average price algorithm trading strategy (VWAP), volume fixed percentage algorithm trading strategy (VP), time weighted average price algorithm trading strategy (TWAP), etc. These high-frequency trading algorithms have been widely used in the financial market [3]. From the 1950s to the present, information science and technology and computer networks have developed by leaps and bounds, and big data materials are becoming more and more abundant. In the past, people used to predict financial market prices based on economics and finance, but now they use a variety of cross disciplinary combination models to predict financial markets, which makes financial market forecasting develop into a unique financial research field. The research content and purpose of this paper can be described as follows: using certain methods to extract the laws and information contained in financial time series to the greatest extent. According to the information reflected by the existing financial time series, a robust hybrid prediction model is established, and the model is used to predict the short-term operation trend of the financial time series, so as to provide a reference for the investment decision making of market investors [4].The securities market always plays the role of a barometer of the economic situation [5]. Both institutional investors and individual investors are trying to find a better investment strategy in order to make a profit in the securities market [6]. Profit is the goal of every investor, but the proportion of investors who can really make profit is very low. Due to transaction commissions and other expenses, trading at the same price is a loss. Therefore, finding an effective prediction model to improve the prediction accuracy and provide reference for investors’ investment decision making is a challenging topic in the industry and academia. The FEPA model proposed in this paper is a combined prediction model, which combines the advantages of empirical mode decomposition, principal component analysis, and artificial neural network. Compared with other financial market prediction methods, FEPA has stronger prediction accuracy and is more sensitive to the risks in the financial market. Therefore, the main references involved in this paper include EMD research status, PCA research status, artificial neural network research status, and so on. In order to make the FEPA combination prediction model proposed in this paper have reference objects, six models such as Arima are introduced as reference models [7]. The hybrid model based on EEMD is widely used in the prediction of related problems, and many scholars have achieved good results in this aspect [8–11]. Based on the idea of decomposition reconstruction synthesis, the FEPA model organically combines EMD, PCA, and back propagation neural network to form a combined prediction model. The algorithm flow of the model is as follows. First, an appropriate length window is used to scroll the intercepted data, and the intercepted data are decomposed into IMF components from high frequency to low frequency by EMD algorithm. Then, the dimensionality of EMD decomposed prediction variable IMF sequence and trend term is reduced by PCA algorithm. Finally, this article mainly applies this model to the two stock markets of China and Australia. China has become an economy with an important influence in the world. The trend of China’s stock market also affects the trend of world stock markets to a certain extent. Therefore, it is of great significance to study the laws of Chinese stocks. Australia is another important developed country, so choosing the Australian stock index is also of great significance for us to explore FEPA’s predictions for financial markets. To sum up, this paper selects CSI 300 index and Australian stock index as the target market [12].


## 2. Related Work

The research on market predictability is closely related to the development of financial theory. The early empirical research on the predictability of market prices promoted the birth of the efficient market hypothesis. Before the efficient market hypothesis was put forward, there were many literatures on the predictability of securities prices. For example, Liang et al. [13] found that the changes of securities prices and commodity prices in European and American markets were random; Nápoles et al. [14] found that the stock price time series in the United States is random and cannot be distinguished from the series generated by a random number series; Tonziello et al. [15] found that the characteristics of stock price behavior are similar to those of particles in fluid and accord with the characteristics of random walk. It is these empirical studies that found the unpredictability of market price that prompted economists to think about the reasons for the random walk of securities price and finally promoted the formation of efficient market hypothesis theory. In recent years, the empirical study of market price predictability has promoted the rise of behavioral finance. For example, Sargun et al. [16] found that the price fluctuation in the U.S. stock market is too large to be explained by fundamental information; When Nguyen and Duong [17] studied the stocks on the New York Stock Exchange, they found that the stocks with the best performance in three to five years will have the lowest market adjusted return in the same period of time; Coba Salcedo et al. [18] found that the dividend price ratio can significantly predict the future yield; and Pembury Smith and Ruxton [19] found that trading strategies based on price momentum over the past 3 to 12 months can obtain excess returns. In addition, many other studies have found that financial markets are predictable. Okewu et al. [20] regards market predictability as a new fact in finance. The abnormal fluctuation of stock market price and more and more market anomalies urge some economists and financiers to re-examine the efficient market hypothesis. It can be said that the research on the predictability of financial markets has once again promoted the rise of behavioral finance. The research on market predictability can provide reference for the government to formulate relevant financial policies, and it is also an important tool for investors to carry out technical analysis. In the past decades, traditional statistical and econometric techniques have been widely used in financial market price forecasting, such as cointegration analysis, linear regression, random walk model, GARCH family model, vector regression, and error correction model. Yang et al. [21] used the Fourier transform of finite dimensional Hilbert space to predict the change trend of stock return. However, Fourier transform is appropriate only when the processed signal is linear and stable. Ma et al. [22] proposed three famous feature selection methods, namely principal component analysis, decision tree, and genetic algorithm. The results show that the combination model of principal component analysis and genetic algorithm and the combination model of principal component analysis, genetic algorithm, and decision tree have the best empirical effect. Henrique et al. [23] proposed that the principal components extracted by the principal component analysis algorithm are orthogonal to each other, and the market shows short-term memory and long-term memory. Senoguchi [24] studied the transmission mechanism of China’s stock market and the U.S. stock market through the principal component analysis method. The results show that China’s stock market is little affected by the global financial crisis and less affected by other market fluctuations than the U.S. market. Huang and Liu [25] designed and ran a Monte Carlo experiment. The results show that the parameter estimation of the affine term structure model (ATSM) based on principal component analysis is robust and can realize error self-correction. Principal component analysis is widely used in the fields of stock index prediction and financial risk evaluation of commercial banks. Based on the traditional support vector machine method, Pang et al. [26] introduced genetic algorithm and principal component analysis to construct PCA-GA-SVM model and used the model to analyze the trend of CSI 300 index and the top five constituent stocks. Zhang et al. [27] constructed a comprehensive evaluation model of financial risk of commercial banks based on RBF neural network and principal component analysis. The results show that the model provides new methods and ideas for financial risk assessment and monitoring of commercial banks. Prokop and Kammann [28] used TOPSIS method, principal component analysis method, and factor analysis method to comprehensively evaluate and rank the operating performance of 53 listed companies whose main business is power industry in Shanghai and Shenzhen stock market.


### 2.1. FEPA Model

As we all know, the price of stocks changes with changes in the market. Therefore, it is impossible to accurately predict the stock market. The FEPA model proposed in this paper combines the three ideas of decomposition, reconstruction, and integration to make predictions. Therefore, this model has the advantage that a single method cannot be compared. The FEPA model has significant advantages in forecasting methods for financial markets, which are mainly reflected in the scope and accuracy of the forecast. The main flow chart of the forecast is shown in Figure1.Figure 1
The main flow chart of the FEPA model.The FEPA model is the time series decomposition algorithm which is developing rapidly and widely used. It includes two processes: decomposition and reconstruction. It is a time-frequency localization analysis method. Principal component analysis is an important innovation of the FEPA model. Principal component analysis can reduce the dimension of decomposed data, extract the main information of decomposed data, and shorten the training time [29]. When using EMD decomposition algorithm in the FEPA model, the information of highest price and lowest price is introduced to improve FEPA model. The empirical results show that applying the interval EMD decomposition algorithm to the FEPA model can effectively improve the prediction effect of the highest and lowest prices, although the prediction effect of the closing price is not much improved. The empirical results show that the interval EMD decomposition algorithm can achieve good results in predicting the change trend of interval end point value.


#### 2.1.1. EMD Decomposition Algorithm

EMD has the advantages of data-driven adaptability, can analyze nonlinear and nonstationary signals, and is not restricted by Heisenberg uncertainty principle. EMD has significant advantages in nonlinear and nonstationary signal analysis. Compared with the traditional time-frequency analysis technology, EMD does not need to select the basis function, and its decomposition is based on the distribution of extreme points of the signal itself. In this paper, a sliding window EMD technology with forward scrolling day by day is proposed. The data are extracted by scrolling day by day with a window of appropriate length, and the extracted data are decomposed by EMD. An important concept of EMD method is instantaneous frequency. Applying instantaneous frequency to signal time-frequency analysis is an innovation of the EMD method. In the past, we were familiar with the concept of signal instantaneous amplitude, but not so familiar with instantaneous frequency. The object of Fourier analysis is stationary signal. Each harmonic obtained by signal decomposition has a fixed frequency, and the frequency does not change with time. The frequency of nonstationary series is not fixed, but changes with time, so it is necessary to use the new parameter instantaneous frequency to describe nonstationary series. Given any continuous time functionX(t), its Hilbert transform Y(t) is calculated as follows:(1)Yt=π3p∫−∞+∞1−t21+t2Xttdt.This transformation exists for all functions. By observing equation (1), it can be seen that Hilbert transform emphasizes the locality of X (t) and realizes the conversion of signal from time domain to time domain, while Fourier transform is often used to realize the conversion of signal from time domain to frequency domain, which is the difference between the two. According to equation (1), an analytical signal T(t) can be constructed:(2)Tt=−i22Yt+iXt=Atei/πωt,(3)At=Xt2+Yt2,ωt=arccosYtXt2+Yt2.From the definition formula (2) of T(t), X(t), and Y(t) are the real and imaginary parts of T(t), respectively. Equation (3) defines the amplitude and angle of T(t). The instantaneous frequency can be defined by the time derivative of the argument.(4)θt=dωtdt.Equation (4) shows that the instantaneous frequency at a given time, the signal is a single valued function of time, and there is only one frequency value. When the number of zero crossings is equal to the number of extreme points, the instantaneous frequency defines the signal as narrowband.According to the above definition, a typical IMF graph is given, as shown in Figure2. IMF is called intrinsic mode function because it represents inherent and inherent oscillation modes. “Oscillation” here generally refers to nonperiodic fluctuation and fluctuation changes. Whether the oscillation is periodic is detected by the significance test.Figure 2
Typical eigenmode functions.With the definition of eigenmode function, Hilbert transform can be performed on each IMF component to obtain the corresponding analytical signalA(t), and then the instantaneous frequency of each component can be obtained. For this purpose, the Fourier transform of A(t) is as follows:(5)Mθ=λ∫−∞+∞Ate2iωtei2atdt=λ∫−∞+∞Ate2iωt+i2atdt.According to the stationary phase method, the frequency at whichW reaches the maximum value shall meet the following conditions:(6)dωtdt+π2θ=0.The instantaneous frequency in equation (6) is obtained by stationary phase approximation. For the case of amplitude gradient, the definition of instantaneous frequency is consistent with that in the classical wave theory.The screening process of EMD is shown in Figure3. EMD data decomposition is sensitive to adding new data and local disturbances, which is determined by the termination condition of IMF sequence decomposition. Wu and Huang proposed a solution to this problem. They suggested fixing the screening times of decomposed data. In order to ensure the stability and convergence of decomposed data, the maximum number of screening is usually 10. As can be seen from Figure 3, iteration cycle and component cycle constitute the two main cycles of EMD screening process. In addition, EMD algorithm has two important parameters: the number of IMF m and the window length of time series W. These parameters are usually set as needed in practical applications. For example, when analyzing financial time series, the number of IMF decomposed is usually 4 to10.Figure 3
Flow chart of EMD screening process.


#### 2.1.2. Principal Component Extraction after EMD Decomposition

Principal component analysis is a statistical method of dimensionality reduction. It uses an orthogonal transformation to transform the original random vector whose components are related to a new random vector whose components are not related. This is represented algebraically as the original random vector. The covariance matrix is transformed into a diagonal matrix, which is geometrically expressed as transforming the original coordinate system into a new orthogonal coordinate system, making it point to thep orthogonal directions where the sample points are spread the most, and then reducing the multidimensional variable system. Dimensional processing enables it to be converted into a low-dimensional variable system with a higher precision, and then by constructing an appropriate value function; the low-dimensional system is further transformed into a one-dimensional system.Through principal component analysis, the matrix multiplication can be expressed as formula (7):(7)T1⋮Tn=λa11⋯a1n⋮⋱⋮an1⋯annX1⋮Xn+χY1⋮Yn.We transform the variableXt into a set of independent variables Yt through orthogonal transformations. The two variables satisfy the following relationship:(8)∏i=1n∬Xt=∏i=1n∬Yt.There are many methods to determine the principal components, and KMO sample measurement is often used. (Kaiser Meyer Olkin) KMO test statistic is an index used to compare simple correlation coefficient and partial correlation coefficient between variables. It is mainly applied to factor analysis of multivariate statistics. The calculation formula of KMO test statistics is(9)KMO=∪ij1+xij2∪inxij2−∪inxij∪jnyij+∪inyij2.


#### 2.1.3. Design of Back Propagation Neural Network

The neural network has generalization ability through training and learning and can establish the mapping relationship between sample input data and output data. The neural network uses training samples to find the inherent law of the sample data mapping relationship, rather than simply memorizing the sample input, so as to correctly predict the missing input-output mapping relationship. Figure4 shows a three-layer back propagation neural network with an input layer, an output layer, and a hidden layer.Figure 4
Structure of back propagation neural network.We useFi to denote the input factor, Yi and Lj to denote the output factors of the hidden layer and the output layer, and Yi and Lj can be expressed as(10)Yi=fxi=fθij−∏ijθijxj,Lj=fyj=fθij−∏ijθijxi.Due to the existence of hidden layer, the nonlinear separable problem is difficult to learn. Back propagation neural network algorithm can improve the speed of multilayer network. A part of the multilayer forwarding network is shown in Figure5. The solid arrow is the function signal, and the dotted arrow is the error signal.Figure 5
Signal flow in multilayer perceptron.


## 2.1. FEPA Model

As we all know, the price of stocks changes with changes in the market. Therefore, it is impossible to accurately predict the stock market. The FEPA model proposed in this paper combines the three ideas of decomposition, reconstruction, and integration to make predictions. Therefore, this model has the advantage that a single method cannot be compared. The FEPA model has significant advantages in forecasting methods for financial markets, which are mainly reflected in the scope and accuracy of the forecast. The main flow chart of the forecast is shown in Figure1.Figure 1
The main flow chart of the FEPA model.The FEPA model is the time series decomposition algorithm which is developing rapidly and widely used. It includes two processes: decomposition and reconstruction. It is a time-frequency localization analysis method. Principal component analysis is an important innovation of the FEPA model. Principal component analysis can reduce the dimension of decomposed data, extract the main information of decomposed data, and shorten the training time [29]. When using EMD decomposition algorithm in the FEPA model, the information of highest price and lowest price is introduced to improve FEPA model. The empirical results show that applying the interval EMD decomposition algorithm to the FEPA model can effectively improve the prediction effect of the highest and lowest prices, although the prediction effect of the closing price is not much improved. The empirical results show that the interval EMD decomposition algorithm can achieve good results in predicting the change trend of interval end point value.


### 2.1.1. EMD Decomposition Algorithm

EMD has the advantages of data-driven adaptability, can analyze nonlinear and nonstationary signals, and is not restricted by Heisenberg uncertainty principle. EMD has significant advantages in nonlinear and nonstationary signal analysis. Compared with the traditional time-frequency analysis technology, EMD does not need to select the basis function, and its decomposition is based on the distribution of extreme points of the signal itself. In this paper, a sliding window EMD technology with forward scrolling day by day is proposed. The data are extracted by scrolling day by day with a window of appropriate length, and the extracted data are decomposed by EMD. An important concept of EMD method is instantaneous frequency. Applying instantaneous frequency to signal time-frequency analysis is an innovation of the EMD method. In the past, we were familiar with the concept of signal instantaneous amplitude, but not so familiar with instantaneous frequency. The object of Fourier analysis is stationary signal. Each harmonic obtained by signal decomposition has a fixed frequency, and the frequency does not change with time. The frequency of nonstationary series is not fixed, but changes with time, so it is necessary to use the new parameter instantaneous frequency to describe nonstationary series. Given any continuous time functionX(t), its Hilbert transform Y(t) is calculated as follows:(1)Yt=π3p∫−∞+∞1−t21+t2Xttdt.This transformation exists for all functions. By observing equation (1), it can be seen that Hilbert transform emphasizes the locality of X (t) and realizes the conversion of signal from time domain to time domain, while Fourier transform is often used to realize the conversion of signal from time domain to frequency domain, which is the difference between the two. According to equation (1), an analytical signal T(t) can be constructed:(2)Tt=−i22Yt+iXt=Atei/πωt,(3)At=Xt2+Yt2,ωt=arccosYtXt2+Yt2.From the definition formula (2) of T(t), X(t), and Y(t) are the real and imaginary parts of T(t), respectively. Equation (3) defines the amplitude and angle of T(t). The instantaneous frequency can be defined by the time derivative of the argument.(4)θt=dωtdt.Equation (4) shows that the instantaneous frequency at a given time, the signal is a single valued function of time, and there is only one frequency value. When the number of zero crossings is equal to the number of extreme points, the instantaneous frequency defines the signal as narrowband.According to the above definition, a typical IMF graph is given, as shown in Figure2. IMF is called intrinsic mode function because it represents inherent and inherent oscillation modes. “Oscillation” here generally refers to nonperiodic fluctuation and fluctuation changes. Whether the oscillation is periodic is detected by the significance test.Figure 2
Typical eigenmode functions.With the definition of eigenmode function, Hilbert transform can be performed on each IMF component to obtain the corresponding analytical signalA(t), and then the instantaneous frequency of each component can be obtained. For this purpose, the Fourier transform of A(t) is as follows:(5)Mθ=λ∫−∞+∞Ate2iωtei2atdt=λ∫−∞+∞Ate2iωt+i2atdt.According to the stationary phase method, the frequency at whichW reaches the maximum value shall meet the following conditions:(6)dωtdt+π2θ=0.The instantaneous frequency in equation (6) is obtained by stationary phase approximation. For the case of amplitude gradient, the definition of instantaneous frequency is consistent with that in the classical wave theory.The screening process of EMD is shown in Figure3. EMD data decomposition is sensitive to adding new data and local disturbances, which is determined by the termination condition of IMF sequence decomposition. Wu and Huang proposed a solution to this problem. They suggested fixing the screening times of decomposed data. In order to ensure the stability and convergence of decomposed data, the maximum number of screening is usually 10. As can be seen from Figure 3, iteration cycle and component cycle constitute the two main cycles of EMD screening process. In addition, EMD algorithm has two important parameters: the number of IMF m and the window length of time series W. These parameters are usually set as needed in practical applications. For example, when analyzing financial time series, the number of IMF decomposed is usually 4 to10.Figure 3
Flow chart of EMD screening process.


### 2.1.2. Principal Component Extraction after EMD Decomposition

Principal component analysis is a statistical method of dimensionality reduction. It uses an orthogonal transformation to transform the original random vector whose components are related to a new random vector whose components are not related. This is represented algebraically as the original random vector. The covariance matrix is transformed into a diagonal matrix, which is geometrically expressed as transforming the original coordinate system into a new orthogonal coordinate system, making it point to thep orthogonal directions where the sample points are spread the most, and then reducing the multidimensional variable system. Dimensional processing enables it to be converted into a low-dimensional variable system with a higher precision, and then by constructing an appropriate value function; the low-dimensional system is further transformed into a one-dimensional system.Through principal component analysis, the matrix multiplication can be expressed as formula (7):(7)T1⋮Tn=λa11⋯a1n⋮⋱⋮an1⋯annX1⋮Xn+χY1⋮Yn.We transform the variableXt into a set of independent variables Yt through orthogonal transformations. The two variables satisfy the following relationship:(8)∏i=1n∬Xt=∏i=1n∬Yt.There are many methods to determine the principal components, and KMO sample measurement is often used. (Kaiser Meyer Olkin) KMO test statistic is an index used to compare simple correlation coefficient and partial correlation coefficient between variables. It is mainly applied to factor analysis of multivariate statistics. The calculation formula of KMO test statistics is(9)KMO=∪ij1+xij2∪inxij2−∪inxij∪jnyij+∪inyij2.


### 2.1.3. Design of Back Propagation Neural Network

The neural network has generalization ability through training and learning and can establish the mapping relationship between sample input data and output data. The neural network uses training samples to find the inherent law of the sample data mapping relationship, rather than simply memorizing the sample input, so as to correctly predict the missing input-output mapping relationship. Figure4 shows a three-layer back propagation neural network with an input layer, an output layer, and a hidden layer.Figure 4
Structure of back propagation neural network.We useFi to denote the input factor, Yi and Lj to denote the output factors of the hidden layer and the output layer, and Yi and Lj can be expressed as(10)Yi=fxi=fθij−∏ijθijxj,Lj=fyj=fθij−∏ijθijxi.Due to the existence of hidden layer, the nonlinear separable problem is difficult to learn. Back propagation neural network algorithm can improve the speed of multilayer network. A part of the multilayer forwarding network is shown in Figure5. The solid arrow is the function signal, and the dotted arrow is the error signal.Figure 5
Signal flow in multilayer perceptron.


## 2.1.1. EMD Decomposition Algorithm

EMD has the advantages of data-driven adaptability, can analyze nonlinear and nonstationary signals, and is not restricted by Heisenberg uncertainty principle. EMD has significant advantages in nonlinear and nonstationary signal analysis. Compared with the traditional time-frequency analysis technology, EMD does not need to select the basis function, and its decomposition is based on the distribution of extreme points of the signal itself. In this paper, a sliding window EMD technology with forward scrolling day by day is proposed. The data are extracted by scrolling day by day with a window of appropriate length, and the extracted data are decomposed by EMD. An important concept of EMD method is instantaneous frequency. Applying instantaneous frequency to signal time-frequency analysis is an innovation of the EMD method. In the past, we were familiar with the concept of signal instantaneous amplitude, but not so familiar with instantaneous frequency. The object of Fourier analysis is stationary signal. Each harmonic obtained by signal decomposition has a fixed frequency, and the frequency does not change with time. The frequency of nonstationary series is not fixed, but changes with time, so it is necessary to use the new parameter instantaneous frequency to describe nonstationary series. Given any continuous time functionX(t), its Hilbert transform Y(t) is calculated as follows:(1)Yt=π3p∫−∞+∞1−t21+t2Xttdt.This transformation exists for all functions. By observing equation (1), it can be seen that Hilbert transform emphasizes the locality of X (t) and realizes the conversion of signal from time domain to time domain, while Fourier transform is often used to realize the conversion of signal from time domain to frequency domain, which is the difference between the two. According to equation (1), an analytical signal T(t) can be constructed:(2)Tt=−i22Yt+iXt=Atei/πωt,(3)At=Xt2+Yt2,ωt=arccosYtXt2+Yt2.From the definition formula (2) of T(t), X(t), and Y(t) are the real and imaginary parts of T(t), respectively. Equation (3) defines the amplitude and angle of T(t). The instantaneous frequency can be defined by the time derivative of the argument.(4)θt=dωtdt.Equation (4) shows that the instantaneous frequency at a given time, the signal is a single valued function of time, and there is only one frequency value. When the number of zero crossings is equal to the number of extreme points, the instantaneous frequency defines the signal as narrowband.According to the above definition, a typical IMF graph is given, as shown in Figure2. IMF is called intrinsic mode function because it represents inherent and inherent oscillation modes. “Oscillation” here generally refers to nonperiodic fluctuation and fluctuation changes. Whether the oscillation is periodic is detected by the significance test.Figure 2
Typical eigenmode functions.With the definition of eigenmode function, Hilbert transform can be performed on each IMF component to obtain the corresponding analytical signalA(t), and then the instantaneous frequency of each component can be obtained. For this purpose, the Fourier transform of A(t) is as follows:(5)Mθ=λ∫−∞+∞Ate2iωtei2atdt=λ∫−∞+∞Ate2iωt+i2atdt.According to the stationary phase method, the frequency at whichW reaches the maximum value shall meet the following conditions:(6)dωtdt+π2θ=0.The instantaneous frequency in equation (6) is obtained by stationary phase approximation. For the case of amplitude gradient, the definition of instantaneous frequency is consistent with that in the classical wave theory.The screening process of EMD is shown in Figure3. EMD data decomposition is sensitive to adding new data and local disturbances, which is determined by the termination condition of IMF sequence decomposition. Wu and Huang proposed a solution to this problem. They suggested fixing the screening times of decomposed data. In order to ensure the stability and convergence of decomposed data, the maximum number of screening is usually 10. As can be seen from Figure 3, iteration cycle and component cycle constitute the two main cycles of EMD screening process. In addition, EMD algorithm has two important parameters: the number of IMF m and the window length of time series W. These parameters are usually set as needed in practical applications. For example, when analyzing financial time series, the number of IMF decomposed is usually 4 to10.Figure 3
Flow chart of EMD screening process.


## 2.1.2. Principal Component Extraction after EMD Decomposition

Principal component analysis is a statistical method of dimensionality reduction. It uses an orthogonal transformation to transform the original random vector whose components are related to a new random vector whose components are not related. This is represented algebraically as the original random vector. The covariance matrix is transformed into a diagonal matrix, which is geometrically expressed as transforming the original coordinate system into a new orthogonal coordinate system, making it point to thep orthogonal directions where the sample points are spread the most, and then reducing the multidimensional variable system. Dimensional processing enables it to be converted into a low-dimensional variable system with a higher precision, and then by constructing an appropriate value function; the low-dimensional system is further transformed into a one-dimensional system.Through principal component analysis, the matrix multiplication can be expressed as formula (7):(7)T1⋮Tn=λa11⋯a1n⋮⋱⋮an1⋯annX1⋮Xn+χY1⋮Yn.We transform the variableXt into a set of independent variables Yt through orthogonal transformations. The two variables satisfy the following relationship:(8)∏i=1n∬Xt=∏i=1n∬Yt.There are many methods to determine the principal components, and KMO sample measurement is often used. (Kaiser Meyer Olkin) KMO test statistic is an index used to compare simple correlation coefficient and partial correlation coefficient between variables. It is mainly applied to factor analysis of multivariate statistics. The calculation formula of KMO test statistics is(9)KMO=∪ij1+xij2∪inxij2−∪inxij∪jnyij+∪inyij2.


## 2.1.3. Design of Back Propagation Neural Network

The neural network has generalization ability through training and learning and can establish the mapping relationship between sample input data and output data. The neural network uses training samples to find the inherent law of the sample data mapping relationship, rather than simply memorizing the sample input, so as to correctly predict the missing input-output mapping relationship. Figure4 shows a three-layer back propagation neural network with an input layer, an output layer, and a hidden layer.Figure 4
Structure of back propagation neural network.We useFi to denote the input factor, Yi and Lj to denote the output factors of the hidden layer and the output layer, and Yi and Lj can be expressed as(10)Yi=fxi=fθij−∏ijθijxj,Lj=fyj=fθij−∏ijθijxi.Due to the existence of hidden layer, the nonlinear separable problem is difficult to learn. Back propagation neural network algorithm can improve the speed of multilayer network. A part of the multilayer forwarding network is shown in Figure5. The solid arrow is the function signal, and the dotted arrow is the error signal.Figure 5
Signal flow in multilayer perceptron.


## 3. Result Analysis

### 3.1. FEPA Model and Empirical Results for Predicting CSI 300 Index

Figure6 shows the IMF components and trend items of the CSI 300 index. It can be seen from the figure that increasing the number of IMF sequence layers of EMD decomposition can better obtain signals and improve the accuracy of prediction.Figure 6
IMF component diagram of closing price of CSI 300 index.In order to measure the effectiveness of the proposed model, there are four main indicators: mean absolute error (MAD), mean absolute percentage error (MAPE), root mean square error (RMSE), and hit rate (DS). The DS gives the correct rate of stock index direction prediction, expressed as a percentage. Because when the stock market price fluctuates up and down, we are more concerned about whether the stock market price rises or falls, rather than the specific predicted value.(11)MAD=λn∏inMi+λLi,MAPE=λn∏inMiLi−Li21+λ2×100%,RSME=λn∏inMi2+λLi2,DS=π2∏inMi,Mi=1,Xi−1Yi−1>0,2,Xi−1Yi−1≤0.In order to verify the prediction ability of the model selected in this paper, the time series of Shanghai and Shenzhen 300 index is used as the empirical data set. MAPE, RMSE, MAD, and DS are used as performance indicators to evaluate the prediction performance of the FEPA model and other models. The predicted performance index values of the FEPA model and other models are shown in Table1.Table 1
Prediction results of each model.
ModelPerformance standardMADMAPERMSEDS (%)ARIMA90.221.87108.2253.11GARCH107.232.12145.6852.88BPNN185.243.86251.2258.27EMD-BPNN65.241.3487.2564.21WD-BPNN75.251.63103.2264.12EMD-LPP-BPNN63.211.3681.0970.23FEPA57.251.3374.2572.11The empirical results show that EMD decomposition time series and dimension reduction of principal component analysis algorithm can improve the prediction performance of neural network. Compared with the WD-BPNN model, the EMD-BPNN model has higher prediction accuracy and smaller prediction error, indicating that the EMD decomposition method is more effective than wavelet decomposition method. Wavelet decomposition method has wavelet base function, and the EMD decomposition method determines the number of decomposition layers according to the internal structure of data. Compared with EMD-LPP-BPNN model, the FEPA model has achieved better results in detecting the two markets, but the performance improvement is not large.


### 3.2. FEPA Model and Empirical Results for Predicting Australian Stock Index

This chapter tests the closing price of Australian stock index. All sample empirical data can be divided into two subsets: training set and test set. The first 1000 data are training sets and the last 250 data are test sets. Australian stock market belongs to the stock market of other developed countries except the United States, which has predicted economic value and good risk controllability. The time span of the sample covers many emergencies and financial crises, so the author believes that this is sufficient for the training model.The EMD decomposition algorithm is used to decompose the Australian stock index into IMF series from high frequency to low frequency and a trend term. The time-frequency characteristics of the Australian stock index time series are found, the input variables of the neural network are optimized, and the prediction accuracy of the neural network is improved. Before EMD decomposition, the forward rolling sliding window is used to extract the sample data. The window length is an important parameter affecting the prediction effect because the sample data are about 5 years long. According to experience, the author sets the range of window length from 150 days to 300 days and uses the dichotomy to test the prediction performance of sample data under different window lengths. It is found that when the window length is 250 days, the FEPA model has the best effect on detecting Australian stock index data. Figure7 shows the IMF component and trend of EMD decomposition of Australian stock index when the window length is 250 days.Figure 7
IMF component and trend of EMD decomposition of Australian stock index.This chapter compares the prediction results of the FEPA model with each reference model. The performance measurement results and performance comparison of the prediction model selected in this chapter to predict the closing price are shown in Table2, respectively. When predicting the closing price of Australian stock index, the hit rate (DS) of the FEPA model decomposition method reaches 72.22%, which is 10.86% higher than that of the EMD-BPNN model and 3.23% higher than that of the EMD-LPP-BPNN model. When the FEPA model predicts the Australian stock index, the hit rate is improved to a certain extent, and the effect is better than other models.Table 2
Prediction results of Australian stock index models.
ModelPerformance standardMADMAPERMSEDS (%)ARIMA133.222.31178.2351.22GARCH104.222.12133.4554.22BPNN121.332.43148.2256.22EMD-BPNN73.221.3487.2264.23WD-BPNN81.231.6798.3563.12EMD-LPP-BPNN68.221.2888.2471.33FEPA58.891.1975.8872.22


## 3.1. FEPA Model and Empirical Results for Predicting CSI 300 Index

Figure6 shows the IMF components and trend items of the CSI 300 index. It can be seen from the figure that increasing the number of IMF sequence layers of EMD decomposition can better obtain signals and improve the accuracy of prediction.Figure 6
IMF component diagram of closing price of CSI 300 index.In order to measure the effectiveness of the proposed model, there are four main indicators: mean absolute error (MAD), mean absolute percentage error (MAPE), root mean square error (RMSE), and hit rate (DS). The DS gives the correct rate of stock index direction prediction, expressed as a percentage. Because when the stock market price fluctuates up and down, we are more concerned about whether the stock market price rises or falls, rather than the specific predicted value.(11)MAD=λn∏inMi+λLi,MAPE=λn∏inMiLi−Li21+λ2×100%,RSME=λn∏inMi2+λLi2,DS=π2∏inMi,Mi=1,Xi−1Yi−1>0,2,Xi−1Yi−1≤0.In order to verify the prediction ability of the model selected in this paper, the time series of Shanghai and Shenzhen 300 index is used as the empirical data set. MAPE, RMSE, MAD, and DS are used as performance indicators to evaluate the prediction performance of the FEPA model and other models. The predicted performance index values of the FEPA model and other models are shown in Table1.Table 1
Prediction results of each model.
ModelPerformance standardMADMAPERMSEDS (%)ARIMA90.221.87108.2253.11GARCH107.232.12145.6852.88BPNN185.243.86251.2258.27EMD-BPNN65.241.3487.2564.21WD-BPNN75.251.63103.2264.12EMD-LPP-BPNN63.211.3681.0970.23FEPA57.251.3374.2572.11The empirical results show that EMD decomposition time series and dimension reduction of principal component analysis algorithm can improve the prediction performance of neural network. Compared with the WD-BPNN model, the EMD-BPNN model has higher prediction accuracy and smaller prediction error, indicating that the EMD decomposition method is more effective than wavelet decomposition method. Wavelet decomposition method has wavelet base function, and the EMD decomposition method determines the number of decomposition layers according to the internal structure of data. Compared with EMD-LPP-BPNN model, the FEPA model has achieved better results in detecting the two markets, but the performance improvement is not large.


## 3.2. FEPA Model and Empirical Results for Predicting Australian Stock Index

This chapter tests the closing price of Australian stock index. All sample empirical data can be divided into two subsets: training set and test set. The first 1000 data are training sets and the last 250 data are test sets. Australian stock market belongs to the stock market of other developed countries except the United States, which has predicted economic value and good risk controllability. The time span of the sample covers many emergencies and financial crises, so the author believes that this is sufficient for the training model.The EMD decomposition algorithm is used to decompose the Australian stock index into IMF series from high frequency to low frequency and a trend term. The time-frequency characteristics of the Australian stock index time series are found, the input variables of the neural network are optimized, and the prediction accuracy of the neural network is improved. Before EMD decomposition, the forward rolling sliding window is used to extract the sample data. The window length is an important parameter affecting the prediction effect because the sample data are about 5 years long. According to experience, the author sets the range of window length from 150 days to 300 days and uses the dichotomy to test the prediction performance of sample data under different window lengths. It is found that when the window length is 250 days, the FEPA model has the best effect on detecting Australian stock index data. Figure7 shows the IMF component and trend of EMD decomposition of Australian stock index when the window length is 250 days.Figure 7
IMF component and trend of EMD decomposition of Australian stock index.This chapter compares the prediction results of the FEPA model with each reference model. The performance measurement results and performance comparison of the prediction model selected in this chapter to predict the closing price are shown in Table2, respectively. When predicting the closing price of Australian stock index, the hit rate (DS) of the FEPA model decomposition method reaches 72.22%, which is 10.86% higher than that of the EMD-BPNN model and 3.23% higher than that of the EMD-LPP-BPNN model. When the FEPA model predicts the Australian stock index, the hit rate is improved to a certain extent, and the effect is better than other models.Table 2
Prediction results of Australian stock index models.
ModelPerformance standardMADMAPERMSEDS (%)ARIMA133.222.31178.2351.22GARCH104.222.12133.4554.22BPNN121.332.43148.2256.22EMD-BPNN73.221.3487.2264.23WD-BPNN81.231.6798.3563.12EMD-LPP-BPNN68.221.2888.2471.33FEPA58.891.1975.8872.22


## 4. Conclusion

Due to my limited energy and rush of time, the research of this article needs to be further deepened, and there are still the following shortcomings: First, the FEPA model needs to be improved. The EMD decomposition algorithm has a shortcoming: there is often aliasing between different oscillation modes. In order to solve this problem, scholars have proposed integrated empirical mode decomposition (EEMD). The basic idea of EEMD is to first add a series of white noise to the target signal, and then perform EMD decomposition of the signal after the white noise has been added, and then repeat the above steps to decompose one by one according to the above steps, but each time the white noise is added to the original signal. The noise is inconsistent, and finally the influence of white noise is eliminated through integrated averaging. Second, we can consider introducing multiscale data input into the FEPA model, and introducing new input methods into the FEPA model, hoping to obtain better prediction performance. Third, there are many investment products in the capital market. In addition to stocks and foreign exchange, there are also oil, stock index futures, gold, and commodities.



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*Source: 5955375-2021-12-26.xml*