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Rouyer et al.: Wavelet analysis of multiple time series 13

where * indicates complex conjugate. We used the properties with the original time series (or with a given
Morlet wavelet, a continuous and complex wavelet null model). This method has already been used for
adapted to wavelike signals, that allows extraction of testing purposes in ecology (e.g. nonlinearity in time
time-dependent amplitude and whose scales are series, recurrence patterns, similarity of rhythms
related to frequencies in a simple way (Mallat et al. between time series, phase analysis), as several null
1998). The relative importance of frequencies for each models can be chosen according to the different algo-
time step may then be represented in the time/fre- rithms employed to create the surrogates (Schreiber &
quency plane that forms the wavelet power spectrum: Schmitz 2000, Cazelles & Stone 2003, Cazelles 2004,
Royer & Fromentin 2006). The surrogate approach has
(
S ( a,τ) = W a,τ) 2 (3) also been used to test the wavelet spectrum using a
x
x
hidden Markov process as a null model (Klvana et al.
where S is the wavelet power spectrum and x is the 2004, Saitoh et al. 2006). Other null models are the
raw time series. The wavelet power spectrum, S x (a, τ), ‘Type 0’ surrogates, equivalent to a white noise
is plotted as a function of time and period in a 2-dimen- assumption, whereas the null model referred to as
sional graph. This representation, classically referred ‘Type 1’ surrogates (or Fourier surrogates), preserves
to as a ‘contour plot’, displays contours as identical the autocorrelation structure of the time series (or
power spectrum values, coded in the figures of this equivalently its power-spectrum). The more complex
paper by different colors. The wavelet transform acts null model, ‘Type 2’ surrogates, preserves both the
as a local filter that directly relates the magnitude of Fourier spectrum and the original distribution of the
the signal to time and thus enables one to track how data (Schreiber & Schmitz 1996, Royer & Fromentin
the frequency components change over time. There- 2006). The Fourier Type 1 and Type 2 surrogates are
fore, wavelet analysis is particularly adapted to inves- very interesting in an ecological perspective, as they
tigation of non-stationary and transient signals. preserve the Fourier spectrum of the original time
Statistical inference of the wavelet spectrum. Point- series, allowing the oscillations that cannot be
wise testing: While analytical tests for significance can produced by an autoregressive process to be tested.
be straightforwardly calculated in Fourier analysis, the However, these classes of surrogates require the
validity of such tests is greatly questioned in the case of length of the time series to be much larger than the
wavelet analysis (Maraun & Kurths 2004). The point- dominant frequency in order to lead to satisfying
wise testing approach, used by Torrence & Compo surrogates, a requirement often difficult to obtain for
(1998), relies on a parametric bootstrap to assess the ecological time series. They also present the disadvan-
significance of areas. A reasonable null model of a tage of introducing spurious low-frequency effects due
given form is first chosen (e.g. a white noise or a first to the phase randomization, but also spurious high-
order autoregressive process, AR[1]), and a large num- frequency effects when the time series are short and
ber of random realisations is produced. Computing the non-stationary (for more details see Schreiber &
wavelet spectrum for each realisation generates an Schmitz 1996).
empirical distribution for each point under the null Surrogates using the slope of the spectrum: We
model hypothesis. The test is then performed by proposed a class of surrogates, the ‘beta surrogates’,
comparing the values obtained for the original time that display a similar variance and autocorrelation
series with the empirical distributions. As for all Monte structure as the original time series and that form a less
Carlo approaches, the choice of the null model used to constrained hypothesis than the Fourier and Type 2
test the wavelet spectrum is central. Making the surrogates. The beta surrogates display the same rela-
assumption of a white noise process is generally not tive distribution of frequencies, i.e. the same slope of
appropriate for ecological data, while an autoregres- the Fourier spectrum, as the original time series; this
sive process (e.g. an AR[1]) can be an acceptable null allows the dominance of low frequencies often
model in some cases, for both ecological and geo- displayed by ecological time series to be taken into
physical data (Steele 1985, Vasseur & Yodzis 2004). account. To do so, we estimated the exponent of a
β
However ecological time series display a large variety power law model 1/ƒ , often called ‘beta’ in the
of autocorrelation structures that neither a white noise, literature, fitted to the power spectrum of the time
nor an autoregressive process could consistently series (Halley 1996), using the multiple segmenting
describe (Arino & Pimm 1995, Cuddington & Yodzis method proposed by Miramontes & Rohani (2002) for
1999, Inchausti & Halley 2002, Halley & Stergiou short time series. Like Cuddington & Yodzis (1999), we
2005). used the spectral synthesis to generate surrogates with
A traditional surrogate approach: The surrogates the previously estimated exponent. In the spectral
can be best suited to test wavelet spectra, as they pro- synthesis, the amplitudes of the Fourier spectrum are
duce synthetic time series that share given statistical scaled according to the estimated spectral exponent of