Wavelet Transform as an Alternative to Power Transformation in Time Series Analysis

Abstract

This study examines the discrete wavelet transform as a transformation technique in the analysis of non-stationary time series while comparing it with power transformation. A test for constant variance and choice of appropriate transformation is made using Bartlett’s test for constant variance while the Daubechies 4 (D4) Maximal Overlap Discrete Wavelet Transform (DWT) is used for wavelet transform. The stationarity of the transformed (power and wavelet) series is examined with Augmented Dickey-Fuller Unit Root Test (ADF). The stationary series is modeled with Autoregressive Moving Average (ARMA) Model technique. The model precision in terms of goodness of fit is ascertained using information criteria (AIC, BIC and SBC) while the forecast performance is evaluated with RMSE, MAD, and MAPE. The study data are the Nigeria Exchange Rate (2004-2014) and the Nigeria External Reserve (1995-2010). The results of the analysis show that the power transformed series of the exchange rate data admits a random walk (ARIMA (0, 1, 0)) model while its wavelet equivalent is adequately fitted to ARIMA (1,1,0). Similarly, the power transformed version of the External Reserve is adequately fitted to ARIMA (3, 1, 0) while its wavelet transform equivalent is adequately fitted to ARIMA (0, 1, 3). In terms of model precision (goodness of fit), the model for the power transformed series is found to have better fit for exchange rate data while model for wavelet transformed series is found to have better fit for external reserve data. In forecast performance, the model for wavelet transformed series outperformed the model for power transformed series. Therefore, we recommend that wavelet transform be used when time series data is non-stationary in variance and our interest is majorly on forecast. 1.0 Introduction In several organizations, managerial decisions are largely based on the available information of the past and present observations and possibly on the process that generate such observations. A time series data provides such information. Time series is used to represent the characterized time course of behavior of wide range of several systems which could be biological, physical or economical. The utility of the time series data lies in the result of the time series analysis. Such analysis will be helpful in achieving the aim for collection of such data which could be for description (exposing the main properties of a series), explanation (revealing the relationship between variables of a series especially when observations are taken on two or more variables), forecasting (prediction of the future values of a series) and control (taking appropriate corrective actions) [1]. To analyze any time series data, time series analysis techniques are adopted. The commonly used techniques are: descriptive technique, probability models technique and spectral density analysis technique. The inference based on the descriptive method and probability models is often referred to as analysis in time domain while inference based on spectral density function is referred to analysis in frequency domain [2,3,4,5]. Bulletin of Mathematical Sciences and Applications Submitted: 2016-09-12 ISSN: 2278-9634, Vol. 17, pp 57-74 Revised: 2016-09-25 doi:10.18052/www.scipress.com/BMSA.17.57 Accepted: 2016-10-24 2016 SciPress Ltd, Switzerland Online: 2016-11-01 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ All these models assume that the error component of the series (et) is normally distributed with zero mean and constant variance   2  or that the series is normally distributed with constant mean (μ) and constant variance   2  . When any study data violates any or all of these assumptions, the series is subjected to transformation. Transformation helps to (i) stabilize the variance of a series, (ii) make the seasonal effect when present additive and (iii) make the data normally distributed [1]. One of the transformations commonly used is the power transformation developed by [6]. [7] noted that the power transformation (i) changes the scale of the original series, (ii) may introduce bias in the forecast especially when data have to be transformed back to its original scale and (iii) often the transformed series have no physical interpretation. [1] argued that transformation alone may not be helpful when variance changes through time in the absence of trend. In such case (i.e., when variance changes through time in the absence of trend), he recommended that a model that allows for changes in variance should be considered. Wavelet method is one such method that allows for changes in variance which has been found to be useful in time series analysis. It involves decomposition, de-nosing and reconstruction of series. Decomposition involves breaking down time series into two main components namely the detail and the smooth components, de-noising deals with the removal of the non significant components of the series, while reconstruction involves recovering of the original series devoid of noise. Unlike the power transformation, wavelet method does not change the scale of the series, poses no problem in its interpretation and does not rely on the assumption of any underlying distribution of the study data. Additionally, wavelet transformation method allow for decomposition of a series without knowing the underlying functional form of the series [8]. Could this lead to an improved model and forecast performance from those based on power transformation? This and other related questions are what this study intends to address. Therefore, the objective of this study is to examine the precision (in terms of goodness-of-fit) and forecast performances of the models for wavelet transformed series while comparing it with models for power transformed time series. 2.0 Literature Review Various transformations exist, but the power transformation developed by [6] is often used. This transformation requires a correct choice of the transformation parameter often denoted as (λ). [4] suggested using a maximum likelihood value for the choice of the value of λ that results in the smallest residual sum of squares. [9] proposed a Bayesian method to choose the value of λ for a given model structure. The correct choice of the transformation parameter (λ), the simultaneous transformation and fitting of the model of a given series are the noticed limitations in the use of [6] power transformation. To remedy these limitations, [10] have shown how to apply Bartlett’s transformation to time series data. Accordingly, they regress the natural logarithms of the group standard deviation ) ,..., 2 , 1 , ˆ ( m i i   against the natural logarithms of the group means ( , 1,2, , ) i X i m  of time series data arranged chronologically in equal groups and determine the slope (β) of the relationship. [11] derived a confidence interval for the index of a power transformation that stabilizes the variance of a time series. They claimed that the confidence interval for the minimum coefficient of variation can also be used to construct confidence interval for any coefficient of variation. [12] used Box-Cox transformation approach to transform a streamflow time series data to turn the non-Guassian heavy tailed distribution to a nearly Gaussian series. [13] applied log-transformation in time series modeling of US macroeconomic data. He demonstrated that the claim previously made concerning improvement in forecast accuracy following bias correction for the transformed data were not generally well founded. [14] in his work applied differencing and Box-Cox transformation on a non-stationary series prior to application of ARMA model. He discovered that the technique was not optimal for forecasting of the study data. Therefore, he recommended a method for modeling time series data with unstable roots and changing parameters. The appropriateness in context of out-of-sample forecast of vector autoregressive models (VAR) when the series is log-transformed was examined by [15]. The 58 BMSA Volume 17