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95% signii cant level, suggesting that they represent signii cant cyclicities.
We have therefore obtained similar results to those obtained using the
periodogram method. However, the Lomb-Scargle method has the advantage
that is does not require any interpolation of unevenly-spaced data, as well as
permitting quantitative signii cance testing.
5.8 Wavelet Power Spectrum
Section 5.6 demonstrated the use of a modii cation to the power spectrum
method for mapping changes in cyclicity through time. A similar modii cation
could, in theory, be applied to the Lomb-Scargle method, which would have
the advantage that it could then be applied to unevenly-spaced data. Both
methods, however, assume that the data are composites of sine and cosine
waves that are globally uniform in time and have ini nite time spans. h e
evolutionary power spectrum method divides the time series into overlapping
segments and computes the Fourier transform of these segments. To avoid
spectral leakage, the data are multiplied by windows that are smooth bell-
shaped curves with positive values (Section 5.3). h e higher the temporal
resolution of the evolutionary power spectrum the lower the accuracy of
the result. Moreover, short time windows contain a large number of high-
frequency cycles whereas the low-frequency cycles are underrepresented.
In contrast to the Fourier transform, the
wavelet transform
uses base
functions (
wavelets
) that have smooth ends
per se
(Lau and Weng 1995,
Mackenzie et al. 2001). Wavelets are small packets of waves; they are dei ned
by a specii c frequency and decay towards either end. Since wavelets can
be stretched and translated in both frequency and time, with a l exible
resolution, they can easily map changes in the time-frequency domain.
We use the functions for wavelet analysis that are included in the Wavelet
Toolbox (MathWorks 2014b). h ere is also, however, a very popular wavelet
toolbox produced by Christopher Torrence and Gilbert P. Compo (1998),
which is freely available online from
http://paos.colorado.edu/research/wavelets/
A wavelet transformation mathematically decomposes a signal
y
(
t
) into
elementary functions ˈ
a,b
(
t
) derived from a
mother wavelet
ˈ(
t
), by dilation
and translation,
