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Rouyer et al.: Wavelet analysis of multiple time series 21

rhythmic properties (e.g. Henson et al. 1998, Keogh & procedure on the SST time series clearly separated
Pazzani 1998, Xiong & Yeung 2002, Cazelles 2004). different areas. However, the areas obtained did not
Other approaches have compared time series at vari- consist of random groups of contiguous locations, but
ous resolutions, but comparing the time series focusing rather reflected different hydrological conditions, as
on their time–frequency properties using the continu- they were consistent with outputs from oceanic mod-
ous wavelet transform as we did has not, to our knowl- els of the Mediterranean Sea. For instance, the sepa-
edge, been done (Lukasik 2000, Keogh et al. 2001). ration found between the eastern and the western
Comparing time series based on their wavelet spectra part of the Mediterranean at the relatively shallow
is similar to comparing a set of images, and many and narrow Sicily Strait (some 450 m depth and 140
methods from this field of research (e.g. face recog- km wide) conforms with the literature, as it separates
nition) have used multivariate approaches coupled to their respective gyral circulation (e.g. Zavatarelli &
wavelet transform techniques to extract features of the Mellor 1995). The combination of the Alboran Sea and
original image (Feng et al. 2000, Gupta & Jacobson the south of the Balearic Basin (Fig. 8b) identifies an
2006). The multivariate method used here proved to be area of more dynamic circulation. The Atlantic water
powerful for indexing images and also for comparing that enters the basin forms anticyclonic gyres in the
spatio-temporal fields (Wu et al. 1996). It also allowed Alboran sea (Fig. 3), and it defines a front at their
us to put the emphasis on the common time–frequency eastern boundary from which mesoscale eddies arise
properties of the time series, as this can be of central and drift into the Balearic Basin (Millot 1985, Tintore
interest to ecologists dealing with non-stationarity et al. 1988). The joining of the Aegean sea with the
(Hastings 2001, Hsieh et al. 2005, Cazelles & Hales northern part of the Levantine Basin (Fig. 8c) sepa-
2006). Transformation, and particularly log-transfor- rates the Rhodes Gyre from the more southwestern
mation, is a common procedure for ecological data. Shikmona and Mersa-Matruh gyres (Pinardi &
This is useful for many analyses (e.g. to stabilize or Masetti 2000). The last level of aggregation roughly
rescale the variance) but it is not a requirement for separates the Gulf of Lions and the Ligurian Sea from
wavelet analysis because it can handle non-stationary the rest of the western Mediterranean (Fig. 8d). This
signals. Such transformations may further be applied area corresponds to a gyre but is also the source —
carefully, as they affect the relative changes in ampli- with the northern Adriatic, that was not covered by
tude between time steps and can thus distort the our dataset — of deep water (Zavatarelli & Mellor
time–frequency patterns detected by the wavelet 1995). If spatial groups were expected from a classifi-
analysis. cation of such a spatially autocorrelated variable, the
Being fairly robust, the method is not expected to fail boundaries obtained well matched the different
in common ecological cases. However, simulated cases hydrological conditions over the region. This showed
showed that wavelet spectra displaying patchy pat- that the procedure was able to detect consistent
terns without a dominant frequency mode and no clear groups over a large number of rather similar wavelet
common areas could lead the procedure to produce spectra.
spurious associations. One could also get counter-
intuitive groupings when a wavelet spectrum shares
different areas with 2 spectra, and when these 2 other CONCLUSIONS
spectra do not clearly share a common area (an artefact
that is also known for cluster analysis computed from a Associations between environmental and ecological
distance matrix between raw time series). In fact, the signals are often transient and difficult to identify.
main limitation of the method is that it does not specify Wavelet analysis offers a powerful way to investigate
the common patterns detected; one has to go back to these associations, but it has so far been restricted to
the wavelet spectra to identify them. However, using univariate or bivariate analyses. Our approach in the
the reconstruction by the first axes of the MCA to iden- present study allows the exterior of the use of wavelet
tify the common patterns between wavelet spectra analysis to multiple ecological and environmental time
may help in detecting such spurious grouping (e.g. series. For instance, this procedure can be used as a
Fig. 7a). basis for investigating potential statistical links
between ecological and environmental time series or
how an environmental variable can affect different
Classification of the Mediterranean sea surface populations. The beta surrogate addresses a second
temperature critical point, i.e. the assessment of whether or not the
associations detected are likely to be an artefact
As expected with such a spatially autocorrelated caused by the intrinsic autocorrelation structure. The
variable, the results obtained by the classification 2 approaches presented in this study might thus form