Page 10 - Rouyer_Fromentin_2008

20 Mar Ecol Prog Ser 359: 11–23, 2008

Fig. 8. Classification of the wavelet spectra for the yearly time series of sea surface temperature in the Mediterranean Sea. The
classification method was applied with the covariance threshold fixed at C = 99% of the total covariance. The cluster tree was cut
to identify (a) 2, (b) 3, (c) 4 and (d) 5 groups of pixels that were mapped and identified by the different shading or hatching.
SST time series were extracted from the COADS dataset on a 2 × 2° grid and covered the period 1900 to 2005

level of aggregation identified an area located in the term fluctuations to reliably fit the model. Following
western Mediterranean, that extended from the Albo- Miramontes & Rohani (2002), and from an empirical
ran Sea (see Fig. 3) up to the Balearic islands (Fig. 8b). perception, a size of 40 points seems to be the lesser
The third level of aggregation divided the eastern bound to apply the beta surrogates. The generation of
β
Mediterranean, with a group that combined the the 1/ƒ process can then be achieved by different
Aegean sea with the northern part of the Levantine techniques. Even if complex techniques proved to be
β
Basin (Fig. 8c). Finally, the last level of aggregation more optimal to generate discrete 1/ƒ processes (Wor-
identified pixels from the Gulf of Lions and the Lig- nell 1993, Kasdin 1995), we used the spectral synthesis
urian Sea, that were separated from the rest of the as it produced consistent results and displayed a good
western Mediterranean (Fig. 8d). trade-off between simplicity, accuracy and computa-
tional speed. The underlying null model assumed is
critical for testing the wavelet spectrum. The beta sur-
DISCUSSION rogates build time series that mimic the slope of the
power spectrum of the original time series; in other
words, the same relative importance of frequencies in
Testing the wavelet spectra the signal. Therefore, the beta surrogates assume a
large range of autocorrelation structures that are not
The class of surrogates presented in this study offers constrained to a reduced frequency band; unlike auto-
a consistent and more powerful approach to the signif- regressive processes, the beta surrogates consistently
icance testing of the wavelet spectra of ecological time test the wavelet spectrum at both low and high fre-
series. The autocorrelation structure of time series is quencies. It can also be of further interest when there
β
here described as a 1/ƒ process, which takes into is important shift in the autocorrelation structure be-
account the ‘spectral redness’ often displayed by real tween different time periods, as the surrogates enable
time series (Lawton 1988, Pimm & Redfearn 1988, Ryb- an assessement of the significance of this change in the
ski et al. 2006). While Fourier Type 1 and Type 2 frequency content of the signal.
exactly reproduce the initial spectrum, the beta surro-
β
gate uses the fit of a 1/ƒ model to generate the surro-
gates. As fitting such models with the classical fast Time series clustering
Fourier transform regression is problematic for short
time series, we used the multiple segmenting method Many approaches have been developed in the field
to overcome this (Miramontes & Rohani 2002). How- of signal processing to compare time series by using
ever, a time series dominated by low frequency fluctu- their raw properties, the fitted parameters of auto-
ations must be long enough to contain sufficient long- regressive moving average (ARMA) models or their