# Long-term Stock Market Forecasting using Gaussian Processes

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3 and y! ℝ, the goal is to predict new y given x using f x such that: y! = f x! + δ! where δ! is a Gaussian noise with mean zero and variance σ!. However, we assume that closing prices in stock market are noise free because true prices are evaluated at closing time (Todd & Correa, 2007). The prior distribution of the observed target y is given by y~𝒩(0, K(X, X)), where, K(X, X) is the covariance matrix between all pairs of training points and X is (n m) matrix of input. In this project, (Gaussian) radial basis function kernel, or RBF kernel is used: k x!, x! = exp ( σ x! x! ! ). (3) The predictive distribution of y can be computed by conditioning on the training data to get p(f(x ) x, D). The joint distribution over y and predictions of x is given by: y K(X, X) f(x ) ~𝒩 0, K(x, X) (2) K(X, x ). K(x, x ) (4) The conditional distribution of (2) allows us to get the predictive distribution of y with the following mean and covariance (Ou & Wang, 2011): f x = K X, x 106! K + σ!! I V! x = K x, x K X, x 107! K+!! y, σ!! I!! K (5) X, x. (6) The main idea of this approach is to avoid representing the whole history as one time series. Each time series is treated as an independent input variable in the regression model (Chapados & Bengio, 2007). For trading year i, there are M! trading days, i = 1,..., N and t = 1,..., M!. The model problem is given M observations from i = 1,..., N 1 trading years and partial trading days from N, {y!! }, t = 1,..., M!, we want to predict the rest of trading days in N, {y!! }, τ = M! + 1,..., M! + H, where M! + H is the last day of trading in N. Also, it is given {x!! } for each series and our objective is to find P({y!! }, M! + 1,..., M! +!!!,...,! H x!!, y!!!!!,...,! ). See Figure 1. Methodology! Figure 1: Illustration of the regression variables (price history from 2002 to the first quarter of 2011) of Starbucks stock. The objective of this model is to predict the "green strip" in 2011.

5 days of year 2011 of size 126 days are reserved for test data. Figure 3 shows the training data and the forecast results for Starbucks stock Figure 3: Top plot: Training set of Starbucks stock for the period from 2002 to the first quarter of Each line represent complete trading year. Meddle plot: Shows the first scenario where forecast made for the rest quarters of 2011 (Q2, Q3 and Q4). Bottom plot: shows the second where training set is the period from 2002 to the second quarter of Forecast made for the third and fourth quarters of Results and discussion The forecast results for the three stocks (HP, Yahoo and Starbucks) are shown in Figure 4, 5, 6. The blue lines show the forecast prices and the black lines show the actual prices. In Figure 4, the results of scenario 1 (top part) shows drop in HP stock prices in Q2, Q3 and Q4 of Also, scenario 2 (bottom part) confirms this drop until the end of Based on that, investors should not buy HP stock in 2011 and if they already did, it is highly recommended to sell it to minimize their loss. Although, the model could not predict the high drop in Q3, it keeps following the trend of the actual prices. Figure 5 shows the forecast price of Yahoo stock. The results of scenario 1 (top part) show slight decrease in Yahoo stock prices in Q2 and Q3 of 2011; however, the price shows some improvement in Q4. The second scenario shows Yahoo stock prices reverse direction in Q4. Investors can take the risk and buy in Q3 or wait until the beginning of Q4. The forecasting model is able to track the trend of this stock most of the time.

6 Figure 4: Top part: Forecast result for HP stock from scenario 1. Bottom part: Forecast result for HP stock from scenario Figure 5: Top part: Forecast result for Yahoo stock from scenario 1. Bottom part: Forecast result for Yahoo stock from scenario 2.

7 The forecasting result for Starbucks stock is sown in Figure 6. Although, the true model shows high fluctuation in 2011, our model keeps following the main trend of the stock. Scenario 1 shows falling in the price until the mid of Q3, however, scenario 2 updates the curve in Q3 to follow the increase at the end of Q2. Both scenarios agree that the mid of Q3 is suitable to buy this stock. If investors own the stock before Q3, it is highly recommended to wait until the end of Q Figure 6: Top part: Forecast result for Starbucks stock from scenario 1. Bottom part: Forecast result for Starbucks stock from scenario 2. In general, this model is able to track the prices of the three stocks. As we know, stock price could be affected by several factors such as political situation and economic conditions, which may cause high fluctuations as shown in some areas of this experiment. As a longterm forecasting model, it is acceptable to not follow these fluctuations. 5 Conclusion and future work In this project, we applied Gaussian processes to perform long-term forecasting in stock market. This technique showed acceptable prediction to three stocks from NASDAQ Stock Market. The experiment showed highly acceptable time to buy and sell over different period of times. Due to the fast computation and the simplicity of this model, investors could use this model to do a long-term investment or to validate their investment decisions. More stocks could be tested on this model from other stock market. References Ayodele, A., Charles, A., Marion, A. & Otokiti Sunday O. (2012). "Stock Price Prediction using Neural Network with Hybridized Market Indicators. Journal of Emerging Trends in Computing and

8 Information Sciences, VOL. 3: 1, 1-9. Chapados, N. & Bengio, Y. (2007). Forecasting Commodity Contract Spreads with Gaussian Process, in 13th International Conference on Computing in Economics and Finance, June 14-16, 2007, Montréal, Quebec, Canada. Groot, P., Lucas, P. & Paul van den Bosch. (2011). Multiple-step Time Series Forecasting with Sparse Gaussian Processes, BNAIC, 1-8. Jeffrey, A. & Kass, D. (2012). The Little Book of Stock Market Cycles (Little Books. Big Profits), Wiley. Mori, H. & Ohmi M. (2005).Probabilistic short-term load forecasting with Gaussian processes. Proceedings of the 13th International Conference on Intelligent Systems Application to Power System (ISAP), November 6-10, 2005, Arlington, Virginia, Ou, P. & Wang, H. (2009). Prediction of market index movement by ten data mining techniques. Modern Applied Science, 3:12, Ou, P. & Wang, H. (2011). Modeling and Forecasting Stock Market Volatility by Gaussian Processes based on GARCH, EGARCH and GJR Models. Proceedings of the World Congress on Engineering, July 6-8, 2011, London, U.K., Preethi, G. & Santhi, B. (2012).Stock market forecasting techniques: a survey. Journal of Theoretical and Applied Information Technology, 46:1, Rasmussen, C. & Nickisch, H. (2006). Gaussian Processes for Machine Learning (GPML) Toolbox. Journal of Machine Learning Research, 11, Todd, M. & Correa, A. (2007). Gaussian Process Regression Models for Predicting Stock Trends. Technical Report on MIT University. Wikipedia: The free encyclopedia. (2013) Stock market. Retrieved April 2, 2013, from World Capital Markets Size of Global Stock and Bond Markets. Retrieved April 1, 2013, from

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