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

Chukwudi Justin Ogbonna,Iheanyi Sylvester Iwueze,C Jeol Nweke,Eleazer C Nwogu

doi:10.18052/www.scipress.com/bmsa.17.57

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.

Highlights

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
Application of Eq 1 will lead to a power transformation which is given by: Yt where Yt is the transformed series at time t, X t is the original value of the series at time t, α, β are the regression coefficients given in Eq 1 and i is zero mean white noise with constant variance
The results show that the root mean square error (RMSE) of 54.4304 for the forecast of the model for power transformed series is greater than the RMSE of 42.8487 for the forecast of the model for the wavelet transformed series for exchange rate

Summary

Introduction

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. 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]. 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]

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