The Forecasting Model of Stock Price Based on PCA and BP Neural Network

Based on Principal Component Analysis (PCA) and Back Propagation neural network, this paper establishes stock forecast model, and takes the Yunnan Baiyao (000538) as example, 29 indicators are selected from stocks technical analysis, and the neural network is input after dimension reduction and further confirms number of hidden layer nodes, learning rate, activation function and training function of the network in accordance with comparison and analysis of Mean Square Error (MSE) and Mean Absolute Error (MAE) in different parameter data experiments. Lastly, the model with steadiness and accuracy is obtained.


Introduction
With the economic development and people's increasing financial and investment awareness, stock investment of high-risk and high-income has become a significant means of financial management.As of May 2018, A-share investors have exceeded 100 million, the total amount of A-shares has already exceeded 3500 and the total amount of A-share reaches 80 trillion yuan.Meanwhile, there are differences between stock market structure in China and that in capitalist countries, such as USA and UK.In the proportion of investment, individual investors are higher than institutional investors, which lead the lack of integrity in China's stock market, and more tend to change with difference of people's psychological state, and its law of fluctuation is hard to grasp.Therefore, the forecast analysis of stock valuation has very attractive commercial application value.In recent year, a series of the analysis method has emerged Average Method, Dow Analysis, Price Analysis and Wave Theory, and ect.(Yang, 2017).However, due to the large amount and complexity of stock market on daily basis, the traditional fundamental analysis of stock (Du, 2006) or technical analysis method (Qiao, 2013) is hard to forecast accurately the dynamic change trends of stock market.Meanwhile, as stock data of abnormal distribution, high noise and unstable characteristics as concerned, the traditional mathematical statistics analysis is too simple and not easy to change, as well as cumbersome processing and easy to cause information loss after standard processing.
In recent years, with the continuous development of data mining technology, more and more researcher began to analyze and mine deeply stock market by using computer intelligent algorithm, so as to master stock's law of ups and downs, predict its corresponding approximate valuation for the future, and further enhance the odds of people's investment decisions (Hao, 2017).
Based on the PCA and BP neutral network, this paper establishes prediction models by using dimension reduction in data pre-processing to eliminate the multicollinearity between the original stock evaluation indicators, reduces the redundant information of network input data, and enhances network working efficiency and accuracy.

Foreign Related Research Status
In the United States, Pati and Shnaprasad (Refenes & Latif, 2015) first proposed the Wavelet Neural Network (WNN) in 1992.In the neural network model, the wavelet function is used as a neuron, and the time-frequency and local characteristics of the wavelet mapping function are utilized as well as the discrete analysis model is constructed by wavelet neural network.Golan & Ziarkow (1995) used rough set theory to analyze the stock historical data of the decade and studied the correlation between stock price and economic index.
Liu & Lee (1997) and others used MATLAB to establish a securities manage-  Ticknor (2013) proposed a neural network model based on Bayesian regularization to predict stock price movements.This method uses a Bayesian normalized network to assign a probability feature to the network weights.The network can automatically and optimally Punish overly complex models that demonstrate their effectiveness by predicting the stock prices of Microsoft and Goldman Sachs.
Panigrahi & Mantri (2015) combined the improved support vector regression machine and decision tree to empirically study stock market historical data and BSE-sensex index, and provide decision support for investors to analyze stock market trends.

Domestic Related Research Status
In China, data mining technology is actually applied to the real economy later than foreign countries.Beat Wuthrich (1998) of Hong Kong University of Science and Technology developed a stock price forecasting system based on data mining technology.The system predicts the trend of the Hang Seng Index by entering real financial data.Today's domestic stock analysis software has predictive functions like Straight Flush, Oriental Wealth, and Great Wisdom, but the data mining functions of these software are relatively simple.Ma, Lan, & Chen, (2007) proposed based on WNN and SVM to use time series stock data to make data mining using the decision of sequence trading rules, and apply the securities comprehensive index to predictive analysis.Zhang (2009) applied the SVM to classify the stock's ups and downs.After analyzing the four factors of quantity relative ratio, turnover rate, internal and external ratio and amplitude, which had a great impact on the results, she shifted her research focus to the prediction of SVM regression on the market index.Guo (2010) proposed an improved Aprior algorithm based on the correlation between stock prices and using the advantages of algorithms in vertical data representation and cross-counting.This algorithm can be applied to the stock analysis simulation system to quickly find the relationship between the rise and fall of stocks, and provide investors with more accurate decision support.Bao (2013) applied the support vector regression of phase space reconstruction theory, Hopfield network algorithm of neural network and correlation vector machine, LSSVR-CARRX model algorithm and financial time series algorithm based on support vector regression and used these techniques to analyze and forecast the stock market.
In the same year, Sun (2013) used the decision tree, neural network and logistic regression model in data mining to study the relationship between financial indicators and stock prices of listed companies.After comparing and evaluating the advantages and disadvantages of these three methods, they can better determine the investment value of the stock.
Chen (2014) used data mining technology to solve the problem of stock picking and investment timing, and analysed the factors affecting stocks' rise and fall through association rules analysis.After clustering stock data, find the best investment portfolio.The stock price is estimated and predicted by applying an improved neural network algorithm.Wang (2010) constructed a KNN model with the same effect as the neural Through the above combing, data mining technology has been widely used in the financial industry, but the research of foreign scholars is mainly aimed at the optimization of specific algorithms and the overseas stock market.Only a small amount of research is available for individual investors.At the meantime, the domestic stock market research is relatively backward and the market complexity is higher.Therefore, the starting point of this paper is to establish a relatively accurate and stable stock forecasting valuation model for A-share investors.

The Forecasting Model Based on BP Neutral Network to
Establish Stock Price

The Indications of Forecasting Evaluation in Stock Price
The paper selects 29 transaction data and technical indicators as evaluation indicators of stock price.

1) Stock price
The Stock Price includes the Opening Price, Closing Price, Highest Price and Lowest Price of the day.

2) Transaction data
The Transaction Data includes the Turnover and Volume.

3) Price Limit
Closing price of the day Opening price of the day Price Limit 100% Stock market closing price yesterday The Moving Average is based on the principle of "moving average" in statistics, which the average value of stock prices over a period of time is linked into a curve, showing the historical fluctuations in stock prices.This paper selects MA1 (5-day average line), MA2 (10-day average line), MA3 (30-day average line), MA4 (60-day average line), MA5 (120-day average line), and MA6 (240-day average line) as indicators.

5) MACD
The MACD formula calculates the difference between the long-term and medium-term Exponential Moving Average (EMA) to determine the market.When the MACD turns from a negative number to a positive number, it is a signal to buy.When the MACD turns from a positive number to a negative number, it is a signal to sell.Calculation formula: DIF line: The difference between the short-term moving average and the long-term moving average.

( ) ( )
DEA line: M-day exponential moving average of the DIF line, in this paper, M = 9, then 9 th .
DIF Difference Exponential Average (DEA) = DIF (Today) × 0.2 + DEA (Yesterday) × 0.8 ( ) The stochastic indicator generally calculates the immature random value RSV of the last calculation period by the highest stock price, the lowest stock price, the closing price and the proportional relationship among the three in a given period, and then in accordance with the method of Exponential Moving Average, the K value, the D value, and the J value are calculated.In this paper, 9 days is a cycle. ( ( ) C n is the closing price for the n day; ( ) L n is the lowest price for the n day; ( ) H n is the highest price for n day.

7) WR
The Williams Overbought/Oversold Index is mainly used to judge whether the stock market is overbought or oversold, in order to analyze the strength of The Bias rate indicates the degree of deviation between the closing and the stock moving average, and thus uses the Bias rate to calculate the probability of returning or rebounding when the stock price fluctuates sharply, and the stock price returns to the original credibility within the normal fluctuation range.This paper uses BIAS.BIAS1 is for 12th and BIAS.BIAS2 is for 24th.

( )
Closing price of the day N_day moving average BIAS 100 N_day moving average ( ) The closing price on the day The closing price before the day ROC 100 Closing price before the day The energy tide is to draw the stock market turnover into a trend line, and speculate on the trading atmosphere of the stock market based on the price changes and the increase and decrease of the volume.
OBVof that day OBV of the day before Today's trading volume ) trend; the popularity index determines the future trend of the stock price through the linkage relationship between the volume and the stock price.

Using PCA to Reduce the Dimensionality of Stock Evaluation Index
This Specific steps are as follows: Set 1 2 , , , P X X X  observe n times and get the observed data matrix as: The results of PCA are affected by magnitude or dimension.Due to the possibly different units of each variable, if the dimensions are changed, the results will H.L. Zhang be different.Therefore, it is necessary to standardize the data of each variable first, and then the analysis is then performed by using a covariance matrix or a correlation coefficient matrix.
Normalization of the original variable matrix: Subsequently, the normalized covariance or correlation matrix, and the eigenvalues and eigenvectors of the covariance matrix are obtained.At the same time, for each i, there is , and ( ) , but also value of ( ) Var Y reaches the maximum; and so on, until all the components p can be obtained.The solution is to find the eigenvalues of the T X X matrix.Set the obtained eigenvalues , and their corres- ponding normalized orthogonal eigenvectors are 1 2 , , , p η η η  , the P principal components: When performing PCA, it is determined whether or not a principal component is retained, generally based on the retained feature vector accounting for more than 85% of the sum of the total feature vectors.However, sometimes it is necessary to consider the contribution value of the selected principal component to the original variable, which can be expressed by the squares of the correlation coefficients and the meters.If the selected principal components are , and q m ≤ are satisfied, then they contribute ( ) to the original variables of i X .

The Predicted Model of Designing BP Neutral Network for Stock Price
During the construction of BP neutral network (Li & Song, 2013) stock price predict models, the important network parameters are needed to confirm, such as the number of neurons of input layer and output layer, the number of hidden layer and number of neurons, the confirmation of learning rate, and the choice of activation function and training function.

The Number of Neurons in Input Layers and Output Layers
This paper selects 29 stock price evaluation indicators.After using PCA to reduce the dimension, the principal components that determined lastly are input H. L. Zhang variables.Meanwhile, the number of the day that needs to predict closing price should be corresponding to output of variable.

The Number of the Hidden Layers and the Number of Neurons
The Robert Hecht-Nielsen in 1989, proved that any continuous function in any closed interval can be similar to 3-layer BP neutrons.Therefore, this paper confirms to establish the model with 3-layer BP neural network.
The number of hidden neurons is playing an important role of predictable accuracy in neural network.In general, more hidden layers nodes can have better performance; however, it can lead to too long training time.At present, there is no ideal analytical formula to determine the rational number.Generally, it can estimate the approximate number by means of empirical formula and determine this amount by actual operation in the end.
Among them, d is the amount of neutrons in the input layer; h is the amount of neutrons in the hidden layer; s is the number of neutrons in the output layer; a is the constant between 0 and 10.

The Learning Rate
The learning rate is playing an important role of training effects and training speed in neural network.Its value range is between 0 and 1, which means that the larger the value is, the greater the modification of link weight for each iteration, the faster the learning rate, and the shorter the training period.However, it is possibly that the excessive learning rate makes the network not converge.On the contrary, if its value is smaller, the studying speed is slower, the operational time will be longer and the memory capacity for algorithm will be larger.Therefore, there must be certain shortcoming with learning rate too large or too small.
The specific value of learning rate takes reference to the empirical formula:

Activation Function
The activation function of hidden layer and output layer has an important influence on the predicted accuracy of BP neural network.The function of newff provides several activation functions in the toolbox of MATLAB neural network.

1) logsig function:
( ) 2) tansig function: ( ) H. L. Zhang 3) purelin function: This paper uses controlling variable method.Under the condition of other same parameters, the error of prediction of different activate function is compared to determine the most suitable activation function.

Training Function
Training function is the most core training algorithm in the neural network, different training function corresponds to different training algorithm.

1) Gradient Descent method
Its principle is to obtain the optimization information from gradient vector and solving the optimal weight by first-order partial derivative.
There provide many gradient descent training functions in the MATLAB neutral network Toolbox, such as Gradient Descent Method (traingd), Gradient Descent Method with Momentum (traingdm), and Adaptive Algorithm (traingda).
2) Quasi-Newton Methods Newton's method is rapidly optimized algorithm based on the expansion of Taylor series, its iterative formula is as follows: ( ) ( ) ( ) ( ) ( ) Among them, H represents Hessian matrix of the error performance function, represents optimization direction of Newton method.If the matrix of Hessian is not positive, its search direction may not be downward direction.Therefore, the improved Quasi-Newton Methods (quasi-Newton) is selected.
( ) ( ) ( ) ( ) ( ) Its principle is to use the first-order partial derivative of the training function and then the inverse of the approximation ( ) H n by another ( ) The corresponding training function in MATLAB is trainbfg.

3) LM algorithm
Using LM (Levenberg-Marquardt) doesn't need to calculate the specific matrix H, while it is shown as an approximate representation as Its descending gradient is expressed as: J is the Jacobian matrix containing the first derivative of the network's weight error function.It modified network weight: When 0 µ = , it expresses Newton method, when µ is very large, it is simi- lar to the gradient descent method with a smaller step size.It is calculated by trainlm in MATLAB.

4) Powell-Beale algorithm
Its principle is to firstly judge the orthogonality of the gradient before and after the network.If it is orthogonal, adjust the threshold and weight of the network back to the negative gradient direction.It is calculated by traincgb in MATLAB.
The paper uses these six algorithms to be trained, such as traingd traingdm traingda trainbfg trainlm traincgb, to observe the prediction accuracy, so as to select the best training function.
In the past five years from May 13, 2013 to May 10, 2018, the suspended data were removed from July 19, 2016 to Dec. 30, 2016 as well as from April 25, 2017 to April 26 with total of 1105 groups of data as sample.1005 groups of data are selected at a random as training set, the other 100 groups of data as testing set.
The data of this paper comes from Yahoo finance.
The data selection of this paper mainly considers two factors: 1) The time span of input variables This paper selects 29 indicators of daily data, including transaction data and technical indicator data of Yunan Baiyao from May 2013 to May 2018.For these five years, this time span is mainly based on the completing data of up and down cycle of general value investment, while generally speaking, the 5 -10 years is the interval of bull and bear markets.From the historical data, we can also conclude that in the year of 2013-2018, our country's stock market experiences a complete up and down cycle, therefore, using the interval as the research one can be better to get rid of the bigger error that bull market or bear market can bring purely in the market fluctuation, and better reflects the long-term development trends of this stock.
2) Getting rid of missing tuple of data This paper uses the recent five-year data of Yunnan Baiyao (000538) from Yahoo Finance, Yunnan Baiyao Group Co., Ltd. has been suspended twice during these five years.From July 19, 2016 to Dec. 30, 2016, due to planning of promotion of mixed ownership reform related stocks into suspension period.Meanwhile, from April 25, 2017to April 26, 2017, due to all unrestricted shares of Yunnan Baiyao fully takeover offer suspension, in order to ensure that data's fluctuation and development trends are not affected by the change during the suspension period, this paper remove the data tuple of this period, to ensure that the data of this period is more representative.

Reduction of Dimension of Main Components Analysis
This paper reduces the dimension of 29 stocks evaluation indicator through PCA by SPSS, the principal components are extracted as input variables of BP neural network stock price forecasting model.The contribution rate of previous five components reaches 88.915%, but in order to further increase information coverage rate, this paper extracts the sixth component, the variance of extracted six main components at the end accounts for 92.179% of principal component variance, enough to describe the fluctuation of Yunnan Baiyao in the latest five years.

Data Normalization
This paper uses normalization function of mapminmax in MATLAB to nor- malize 6 principal components and one closing price to prevent network output error from being large due to difference of data magnitude or dimension.

Hidden Layer and Node of Hidden Layer
In accordance with empirical formula, the approximate range of number in hidden layer neurons is between 2 and 13.Through trial and error to compare prediction accuracy of different node numbers, the number of hidden layer nodes is finally determined shown in Table 2.
Experiments show that the MSE of BP neural network will decrease first and then rise with the increase of the number of nodes.By comparing the MSE and MAE of different implied nodes, it is found that the network error is the smallest when the number of hidden nodes is 6 in this case.Therefore, the BP neural network stock price prediction model established in this paper determines that the number of hidden layers is 2 and the number of hidden layer nodes is 6.

The Number of Neurons of Input Layer and Output Layer, and
Learning Rate The input variables are the six principal components after the dimension reduction of the PCA, and the output variable is the stock price of the next day.
Based on the empirical formula determined by the learning rate mentioned above, the initial learning rate is finally determined to be 0.4082.

Activation Function
According to the three activation functions mentioned above, and applied to the hidden layer and the output layer respectively, comparing the different activation functions through multiple experiments leads to the MAE of the prediction result to determine the selected activation function shown in Table 3.According to the results, the paper finally selects logsig as the hidden layer activation function, purelin as the output layer activation function with picking the smallest error.

Results Analysis
This paper mainly evaluates the prediction results according to MSE (Mean Square Error) and MAE (Mean Absolute Error).

(
) Among them, i Y represents the actual output of the model; i O represents the expected output of the model.The smaller the MSE, the higher the accuracy of the model.
After the parameters of the model are determined, the data is put into in MATLAB and the model is trained.The operating results show that the model converges after 195 iterations, and the test error is close to the target setting accuracy of 1.0e−6.The MSE is 6.63 and the MAE is 1.98, which has a good fitting effect.The BP neural network stock price forecasting model structure, BP neural

Conclusion
In this paper, the stock price forecasting model is established based on BP neural network.From the many factors affecting the stock price, the opening price, the closing price, the highest price, the lowest price, the trading volume, the turno- in studying development trends of stock price: K-line Graph Analysis, Moving ment system and use it to analyze large amounts of securities data.Tsaih, Hsu, & Lai (1998) and others used clustering methods and visualization techniques to find the best investment opportunity for stocks.Povinelli, Cai, & Song (2000)  proposed to extract the characteristics of the intensity of the event based on the interest of the digger and construct a temporal mode feature function.Dose & Cinacotti (2005)  applied data mining techniques to find the best way to derive stock portfolios, optimize the weight ratio of individual stocks in asset allocation, and thus passively manage funds.Fiol-Roig,Miro-Julia, & Isern-Deya (2010) and others used data mining technology to evaluate past stock prices, and at the same time, to obtain useful vant stocks.
network and time series according to the time series.This model is based on the model training of the closing price of the first n trading days of the stock to obtain the closing price of the first n + 1 trading days.The closing price of the trading day, which adjusts the weight according to the actual situation of each dimension, thereby improving the accuracy and reliability of the model.
The Relative Strength Index analyzes the market's intentions and strengths by comparing the average closing gains and the average closing declines over a period of time to determine future market trends.This paper selects RSI1 for the 6-day line, RSI2 for the 12-day line, and RSI3 for the 24-day line.The sum of the gains in the closing day of U n = ; the sum of the closing de-clines in V n = (absolute index);The price rate of change is based on the closing price of a certain day before a period of time, and the speed of the closing price of the day is used to judge the strength of buying and selling power, and then analyze the future development trend of the stock price.
paper predicts that the stock price selects 15 items of 29 different stock transaction data and technical evaluation indicators.Each evaluation index describes and predicts the fluctuation forms and trends of stock from different angle in the past period of time.Each evaluation index includes part of information, but different indicators would inevitably exist multiple collinearity, that is, the amount of information covered by all the indicators overlaps to some large extent.Meanwhile, overlarge network input also affects the operating convergence efficiency of neutral network and predictable accuracy.Therefore, this paper uses PCA (Zhang & Dong, 2013) to reduce dimensionality of 29 stocks trading data indicators and technical evaluation indicators.
min x is the minimum value of data, max x is the maximum of data.
According to the six training functions commonly used in MATLAB mentioned above, comparing the different training functions through multiple experiments leads to the comparison of the MAE and MSE of the prediction results to determine the selected training function shown in Table 4.After experimental testing, this paper finally selects traincgb as a training function with picking the smallest error.

Figure 1 .
Figure 1.BP neural network stock price forecasting model structure.

Figure 2 .
Figure 2. BP neural network stock price prediction model output and actual output ccomparison.

Figure 3 .
Figure 3. BP neural network stock price prediction model MAE.

Figure 4 .
Figure 4. BP neural network stock price prediction model scatter plot.
ver, the price change, the MA, MACD, KDJ, BIAS, RSI, ROC, OBV, WR, a total of 29 indicators are selected.PCA was used to reduce the 29 indicators to a 6-dimensional input network.Taking the data of Yunnan Baiyao (000538) for H. L. Zhang the past 5 years as an example, the data experiment is carried out.After experimentally comparing the MSE and MAE of different parameters, the number of hidden layer nodes in the model are finally determined as 6, the initial learning rate is 0.4082, the activation functions of the hidden layer and the output layer are respectively logsig and purelin and training function is traincgb, and finally the model is established.After testing, the model has good precision, MSE is 6.63 and the MAE is 1.98.Although the test results in this case are good, there are still some problems, and further improvements and in-depth research can be carried out in the future.1)The selected indicators are not comprehensiveThis paper only selects 29 indicators from many stock price evaluation indicators, but in fact, there are far more than 29 factors affecting stock prices.In the follow-up study, the fundamental indicators of the technical surface can be further explored.2)The data is not rich enough In this paper, only 1105 groups of data of Yunnan Baiyao (000538) stocks from May 13, 2013 to May 10, 2018 are selected.If the data volume is further increased, two or even more alternate regions of the bull and bear market selected as the sample size can further improve model prediction accuracy.3) Further optimize model parameters In this paper, the existing activation function and training function are selected.If the parameters of the model are further optimized in the subsequent research, the prediction accuracy of the model can be further improved.

Table 1 .
29 forecasting indicators of stock price.
shown in Table 1.Among them, the market trend indicator is mainly used to indicate the current stock market development trend; the market energy indicator is mainly used to determine the key turning point of the stock market H. L. Zhang DOI: 10.4236/jfrm.2018.74021375 Journal of Financial Risk Management

Table 2 .
MSE and MAE corresponding to the number of different hidden layer nodes.

Table 4 .
Comparison of 6 training functions MAE and MSE.