25-10-2012, 03:58 PM
Application of the cross wavelet transform and wavelet coherence to
geophysical time series
Application of the cross wavelet.pdf (Size: 204.88 KB / Downloads: 30)
Abstract.
Many scientists have made use of the wavelet
method in analyzing time series, often using popular free
software. However, at present there are no similar easy to
use wavelet packages for analyzing two time series together.
We discuss the cross wavelet transform and wavelet coherence
for examining relationships in time frequency space between
two time series. We demonstrate how phase angle
statistics can be used to gain confidence in causal relationships
and test mechanistic models of physical relationships
between the time series. As an example of typical data where
such analyses have proven useful, we apply the methods to
the Arctic Oscillation index and the Baltic maximum sea
ice extent record. Monte Carlo methods are used to assess
the statistical significance against red noise backgrounds. A
software package has been developed that allows users to
perform the cross wavelet transform and wavelet coherence
(http://www.pol.ac.uk/home/research/waveletcoherence/).
Introduction
Geophysical time series are often generated by complex
systems of which we know little. Predictable behavior in
such systems, such as trends and periodicities, is therefore
of great interest. Most traditional mathematical methods
that examine periodicities in the frequency domain, such as
Fourier analysis, have implicitly assumed that the underlying
processes are stationary in time. However, wavelet transforms
expand time series into time frequency space and can
therefore find localized intermittent periodicities. There are
two classes of wavelet transforms; the Continuous Wavelet
Transform (CWT) and its discrete counterpart (DWT). The
DWT is a compact representation of the data and is particularly
useful for noise reduction and data compression
whereas the CWT is better for feature extraction purposes.
Methods
The Continuous Wavelet Transform (CWT)
A wavelet is a function with zero mean and that is localized
in both frequency and time. We can characterize a wavelet
by how localized it is in time (1t) and frequency (1! or the
bandwidth). The classical version of the Heisenberg uncertainty
principle tells us that there is always a tradeoff between
localization in time and frequency. Without properly defining
1t and 1!, we will note that there is a limit to how small the
uncertainty product 1t·1! can be. One particular wavelet,
Summary
The CWT expands a time series into a time frequency space
where oscillations can be seen in a highly intuitive way. The
Morlet wavelet (with !0=6) is a good choice when using
wavelets for feature extraction purposes, because it is reasonably
localized in both time and frequency. From the CWTs
of two time series one can construct the XWT. The XWT exposes
regions with high common power and further reveals
information about the phase relationship. If the two series
are physically related we would expect a consistent or slowly
varying phase lag that can be tested against mechanistic models
of the physical process. WTC can be thought of as the local
correlation between two CWTs. In this way locally phase
locked behavior is uncovered.