were used as features for a prediction stage based on combined neural networks. ..... calculating the arithmetic means of the observed values of the series.
Financial Time Series Prediction Using Exogenous Series and Combined Neural Networks Manoel C. Amorim Neto, George D. C. Calvalcanti and Tsang Ing Ren Abstract—Time series forecasting have been a subject of interest in several different areas of research such as: meteorology, demography, health, computer and finance. Since it can be applied to various practical problems in real world, techniques to predict time series have been a topic of increasing research activities, especially in the financial sector that has a great interest in the forecast of the stock market. In this article, we are interested in the forecast of the time series related to the Brazilian oil company, Petrobras (PETR4). A methodology based on information obtained from exogenous series was used in combination with a neural network to predict the PETR4 stock series. Exogenous series were selected by analyzing the correlation between the series with the Petrobras stocks series. In this way, the prediction was obtained by not just using the previous values of the series but also by using information external to the PETR4 series. The values of the selected series were used as features for a prediction stage based on combined neural networks. To evaluate the performance of the system classical measurements were used, however we also introduce a new performance index called Sum of the Losses and Gains (SLG). I.
A
INTRODUCTION
series can be defined as a set of sequential observations, of a variable of interest, recorded over a define period of time. Another property of the time series that we are interested in, is that they are discrete and have an equal interval of time in between recorded observations. In general, time series are a subject of research interest in various area of knowledge such as: in economy (stock prices, unemployment rate, and industrial production), in epidemiology (rate of cases of an infectious diseases), in medicine (electrocardiogram, and electroencephalogram), and in meteorology (temperature, wind velocity, and pluviometric precipitation). An important characteristic of these types of data is that the observations are dependent in time, meaning that the observation done in time t depends on previous observation. In time series prediction, researchers try to identify structure and pattern in the data, so to construct a model to predict future patterns of the series. The estimation of the future values of the series is accomplished using past observations either from the series of interest or from exogenous data. One of the most popular linear models of time series is the Box & Jenkins model also known as ARIMA [3]. Non-linear models have also been proposed, TIME
Manoel C. Amorim Neto , George D. C. Cavalcanti and Tsang Ing Ren are with Center of Informatics (CIn), Federal University of Pernambuco (UFPE), Av. Prof Luis Freire s/n, Cidade Universitária, Cep: 50.740-540 Recife PE Brazil. E-mail:{mcan,gdcc,tir}@cin.ufpe.br. Site: www.cin.ufpe.br/~viisar.
examples of such models are: bilinear, exponential autoregressive, threshold autoregressive, smooth transition autoregressive, autoregressive with time dependent coefficients [4], autoregressive conditional heteroscedasticity (ARCH) and general autoregressive conditional heteroscedasticity (GARCH) [5] among other models. In the last years, artificial neural networks (ANN) have been successfully used to model time series [14][15]. ANN has some important characteristics, such as: learning and generalization from a dataset, and universality in function approximation of continuous linear and non-linear multivariable. These properties make ANN an interesting approach to model and predict non-stationary series [1]. Furthermore, ANN handles well noise data and it is able to predict nonlinear system, which is the type of system that we are interested to predict, the stock market. Among the various types of ANN like Multi-Layer Perceptron (MLP), Recurrent Neural Networks, Kohonen SOM, Hopfield Neural Networks and ART, the MLP neural networks are the most used ANN for time series prediction [1][2]. However, other types of ANN (radial based function and waveletbased) have been also applied [6]. Stock Market is a complex system composed of many investors selling and buying financial products, in the form of securities. Here, we are interested in the prediction of stocks the biggest Brazilian oil Company, Petrobras, since it strongly affect BOVESPA, the São Paulo stock exchange (Bolsa de Valores de São Paulo). The Petrobras stock index is named PETR4, and we analyzed the time series over a period of six years. In this article, we analyzed several other series that are correlated to the PETR4 series and used these information as exogenous series in the prediction procedure. An important parameter in the process to evaluate the best configuration of the technique was the performance measurement. The evaluation procedure was implemented using well established measurements such as: mean square error (MSE), mean absolute percentage errors (MAPE) and average relative variance (ARV) among other metrics. However, we propose a new performance measurement Sum of the Losses and Gains (SLG), which is more interesting and appealing for the investment sector. This article is organized as follows: Section II presents the importance of the stock market and the Brazilian stock market (BOVESPA). Section III describes the time series data, its normalization procedure and the exogenous time series used. Section IV explains the performance measurements used and introduces a new measure showing its characteristics and advantages. In Section V, we describe the experiments and demonstrate the results, showing the improvements of the proposed method. In Section VI some discussion and concluding remarks are presented.
II. THE STOCK MARKET According to Adam Smith, “A society cannot be wealth and happy, if the great majority of its member are poor and miserable” [16]. The economic developments of a society are indicated by the evolution of indexes such as: access to health care system, education, basic infrastructure and social equality. In theory, the continuous expansion of the production capacity, that is associated to investment in capital and human resources, is a precondition to a sustainable development. On the other hand, this expansion is a result of saving [16]. The main function of the capital market is the trade of stocks with the purpose to finance development, which produces and nourishes the market itself. In this way, a third function is attributed: the market of its own sources of incomes [17]. The monetary market, as a whole, is important for the economic development. However, when the economy and the market develops, the market of the source of capital emerges, which are the stock market, debt titles and real estate market. The monetary capital allows the investors to diversify their portfolio of investment, adjusting the risk according to the profile of the investor [18]. Institutional investors and financial institutions in general can designate a great part of its wealth to investment plans that have a high liquidity. Therefore, allowing the banks to diversify its exposition to risks among several different sectors and companies, so reducing the risk of the financial system. Globalization is a trend that allows an intense interchange between countries. Consequently, it is common nowadays that the stock market of an emergent country like Brazil attain an increasing importance in the international scenario. Today the stock market is not only an important source of corporation finance but also an individual capitalization resource. When investing in a portfolio, the investor wishes to obtain a large return in order to compensate the risks associated, in other words, the objective is to minimize risk and maximize capital returns. Hence, a prediction method is most useful and neural network is well-suited for this kind of optimization procedure. Currently, the Brazilian stock market, which is also known in the World Federation of Exchange (WFE) by São Paulo SE, has a global importance. From the 51 stocks monitored by WEF, BOVESPA is in the eighth position among the biggest stock market in a ranking for developing countries by capitalization terms and stock values. One of the biggest companies in the BOVESPA stock market is the Petrobras oil company stocks, which makes it an ideal stock to be analyzed. Stocks are titles that represent fractions of the capital from a determined company. Basically, there are two types of stocks: the ordinary stock, which concedes the right to vote in the general company assembly, and the preferential stocks, which have priority in the profit of the dividends but does not concede the right to vote. Also the preferential
stocks have precedence in the refund of the capital in case of company bankrupt, and they are more frequently negotiated. For the professional investor to understand the behavior of a stock, at least five series are necessary: 1) The highest value that the stock was negotiated in a certain day. 2) The lowest value that the stock was negotiated during the same day. 3) The value of the first negotiation of the day: opening price. 4) The value of the last negotiation of the day: closing price. 5) The business volume of the stock during the same day. The closing prize is the series that is really important, since most of the professional investors and financial institutions take action based on its value. III. DATA SECTION AND EXOGENOUS TIME SERIES In applications using neural networks with the objective of stock market time series prediction, the choice of the input values can be obtained by using knowledge in economic theory to determine the relevant exogenous variables and the common procedures utilized in the analyzes of time series, for example the autocorrelation and partial autocorrelation function [5]. Since we are interested in the prediction of the stock quotation PETR4 using information external of the series, some exogenous time series were chosen to be evaluated based on economic analyzes of its relationship with the Petrobras Company. Also, we conducted interviews with specialists of the area to obtain more insights and knowledge of which exogenous series should be chosen. The exogenous time series analyzed were: 1) IBOVESPA (IBOV) – The São Paulo stock market index. 2) SP500 – Standard & Poor’s index calculated over 500 stocks traded in the News York Stock Exchange (NYSE) and American Stock Exchange (AMEX). It represents 80% of the traded stocks at NYSE and represented by SP500. 3) DAX – Stock index that groups 30 Germany stocks form the Frankfurt Stock Market – FWB (Frankfurter Wertpapierborse). 4) NYSE:PBR – Stock quotation of the American Depositary Receipt (ADR), which is an indirect way for foreign stocks to trade on American exchanges, related to Petrobras. 5) Dollar – Exchange rate between Dollar and Reais (Brazilian currency). 6) Euro – Exchange rate between Euro and Reais. 7) BRENT – Quotation from the Light Sweet Crude Oil Futures, NYMEX (New York Mercantile Exchange) stock related energy products, metals and other commodities. This quotation is known in the financial market as the reference quotation for the oil barrel.
The quotation of a stock is represented by a set of 5 series that compose the time series of the company that we are interesting in predict. In the case of PETR4, that represents the preferential stock of Petrobras. The 5 series are the one that professional investor uses to try to understand the behavior of a stock as listed in the end of Section II. Once the 5 different series and the exogenous series were defined, a normalization process is necessary to correct possible discontinuities due to stock split. Stock split is not an uncommon procedure in the financial market. The price of the stocks is adjusted after market capitalization. Figure 1 shows the closing series for the PETR4 stock values over a period of 6 years.
Figure 2. The autocorrelation plot of PETR4 closing stock market series versus lags.
mt = (2q + 1) −1 ∑ j =− q X t − j , q + 1 ≤ t ≤ n − q (2) q
Figure 1. Plot of the daily PETR4 closing stock market series. The plot shows the time series over a 6 year period from August 2002 until July 2008.
where mt correspond to the estimated tendency; 2q + 1 to the size of the window; Xt, the original time series; and n the number of samples in the series. Since the seasonal component is equal to zero and having the estimation of the tendency component, the values of the stationary residues was obtained by the equation Yt =Xt- mt, which is shown in Figure 3.
A graphic analyzes of the series shows the deterministic components such as, tendencies, seasonality, and noise according to the classical decomposition model based in Equation 1 [5].
X t = mt + St + Yt
(1)
where mt represents a slow changing function, known as the tendency component; St represents the seasonal component and Yt the noise or residue. To verify the existence of a tendency in the series a spectral analysis was applied on the PETR4 closing series [5]. In this case, it is observed the peak of maximum amplitude in frequencies close to zero which corresponds to an infinite period. This characteristic determines a behavior along the whole series, which means that the PETR4 closing series have tendency mt following Equation 1. An autocorrelation analyzes was used to detect the seasonal component St. Figure 2 shows the autocorrelation plot of the PETR4 closing series. It can be observed that the maximum value was obtained at lag 0, and the values shows a linearly decay increasing on both side of the curve. This indicates that there is no sign of cyclical movements or seasonality in the series, so in the case St is equal to zero. To estimate the tendency component mt, it was applied a moving average filter [5], with window of size 5 (q=2), according to Equation 2.
Figure 3. The stationary residues plot of PETR4 closing stock market series versus lags.
To be able to identify the relevant delay, it is necessary to analyze the time dependency of the stationary residue. Figure 4 shows the estimative of the autocorrelation coefficients for the series residue Yt. The existence of coefficients outside the confidence interval represented by the limit of ± 1.96 / n , indicates that the series Yt, in the considered period (August 2002 until July 2008), has a linear dependence of its past values. The characteristics revealed by the sample autocorrelation function [5], shown in Figure 4, suggest that lags 2, 3 are relevant for the predictor in this case a neural network [19]. This lags corresponds to data PETR4t-1 and PETR4t-2. However we performed experiments that indicates also the importance of lag 1, which corresponds to PETR4t, which makes sense in a stock quotation prediction, since investors use this data to guide their next transaction, so PETR4t, PETR4t-1 and PETR4t-2 can be used as input of the ANN for the prediction of PETR4t+1.
IV. PERFORMANCE MEASUREMENT
Fig. 4. The autocorrelation of the residues of the PETR4 closing series versus lags.
The seven exogenous series were analyzed using a correlation function between the series and the PETR4 series. Table 1 shows the correlation coefficients between the series with a delay of 0 and +1. The series SP500 and EURO shows a correlation coefficient below 0.8 they were than discarded as an input for the neural network. Table 1. Correlation coefficients between the PETR4 series and the exogenous series.
Exogenous Series
Correlation Correlation coefficients with 0 coefficients with +1 delay of PETR4 delay of PETR4
SP500
0.7921
0.7937
DOLLAR
0.9594
0.9591
DAX
0.8734
0.8742
EURO1
-0.0817
-0.0794
IBOV
0.9669
0.9668
BRENT
0.955
0.9537
NYSE:PBR
0.9282
0.9288
For the other 5 related series that compose the PETR4 time series, the same kind of evaluation was carried out, as shown in Table 2. The series Volume PETR4 was discarded since the correlation coefficients show a very low value. Table 2. Correlation coefficients between the PETR4 series and its related series. Related PETR4 series
Correlation coefficients closing PETR4
Opening PETR4
0,9992
Highest PETR4
0,9996
Lowest PETR4
0,9997
Volume PETR4
-0,3567
1 The EURO series were available starting from 24/04/2006, so in this case the coefficients were calculated between the period 24/04/2006 until 31/07/2008.
A performance procedure is necessary to appropriate evaluate the results obtained by the proposed neural network prediction system. The pre-processing of the exogenous data that are useful to improve the stock market prediction performance is also evaluated. The performance measurements are defined on the base of the prediction errors, which is established as the difference between the real value of the series (target or objective of the prediction) and the predicted value (the output of the neural net). Therefore, they are presented by the following equation:
e t = (target t − output t )
(3)
where targett is the desired output of the prediction model at time t, and outputt is the output of the neural network at time t. The performance measurements used to evaluate the modes are described below. A. MSE - Mean Square Error The Mean Square Error is the most common metric used to analyze neural network performance and it is defined by the equation: MSE =
1 N
N
∑ ( target
t
− output t ) 2
(4)
t =1
N is the number of data values of the time series. The lower the value of MSE better is the result of the prediction. Even though this measurement is quite common as performance parameter, it does not provide a complete and confident idea about the accuracy of the model [7]. Therefore, other metrics were also used to improve confidence on the proposed models. B. MAPE - Mean Absolute Percentage Error The MAPE measure describes the errors in percentages which is an advantage in relation to the MSE measure, since it does not depend on the values or the scale of the time series simplifying its usage. MAPE is defined as: MAPE =
100 N
N
∑ t =1
target t − output t output t
(5)
The lower the value of MAPE closer is the desired results from the prediction method. C. U of THEIL or NMSE – Normalized Mean Square Error The measurement U of THEIL which is also known as NMSE [8] determine the relationship between the model performances compared to a random walk model. U of THEIL is described by the equation:
∑ (target − output ) ∑ (output − output ) N
THEIL =
2
t =1 N
t
t =1
t
t
(6) 2
t −1
F. SLG – Sum of Losses and Gains
if U of THEIL is equal to one, the tested model has the same performance of a random walk. If U of THEIL is greater than one, the predictor has a performance worse than the random walk. And if U of THEIL is lower than the value one the predictor is better than a random walk model. The closer the value of the U of THEIL is to zero better is the results of the predictor. Also in this case the model just shows an accepted result if the U of THEIL is at least less than one. D. POCID – Prediction On Change In Direction The POCID measurement demonstrate the percentage of the number of correct decisions when predicting whether the value of the time series will increases or decreases in the next time interval. POCID is defined as:
POCID = 100
∑
t =1
Dt
N
(7)
The values that POCID assumes are in between 0 to 100, so that, the closer the values are to 100, the better is the model prediction. This measurement is important when applied to the stock market, because a correct prediction on the direction of the series of the stock quotation affects directly the financial gains and losses on the investment. E. ARV – Average Relative Variance ARV determines the gain in the performance of the model in relation to another model that performs prediction by just calculating the arithmetic means of the observed values of the series. This measurement is defined by the equation: N
t =1 N
(outputt − target t ) 2
t =1
∑ SLG =
N L t =1 t
N
(9)
+ var, if (target t − target t -1 )(output t − output t -1 ) > 0 Lt = − var , otherwise
where var is equal to|(targett – targett-1)|.
1, if (target t − target t -1 )(output t − output t -1 ) > 0 Dt = 0, otherwise
∑ ∑
For the investors the predicted information of increase or decrease of a stock is used to decide a stock operation (buy/sell). The proposed measurement is determined by the summation of the losses and gains of the model considering the direction of the prediction. If the model correctly predicts the direction of the series, the absolute value |targett-targett-1| is added; otherwise if the predictor misses the value is subtracted. The usage of the value |targett-targett-1| is justified because once the operation is performed the investor neglects the predicted values and uses the real values corresponding to his gains or losses. The SLG measurement is defined by the equation:
the value Lt is defined as: N
having the value of Dt determined by:
1 ARV = N
prediction. Therefore, the model is practical if the values of ARV is less than 1, and the closer the value is to 0 it means that the predictor tends to be perfect.
(8)
(output t − target) 2
target is defined as the time series means of the desired output. If AVR is equal to 1, the predictor model has the same performance as the mean values of the series. If the ARV value is greater than 1, the predictor model is worse than the mean value of the series. And, if ARV is less than 1, then the predictor is better than having the mean values as the
If the value of SLG is less or equal to zero, the proposed model has a weaker performance, since it can cause financial losses. The greater the value of SLG assumes the better predictor performs. V. EXPERIMENTS AND RESULTS A. Usage of the Exogenous Series The experiments were performed using different architectures of the MLP neural network. A set of two experiments were performed, the first one using 3 input variables, corresponding to the past 3 closing values of the series PETR4, i.e. values of time t, t-1, and t-2. The second model uses 11 input variables, 3 variables used in the previous experiments and 8 variables from the exogenous that were defined in Section 3. Table 3 shows all variable used for the second experiment the output value of all the experiments was defined as Closing PETR4 t+1. On the total 1488 samples were used in these experiments corresponding to the period of August 01, 2002 to July 31, 2008. In the experiments, the whole set of numbers were divided into 3 parts, 60% for the MLP neural network training, 20% for the validation and 20% as test set. The learning algorithm used was the Backpropagation Levenberg-Marquart [9] to optimize the network weights. For each architecture, different numbers of hidden neurons were tested. In total this number varied from 25 to 50 (total of 26 different ANN), executed 10 distinct times, the average result was used to evaluate the performance.
Table 3. Input variable used for the neural network. (PETR4 and exogenous series)
Table 5. Average and standard deviation (µ ±σ) of the best performances in the second experiments.
Input variable
Training Set
Validation Set
Test Set
1
Closing PETR4 t
MSE
0.043697±0.003087
0.156676±0.005931
0.827401±0.076608
2
Closing PETR4 t-1
MAPE
0.014066±0.001053
0.014323±0.000364
0.017849±0.000984
3
Closing PETR4 t-2
THEIL
0.792461±0.108781
0.966696±0.112369
0.745869±0.136815
4
Opening PETR4
POCID
58.464126±2.890382
5.912162±1.957323
56.161616±3.076916
ARV
0.000564±0.000569
2.042147±3.392381
0.000084±0.000103
SLG
0.043700±0.010078
0.004436±0.012156
0.172051±0.064053
5
Highest PETR4
6
Lowest PETR4
7
DOLLAR
8
DAX
9
IBOV
10
BRENT
11
NYSE:PBR
B. Combination of the Best Neural Networks Based in SLG
In the first set of experiments just the 3 first variables from Table 4 were used, that represent the variable with time delays that has a relevant influence for the prediction as shown in Section III. From all the 26 different variations of hidden layers the best result was obtained with hidden layer equals to 38 neurons, since configuration leads to the lowest MSE value. These results show that a MAPE with value 2.08% of error, the value of POCID above 50% corresponds to a prediction better than random choice. The value of THEIL below 1 indicates that prediction is better them a Random Walker model, and the value of ARV shows that the result is better than moving average. Table 4. Average and standard deviation (µ ±σ) of the best performance in the first experiments. Training Set
Validation Set
Test Set
MSE
0.0462±0.0006
0.1576±0.0024
1.0867±0.2980
MAPE
0.0151±0.00009
0.0142±0.0001
0.0208±0.0025
THEIL
0.8456±0.0482
0.9091±0.0763
0.9049±0.2725
POCID
51.5807±0.3601
51.5878±0.6576
52.8282±2.1614
ARV
0.0008±0.0004
0.1360±0.1024
0.00005±0.00007
SLG
0.0153±0.0022
0.0058±0.0050
0.0873±0.0537
In a second experiment we applied all 11 input values from PETR4 series and the exogenous series as shown in Table 3. The best architecture was obtained using 45 hidden neurons and the general results are shown in Table 5. These results are better than the first experiments as we would expect – five of the performance index showed better results. The ARV in this case was slight worse, however this result is normally considered good if the value is below 1. The obtained vary is very close to zero, which implies in a good prediction. Table 6 shows a comparison result in terms of percentage between the two experiments, showing that the additional information from the exogenous series have a great impact in the prediction.
For time series prediction, the best results are obtained using a combination of different prediction models instead of a simple selection of those that presents best performance individually. The combination of specialist is capable to aggregate knowledge to obtain a global decision supposedly better than any one obtained isolated [1]. In these experiments, we choose eight MLP neural networks that presented the best performance under the SLG measurement, since this index is related to the best financial return in terms of profits and losses. The 8 neural networks chosen were obtained from the results of the second experiments, the neural networks differs by the number of hidden neurons.
Table 6. Comparison results between the two experiments. Results shown for the test set. Performance
Experiment 1
Experiment 2
Comparison
MSE MAPE THEIL POCID ARV SLG
1.086715 0.02084 0.904944 52.828283 0.000058 0.087391
0.827401 0.017849 0.745869 56.161616 0.000084 0.172051
+23.8622% +14.3522% +17.5784% +6.3098% -44.8276% +96.8750%
Two different approaches were used to combine these networks, a simple average and another neural network. The simple average combination method and methods based in regression are, in fact, the most discussed in the literature. Comparative studies using different rules of combination had shown that the combined simple average of the selected neural networks is one of the simplest and most effective methods [12]. Table 7 shows the results of this procedure compared to experiment 2. Another way to combine classifies is instead of using a simple average, we used another neural network of the same type so that the output of the 8 neural network is the input of the decision neural network.
Table 7. Comparison results between Experiment 2 and Combined neural network using simple average. Average Experiment 2
Simple Average
Comparison
MSE
0.827401
0.534453
+35.4058%
MAPE
0.017849
0.015194
+14.8748%
THEIL
0.745869
0.616343
+17.3658%
POCID
56.161616
57.575758
+2.5180%
ARV
0.000084
3.8315e-8
+99.9544%
SLG
0.172051
0.253047
+47.0767%
C. Final Results Since the results of the best results were obtained using neural network, we compared these results with experiment 2, as shown in Table 9. It is observed that the combined NN obtain better results in all cases. Note that the performance measurement SLG obtained a gain of 67.53% in relation to experiment 2 and MSE obtained an improvement of 37.30%. Table 9. Comparison results between Experiment 2 and Combined neural network, Results shown for the test set. Average Experiment 2
Figure 5 shows the committee machine based on a MLP neural networks, having 8 neuron of the input layer fed with the values corresponding to the respective output of the 8 prediction models selected in the previous section. Note that the neural network shown in Figure 5 does not have an intermediary layer.
Combined neural Comparison networks
MSE
0.827401
0.518795
+37.30%
MAPE
0.017849
0.014779
+17.20%
THEIL POCID
0.745869 56.161616
0.589597 59.59596
+20.95% +6.12%
ARV
0.000084
0.000003
+96.43%
SLG
0.172051
0.288232
+67.53%
To conclude the experiments, we compared the results of combined NN to experiment 1, that represents the traditional model and have as input variable only temporal delayed information. The results are shown in Table 10, note that the comparison shows a great improvement for all different performance measurement. The gain obtained from the SLG metric represents the sum of gains and losses, which demonstrated the best benefit, with an increment of 229.82% of profit. Fig. 5. Block diagram of the committee machine based on a MLP Neural Networks.
Table 8 shows a comparison of the results between simple average and the combined neural networks. We can observe that except for the ARV, the combined neural networks shows better performance. Note that SLG obtained an improvement of 13.90% as a result of the combined NN method. Table 8. Comparison results between Simple Average and Combined neural network. Simple Average
Combined neural networks
Comparison
MSE MAPE THEIL POCID ARV
0.534453 0.015194 0.616343 57.575758 3.8315e-8
0.518795 0.014779 0.589597 59.59596 0.000003
+2.9297% +2.7313% +4.3395% +3.5088% -98.7228%
SLG
0.253047
0.288232
+13.9045%
Table 10. Comparison results between Experiment 1 and Combined neural network. Results shown for the test set. Average Experiment 1
Combined neural networks
Comparison
MSE MAPE THEIL
1.086715 0.02084 0.904944
0.518795 0.014779 0.589597
+52.2603% +29.0835% +34.8471%
POCID
52.828283
59.59596
+12.8107%
ARV SLG
0.000058 0.087391
0.000003 0.288232
+94.8276% +229.8189%
VI. CONCLUSIONS In conclusion, we presented a methodology based on information obtained from exogenous time series that uses a combined neural network to predict the PETR4 stock series. The Exogenous series showed to be important in improving the predictive capability of the neural network. They were selected by analyzing the correlation between the series with the Petrobras stocks series and also using insights and
knowledge of stock market specialist. In this way, the prediction was obtained not just by using the previous values of the series but also by using information external to the PETR4 series. Another contribution of this work was to introduce a new performance measurement defined as Sum of the Losses and Gains (SLG). This metric is obtained by the summation of the losses and gains of the model considering the direction of the prediction. This information is more relevant for a financial investor since it better describe his real investment returns. Even though we applied this method to just one time series, the Petrobras stock, we believe that the method will perform well in any other time series, constrained to the exogenous series used. For future investigation, we plan to use the method on other types of time series and investigate different approaches on how to define and choose the exogenous series related to the time series of interest.
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