Stock Market Indices Prediction Using Time Series Analysis

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1 Stock Market Indices Prediction Using Time Series Analysis ALINA BĂRBULESCU Department of Mathematics and Computer Science Ovidius University of Constanța 124, Mamaia Bd., , Constanța ROMANIA IULIA ILIE Master Student - Department of Mathematics and Computer Science Ovidius University of Constanta 124, Mamaia Bd., , Constanta ROMANIA iulia_ilie1988 yahoo.com Abstract: - In this paper we present two non-parametric approaches used for time series analysis and modeling for a financial time series: the DJIA - stock index open values. We used two recently developed algorithms and methods for time series prediction, Gene Expression Programming and Neural Networks because they are suitable for the series that present high variability, as in the present situation. After using these approaches we managed to obtain models which explain 92% of the variance in the case of GEP and 98% in the case of Multilayer Perceptron. Key-Words: - Gene Expression Programming, Neural networks, time series, DJIA, evaluation, accuracy 1 Introduction Planning the future is very necessary for the business companies, governmental institutions, and for every economy. As like every business is doing planning for possibilities of its financial resources and sales and for the profit maximization. Having the basis of time series analysis, businessmen can predict the changes in economy and their business environment. One of the main applications of time series analysis and forecasting is stock market analysis which uses mostly stock indices. The evolution indices are convenient gauges of the stock market that indicate the direction of the market over a period of time. By their using one can compare how well individual stocks and mutual funds have performed against comparable market indicators for the same period. Since stock market prices do not follow random walks [8] and almost never satisfies the stationarity hypotheses [10], traditional parametric approaches are not enough in finding accurate models. Therefore, innovative methods have been proposed for these time series [11] [12], including nonparametric methods and artificial intelligence [1]. 2 Problem formulation In the following we present the solution proposed by us for solving the following problem: finding a nonparametric method for developing a good model (in term of explained variance) for the analyzed time series, here, the Dow-Jones Industrial Average (monthly open values) registered in the period The best model is the one having the smallest error between the actual data and the predicted one. Two approaches we decided to use in this purpose are Gene Expression Programming and Neural Networks and they will be described in the following paragraphs. 3 Methods Two approaches have been proposed for solving the given problem: Gene Expression Programming and Neural Networks. They will be described in the following subsections. 3.1 Gene Expression Programming Gene Expression Programming is a programming technique which imitates natural, biological ISBN:

2 evolution and it combines features from Genetic Programming (GP) and Genetic Algorithms. This programming technique was introduced by Candida Ferreira, and it is a much more elegant and powerful tool for developing programs which try to describe different phenomena than Genetic Programming [5] The best new feature Gene Expression Programming posses is that it contains a system for encoding expressions which enables the developer to add mutations and let the program evolve on its own but by being sure that in the end the resulting new program will be valid. The working diagram of a gene expression algorithm (GEA) is shown in Fig.1. desired diversity and randomness in the analyzed population. One important aspect of GEP is that not all members of a generation are the object of alterations done through mutation, inversion, transposition and recombination but only the ones which have been previously randomly selected with the help of genetic operators. The rest of the specimens of a generation are replicated, meaning that their genomes will be copied and forwarded to the new generation. These operators randomly select the chromosomes to be modified. Thus, in GEP, a chromosome might be modified by one or several operators at a time or not be modified at all. By randomly selecting which member to be altered, one chromosome might sustain modifications through the action of one, more operators or even none at all. Fig. 1. The workflow of a gene expression programming algorithm [14] The algorithm begins with the selection of an initial population for which we have to randomly generate chromosomes. In the next step these generated chromosomes are evaluated for fitness to characteristics set in the beginning. Then, a selection is made for the specimens which are compliant with the fitness criteria in order to be altered trough evolution, leading to new, enhanced traits in the following generation. The same developmental process is then applied to the members of this new generation: generate the chromosomes, test for fitness and then select the best members for altering. This part of the algorithm is repeated for a specified number of times or until the criteria for finding a solution has been met. The alteration done to certain members from a generation is done with the help of pre-established genetic operators with which we can obtain the 3.2. Neural Networks A neural network is an interconnected group of natural or artificial neurons which uses a mathematical or computational model for data analysis and manipulation. An artificial neural network is usually an adaptive system that changes its structure according to internal or external stimuli existing through the network. In recent times, it has also become known as a practical technology, with many successful applications in different areas [9]. One of the widest used types of neural networks is the multilayer perceptron network (MLP). Its basic architecture is presented in the following diagram (Fig. 2). The network has an input layer, one hidden layer and an output layer. The input layer is a vector of predictor standardized variables values. Fig. 2. MLP with three neurons in each layer [14] Each neuron in the hidden layer will then take its value from the input layer. A bias is also sent to ISBN:

3 each of the hidden layers, is multiplied by a weight and added to the sum going into the neuron. In the hidden layer, linear combinations weighted input values are produced ( u j ) and are sent into a transfer function, σ, obtaining as results the values h j, that are forwarded to the output layer. Arriving at the neurons in the output layer, the values from the hidden layer are again weighted and combined, producing new values ( v j ) fed again into a transfer function σ, the resulting values being the output of the network. The MPL learning algorithm is the following [15]: Initialize the network, with all weights set to random numbers between -1 and +1. Present the first training pattern, and obtain the output. Compare the network output with the target output. Propagate the error backwards. Compute the error between the results and the target data. Repeat the previous 4 steps for each pattern in the training set until we complete one era. Shuffle the training set pattern randomly Repeat the algorithm for more eras until the error doesn t change anymore. Setting a training pattern requires the following procedures [3]: Selecting how many hidden layers to use in the network. Deciding how many neurons to use in each hidden layer. Finding a globally optimal solution that avoids local minima. Converging to an optimal solution in a reasonable period of time. Validating the neural network to test for overfitting. the randomness hypothesis was also rejected after performing the rank correlation test; the presence of break points was proved after conducting Hubert s segmentation algorithm [6] (Table 1); the long range dependence hypothesis was not rejected after computing the Hurst coefficient by R/S method [4] [13] (Fig.4). Fig.3. DJIA series (monthly, open values) registered in the period January 1896 December 2009 Table 1. the results of segmentation procedure of Hubert 4 Problem Solution For the series of monthly DJIA open values (Fig. 3), we have done a preliminary analysis consisting of testing the following hypotheses: normality, stationarity, autocorrelation, the existence of break points and long range dependence. After applying the tests we received the results: the normality hypothesis was rejected by Kolmogorov Smirnov and Shapiro Wilks tests; the KPSS test [7] rejected the stationarity hypothesis; Fig.4. Evaluation of Hurst coefficient for initial series by R/S method The high variability, especially at the end of the study period, and the existence of a big number of break points, determined us to take logarithms of ISBN:

4 values of the initial series ( x t ) and to apply the modeling techniques described in Section 3 on the transformed ones, y t ln( x t ) (denoted by ln(djia) in the following figures). The same statistical tests have been performed by using ( y t ), the results being similar to those from [2] for the DJIA close data. Before running the Gene Expression Programming Algorithm over the new data, the series trend has been determined. It is of exponential type, having the equation: multiplication, division. The following rates have been set: for mutation, 0.1 for inversion, IS, RIS and Gene transposition and 0.3 for recombination rate. The data was divided in two subseries: one for training (1000 values) and one for the model validation (355 values). The model and the corresponding prediction error are presented in Figs. 7, 8. In Fig 7, we also remark the prediction forecasts, done on the basis of the GEP model. y e t t where t is the time (Fig.5). The variance explained by it is of %., Fig.7. Model for ( y t ) Fig.5. Exponential trend of ( y t ) Fig.8. The prediction error in GEP model Fig. 6. Settings for GEP algorithm The next step we conducted the fitting procedure. The settings from Fig. 6 were used. The linking function was the addition, and the functions used in expression: addition, subtraction, The following results have been obtained: For the training data: - the variance in input data = ; - residual variance after model fit = ; - proportion of variance explained by model = %; - correlation between actual and predicted = %; For the validation data: - the variance in input data = ; - the residual variance after model fit = ; - proportion of variance explained by model %. - correlation between actual and predicted = ; ISBN:

5 The model quality is good enough since the proportion of variation explained by model is greater than %, and the correlation between actual and predicted values, very close to 1. Since the previous result could be improved, the MLP algorithm was also run. As in the GEP case, the trend was first subtracted from y ) (Fig. 9). ( t After performing the curve fitting (Fig.10), the residual variance obtained was equal to and the correlation between actual and predicted values has was of ( %). The prediction quality can be observed in Figs. 10 and 11. In the last one, where the predicted values are plotted versus the registered ones, the distribution of the predictions, very close to the first bisectrix of the axes, confirms the model accuracy. Fig.9. Transformed series Ln(DJIA), after the trend subtraction Then, the following settings have been done to run the MLP algorithm: The predictor variable was chosen to be the difference of the first order of ( y t ), i.e. ( yt 1 yt ). The number of layers was 3 (1 hidden) and the number of neurons in the hidden layer: 2. The activation function for the hidden layer was the logistic one and that for the output layer was the linear one. The conjugate gradient method has been used for training the network. Fig. 11. The chart of the predicted values versus the actual ones 5 Conclusion After conducting the tests over the analyzed financial data it was clear that trying to find a good fitting model through traditional, parametric methods would be futile. The models proposed by us fit well the DJIA series, the second one performing better on the validation stage, and consequently, in the prediction one. Considering that the methods we worked with gave good results for the analyzed series, we are encouraged to use them for solving other problems for which classical approaches fail. Fig.10. Actual data, the predicted ones and the forecast in MLP References: [1] E. M. Azoff, E.M. Azoff, Neural Network Time Series Forecasting of Financial Markets, New York, John Wiley & Sons, 1994 [2] A. Bărbulescu, E. Băutu, A hybrid approach for modeling financial time series, International Arab Journal of Information and Technology, vol. 9, No. 4, July 2012, in print [3] A. Bărbulescu, E. Pelican, On the Sulina Precipitation Data Analysis Using the ARMA models and a Neural Network Technique, Recent Advances in Mathematical and Computational Methods in Science and Engineering, Part II, 2008, pp ISBN:

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