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Forecasting the Time Series Data Using ARIMA with Wavelet Journal of Computer and Mathematical Sciences, Vol.6(8),430-438, August 2015 ISSN 0976-5727 (Print) (An International Research Journal), www.compmath-journal.org ISSN 2319-8133 (Online) Forecasting the Time Series Data Using ARIMA with Wavelet Jatinder Kumar, Amandeep Kaur and Pammy Manchanda Department of Mathematics, Guru Nanak Dev University, Amritsar, INDIA. email: [email protected], [email protected] (Received on: August 18, 2015) ABSTRACT Wavelet transform has got very high attention in various fields such as mathematics, signal processing, engineering, physics, economics and finance. This paper illustrates an application of wavelet transform in the time series analysis. A novel technique based on wavelet transform and ARIMA model to forecast the time series data is proposed. Time series data for closing prices of the energy sector companies, GAIL and ONGC, is used for this study. It is observed that Wavelet Transform combined with ARIMA model gives better results at forecasting than the direct use of ARIMA model on the data. Keywords: Forecasting, GAIL and ONGC data, wavelet transform, ARIMA models. 1. INTRODUCTION Forecasting is the process to predict future situations based on past and present data and therefore, it is helpful in planning and future growth. Time series forecasting is widely used in a variety of fields such as economics, business, engineering, natural science and meteorological sciences. Stock market forecasting is required for the investors as it is an important issue in investment decision making. There are a number of time series forecasting models which tell us about the nature of the system generating time series by analyzing the historical data. These models are very helpful in forecasting optimally and understanding dynamic relationship between different variables. These include basic models such as a linear regression, simple moving average and exponential smoothing and some advanced models such as autoregressive moving average (ARMA), its extension autoregressive integrated moving average (ARIMA) and neural network. All the time forecasting models can be divided into two forms- stationary and non stationary. In stationary models, the statistical properties mean and variance remain constant while in non stationarity, these properties are time- dependent. Mostly, the real world time series data is non stationary as it contains extreme variations and these fluctuations occur with high frequency. So wavelet methods are most August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 431 Jatinder Kumar, et al., J. Comp. & Math. Sci. Vol.6(8), 430-438 (2015) suitable for such type of time series data. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities. The wavelet transform decomposes the data (signal) in terms of both time and frequency, allowing us to effectively analyze the main frequency component and to extract local information from the signal. It gives good results in various fields such as signal processing, image coding and compression, and in certain areas of mathematics, as in solutions of partial differential equations or numerical analysis. The wavelet transform is also very effective for the time series analysis. In9, Ramsey provides some important properties of the wavelets and discusses its applications in both economics and finance. S. Yousefi et al.10 describe a wavelet based prediction procedure for oil market and show that it outperforms the future market in average. Gencay et al.8 discussed the use of wavelets in economics and finance with many illustrations and examples. Davidson et al.5 show that wavelet analysis is effective in describing the unstable variance structure and general features of the commodity prices such as structural breaks. Kumar, J. et al.6 describe the concept of neuro-fuzzy with wavelet decomposition for stock market. A. J. Conejo et al.4 discuss the role of ARIMA with wavelet in electricity market forecasting. The stochastic models for forecasting oil prices are discussed by Mohammed et al. in7. This paper focuses on the month-ahead price forecast of a daily GAIL and ONGC (companies from energy sector) prices. Gas and Oil sectors are about as important to developed country as agriculture. These sectors play a very significant role in country's economy being the biggest contributors to both the central and state treasuries. India is the fourth-largest energy consumer of oil and gas in the world. Today to meet its growing petroleum demand, India is investing heavily in oil fields. Price fluctuations in oil and gas affect largely country's economy. There are some companies that deal with oil and gas section. In this paper we discuss two companies: Gas Authority of India Limited (GAIL) and Oil and Natural Gas Corporation Limited (ONGC). GAIL is the largest natural gas processing and distribution company in India and ONGC is an Indian multinational oil and gas company. The wavelet transform converts time series into constituent series which show more stable variance and no outlier and so it can be predicted more accurately. That is why, we use the wavelet transform as preprocessor in the procedure explained in this paper. For this purpose we take daily closing price data of Gas and Oil sector. The paper is structured as follows; section 2 and 3 describe the basic introduction and properties of wavelet and ARIMA model respectively. In section 4 forecasting procedure and result are explained. Finally, conclusion is described in section 5. 2. WAVELETS Fourier transform is an alternative representation of the original time series such that it summarizes information in the data as a function of frequency and therefore does not preserve information in time. This transform is good when working with stationary time series. But most financial time series are non stationary and exhibit quite complicated patterns over time such that trends, abrupt changes etc. The Fourier transform cannot efficiently capture August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org Jatinder Kumar, et al., J. Comp. & Math. Sci. Vol.6(8), 430-438 (2015) 432 these events. Wavelet transform overcomes most of the limitations of this transform. This combines information from both time-domain and frequency-domain and is also very flexible. Also in wavelet analysis, there are fewer coefficients compared to the Fourier analysis. The wavelet transform utilizes a basic function, ∈ (ℝ) , called wavelet or mother wavelet that is stretched and shifted to capture features that are local in time and local in frequency. This function satisfies the following properties: () (i) = ∫ < ∞ where is the Fourier transform of ψ. This condition, called admissibility condition, ensures that (ω) goes to zero quickly as ω→ 0. In fact, to guarantee that < ∞, it is necessary (0) = 0 which is equivalent to ( ) ∫ = 0 (2.1) (ii) Wavelet function must have unit energy. That is | | ∫ () = 1 (2.2) Equations (2.1) and (2.2) imply that at least some coefficients of the wavelet function must be different from zero and these departures from zero must cancel out. By combining several combinations of shifting and stretching of the mother wavelet, the wavelet transform is able to capture all the information in the time series and associate it with specific time horizons and locations in time. Wavelets can be defined in terms of the sequence of a pairs of filters or in terms of functions created through splines that satisfy certain properties. Here we will define precisely 'father' and 'mother' wavelets. Father wavelets generate the scaling coefficients and represent the very long scale smooth component of the signal whereas mother wavelets generate the differencing coefficients and represent deviations from the smooth component. Father wavelet acts as a low pass filter and the mother wavelet acts as a high pass filters. The application of both the father and mother wavelets allows separating the low-frequency components of a time series from its high-frequency components. For any suitable choice of function (. ) ∈ (ℝ), we define the corresponding father and mother wavelets: ,() = 2 (2 − ) ; ∈ ℤ, ∈ ℤ ( ) ∫ = 1 And ,() = 22 − ; = 1, 2, … , ( ) ∫ = 0 August, 2015 | Journal of Computer and Mathematical Sciences | www.compmath-journal.org 433 Jatinder Kumar, et al., J. Comp. & Math. Sci. Vol.6(8), 430-438 (2015) Here , is the father wavelet and , is the mother wavelet. Given this family of basis functions, we can define a sequence of coefficients ( ) , = ∫ ,() ( ) and , = ∫ ,() ; = 1, 2, … , where the , are the coefficients for the father wavelet, known as, “smooth coefficients", and , are "detail coefficients" obtained from mother wavelets. So from the coefficients, the function f (.) can be represented by () = ∑∈ℤ ,,() + ∑ ,,() + ⋯ + ∑∈ℤ∈ℤ ,,() or f(t) can be represented as () = + + + ⋯ + Where = ∑∈ℤ ,,() and = ∑∈ℤ ,,() ; = 1, 2, … , As the Discrete wavelet transform (DWT) represents a time series in terms of the coefficients that are associated with particular scales, so it is effective tool for the time series analysis. By applying the DWT to signal f(t), the signal f(t) is decomposed into different scales of resolution. The inverse wavelet transform reconstruct the signal from its wavelet coefficients. There are some software available for the applications of the wavelet transform. We use the wavelet toolbox from MATLAB. Applying DWT
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