Some New Methods for Prediction of Time Series by Wavelets

Extended Abstract. Forecasting is one of the most important purposes of time series analysis. For many years, classical methods were used for this aim. But these methods do not give good performance results for real time series due to non-linearity and non-stationarity of these data sets. On one hand, most of real world time series data display a time-varying second order structure. On the other hand, wavelets are well situated for forecasting non-stationary time series since wavelets are local which is very important property to analyze non-stationary time series. To extend stationary processes, slow changes are allowed over time of the second order structure of time series or equivalently the amplitude in spectral representation is allowed to depend on time the spectral representation. Dahlhaus (1997) proposes a minimum distance estimation procedure for non-stationary time series models. Ombao et al. (2002) defined non-stationary processes by changing covariance stationary processes over time. Instead of using windowed Fourier transform (as in Priestley 1965), Nason et al. (2000) introduce the Locally Stationary Wavelet (LSW) processes, using the (non-decimated) wavelet transform and the rescaled time principle. Fryzlewicz et al. (2003) suggest an algorithm to predict LSW processes. This algorithm was used by Van Bellegem and Von Sachs (2002) to model financial log-return series. Forecasting LSW processes arrives to a generalization of the Yule-Walker equations, which can be solved numerically by matrix inversion or standard iterative algorithms such as the innovations algorithm. In the stationary case, these equations reduced to ordinary Yule-Walker equations. In all the above articles, the point wise prediction of discrete time series has been considered, but functional prediction (prediction of an interval) of time series can be considered instead of a point wise prediction since for continuous time series, interval prediction is more suitable than point wise prediction. Antoniadis et al. (2006) propose functional wavelet-kernel smoothing method. This method uses interpolating wavelet transform which is not most popular wavelet transform. The predictor may be seen as a weighted average of the past paths, associating more weight on those paths which are more similar to the present one. Hence, the ‘similar blocks’ are to be found. To summarize, this method is performed in two following steps: 1.We should find within the past paths the ones that are ‘similar’ or ‘close’ to the last observed path. This determines the weights; 2.then we use locally weighted average using obtained weights to predict time series. In this article, after describing mentioned methods, we suggest some extensions to the functional wavelet-kernel method to forecast time series by means of wavelets and then we compare that with several prediction methods. We propose to use two different types of wavelet transform instead of interpolating wavelet transform: discrete wavelet transform (DWT) and non-decimated wavelet transform (NDWT). The first one is an orthogonal wavelet transform while the second one is a redundant transform. These transformations are applied more than interpolating wavelet transform and can be used easily in most of mathematical programming software as S-Plus and MATLAB. We consider the following methods: the methods proposed by Fryzlewics et al. (2003) and Antoniadis et al. (2006), the classical autoregressive model and our two proposed methods. Then, we compare these methods by simulation and real data. We simulate the data from AR(7) (stationary data) and AR(7) contaminated by a sinusoid (non stationary data). We also consider two real data set; Electricity Paris Consumption and El-Nino data. In our comparison, our methods give better results than other compared methods. In this paper we also show that mean square prediction error converges to zero under some conditions when the sample size is large. Reference Antoniadis, A., Paparoditis, E. and Sapatinas, T. (2006). A functional wavelet-kernel approach for continuous-time prediction. J. R. Statist. Soc. B, 68, 837-857. Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist., 25, 1-37. Fryzlewicz, P., Van Bellegem, S. and von Sachs, R. (2003). Forecasting nonstationary time series by wavelet process modeling. Annals of the Institude of statistical Mathematics, 55, 737-764. Nason, G.P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of evolutionary wavelet spectra. J. Roy. Statist. Soc. Ser. B, 62, 271-292. Ombao, H., Raz, J., von Sachs, R. and Guo, W. (2002). The SLEX model of a nonstationary random process. Ann. Inst. Statist. Math., 54, 171-200. Priestley, M. (1965). Evolutionary spectra and nonstationary processes. J. Roy. Statist. Soc. Ser. B, 27, 204-237. Van Bellegem, S. and Von Sachs, R. (2002). Forecasting Economic Time Series using Models of Nonstationary (Discussion paper NO. 0227). Institut de statistique, UCL.