Forecasting Time Series via Discrete Wavelet Transform

Miguel A. Ariño
IESE, Universidad de Navarra
AArino@iese.es

Abstract

Our purpose in this communication is to present a methodology for forecasting univariate time series. This methodology combines standard forecasting techniques with ``wavelet methodology". The recently developed wavelet theory has proven to be a useful tool in the analysis of some problems in engineering and related fields. However, the potential of this theory for analyzing economic problems has not been fully exploited yet. The communication presents one of its many possible applications in this field. As an example we will apply the methodology to forecast car sales in the Spanish market and compare the results with those given by standard forecasting techniques. Roughly speaking, using wavelets, we decompose a time series ( ) in its trend ( ) and its seasonal component ( ). The way of decomposing the series follows the method by in Arino and Vidakovic (1995) and is outlined later in this paper. Then we apply standard forecasting techniques to each component to obtain a forecast of the original series ( ).

If

is the vector of coefficients of the discrete wavelet transformation of a data set of size 2n (i.e., d = W x), the energy of d at level j is defined as

The scalogram of d is the vector of energies

The scalogram of the discrete wavelet transform of a time series is used to decompose the series into cycles of different frequencies.

If at a low level j of the wavelet decomposition of x, most of the coefficients are ``big", it means that a long-period, low-frequency cyclic component is present in x. The length of the period need not be constant, because some particular of that level can be ``small". On the contrary, if at a high-level j, most of the coefficients are ``big", a short-period, high-frequency cyclic component is present in x.

In the analysis of economic time series, it is usual to find a 12-month period seasonal component and a long-term trend. Thus, it is reasonable to find two peaks in the scalogram of the wavelet coefficients d of an economic data set x. Splitting d into and as explained below, and applying to the split parts the inverse wavelet transform , the two components y and z of can be identified. Normally, y represents a long-term trend of the economic data set. This component should be somehow related to long-term trends of other economic magnitudes, all of them defining the business cycle of an economy. z usually represents the 12-month seasonal behavior of the economic time series. It should be easier to forecast y and z separately, than the whole series x, since x presents a mixed cyclical behavior while y and z present, by construction, its specific periodicity: z should have a 12-month periodicity and y should be quite smooth, as it reflects the long-term trend of the series x.

The paper ends with applications of the methodology to a particular data set.