Page 2 - Rouyer_Fromentin_2008

12 Mar Ecol Prog Ser 359: 11–23, 2008

Unlike the Fourier transform which decomposes the the local covariance and correlation between 2 non-
time series into a sum of sine and cosine functions that stationary signals (e.g. Cazelles et al. 2008). However,
are not resolved in time, wavelet analysis uses a set of these methods only enable the study of associations be-
functions locally defined in both time and frequency tween pairs of time series, and their use remains limited
domains. Wavelet analysis has spread into many fields in the case of large datasets. In addition, wavelet analy-
of research, such as signal processing, geophysics and ses yield outputs in both time and frequency domains
climatology, and is becoming more and more popular and comparing wavelet spectra of numerous time series
within ecology (e.g. Bradshaw & Spies 1992, Grenfell quickly turns into a very complex issue. When dealing
et al. 2001, Klvana et al. 2004, Keitt & Urban 2005, Keitt with large datasets of time series, a classical and useful
& Fischer 2006, Ménard et al. 2007). Nonetheless, approach in ecology is to use hierarchical clustering. A
wavelet analysis still displays 2 shortcomings with matrix of dissimilarities between the time series is con-
respect to its use in ecology. First, the time and structed, over which a clustering algorithm is applied.
frequency locations of the wavelet spectra are not The dissimilarity matrix can either be performed on the
uncorrelated, and the statistical inference is therefore raw properties of the time series or on their power spec-
difficult (Maraun & Kurths 2004). If standard boot- trum, according to whether interest is more in the tempo-
strapping methods allow testing the wavelet spectra ral or frequency aspects. We present here an approach
with various re-sampling procedures, the underlying that combines both time and frequency domains, as the
null hypotheses are most often invalid for ecological matrix of dissimilarities used for clustering is constructed
time series. For instance, in a number of cases, white from the comparison between pairs of wavelet spectra.
noise and red noise are not supported by the observa- The wavelet spectra are compared using a procedure
tions. Second, although wavelet analysis is a powerful based on the maximum covariance analysis (MCA), a
univariate or bivariate analysis, it remains limited to multivariate method that was originally used to compare
analysing a large number of (shorter) time series; this spatio-temporal fields (Bretherton et al. 1992). Both the
is, however, a situation commonly encountered by surrogate and the clustering approaches are applied in
ecologists. this paper to several datasets of time series, either
We present 2 approaches to circumvent these limita- simulated or real, to illustrate their applicability. Wavelet
tions and to extend the use of wavelets within the field analysis and the proposed approaches applied to ecolog-
of ecology. First, we propose to test the wavelet spectra ical questions illustrate how these methods can bring
using surrogates. The surrogates have been intro- fruitful insights to the study of coupling between envi-
duced by Theiler et al. (1992) to determine the consis- ronmental and biological fluctuations.
tency of experimental time series with various null
hypotheses of simple systems. Each surrogate is
consistent with a specific null hypothesis about the MATERIALS AND METHODS
underlying system (e.g. a random variable or an
autoregressive-like process) while retaining some of Wavelet analysis. The wavelet methodology is well
the statistical features of the original time series (e.g. suited for signals whose frequencies change with time.
mean and variance, power spectrum). Statistics are This is because this methodology enables description
then computed for the original data and a large set of of the variability of a time series in both time and
surrogates, thus allowing testing of the original data frequency domains, and it can cope with aperiodic
against an empirical distribution consistent with the components, noise and transients (Daubechies 1992,
null hypothesis (Royer & Fromentin 2006). We propose Lau & Weng 1995, Torrence & Compo 1998). The
a class of surrogates that models the underlying statis- wavelet transform is based on the convolution product
tical structure of the time series as 1/ƒ noise (Halley between the time series and a mathematical function,
1996). Such a model is rather convenient as it allows for the so-called ‘daughter wavelet’. For a given set of pa-
the ‘more time, more variation’ effect displayed by rameters a (scale parameter related to frequency) and τ
many ecological time series (Lawton 1988, Inchausti & (translation parameter related to time position), the
Halley 2002, Vasseur & Yodzis 2004). This class of sur- wavelet functions Ψ, are defined at time t as follows:
rogates is thus adapted to ecological time series, as it 1 ( )
t −
τ
t=
is, in addition, designed to deal with short time series. ψ a, τ () ψ a (1)
Second, understanding ecological phenomena using a
time series often requires the analysis of large datasets. The wavelet transform, W, is then defined as a con-
Wavelet cross analyses allow investigation of the associ- volution integral of the time series with the wavelet
ation between 2 signals by extending the wavelet trans- function:
form to bivariate cases. The wavelet cross-spectrum and 1 t − τ
∫
()
(
( )
the wavelet coherence are respectively used to quantify Wa,τ) = ∫ xt ψ* ( ) d t= x t ψ* ( a,τ) t d (2)
x
a a