As part of the transportation industry, the aerospace industry has become an indispensable part of human life. Aerospace is the most active and influential field of science and technology in the 21st century. The outstanding achievements in this field mark the level of development of human civilisation. It also reflects the level of comprehensive national strength and science and technology of a country.
In the aerospace sector, the most representative industries are aerospace manufacturers (e.g. Boeing, Airbus) and aerospace operating sector (e.g. Alaska Airlines, United Airlines). To expand financing, invest in new projects or new businesses, and increase market competitiveness, many companies choose to be listed on stock market. This is because that entering the stock market is a straightforward approach to raising more capital for aerospace companies, expanding international market share, gaining higher public trust, and creating more transparent and measurable company values.
The price of stock markets is the critical indicator of economic growth in a country. In 2015, there were more than 60 stock exchanges in the world. At the mid-2017, the total stock market value size in the world was around $76.3 trillion US dollars. The stock price prediction is trying to confirm the future values of the stocks or other financial tools traded in the stock exchanges (Malkiel, 1991) . The investors and traders are interested in share price prediction because accurate prediction may produce high profit through stock trading. However, the share prices of aerospace companies fluctuate significantly due to many cyclical and non-cyclical factors. Each factor has a significant impact on profit, but these elements are not synchronised. Due to the uncertainty and complexity of the stock market, the prediction of share prices is a critical challenge, and thus the stock trading is a high-risk financial activity. Therefore, a slight improvement of share price prediction could bring a substantial return of rate for investors.
Although financial market analysis requires knowledge, intuition, and experience, the automation process has been growing steadily because of the availability of large finance data and the development of machine learning technology. There are two conventional methods for stock price prediction: fundamental analysis and technical analysis.
Fundamental analysis is to concern the stock itself (Harrington, 2003) , while technical analysis is to predict the future trend of the transaction values and volumes by researching the past market data without concerning any fundamental aspect and financial condition ratios of the company itself (Kirkpatrick & Dahlquist, 2015) .
Correspondingly, the two sets of data, used in the two analysis methods, are called fundamental stock data and technical stock data, respectively. Moreover, various machine-learning techniques have been widely used in the finance industry.
This research is to improve the accuracy of share price prediction in aerospace industry using advanced machine learning techniques, to provide decision-makers with references about their stock price as well as various factors that could affect the stock price in future for their economic strategies and business activities. We develop a hybrid approach with two stages of PCA and recurrent neural network for stock price prediction.
Two types of aerospace industries are studied based on their history data. One is a manufacturing company, denoted as MFG, and the other is an operating company, denoted as OPR. We select 15 fundamental features and 15 technical features, respectively. The three groups of experiments will be conducted based on the different data types: fundamental data, technical data and their combination.
The rest of the paper is organised as follows: Section 2 provides a brief literature review on the research of computational finance, especially the methods and techniques on share price analysis and prediction. Section 3 describes the hybrid model for share price prediction. Section 4 provides the experiments and comprehensive evaluation. Finally, Section 5 provides conclusions and future work.
Fundamental analysis is the process of looking at a business at the most basic or fundamental financial level. It examines the key ratios of a business to determine its financial health. Fundamental analysis can provide an idea of the value of what a company's stock should be. Hence, fundamental analysis concerns the stock itself, such as the assets, liabilities, and incomes of the target companies by analysing their financial statements, as well as the ratios of previous performance, such as Price-to-Earnings Ratio (P/E ratio) (Harrington, 2003) , which is the ratio for valuing a company that measures its current share price, related to its earnings per-share (EPS). Fundamental analysis is based on the belief in the business needs for the capital to keep operating. If the company runs well, it should obtain additional capital awards which will lead to stock price soared (Fundamental Focus, 2011) . Fundamental analysis is conducted from the global economy firstly and then national economy before analysing a specific industry and a specific company. It is a top-down process. As the fundamental analysis is a relatively reasonable and objective method, it is used extensively (Harrington, 2003) .
Technical analysis is to predict the future stock price and makes the trade decisions of financial derivatives based on the position and the seasonal change theory of the market trend. The principles of technical analysis are that the price will reflect all the relevant information of the market and the history trend will repeat itself (Edwards et al., 2012) .
Both analyses can predict stock price to some extent. The main difference between these two methodologies is that the datasets of fundamental analysis could update very slow.
For example, return on assets (ROA) and return on equity (ROE) are likely to upgrade every three months, rather than daily as with technical analysis dataset, e.g. stock price data (Larkin, 2014) .
For the fundamental analysis, there are lots of indicators in stock analysis processes. 42 features based on the fundamental analysis were extracted by different analysts for the prediction of share price. The 5 feature vectors, such as profitability, growth, liquidity, solvency and operational efficiency, were used for tackling the value stock analysis of IT stocks in Taiwan. The profitability vector includes ROA, operational gross profit, operational profit, net profit after tax; the growth vector includes net profit after tax growth rate, ROA growth rate, total assets growth rate, revenue growth rate, gross profit growth rate; the liquidity vector includes quick ratio, liquidity ratio, cash ratio; the solvency vector includes debt ratio, interest coverage ratio; and operational efficiency vector includes asset turnover rate, inventory turnover rate, average days for sales (Shen & Tzeng, 2015a) . Besides, the rate of return, the number of transactions, gross profit, gross loss, the number of profitable transactions, the number of consecutively profitable transactions, the number of unprofitable consecutive transactions, Sharpe ratio, the average coefficient of volatility, the average rate of return per transaction, the value of the evaluation have been used in the decision system on FOREX market (Korczak et al., 2016) . In order to select optimal stocks from the stock market and predict the future price trends, Chen et al. (2017) developed an improved fundamental approach with 14 features that were widely used as indicators in the analysis of finance and investment, such as long-term funds to fixed assets, current ratio, interest guarantee, average inventory turnover, average collection turnover, fixed assets turnover, total assets turnover, return on total assets, total stockholders' equality, operating income to capital, pre-tax income, net income, earnings per share, and the price-earnings ratio. More features can be extracted from the daily stock market data. Moving averages and relative strength index (RSI) have been used in a mobile app to classify stock market (Larkin, 2014) . To support investment decision, night days K value (KD), psychology indicator (PSY), moving average convergence and divergence (MACD), and RSI have been applied in a rough set approach (Shen & Tzeng, 2014; Shen & Tzeng, 2015b) . Besides, to determine the future share price predictability of Hong Kong, South Korea and Singapore, such features as 20 days simple moving average, 20 days exponential moving average, moving average crossover, MACD, Kaufman adaptive moving average, and the most optimised moving average have been used to analyse the increasing profitability (Phooi M'ng, 2018) . For the Morgan Stanley Capital International Emerging Market, in terms of moving average, relative strength index, and moving average convergence divergence, the prediction of transaction costs, using technical analysis, provided evidence against the efficient market hypothesis for emerging market index (Metghalchi et al., 2019) . Technical analysts always consider the influencing factors from various aspects. With the continuous increase in the number and spread of social web media, the stock market volatility is affected by information release, dissemination and public acceptance .
To take full advantage of the strengths of advanced machine learning techniques to produce broader impacts, effective practical implementations of predictive systems must incorporate the use of innovative technologies. Stock prices prediction can be transferred to two types of problems: (1) decision making or classification problems for price trend prediction, such as fuzzy rule-based systems (ElAal et al., 2012) , neural networks (ElAal et al., 2012 , Lertyingyod & Benjamas, 2017 , and random forests with imbalance learning (Zhang et al., 2018) , and (2) time series prediction (TSP) problems for price value prediction. Various machine learning techniques have been applied for TSP problems (Jadhav et al., 2015 ,He & Qin, 2010 . Especially, neural networks (e.g. recurrent neural networks and feed-forward neural networks (Chandra & Chand, 2016; Khare et al., 2017)) have been proven to be very promising for solving TSP problems in the literature.
Classic artificial neural networks (ANNs) have been widely used for predicting stock price (Khare et al. 2017 , Göçken et al. 2019 , digital content stocks (Chang, 2011) , and the closing price of PETR4 stocks (Andrade De Oliveira et al., 2011) . Various types of neural networks were developed for the stock price prediction, such as Bat-neural networks based on the quarterly data for eight-year DAX stocks (Hafezi et al., 2015) , nonlinear autoregressive with exogenous inputs (NARX) neural networks based on the data from the NASDAQ stock AAPL (Wei & Chaudhary, 2016) , Recurrent Neural Networks based on the data from NASDAQ stock exchange (Chandra & Chand, 2016) , and Radial basis function (RBF) neural network on the average prices of previous 8 months (Liu, 2018) , Recently, deep learning techniques have been widely used for stock price prediction, for example, the deep direct reinforcement learning algorithm (Deng et al., 2017) , Convolutional neural networks (CNN) (Cao & Wang, 2019) and deep Q-learning (Jeong & Kim, 2019) . Support vector machine (SVM) was also widely used in this area. For example, Long et al. (2019) developed SVM based on different kernels to predict stock price trend based on based on financial news semantic and structural similarity.
Hybrid models always perform better than single machine learning models, and the neural networks and SVMs are often an important component in many hybrid models, for example, the combination of neural network and decision tree for the prediction of digital game content stocks price (Chang, 2011) , the integration of gray algorithm and RBF neural network, trained by four different learning strategies (Lei, 2018) , fuzzy time series analysis with neural networks for the forecast of the Taiwan Stock Exchange Capitalization Weighted Stock Index (Yolcu & Alpaslan, 2018) , the integration of piecewise linear representation (PLR) and support vector machine (SVM) (Luo & Chen, 2013) , and the combination of CNN and SVM (Cao & Wang, 2019) .
Evolutionary optimization with wrapper techniques are often used to optimize machine learning models or select features for stock price (trend) prediction. For example, Harmony Search and Genetic Algorithm was used to find the best structure of neural network (Göçken et al., 2016) , neural network with greedy algorithm based on the feature reduction with wrapper techniques for the prediction of stock price trend (Lertyingyod and Benjamas, 2017) , simulated annealing algorithm was used to optimise feature space and model parameters (Torun & Tohumolu, 2011) , and a Markov decision process was incorporated on genetic algorithms to develop stock trading strategies (Chang & Lee, 2017) .
Many other techniques were used to improve the performance of computational finance.
For example, principal component analysis (PCA), as data pre-processing, was used to reduce the dimensions of feature space to improve prediction accuracy (Lin et al., 2009) , and it was applied to clean up the original data set and produce a new data structure (Chen & Hao, 2018) , an analytic hierarchy process was used for feature ranking and selection for a weighted kernel LS-SVM (Marković et al., 2017) , and the information gain oriented feature selection was run before the Genetic Algorithm with SVM in wrapper for credit scoring (Jadhav et al., 2018) .
This research focuses on the classic TSP problem for stock price prediction. It is to explore the advanced methods of share price prediction in the aerospace industry by using some historical finance data. A set of features, 1, … , , can be extracted from raw date to feed into a proposed model to predict the future close price, y.
Due to the uncertainty and dynamics of stock markets, the TSP problem is a strong nonlinear regression problem. The value of the time series at time t + τ is predicted or forecast based upon the properties of the historical time series t, t -1, ….., t -n+1 (where n is the number of time steps within the duration of the time series). The non-linear regression can be expressed as shown in Equation (1).
To solve this problem and accurately predict the future share price, two challenges need to be overcome. The first one is to choose proper features, and the other is to develop an appropriate model that represents the function f(*), mapping the non-linear relationship between the historical data and the future stock price to be predicted, and it should be robust for the uncertainty and complexity of data. There are hundreds of finance features for share price prediction and each feature has different influence levels in various industries, even has different influence levels for different companies in the same industry sector. The parameter τ could be different for different industries, due to their different data properties.
After decades of stock research, more than 100 indicators and ratios have been developed for fundamental and technical analysis, respectively. In this research, data is collected from the Bloomberg databases, which is the most popular stock database. Tables A1 and A2 in Appendix. The collected raw data will go through two stages of data processing.
For the fundamental features, data is collected every quarter after the companies publishing their financial reports. For the technical features, data is collected every trading day, synchronised with the close price. A sample is the data on every trading day.
To compare the impact of two sets of features on technical analysis, the quarterly fundamental data is transformed to daily data by using the univariate linear processing, which is decided by the beginning and the ending values of the feature in the quarter, as shown in Fig. 1 , where, assume all the change of a fundamental feature in a quarter is linear.
When processing the technical data, to keep the integrity of the sample data, the samples that miss some feature values are deleted. At the same time, the corresponding trading day samples transferred from the fundamental data are also deleted to maintain the consistency of the fundamental data and technical data.
To overcome the curse of dimensionality, the technique of classic Principal Components Analysis (PCA) is used. The idea of PCA is to project n-dimensional data onto kdimensional (n>k) hyperplane, thus minimising the projection distance from each sample point to the hyperplane and maximise the variance. The implementation of PCA mainly includes five steps (Groth et al., 2013) :
(i) Standardise the data samples by mean normalisation;
(ii) Calculate the covariance matrix of the data;
(iii) Find the eigenvalues and the eigenvectors of the covariance matrix above;
(iv) The obtained eigenvectors are combined according to the size of the eigenvalues to form a mapping matrix, and the largest number of the top k rows or the top k columns of the mapping matrix are extracted as the final mapping matrix;
(v) Mapping the original data with the mapping matrix of step (iv) to achieve the purpose of data dimensionality reduction. Fig. 2 , where,
ℎ ( ) is the hidden state vector with n hidden layers in the time-step t;
( + ) is the output on the τ th day after current time t;
τ is the trading days of delay 1 < τ ≤ t is the direct weight from the input layer at time t to the hidden layer at time t;
is the weight from the hidden layer at time t to the output layer at time t+τ;
−1 is the weight from the output at time t-1+τ to the hidden layer at time t, which means the historical memory; 
There are many different algorithms for training an RNN. To obtain good performance of RNNs, three training algorithms are investigated, such as Levenberg-Marquardt
Algorithm (Ranganathan, 2004) , Bayesian Regularisation BP Algorithm (Mahajan et al., 2015; Burden & Winkler, 2008) and Scaled Conjugate Gradient Algorithm (Zhou & Zhu, 2014 ; (Møller, 1993) . (LM) is the most widely used optimization algorithm. LM, as the damped least-squares method, is an estimation method for the regression parameter least squares estimation in the nonlinear regression. This method combines the Gauss-Newton algorithm and the Gradient Descent algorithm and converts the nonlinear least squares problem into a series of linear least squares problems, then solves the problem by iteration. Hence, it outperforms simple gradient descent and other conjugate gradient methods in a wide variety of problems. The details of LM is provided in (Ranganathan, 2004) .
Bayesian Regularisation nets have been proved to be more robust than standard BP nets (Burden & Winkler, 2008) . The idea of the BR Algorithm is to retain all the features but to avoid the excessive influence of a specific feature by reducing the underlying population parameter θ, which obeys the Gaussian distribution, to improve the problem of overfitting. By using the Maximum A Posteriori (MAP) method to estimate the parameter θ, and then obtain the cost function. Denote the prior distribution before the training set being trained and calculate the posterior distribution through the Bayesian formula. It is considered that the value of θ should maximise the MAP so that the cost function will be minimum. The details of the BR-BP algorithm was provided in (Mahajan et al., 2015) .
The performance of the Scaled Conjugate Gradient (SCG) Algorithm is benchmarked against that of the standard BP algorithm (Møller, 1993) . The basic idea of the SCG is to combine the conjugate property with the steepest descent method. It constructs conjugate directions of two-conjugates by using the gradients at the known points. The minimum value in one direction is obtained by searching and optimising along the direction. It does not affect the minimum value in the obtained direction when searching the minimum value in other directions. After finding the minimum values for all directions, the minimum value of the -dimensional problem is obtained. The details of SCG is provided in (Zhou & Zhu, 2014) .
The mean square error (MSE) on the testing data is used to evaluate the prediction accuracy of different models. The smaller the value of MSE, the better the accuracy of the prediction model.
The aim of this research is to analyse the share price of the aerospace industry. There 
The experiments are carried out on a laptop with Intel Core i5-7200U@2.50GHz, The two termination criteria and the learning rate of the developed RNN are set as below:
1) The training ends when it reaches convergence;
2) If it does not converge within 1000 epochs, automatically ends at the 1000th epoch;
3) The learning rate is 0.02; the number of hidden layer is 1. Fig. 4 (1) -(3) illustrate the results of PCA on the MFG data. Fig. 4 (1) plots several lines connected to form the 2D and 3D graphs for fundamental data, while Fig. 4 ( 2) and (3) are scatter plots for technical features and mixed features, respectively. This is because that fundamental data is transferred from quarterly to daily through univariate linear function, while technical data is the actual daily financial data gathered from the database. Fig. 4 (3) is similar to Fig. 4 (2) in both 2D and 3D graphs. This indicates that the fundamental data of MFG has a small influence on the results of PCA, when it is combined with technical data. In PCA, the loss of detailed information is inevitable. The variance explained ratio is used to indicate the cumulative contribution rate of the first few principal components extracted by the PCA. Table 1 shows the variance explained the ratio for MFG. that BR always converges very quickly, but the adjustment process is slow. Namely, the MAP in the Bayesian algorithm finds a set of approximate optimal parameters very fast, and then the fine adjustment of weights is slowly. 
The most accurate prediction of the share price of MFG is produced by the RNN on the experimental parameters: "BR Algorithm, Mixed Features, 15 Neurons and 5 Delays"
with the average MSE of 3.482006. Fig. 6 shows that the errors are increased from time-step 1000 to 1200. It means that although the RNN obtained by BR has the best performance, the accuracy is reduced from June 16 2017 to April 4 2018. Namely, the dynamics of the stock market prices has changed very much in the past year. Fig. 7 (1) shows a unique trend for the fundamental data of OPR, which is different to Fig. 4 (1) for the fundamental data of MFG, and Fig. 7 (2) looks more compact for the technical data of OPR than Fig. 4 (2) for the technical data of MFG. Furthermore, the most significant difference is the result of PCA on the mixed data of OPR. As can be seen in Fig. 7 (3), the data in the 3D graph is divided into two clusters for the mixed data of OPR, deferring to the 3D graph in Fig. 4 (3) for the mixed data of MFG. This means both fundamental data and technical data affected the results of PCA for OPR. Namely, fundamental data could play different roles for the stock price prediction in aerospace manufacturer and operator.
(1) PCA with Fundamental Features of OPR 
For OPR, the best RNN is produced with the experimental parameters: "SCG Algorithm, Mixed Features, 5 Neurons and 15 Delays" with the average MSE of 1.319737. Fig. 9 shows the model outputs and errors for OPR. The error for OPR is evenly distributed and smaller than that for MFG, as in the SCG algorithm, every direction has a conjugated direction and all the directions could be searched.
The best results of the PCA and RNN for the share price prediction of MFG and OPR have been shown in the last section. The experimental outcomes will be further assessed with more dimensions as follows.
The benefits of PCA lie in the running time reduction, data visualisation and compression to free up the computer memory space. But, the input data of RNN is the initial data without PCA processing. Tables 3 and 4 show the assessment results for MFG and OPR, respectively. From both tables, the average MSE with PCA is lower than average MSE without PCA. This means that the prediction accuracy with PCA is better than without PCA. Moreover, the average running time with PCA is lower than or similar to that without PCA. The results have proved that PCA can improve both efficiency and accuracy of the prediction model. To sum up, from the average value of MSE and running time, LM is the best algorithm for MFG while SCG is the best one for OPR. Even BR performs very well in the single test for MFG, it costs a lot of running time and has a high average MSE when the convergence threshold of BR is set to the same threshold as in LM and SCG. Hence, the convergence threshold should be refined.
The experiments on the technical analysis features obtain the lowest average MSE for MFG, regardless of algorithms, and the experiments on the fundamental analysis features obtain the best for OPR, as shown in Fig. 12 and Fig. 13 . For the MFG Company, as the dynamics of information is approximated as a linear variation, fundamental data misses details of daily information, so the accuracy of the prediction is not as good as the technical data, which represents daily information.
Besides, even though fundamental data occupies a small percentage of the influence on mixed data, these filled fundamental features may conflict with the technical features, thus producing negative impact on the prediction of stock price. Hence, the prediction accuracy of the model on mixed features is reduced.
However, for the OPR Company, the stability of historical share price is poor, as shown in Fig. 9 . That will cause a high fluctuation of the prediction accuracy if the technical data is used. The prediction accuracy can be improved and the high fluctuation could be mitigated by using mixtures of fundamental and technical features. This indicates the quarterly data has the positive impact on the prediction accuracy when the stock market is strongly dynamic. This might be because that the linear approximation of daily information from quarterly data could compromise with the dynamics of technical daily data.
Briefly, technical features can be selected when the share price is stable, while fundamental features are better when the share price has a high fluctuation.
Many parameters can be adjusted for improving the prediction accuracy of a neural network. Appropriately adjusting the parameters plays a vital role in share price prediction. The method of trial and error is used to find the most suitable parameter values of RNN on different feature sets for different companies. Therefore, the best-performed RNN is with 15 hidden neurons for OPR. neurons could be small, and when the nonlinearity of the training data is strong, the number of neurons could be large. When the number of hidden neurons is not enough, the trained model cannot well represent the mapping function between the data and the predicted value, the prediction accuracy will be not good enough. When the number of neuron is larger than the best number, the prediction accuracy on the test data set could be dropped, as the trained model might over-fit with the noise in the training data.
The delays is another important parameter of RNN to be adjusted for improving the performance of RNN. It also determines the time looking back of the history data for the prediction of share prices. As to the financial analysis, the most commonly used indicators are 5-days moving average, 10-days moving average and 15-days moving average. Correspondingly, the delays are set to 5, 10 and 15 delays. From Table 11 and Fig. 16 , it can be seen that the MSE of RNN outputs for MFG is increasing as the number of delays of RNN is increasing. Fig. 17) . Similar with the case of MFG, the running time of RNN is increasing, due to the increasing complexity of RNN (Table 12) . Fig. 16 with Fig. 17 , for MFG, the short-term historical data could have positive impacts on its share price because the historical data is stable and do not need to consider long-term historical data by data processing of fundamental data. In contrast, for OPR, the long-term historical data could have positive impacts on its share price, as the long-term historical data could compensate for the high fluctuation of short-term historical data of OPR.
In brief, the relationship between MSE and the number of delays varied with the stability of the historical data of different companies. Hence, the delays should be adjusted to the optimal value for different companies.
The research developed a hybrid approach to predicting share price of aerospace industry companies by using the combination of PCA and RNNs, trained with LM, BR, and SCG algorithms, respectively. The proposed approach could help create automatic prediction of share prices for different industries, not only aerospace industry sectors, but also other industry sectors. The developed approach to predicting stock prices could provide decision-makers with a reference or evidence for their economic strategies and business activities, thus helping the financial industry to increase the return on investment. Although the research was done based on the data in aerospace industry at pre-COVID-19 time, the developed approach can be used for the share price analysis of aerospace industry at post COVID-19 time.
The combination of fundamental and technical analysis in this research fills the gap in the studies of stock prices that used to use only fundamental or technical analysis individually. However, simply combination of technical and fundamental data may not be good enough for the prediction of stock price for aerospace industries. Hence, feature selection with optimisation algorithms will be future work. Currently, COVID-19 is severely influencing aerospace industry, so, we will investigate the prediction of aerospace share price at post-COVID-19 time. Table. A2. 