文档介绍：Wavelet-based Forecasting of
Short and Long Memory Time
Series
O. Renaud, J.-L. Starck and F. Murtagh
No
Cahiers du département d’économétrie
Faculté des sciences économiques et sociales
Université de Genève
Mai 2002
Département d’économétrie
Université de Genève, 40 Boulevard du Pont-d’Arve, CH -1211 Genève 4
/metri/
Wavelet-based Forecasting of Short and Long Memory Time
Series
O. Renaud,∗ J.-L. Starck†and F. Murtagh‡
Abstract
A wavelet-based forecasting method for time series is introduced. It is based on a
multiple resolution position of the signal, using the redundant “`atrous” wavelet
transform which has the advantage of being shift-invariant.
The result is a position of the signal into a range of frequency scales. The
prediction is based on a small number of coefficients on each of these scales. In its simplest
form it is a linear prediction based on a wavelet transform of the signal. This method uses
sparse modelling, but can be based on coefficients that are summaries or characteristics
of large parts of the signal. The lower level of the position can capture the long-
range dependencies with only a few coefficients, while the higher levels capture the usual
short-term dependencies.
We show the convergence of the method towards the optimal prediction in the au-
toregressive case. The method works well, as shown in simulation studies, and studies
involving financial data.
Index Terms
Wavelet transform, forecasting, resolution, scale, autoregression, time series, model
1 Introduction
The wavelet transform has been proposed for time series analysis in many papers in recent
years. Much of this work has focused on periodogram or scalogram analysis of periodicities
and cycles. For financial time series prediction, Bjorn (1995), Moody and Lizhong (1997) and
Soltani et al. (2000) discussed the use of the wavelet transform in th