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Vol. 359: 11–23, 2008 MARINE ECOLOGY PROGRESS SERIES
doi: 10.3354/meps07330 Mar Ecol Prog Ser Published May 5

Analysing multiple time series and extending
significance testing in wavelet analysis

1
Tristan Rouyer 1, 2, *, Jean-Marc Fromentin , Nils Chr. Stenseth 2, 3 , Bernard Cazelles 4, 5
1
IFREMER, Centre de Recherche Halieutique Méditerranéenne et Tropicale, Avenue Jean Monnet, BP 171, 34203 Sète cedex,
France
2 Centre for Ecological and Evolutionary Synthesis (CEES), Department of Biology, University of Oslo, PO Box 1066,
Blindern, 0316 Oslo, Norway
3
Institute of Marine Research, Department of Coastal Zone Studies, Flødevigen Research Station, 4817 His, Norway
4
CNRS UMR 7625, École Normale Supérieure (ENS), 46 rue d’Ulm, 75230 Paris cedex 05, France
5
IRD, UR 079, GEODES Centre IRD Ile de France, 32 avenue Henri Varagnat, 93143 Bondy cedex, France

ABSTRACT: In nature, non-stationarity is rather typical, but the number of statistical tools allowing
for non-stationarity remains rather limited. Wavelet analysis is such a tool allowing for non-
stationarity but the lack of an appropriate test for statistical inference as well as the difficulty to deal
with multiple time series are 2 important shortcomings that limits its use in ecology. We present
β
2 approaches to deal with these shortcomings. First, we used 1/ƒ models to test cycles in the wavelet
spectrum against a null hypothesis that takes into account the highly autocorrelated nature of
ecological time series. To illustrate the approach, we investigated the fluctuations in bluefin tuna trap
catches with a set of different null models. The 1/ƒ β models approach proved to be the most
consistent to discriminate significant cycles. Second, we used the maximum covariance analysis to
compare, in a quantitative way, the time–frequency patterns (i.e. the wavelet spectra) of numerous
time series. This approach built cluster trees that grouped the wavelet spectra according to their
time–frequency patterns. Controlled signals and time series of sea surface temperature (SST) in the
Mediterranean Sea were used to test the ability and power of this approach. The results were
satisfactory and clusters on the SST time series displayed a hierarchical division of the Mediterranean
into a few homogeneous areas that are known to display different hydrological and oceanic patterns.
We discuss the limits and potentialities of these methods to study the associations between ecological
and environmental fluctuations.

KEY WORDS: Non-stationarity · Multivariate time series · Wavelet clustering · Wavelet significance
testing · Surrogates · Maximum covariance analysis
Resale or republication not permitted without written consent of the publisher

INTRODUCTION logical time series do not meet such requirements and as
growing evidence supports recognition of the impor-
Following the work of Steele (1985), Pimm & Redfearn tance of transient dynamics in ecological processes
(1988) and Lawton (1988), the time–frequency proper- (Hastings 2001, Cazelles et al. 2008), the spectral proper-
ties of a signal have become of major interest to ecolo- ties are not always well suited to analyse ecological time
gists (e.g. Petchey et al. 1997). In nature, non-linear and series. Wavelet analysis (Daubechies 1992) is a time
non-stationary processes are the rule rather than the scale and/or time–frequency decomposition of the signal
exception (Stenseth et al. 1998, Hsieh et al. 2005), and that overcomes these problems and provides a powerful
many classical tools for time series analysis, such as tool for analysing non-stationary, aperiodic and noisy
Fourier analysis, require stationarity (or more often signals often found in ecological time series (Torrence &
second-order stationarity, Chatfield 2004). As many eco- Compo 1998).

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