Signal processing applications of the Fourier-transform encountered several
dif-…culties. Fourier-analysis is unable to detect non-stationarity, and its global nature prevents it from giving information about the local (in time) behaviour of the series. One proposed solution was the windowed Fourier transform that essentially is a slicing up in time if the window is simple. The formula is
fb(!; ) = Z1
1
f(t)w(t ) exp( 2 i!t)dt;
where w(t ) is a "window" function. However it is not self-adapting, one has to discover the appropriate shape of the window function. The wavelet transform can enable us to rectify these problems at the cost of increased free-dom of choice: whereas the Fouruer transform is essentially unique there are an in…nite number of substantially di¤erent wavelet transforms.
6.7.1 The wavelet transform
The wavelet transform provides us with a decomposition of a time series into scale and time components, while the Fourier-transform gives only frequency decomposition, and the Wold Representation Theorem only time decomposition.
As a simpli…cation one could say that the wavelet transform expresses how much a time series changed around a certain date at di¤erent scales. It has been likened to a prism through which one can observe the properties of an object (the time series in our case) otherwise obscured. It is customary to relate it to the Fourier-transform that assumes a similar task, but relies on the assumption of homogeneity (stationarity), and does not account for local (localized in time) changes. In the role of prism wavelets have been proved to improve on Fourier-analysis, at least in the life and earth sciences. In other words, to characterize complex and non-stationary systems this methodology has advantages.
6.7.2 Continuous wavelet transform
It starts from the windowed Fourier transform, but replacesw(t ) exp( 2 i!t) with some (time dependent) …lter:
W(s; ) =
where (t) is called a wavelet n having (somewhat simpli…ed) properties trans-form is replaced with scale (s):The wavelet transtrans-form is a convolution:
(f g)( ) = Z
f(t)g( t)dt;
for any scales.
Continuous wavelets are highly redundant transformations, when calculated from an actual time series the computation produces a matrix with much more entries than the original series. They must be distinguished from discrete wavelets that speci…cally strive for data compression and are used much less in research than in engineering. In economic applications the most commonly used mother wavelet is the Morlet wavelet.
What kind of statistics can we derive from the wavelet transfom to analyze data? The Wavelet Power Spectrum (WPS) is the squared wavelet transform.
WPS …gures can be created with the following interpretation: a point with abscissa (time period), and ordinate (scale) expresses the power attributable to that time and scale. The integral of the WPS equals the variance of the time series, thus the WPS can be interpreted as producing variance decomposition.
If we have two series the cross wavelet transform is de…ned as the conjugate product of the two individual transforms. From this one can de…ne the Wavelet Coherency (WC) measure which is similar to the cross-autocovariance function, but having also a time dimension. The cross wavelet transform makes possible the calculation of phase di¤erences, establishing lead-lag relationships between the series. We can calculate the most powerful time and the most powerful scale statistics, where the WC values are averaged time- and scale-wise, and then arg-maxed according to time and scale, respectively.
6.7.3 The orthogonal wavelet transform
It starts from the Fourier-series, and looks for an orthogonal representation of x(t):
x(t) = X1 j;k= 1
W(j; k)'(t; k; j);
wheref'(t; k; j)gis an orthonormal set of functions. It is the inverse of the wavelet transform resulting in the coe¢ cientsW(j; k).
The construction of this orhogonal representation starts with a mother wavelet and a father wavelet. The simplest is the Haar mother wavelet
(t) =
and the Haar father wavelet:
(t) = 1;0 t <1 0; else The daughters are de…ned as
j;k(t) = 2j2 (2jt k);
whereas the scaling functions as:
k(t) = (t k):
Then the unionf k(t)g [ j;k(t) for all integersj; kforms an orthonormal basis ofL2.
For practical purposes it is important that it can be proved that there is a general method for …nding an orthonormal basis, starting from an appropriate father or mother wavelet.
The discrete orthogonal wavelet transform in practice For …nite data we have to choose a …nite basis. We assume thatn= 2Jand make the restriction k= 2j.
For instance the 8-sample Haar-wavelet (without normalizing constants) looks like:
The …rst wavelet in this case is
1;1(t) = The wavelet transform (without normalizing constants) is
x2 x1 x4 x3
x6 x5 x8 x7
(x3+x4) (x3+x4) (x7+x8) (x5+x6)
(x5+x6+x7+x8) (x1+x2+x3+x4) (x1+x2+x3+x4+x5+x6+x7+x8):
The orthogonal wavelet transform enables multiresolution analysis:
x=w0+w1+:::+wJ+vJ;
where eachwj is in the orthogonal complement of Vj (wherevj 2Vj:) The wavelet transform and its inverse can be written in matrix form as
xw = W x
x = WTxw=X
(WT)ixwi ; providing a scale-wise decomposition:
x=X
j
xj:
This is the multiresolution analysis in practice. Its main use is data com-pression in computer science, but it is also used for data de-noising.
A problem with the orthogonal wavelet transform is that it is sensitive to
"initial" conditions, and it requires "decimated" data. A possible solution is the Maximal Overlap DWT (MODWT). It does not restrict observations tok= 2j, however it is not orthogonal and is redundant. Its construction requires arti…cial data. But reconstruction is possible, and gives a multiresolution analysis with preserving variance. Both the orthogonal wavelet transform and the MODWT have been used in economics for estimating regressions at di¤erent scales.
6.8 Literature
Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: forecasting and control. John Wiley & Sons.
Hamilton, James Douglas. Time series analysis. Vol. 2. Princeton, NJ:
Princeton university press, 1994.
Kirchgässner, Gebhard, Jürgen Wolters, and Uwe Hassler. Introduction to modern time series analysis. Springer Science & Business Media, 2012.
Percival, D. B., & Walden, A. T. (2000). Wavelet methods for time series analysis (Vol. 4). Cambridge university press.
Shumway, Robert H., and David S. Sto¤er. "Time series regression and exploratory data analysis." Time series analysis and its applications. Springer New York, 2011. 47-82.