VIEWS: 22 PAGES: 5 CATEGORY: Computers & Internet POSTED ON: 6/2/2011 Attribution-NonCommercial-NoDerivatives
Stock Price Forecasting Using Exogenous Time Series and Combined Neural Networks Manoel C. Amorim Neto, Victor M. O. Alves, Gustavo Tavares, a Lenildo Arag˜ o Junior, George D. C. Cavalcanti and Tsang Ing Ren Abstract— Time series forecasting is useful in many re- autoregressive conditional heteroscedasticity (GARCH) [1] searches areas. The use of models that provide a reliable among other models. prediction in ﬁnancial time series may to bring valuable proﬁts Artiﬁcial neural networks (ANN) for time series prediction for the investors. An intelligent agent can be built from a suitable prediction model, to make operations in stock market have been successfully used in the last years, because of daily. Furthermore, even that the investor had caution about some interesting features such as universality in function the use of an automatic agent to make operations he can to approximations, robustness and fault tolerance [4]. For these use the prediction model as a valuable decision support. A reasons, neural networks are considered useful to build methodology based on information obtained from exogenous models for prediction of non-stationary time series [4]. series was used in combination with a neural network to predict stock series. Exogenous series were selected by analyzing the Furthermore, ANN handles well noise data and it is able correlation between the series with the stocks series used. In to predict nonlinear systems, which are the type of systems this way, the prediction was obtained by not just using the that we are interested to predict, the stock market. Among the previous values of the series but also by using information various ANN models, the most used in literature is multilayer external to the main series. Additionally, the best trained neural perceptron (MLP) [5]. Radial basis function (RBF), wavelet- networks were used in a combination to improve the prediction capacity of single networks. To evaluate the proposed models based and recurrent neural networks have been also applied for prediction, some known metrics were used plus a proposed with success [6]. one - Prediction in Direction and Accuracy (PDA), which uses Stock Market is a complex system composed of many some features to determine if a model has a great accuracy and investors selling and buying ﬁnancial products in form of trend in prediction. Through this novel metric, we have used securities. Here, we are interested in the prediction of stocks an evolutionary algorithm to choose the best trained models in order to obtain better results. Experiments with two of the of the biggest Brazilian oil Company, Petrobras, and one most important Brazilian companies’ stock quotes have shown of the biggest miners companies of the world, Vale do Rio the usefulness of the proposed prediction system to generate Doce. The Petrobras stock index is named PETR4 and the proﬁts in investments. Vale do Rio Doce is named VALE5. These time series were I. I NTRODUCTION analyzed between the years of 2003 and 2009. In this paper, a comparison between two models of ANN, Time series are sets of variables observed over a deﬁned named MLP and RBF networks, both with and without ex- period of time. These observations may be discrete or con- ogenous time series are presented. Additionally, we propose tinuous and they are taken in an equal time interval [1]. a novel performance metric to select the best trained models, There are many research areas involving time series anal- which aims to maximize trend prediction and accuracy. The ysis, such economy, physics, engineering, social sciences, propose metric was used for selection of the best trained computing, biology, medicine, meteorology and others. networks to be combined in a combination machine. Perhaps the most applied analysis of a time series is in pre- This paper is organized as follows. Section II describes diction. The prediction can be made using past observations brieﬂy the stock market and the exogenous time series used. of the series that will be forecast or even other time series. Section III presents the performance metrics which were These different ones used to predict the main are know as used and the novel introduced metric. Section IV presents Exogenous Time Series. the proposed methods for combining neural networks in a There are two types of models in time series predic- combination machine. Section V describes the experiments tion: linear and non-linear. A known linear method is the and results obtained. Finally, the Section VI presents the ARIMA, proposed by Bob and Jenkins [2]. Some examples conclusions and ﬁnal remarks. of non-linear models are: bilinear, exponential autoregressive, threshold autoregressive, smooth transition autoregressive, II. T HE S TOCK M ARKET AND E XOGENOUS T IME S ERIES autoregressive with time dependent coefﬁcients [3], autore- The main function of the capital market is the trade of gressive conditional heteroscedasticity (ARCH) and general stocks with the purpose of ﬁnance development, which in its Manoel C. Amorim Neto, Gustavo Tavares, Victor M. turn produce and nourish the market itself. On this way, a O. Alves are with the Facilit Technology Company, Brazil, third function is attributed: the market of its own sources of {manoel,gustavotavares,victor}@facilit.com.br. Site: www.aistocktrend.com incomes [7]. The monetary market, as a whole, is important George D. C. Cavalcanti and Tsang Ing Ren are with the Center of Infor- matics, Federal University of Pernambuco, Brazil, {gdcc,tir}@cin.ufpe.br. for the economic development. However, when the economy Site: www.cin.ufpe.br/∼viisar and the market develops, the market of the source of capital emerges, which are the stock market, debt titles and real III. P ERFORMANCE M EASUREMENT OF P REDICTION estate market. M ODELS Globalization is a trend that allows an intense interchange There are several metrics used to evaluate models of between countries. Consequently, it is common nowadays time series forecasting. In this paper we have employed ﬁve that the stock market of an emergent country like Brazil metrics that are commonly used in literature: MSE, MAPE, attain an increasing importance in the international scenario. POCID, THEIL (or NMSE) and ARV. Additionally, it was Today the stock market is not only an important source used SLG, which was proposed by Amorim Neto [8], and of corporation ﬁnance but also an individual capitalization a novel metric proposed in this work, named Prediction in resource. When investing in a portfolio, the investor wishes Direction and Accuracy (PDA). to obtain a large return in other to compensate the risks A simple measure to evaluate the accuracy of a forecasting associated, in other words, the objective is to minimize risk model is the diference between the expected value and the and maximize capital returns. Hence, a prediction method is output value of model. From Equation 1, Tt is a expected most useful and a neural network is well-suited for this kind value and Yt is the output of the forecast model, and et is of optimization procedure. the calculed error, both at time t. Consider this measure as Currently, the Brazilian stock market, which is also known a basis for the others. a in the World Federation of Exchange (WFE) by S˜ o Paulo SE, has a global importance. From the 51 stocks monitored et = |Tt − Yt | (1) by WEF, BOVESPA was in eighth position among the biggest stock market in the world in terms of capitalization The performance measurement metrics used in this work and stock values, in a ranking for developing countries. Two are brieﬂy described here. Consider for every metric: Tt as of the biggest companies in the BOVESPA stock market are the desired output of the forecasting model at time t and the Petrobras oil company and Vale do Rio Doce, which Yt as the output of the proposed model and N as the total makes them ideal stocks to be analyzed. For the professional amount of available patterns. investor to understand the behavior of a stock, at least ﬁve A. MSE (Mean Squared Error) series are necessary: The Mean Squared Error is the most known metric to 1) The highest value that the stock was negotiated in a evaluate the performance of forecasting models. It is deﬁned certain day. as: 2) The lowest value that the stock was negotiated during the same day. N 1 3) The value of the ﬁrst negotiation of the day: opening M SE = (et )2 (2) price. N t=1 4) The value of the last negotiation of the day: closing B. MAPE (Mean Absolute Percent Error) price. The Mean Absolute Percent Error measure the accuracy 5) The business volume of the stock during the same day. of model in percentage. It is deﬁned as: The closing prize is the series that is really important, since most of the professional investors and ﬁnancial institutions 1 N et take action based on its value. M AP E = (3) N t=1 Yt From the methods for forecasting time series, the choice of the input variables is an important step. In this work, A lower value of MAPE is the desired result from a we are interested in the prediction of the stocks quotations prediction method. of PETR4 and VALE5. To predict these stock values, we C. THEIL or NMSE (Normalized Mean Squared Error) have used exogenous time series that were chosen based The Normalized Mean Squared Error evaluate the relation- on the autocorrelation analyzes, similarly to work done ship of the model with the random walk model. Equation 4 previously [8]. deﬁnes this value. For the Petrobras Company (PETR4) the exogenous time series utilized were: Dollar, IBOV, CLF, NSY:PBR, DAX N and SP500. t=1 (et )2 T HEIL = N (4) Dollar time series is the Brazilian Real quotation converted t=1 (Yt − Yt−1 )2 to United States Dollar. IBOV is the BOVESPA quotation. When THEIL is equal to one, the proposed model is CLF is the Crude Light Oil Future quotation. NSY:PBR is equivalent to random walk model. The random walk model the quotation of Brazilian Oil. DAX is the German stock proposes that the time series future value is equal to the market index. SP500 is the S&P 500 index. current value. If THEIL is lower than one, then the proposed For the Vale do Rio Doce Company (VALE5) the exoge- model is better than random walk model. If THEIL is greater nous time series used were: Dollar and IBOV. This stocks than one, then the proposed model has a performance worse were chosen based on economic analyzes [8]. than random walk model. D. POCID (Prediction On Change In Direction) POCID is the percentage of the correct trend of the model N Gt t=1 relative to the trend of expected value. This metric is deﬁned P DA = (10) N by Equation 5. where Gt is deﬁned in Equation 11 : N t=1 Dt P OCID =100 (5) 1− ret , if (Dt = 1) and ret < remax , N remax 0, if (Dt = 1) and ret ≥ remax , The value of Dt is deﬁned by Equation 6 Gt = −1 + reret , if (Dt = 0) and ret < remax , (11) max −1, if (Dt = 0) and ret ≥ remax 1, if (Tt − Tt−1 )(Yt − Yt−1 ) > 0, where Dt is deﬁned by Equation 12, ret = Tt and e Dt = 0, otherwise. (6) t remax = 0.02. This constant value is the relative maximum E. ARV (Average Relative Variance) error accepted by the prediction. In this case, the maximum tolerance is 2% error. The Average Relative Variance evaluates the relationship of the model with the other model, which proposes that the time series future value is equal to the arithmetic mean of 1, if (Tt − Tt−1 )(Yt − Yt−1 ) > 0, Dt = 0, otherwise. (12) the past values. It is deﬁned as: N If the models have a right prediction in direction (Dt = t=1 (et )2 1) and the relative error is lower than maximum error then ARV = N (7) ret t=1 (Yt − T )2 1 − remax is added; if the models have a right prediction in direction (Dt = 1) but the relative error is greater or equal When ARV is equal to one, the proposed model is equiv- than the maximum error then nothing is added; if the model alent to the mean of past values. If ARV is lower than one, has a wrong prediction in direction (Dt = 0) and the relative then the proposed model is better than the mean of past ret error is lower than maximum error then −1+ remax is added; values. If the ARV is greater than one, the proposed model if the model have a wrong prediction (Dt = 0) in direction has a performance worse than mean of past values. and the relative error is greater or equal than maximum error F. SLG (Sum of Losses and Gains) then −1 is added. After the summation, the mean is calculed. The SLG was proposed by Amorim Neto [8] and was IV. N EURAL N ETWORKS C OMBINATION inspired by POCID. It deﬁned as the mean of the losses and gains of the model. The SLG measurement is deﬁned by Neural Network is a stochastic mathematical model that Equation 8: aims to simulate the functionality of a biological network. A Neural Network is formed by a set of connected neurons N organized in layers. Each neuron can be considered a com- t=1 Lt SLG = (8) putational processing unit. There are several kinds of Neural N Networks [4], and Multi-layer Perceptron (MLP) and Radial In Equation 8, the value of Lt is deﬁned by Equation 9 Basis Function (RBF) were used in this paper. The training of MLPs using exogenous time series im- + |(Tt − Tt−1 )| , if (Tt − Tt−1 )(Yt − Yt−1 ) > 0 proves the ability of the model in forecasting, as it has Lt = − |(Tt − Tt−1 )| , otherwise. been was demonstrated by Amorim Neto [8]. Additionally, (9) a combination of MLPs trained with exogenous time series improves the single MLP performance [8]. SLG less than zero indicates ﬁnancial losses. A combination of neural networks is an architecture which G. Prediction in Direction and Accuracy (PDA) uses a set of trained models and combines the outputs of these models, in the same input, in a unique system. The PDA is the novel metric proposed in this paper. The combination architecture used here is depicted in Figure 1. objective is to beneﬁt models of forecasting with the better This paper presents two ways to choose the neural net- behavior in trend and accuracy. This is possible by the works which will integrate the combination: (i) the selection maximization of POCID and the minimization relative error. of the best networks through PDA, and (ii) selection through The accuracy of a model is measured by maximum relative an evolutionary algorithm. The details are in the Section V. error. The best behavior in trend and the most accurate model will have a higher value of PDA. In other words, V. E XPERIMENTS the higher the value of PDA implies in a better model. It is a improvement of the SLG metric. This model is This Section describes all experiments performed to eval- mathematically described in Equations 10 and 11. uate the prediction metric and methods describe above. TABLE II D ISTRIBUTION OF STOCK QUOTES PER PATTERN IN THE DATABASES WITH EXOGENOUS . PETR4 database VALE5 database Stock Lag Stock Lag PETR4 close -1 VALE5 close -1 PETR4 close -2 VALE5 close -2 PETR4 close -3 VALE5 close -3 PETR4 open -1 VALE5 open -1 PETR4 highest -1 VALE5 highest -1 PETR4 lowest -1 VALE5 lowest -1 Dollar close -1 Dollar close -1 DAX close -1 IBOV close -1 Fig. 1. Combination architecture used in this work. IBOV close -1 CLF close -1 TABLE I SP500 close -1 D ISTRIBUTION OF STOCK QUOTES PER PATTERN IN THE DATABASES NSY:PBR close -1 WITHOUT EXOGENOUS . PETR4 database VALE5 database Stock Lag Stock Lag First, two experiments using the main time series, PETR4 PETR4 close -1 VALE5 close -1 without exogenous, were done. One was using MLP and PETR4 close -2 VALE5 close -2 PETR4 close -3 VALE5 close -3 another using RBF. The training of the MLP was made with a variation of hidden neurons in [20 . . . 60], ten times for each number, generating 410 trained MLPs. The training of RBF was made with a variation in [10 . . . 100], generating A. Databases 910 trained RBFs. We have used the validation dataset to Two databases were used for the evaluation of the pro- choose the best network conﬁguration, and then use it in a posed methods: PETR4 stock quotes dataset and VALE5 test set. stock quotes dataset. Besides, all exogenous time series Afterwards, two more experiments were done, but now described in Section III were used to complement the main including the exogenous time series. Again, 410 trained series. MLPs and 910 trained RBFs were obtained. The experiments were performed using two groups of In the end, following Equation 13, where X is the total datasets: dataset without exogenous and dataset with ex- number of trained networks, the N bet networks (trained ogenous time series. Both PETR4 and VALE5 had the two with exogenous) were chosen for combination (according to dataset groups. Table I shows the stock quote distribution for validation dataset). each database without the exogenous time series and Table II shows the distribution with exogenous time series. The ”lag” N = round(log2 (X)) (13) notation is equivalent to the time t of the series, i.e., lag −1 corresponds to the stock quote on previous day; lag −2 We have seven metrics to choose the bests networks for corresponds to the stock quote on two days past; and lag 0 combining. There are a lot of combination possibilities, and corresponds to the stock quote from the current day. we have used two approaches: (i) the combination of the N best networks based on P DA metric and (ii) a genetic B. Experimental Setup algorithm (GA) for this task, where the variable to be Before the experiments, the databases were organized in maximized was P DA. three datasets: training, validation and test sets. The training Genetic Algorithm is a evolutionary technique which aims set was used for the learning of the neural networks. The val- to get optimization by evolution, through some operators, idation set was used for tuning of some training parameters. such mutation and crossover. In this method, each possible The training was performed varying other parameters such solution to the problem, that must be optimized, is repre- as: the number of hidden neurons for MLP and the spread sented by a chromosome. of the RBF, resulting in a large number of experiments and In general, these results show that the RBF has a better neural networks trained. From 1, 500 days of stock quotation performance than MLP according to the proposed metric. in the database, 1, 200 were used for training and validation In the experiments with combination in PETR4 database, and the last 300 days for testing. For each experiment, the the genetic algorithm found the following metrics combina- training and validation sets were divided randomly, with 900 tion: POCID + THEIL + ARV + PDA for MLP and MSE for training and 300 for validation. However, the test dataset + MAPE + THEIL + ARV + PDA for RBF. For VALE5 remained the same. The experiments were done as follows. database, the GA found these metrics combination: POCID TABLE IV + PDA for MLP and MSE + MAPE + THEIL + ARV + PDA B EST RESULTS WITH COMBINATION BY THE BEST NETWORKS RANKED for RBF. Table III shows the combination results generated BY PDA METRIC (X(σ)). by GA. In both databases, the combination of MLP was better than PETR4 database RBF, according to PDA metric. In fact, for both databases, Metric MLP RBF the MLP combination presented the best combination of MSE 0.6056 (0.01061) 0.61335(0.010417) accuracy and trend prediction than RBF combination. MAPE 0.015732 (0.00015777) 0.016285(0.000224) The results of the combination of best PDA networks can POCID 81.5436 (0.47457) 79.4295(0.65319) be seen in Table IV. As in GA combination, this results SLG 0.72031 (0.021801) 0.69428(0.017808) shows that MLP is the best choice. For both databases, THEIL 0.26983 (0.0035511) 0.29633(0.0038447) the MLP combination presented the biggest PDA value, ARV 0.0020787 (3.5254e-005) 0.002107(3.6346e − 005) indicating that this combination had better accuracy and PDA 0.27865 (0.004667) 0.2365(0.011121) trend prediction than RBF combination. Even when RBF VALE5 database outperforms MLP in some metric, the performance of both Metric MLP RBF are very close considering this metric. MSE 0.72078(0.024093) 0.59518 (0.013024) MAPE 0.015687(0.00027368) 0.015478 (0.00025494) TABLE III POCID 82.6756 (0.41113) 80.9365(0.49857) B EST RESULTS WITH COMBINATION BY GENETIC ALGORITHM (X(σ)). SLG 0.83816 (0.010863) 0.80828(0.010803) PETR4 database THEIL 0.33828(0.011696) 0.26519 (0.0069003) Metric MLP RBF ARV 0.0017837(6.0259e − 005) 0.0014727 (3.2269e-005) MSE 0.61812 (0.0063709) 0.62212(0.014382) PDA 0.26401 (0.0080416) 0.22843(0.0074966) MAPE 0.015824 (7.8961e-005) 0.016028(0.00015723) POCID 81.4094(0.17329) 81.5436 (0.27399) SLG 0.70212(0.010981) 0.70862 (0.003683) be more suitable because genetic algorithm have a high THEIL 0.27291 (0.0033154) 0.28875(0.0052983) computational cost. ARV 0.0021179 (2.153e-005) 0.0021358(4.9668e − 005) PDA 0.28546 (0.0054852) 0.26451(0.0044274) Also, the proposed metric is a natural evolution of SLG that aims to improve the rank of network based on accuracy VALE5 database as well, instead of trend exclusively. The results obtained Metric MLP RBF showed some relation between this metric and other accu- MSE 0.71341(0.021186) 0.6008 (0.017369) racy/trend metrics. In other words, when PDA increases, the MAPE 0.015662(0.00017252) 0.015511 (0.00027798) accuracy/trend has also an improvement. POCID 83.1104 (0.47951) 81.0033(0.56406) SLG 0.83871 (0.012075) 0.80091(0.012646) R EFERENCES THEIL 0.33196(0.0093126) 0.26268 (0.0055378) [1] Brockwell, P. J. and Davis, R. A. Introduction to Time Series and ARV 0.0017654(5.3089e − 005) 0.0014873 (4.29e-005) Forecasting. New York, USA : Springer Verlag, 1996. PDA 0.27251 (0.0097393) 0.23946(0.012403) [2] Chu, Ching W., Ching Z. and Guoqiang P. A comparative study of linear and nonlinear models for aggregate retail sales forecasting. International Journal of Production Economics, pp. 217-23, 2003. [3] De Gooijer, Jan G., Jan K. and Kuldeep. Some recent developments in non-linear times series modeling, testing and forecasting. Prentice Hall, VI. C ONCLUSION 1998. [4] Haykin S. Neural Networks: a Comprehensive Foundation. Second This paper presented a comparison between MLP and Edition. International Journal of Forecasting, vol. 8, pp. 135-156, 1992. [5] Charkha, Pritam R. Stock Prediction and Trend Prediction using Neural RBF neural networks using PETR4 and VALE5 time series Network. First International Conference on Emerging Trends in Engi- with and without exogenous data. It also introduced a new neering and Technology, pp. 592-594, 2008. performance metric for selection of trained networks to [6] Ferreira T. A. E., Vasconcelos G. C. and Adeodato P. J. L. A New Intel- ligent System Methodology for Time Series Forecasting with Artiﬁcial combine in combination machines. Experiments were made Neural Networks. Neural Process Letters, vol. 28, pp. 113-129, 2008. to verify the usefulness of the proposed metric and the [7] Schumpeter J. A. The Theory of Economic Development: An Inquiry proposed combinations. into Proﬁts, Capital, Credit, Interest, and the Business Cycle. Transac- tion Publishers, 1982. The experiments showed that: (i) without combination, [8] Amorim Neto M. C., Calvalcanti G. D. C., Ren T. I. Financial time RBF outperforms MLP in general; (ii) with combination, series prediction using exogenous series and combined neural networks. MLP makes an improvement in performance and overcomes International Joint Conference on Neural Networks, pp. 2578-2585, 2009. RBF; (iii) the proposed novel metric is useful for network selection based on the main metrics for ﬁnancial investments, especially it is suitable for minimization/maximization algo- rithms, as used in a genetic algorithm. The two proposed combination methods had similar gains in prediction. However, the selection by the best PDA can