Forecasting of Dow Jones Industrial Average by Using Wavelet Fuzzy Time Series and ARIMA

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1 Forecasting of Dow Jones Industrial Average by Using Wavelet Fuzzy Time Series and ARIMA Muhamad Rifki Taufik 1, Lim Apiradee 2, Phatrawan Tongkumchum 3, Nureen Dureh 4 1,2,3,4 Research Methodology, Math and Computer Science Department Faculty of Science and Technology, Prince of Songkla University, Thailand. 1 m.rifki.taufik@gmail.com Abstract The Dow Jones Industrial Average (DJIA) is one of the oldest, single most-watched indices in the world and includes companies such as General Electric Company, the Walt Disney Company, Exxon Mobil Corporation and Microsoft Corporation. DJIA had strong affect for stock market in the world. This study conducted to forecast DJIA by using Wavelet Fuzzy Time Series and Autoregressive Integrated Moving Average (ARIMA). The data were obtained from 1/29/1985 until 12/14/2016. The result was comparing between both methods by calculating the MAPE which measured the accuracy of prediction and ARIMA had 2.6% and Wavelet Fuzzy Time Series had 1.4% of MAPE. Keywords: ARIMA, DJIA, Fuzzy Time Series, Wavelet. 1 Introduction In the globalization, stock market had strong role for fundamental policy and economic development. Inflation and increasing of exchange rate could be caused by stock market movement. Numerous factors influence stock market performance, including political events, general economic conditions, and trader expectations (Abbasi et al 2015). Based on Clements, economics fundamentals had recently become so important (Clements et al, 2009). One of the well-known stock market and the oldest one was Dow Jones. In 1884 Charles Dow, one of the founders of Dow Jones & Co, created the first stock market index which published by Barron s and The Wall Street Journal. The index used price weights instead of conceptually superior market valuation weights. The Dow Jones Industrial Average (DJIA) was becoming the most quoted stock market index in the world. The index changes were believed be representative of entire stock market index (Shoven & Sialm, 2000). Forecasting on time series data apparently has important part. Predicting financial market changes is an important issue in time series analysis, receiving an increasing attention in last two decades. Financial time series prediction is an important subject for many financial analysts and researchers as accurate forecasting of different financial applications play a key role in investment decision making (Grigoryan, 2015). A time series is a set of well-defined data items collected at successive points at uniform time intervals (Mondal, Shit, & Goswami, 2014). Time series analysis is an important part in statistics, which analyzes data set to study the characteristics of the data and helps in predicting future values of the series based on the characteristics. Nowadays forecasting being important part of researchers could improve the domain field of existing predictive models. Not only Investment decisions and planning ability but also develop effective strategy about their daily and further effort (Adebiyi et al, 2014). Investors wish that they hold the forecasting method which can guarantee easy profiting and minimize risk on their stock market investment. That is the reasons why forecasting of stock price becomes an important topic in finance and economics over year to get better predictive models.

2 2 Literature review Lately, some predictive models have been presented especially in stock market movement. Nowadays researchers are interpreting and summarizing the models in different way to find out better result (Granger, 1992). The better prediction apparently is still improving which the best predictive models is not developed yet. If the time series analysis model is divided into two categories; linear and nonlinear model, then in many cases it will be difficult to determine whether a time series model belongs to a linear model or a nonlinear model and then it is also difficult to determine which model should be used in the study. In practice, there is few time series with pure linear or nonlinear feature. There is no single model that can adapt to all situations and solve the problem (Jia, Wei, Wang, & Yang, 2015). Autoregressive Integrated Moving Average (ARIMA) models have been used in time series prediction of stock market. ARIMA are from statistical models perspectives. ARIMA models are known to be robust and efficient in financial time series prediction. Jaret (2011) demonstrated the usefulness of ARIMA Intervention time series analysis as both an analytical and forecast tool. Pual et al (2013) also examined empirically the best ARIMA model for forecasting. Hence, ARIMA (2, 1, and 2) is found as the best model for forecasting the Square Pharmaceuticals Limited (SPL) data series. A general ARIMA formulation is selected to model the price data. Jadhav et al (2015) did a selection which was carried out by careful inspection of the main characteristics of the hourly price series. In most of the competitive electricity markets this series presents: high frequency, nonconstant mean and variance, and multiple seasonality (corresponding to daily and weekly periodicity, respectively), among others. The second model is combination between Wavelet Transformation and Fuzzy Time Series Analysis. Since the 2000s, wavelet decomposition has been combined with time series models as a preprocessing method. Wavelet decomposition (or wavelet transform) decomposes time series data into approximation and detail components, so that different forecasting models can be applied to each component (Jin & Kin, 2015). This property can improve the performance of forecasting. The validity of this approach has been proved in various studies. Since fuzzy time series model was proposed by Song and Chissom in 1993, there are many forecasting models have been developed to deal with the forecasting problems (Qiu, Zhang, & Ping, 2015). Qiu et al did a test of the effectiveness of the Fuzzy model, the proposed method is demonstrated on the procedure of forecasting enrollments as well as its experiment on forecasting the close price of Shanghai Stock Exchange Composite Index. Empirical analyses show that the proposed method gets a higher average forecasting accuracy rate than the existing methods. Rostamy et al (2013) did a research with the major purpose was to predict the total stock market index of Tehran Stock Exchange, using a combined method of Wavelet transforms, Fuzzy genetics, and neural network in order to predict the active participations of finance market as well as macro decision makers. ARIMA and WFTS are interesting models based on author. Both performance give acceptable predictive models. This paper showed the comparison between two models in current period. The accuracy of prediction had been measured by Mean Absolute Percentage Error (MAPE).

3 3. Data Management and Methodology Data management Dow Jones Industrial Average (DJIA) data had been obtained from yahoo finance websites. The data structure is as time series data which consisted of 5 work-days from January 29 th 1985 until December 14 th The data provided some type of prices such as open, high, low, close, volume and adjusted price. The data that been chosen is adjusted closing price column. An adjusted closing price is a stock's closing price on any given day of trading that has been amended to include any distributions and corporate actions that occurred at any time prior to the next day's open. Autoregressive Integrated Moving Average Figure 1 DJIA Time Series Plot ARIMA is a basic linear forecasting model, which uses a lagged series. Because of its simplicity and good performance, ARIMA has been applied to many time series analyses (Jin & Kin, 2015). ARIMA model is derived by general modification of an autoregressive moving average (ARMA) model. Based on Mondal et al (2014) ARIMA models are generally used to analyze time series data for better understanding and forecasting. This model was calculated using R i version. Initially, the appropriate ARIMA model has to be identified for the particular datasets and the parameters should have smallest possible values such that it can analyze the data properly and forecast accordingly. This model type is classified as ARIMA (p,d,q) where p denotes the partial autocorrelation parts of the data set, d refers to differencing part of the data set and q denotes autocorrelation parts of the data set and p, d, q is all nonnegative integers. ARIMA (p,d,q) forecast 30 days based on previous data pattern. Wavelet The performance of the wavelet transformation was superior in all cases, whereas the application of detail components in the forecasting was only able to yield a small improvement in forecasting performance (Jin & Kin, 2015). Therefore, forecasting with only an approximation component would be acceptable, in consideration of forecasting efficiency. Discrete Wavelet Decomposition is a preprocessing method that projects a time series onto a collection of orthonormal basis functions. This transformation is applied to DJIA time series data to obtain further information from the time-domain original data. First After applying DWD to the data, data was analyzed signals by decomposing them into various frequencies or levels. In simple, the wavelet transformation is used to transform a function with the time domain into the frequency

4 domain. This study conducted Wavelet Transformation used Haar Wavelet data as training data was used to decompose in some levels. Number of level we got from this formula: ( ) Wavelet Fuzzy Time Series This model combined between Wavelet Transformation and Fuzzy Time Series on decomposition part. The purpose of wavelet fuzzy models was to predict the data of the components Discrete Wavelet subseries (DWS) obtained by using DWT on original data. Each DWS form of time series data and had a different effect and frequency with the original data. This model was calculated using matlab R2013a version. Training data as around 8000 observation forecast 30 days further data as testing data. DWS corresponding selected as input into this model. The authors use this model based on the research of Septiarini et al (2016). MAPE (Mean Absolute Percentage Error) The MAPE (Mean Absolute Percent Error) measures the size of the error in percentage terms. It is calculated as the average of the unsigned percentage error. This method is useful for evaluating the accuracy of prediction. MAPE indicates how big error in the prediction comparing with real value. MAPE also has ability to evaluate the precision of techniques or models as percentage of mean absolut error. MAPE is calculated as following formula : 4. Result and Discussion 4.1 ARIMA Differencing a Time series ARIMA models were defined for stationary time series. Therefore from the Figure 1, the data started off with a non-stationary time series, time series data first needed to difference the time series until the data was obtained a stationary time series. Then the data had to difference the time series d times to obtain a stationary series, then we got an ARIMA (p,d,q) model, where d is the order of differencing used. DJIA times series plot of adjusted price consist of 8038 records was showed in Figure 1. The figure identify was that this time series looked like random fluctuation in over time. Then, the time series of first differences as Figure 2 to be statuinary in mean and variance, and so an ARIMA (p, 1, q) model was probably appropriate for the DJI data. The next step was to figure out the values of p and q for the ARIMA model. Figure 2 First order Differencing

5 Selecting a Candidate of ARIMA Model After transforming the DJIA data to be stationary time series by differencing one time, the next step is to select the appropriate ARIMA model, which means finding the value of most appropriate values of p and q. The data was examined the correlogram and partial correlogram. As Figure 3, plot of correlogram that the autocorrelation at lag 1, 5, 12, 15 and lag 18 (-0.054, 0.039, , and ) exceeded the significance bounds. There were 5 lags out of significant border. Figure 3 ACF Figure 4 Partial ACF Then as Figure 4, the partial correlogram shows that the partial autocorrelations at lags 1, 2, 5, 10, 12, 15, 16 and 18 exceeded the significance bounds, and those were negative. Then we got 8 lags which outside the bounds. Probably the appropriate ARIMA model was (8, 1, 5) Wavelet Wavelet Level The number of training data was 8000 records. Then the data was filtered into two part as decomposition and approximation. After calculating the formula, then the rounding levels were consist of 12 levels.

6 Figure 5 Wavelet Decomposition The data had been separated into 12 levels called Discrete Wavelet Decomposition. Every level was measure the association using Pearson Correlation Coefficient, then two strongest correlation which more than 0.4 were D12 and A12. Both series had been combined to be Discrete Wavelet Series (DWS). From DWS, 8000 new records put in Fuzzy Inference System as training data. Universal set was in interval Max and min values of Discrete Wavelet Series. Referred on Chang s Method, universal set had to been separated as seven membership function using Gaussian membership function. Figure 6 Seven Membership Functions Mamdani method was used on this Fuzzy Inference System (FIS). FIS needed rules which the rules had been created from DWS, then created seven rules. Last the defuzzification forecast the 30 records based on rules then the obtained data re-transform to be actual data as predicted data Comparison Predicted data as forecasting 30 days record of Dow Jones Industrial Average showed both model had small MAPE. MAPE for ARIMA was 2.6% and MAPE for WFTS was 1.4%. In this case, ARIMA and WFTS could predict next 30 days record with acceptable performance. 5. Conclusions Autoregressive Integrated Moving Average (ARIMA) and Wavelet Fuzzy Time Series (WFTS) were acceptable predictive model. ARIMA model had bigger error than WFTS, but based on prediction the

7 ARIMA model might be not appropriate to predict long term period of time series data because the forecasting mostly constant which would give bigger error for longer time. WFTS had good pattern track, it gave chance to predict longer period of time. Acknowledgments This study was supported by the Higher Education Research Promotion and The Thailand s Education Hub for Southern Region of ASEAN Countries Project Office of the Higher Education Commission. We would acknowledge the Department of Mathematics and Computer Science of Prince of Songkla University for providing the plotform for this study. References Adebiyi, A. A., Adewumi, A. O., & Ayo, C. K. (2014). Stock Price Prediction Using the ARIMA Model. 16th International Conference on Computer Modelling and Simulation (pp ). UK: Computer Society. Clements, M. P., Milas, C., & Dijk, D. v. (2009). Forecasting returns and risk in financial markets using linear and nonlinear models. Science Direct: International Journal of Forecasting, Granger, C. W. (1992). Forecasting stock market prices: Lessons for forecasters. International Journal of Forecasting, Grigoryan, H. (2015). Stock Market Prediction using Artificial Neural Networks: Case Study of TAL1T, Nasdaq OMX Baltic Stock. Database Systems Journal vol. VI, Jadhav, S., Kakade, S., Utpat, K., & Deshpande, H. (2015). Indian Share Market Forecasting with ARIMA Model. International Journal of Advanced Research in Computer and Communication Engineering, Jia, C., Wei, L., Wang, H., & Yang, J. (2015). A Hybrid Model Based on Wavelet Decomposition- Reconstruction in Track Irregularity State Forecasting. Hindawi: Mathematical Problems in Engineering, Jin, J., & Kin, J. (2015). Forecasting Natural Gas Prices Using Wavelets, Time Series, and Artifcial Neural Networks. Plos, Mondal, P., Shit, L., & Goswami, S. (2014). STUDY OF EFFECTIVENESS OF TIME SERIES MODELING (ARIMA) IN FORECASTING STOCK PRICES. International Journal of Computer Science, Engineering and Applications (IJCSEA), Rostamy, A. A., Abbasi, N. M., Aghaei, M. A., & Fard, M. M. (2013). Forecasting Stock Market Using Wavelet Transforms and Neural Networks: An integrated system based on Fuzzy Genetic algorithm (Case study of price index of Tehran Stock Exchange). International Journal of Finance, Accounting and Economics Studies, Septiarini, T.W, Abadi A.M & Taufik M.R Apprication of Wavelet Fuzzy Model to Forecast the Exchange Rate IDR to USD. International Journal of Modeling and Optimization, Shoven, J. B., & Sialm, C. (2000). The Dow Jones Industrial Average: The Impact of Fixing its Flaws. THE JOURNAL OF WEALTH MANAGEMENT, 9-18.