Stock Market Forecasting Using Machine Learning Algorithms

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2 In addition to stock markets, commodity prices and foreign currency data are also listed as potential features, as different financial markets are interconnected. For instance, slowdown in US economy will definitely cause a drop in US stock market. But at the same time, USD and JPY will increase with respect to its peers as people seek for asset havens. Such interplay implies the underlying relationship between these financial products and the possibility of using one or some of them to predict the move of the other ones. B. Data collection The data set used in this project is collected from [3]. It contains 16 sources as listed in Table I and covers daily price from 4-Jan-2 to 25- Oct -212: Since the markets are closed on holidays which vary from country to country, we use NASDAQ as a basis for data alignment and missing data in other data sources is replaced by linear interpolation. Stock Currency Commodity TABLE I. DATA SOURCE NASDAQ, DJIA, S&P 5, Nikkei 225, Hang Seng index, FTSE1, DAX, ASX EUR, AUD, JPY, USD Silver, Platinum, Oil, GoldC:\Dropbox\CS229 Project C. Feature selection In this project, we focus on the prediction of the trend of stock market (either increase or decrease). Therefore, the change of a feature over time is more important than the absolute value of each feature. We define x i(t), where i {1,2,..., 16}, to be feature i at time t. The feature matrix is given by where F = (X 1, X 2,..., X n) T (1) X t = (x 1(t),x 2(t),...,x 16(t)) (2) The new feature which is the difference between two daily prices can be calculated by δx i(t) = x i(t) x i(t δ) (3) δ X (t) = X (t) X (t δ) = ( δ x 1(t), δ x 2(t), δ x 16(t)) T (4) δ F = ( δx(δ + 1), δx(δ + 2),..., δx(n)) (5) Due to the difference in market value and basis of each market, the differential values calculated above can vary in a wide range. To make them comparable, the features are normalized as following: and the normalization can be implemented as: (6) As discussed above, the performance of a stock market predictor heavily depends on the correlation between the data used for training and the current input for prediction. Intuitively, if the trend of stock price is always an extension to yesterday, the accuracy of prediction should be fairly high. To select input features with high temporal correlation, we calculated the autocorrelation and cross-correlation of different market trends (increase or decrease). The results shown in Figure 2 use NASDAQ as the base market. Cross-correlation with respect to NASDAQ NASDAQ S&P5 DJIA Hang Seng Nikkei225 FTSE1 DAX ASX Gold PM Silver Platinum PM Palladium PM Oil AUD Euro JPY Delay Fig. 2. Autocorrelation and cross-correlation of market trends using NASDAQ as a base. It can be seen from the graph that the autocorrelation of NASDAQ daily trend is only non-zero at the origin based on which we may conclude that the trend of NASDAQ daily index is approximately a Markov process. Hence, past data of NASDAQ will not provide much insight to its future movement. The same conclusion could be made on many other data sources whose cross-correlation with NASDAQ is close to zero. Although the trend of DJIA and S&P5 have strong correlation with NASDAQ given the data are on same day, they are not available by the time they are needed for prediction. However, data sources such as DAX AUD and some other markets are promising features for our predictor built by machine learning algorithm as they have relative high correlation with NASDAQ at the origin and their data is available before or at the beginning of the US market trading time. This observation actually confirms our forecast principle, as discussed in previous sections, about the inter-connection between global markets and how the information reflected by their movements can be beneficial to the prediction of US stock markets. In addition to the correlation of daily market movements, it is also worth investigating the correlation of market trends in longer term, which may also offer valuable information for predicting future price [4-5]. To study this, δ in Eq. (3) is varied from 1 day to 5 days, and part of the results are plotted in Figure 3. As can be seen from the graph, temporal correlations between the markets increase with the time (7)

3 window δ over which the trends of stock indices are calculated. One explanation for this phenomenon is that the calculation in Eq. (3) makes the time spans of the outputs overlap with each other and therefore increase the temporal correlations. Moreover, the operation also implicitly conduct an averaging on the data within the interval which effectively removes the noise and the underlying correlation between the markets become clearer. Normalized correlation Fig. 3. Autocorrelation and cross-correlation of market trends with different time spans. Given all possible features, we implemented forward feature selection algorithm to select the features that contribute most to the accuracy of prediction using different machine learning algorithms. Detail results are presented in the next section. As expected, a combination of daily market trend and long term movement provide the best result. III. 4 Time window size A. Trend Prediction 2 EXPERIMENTAL RESULTS AND DISCUSSION 1) Single Feature Prediction In section 2, we used cross-correlation to estimate the importance of each feature. To verify the information given by correlation analysis, we use individual feature to predict daily NASDAQ index trend. The prediction accuracy by each single feature is shown in Figure Relative delay 5 The result of this experiment supports the analysis of crosscorrelation. Hence, we are convinced that index value of other stock markets and commodity prices can provide useful information in the prediction process. 2) Long Term Prediction Besides daily movement, sometimes we also care about prediction results for longer terms. Here, we define our problem as predicting the sign of difference between tomorrow s index value with respect to that of certain days ago. We use SVM as a training model and the prediction accuracies under different time spans are shown in Figure 5. Fig. 5. Accuracy of Long Term Prediction From the figure, we can see that prediction accuracy increases when time span becomes longer. This is because that the longer the period is, the more information we have and the higher resistance of our prediction to noise. At last, we can reach 85% accuracy when time spam gets longer than 3 days. Actually, we can re-paraphrase this problem as estimation of { }, where. This corresponds to the after-mentioned regression problem of daily stock market movements. 3) Multi-Feature Prediction Using the features described in section 2, we compare the prediction accuracies of SVM algorithm and MART (a decision tree based boosting algorithm). The prediction results are shown in Table 2. TABLE II. ONE-DAY PREDICTION ACCURACY Top 4 Features All Features SVM 74.4% 63.1% MART 7.3% 73.9% Fig. 4. Prediction accuracy by single feature From the figure, we can see that DAX yields the best results, 7.8% accuracy of prediction. Prediction accuracies of Australian dollar, FTSE and oil price are also relatively high, reaching 67.2%, 66.4% and 65.2% respectively. From Table 1, we can see that accuracies from SVM and MART learner can reach as high as 74%. This daily trend prediction accuracy is higher than most of models and the values reported on financial analysis websites. In addition, we note that SVM algorithm is very sensitive to the size of training data. When the size of training set is not large enough, hyper-plane found by SVM algorithm might not be able to split the data properly. Thus, feature selection is essential when using SVM. In contrast, Multiple Additive Regression Trees (MART) algorithm requires less training data and prefers high dimensional feature set.

4 To test the generality of our model, we used the same algorithm to predict the other two US stock markets. The results are shown in Table 3 below. TABLE III. PREDICTION ACCURACY ON ALL US STOCK MARKETS NASDAQ DJIA S&P 5 Accuracy 74.4% 77.6% 76% As can be seen, all entries in the table are high. This shows that our model can be applied to all US stock markets. Actually, the idea of using time delay between different stock markets can be also utilized to predict other indexes. B. Regression Compared to stock trend, the exact increment in stock index may provide more information for investment strategy. This means the classification problem now evolves to a regression problem. To judge the performance of our model, we use square root of mean square error (RMSE) as criteria, which is defined as To build up the multiclass classifier, we firstly need to define the width of the central region. To evaluate how well our classifier is, we introduced the concepts of precision and recall, which are defined as In the equations above, tp, fp and fn stand for true positive, false positive and false negative respectively. The precision and recall values against different central region widths are plotted in Fig. 6. In the figure, Recall for positive class reflects the proportion of predicted rising days among all rising days while precision indicates the hit rate among rising predictions. Thus, recall directly impacts the frequency of trading and precision impacts the profit / loss at each time. Taking the product of the two, we calculate the F1 score which is defined as (7) (8) (9) (6) We use linear regression, generalized linear model (GLM) and SVM algorithm to predict exact value of daily NASDAQ movement. The RMSE values for different algorithms are reported in Table 4 below. TABLE IV. STOCK INDEX REGRESSION ACCURACY Baseline SVM Linear GLM RMSE The baseline predictor in the table is formed by a zeroorder hold filter. From the table, we can see that SVM gives the most accurate prediction. The RMSE given by SVM is 21.6, only half of the average fluctuation, C. Multiclass Classification In previous part, we explored various methods to improve prediction accuracy and minimize square root mean square error. These efforts can be directly used to maximize the trading profit. However, besides maximizing profit, another aspect of our task is to minimize trading risk. In this part, we will use the SVM regression model and start from the basic intuition in SVM algorithm. In SVM, the further distance between the point and hyperplane is, more confident we are for the prediction we made, whereas, our prediction cannot be very accurate when the point is close to hyper-plane. To minimize the trading risk, we can pick out these risky points and ignore their prediction labels. Thus, we need to classify the original data into at least three classes, negative, neutral and positive. This is the intuition that leads to the prototype of our multiclass classification model. Fig. 6. Precision and recall vs. central window widths for positive and negtive class The value of F1 scores for positive and negative classes are shown in Fig. 7. As can be seen from the graph, F1 scores for both class are relatively high around and 5. Therefore, we shall choose window size around when trading fees / tax are small enough to be ignored. Otherwise, we should choose 5 as the optimal window size. Fig. 7. F1 score vs. central window size for positive and negative class

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