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Journal ArticleDOI

Analysing multiple time series and extending significance testing in wavelet analysis

TL;DR: This work used 1/ƒ β models to test cycles in the wavelet spectrum against a null hypothesis that takes into account the highly autocorrelated nature of ecological time series and used the maximum covariance analysis to compare the time-frequency patterns of numerous time series.
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 2 approaches to deal with these shortcomings. First, we used 1/ƒ β 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.

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Citations
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Journal ArticleDOI
TL;DR: The basic properties of the wavelet approach for time-series analysis from an ecological perspective are reviewed, notably free from the assumption of stationarity that makes most methods unsuitable for many ecological time series.
Abstract: Wavelet analysis is a powerful tool that is already in use throughout science and engineering. The versatility and attractiveness of the wavelet approach lie in its decomposition properties, principally its time-scale localization. It is especially relevant to the analysis of non-stationary systems, i.e., systems with short-lived transient components, like those observed in ecological systems. Here, we review the basic properties of the wavelet approach for time-series analysis from an ecological perspective. Wavelet decomposition offers several advantages that are discussed in this paper and illustrated by appropriate synthetic and ecological examples. Wavelet analysis is notably free from the assumption of stationarity that makes most methods unsuitable for many ecological time series. Wavelet analysis also permits analysis of the relationships between two signals, and it is especially appropriate for following gradual change in forcing by exogenous variables.

586 citations


Cites background or methods from "Analysing multiple time series and ..."

  • ...The wavelet power spectra could be compared with procedures based on multivariate methods like maximum covariance analysis that was originally used to compare spatio-temporal fields (Rouyer et al. 2008)....

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  • ...At the other extreme, the surrogate by Theiler et al. (1992) mirrors a ‘‘hard’’ null-hypothesis that should be rejected with difficulty in the case of wavelet analysis because these bootstrapped series and the raw series share the same autocorrelation function (see Rouyer et al. 2008)....

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Journal ArticleDOI
06 Nov 2008-Nature
TL;DR: It is shown that winter weather and snow conditions, together with density dependence in the net population growth rate, account for the observed population dynamics of the rodent community dominated by lemmings in an alpine Norwegian core habitat between 1970 and 1997, and predict the observed absence of rodent peak years after 1994.
Abstract: Norwegian lemmings (Lemmus lemmus) are well known for their population cycles, which are thought, at their peak, to influence other ecosystem components. In fact the role of the physical environment — climate included — in determining rodent cycle dynamics has remained largely a matter of conjecture. Now from a combination of long-term (1970–2007) data on rodent density, bird densities and field estimates of snow pack conditions together with meteorological data, a clearer picture of the lemming cycle has been obtained. What emerges is a marked shift away from the familiar 3–5-year rodent cycles to an aperiodic, mostly low-amplitude state, which can be explained and predicted by the between-year variations in winter climate. There is strong evidence for the hypothesis that climate effects on rodent dynamics are transmitted to other parts of the ecosystem. The population cycles of rodents at northern latitudes have puzzled people for centuries1,2, and their impact is manifest throughout the alpine ecosystem2,3. Climate change is known to be able to drive animal population dynamics between stable and cyclic phases4,5, and has been suggested to cause the recent changes in cyclic dynamics of rodents and their predators3,6,7,8,9. But although predator–rodent interactions are commonly argued to be the cause of the Fennoscandian rodent cycles1,10,11,12,13, the role of the environment in the modulation of such dynamics is often poorly understood in natural systems8,9,14. Hence, quantitative links between climate-driven processes and rodent dynamics have so far been lacking. Here we show that winter weather and snow conditions, together with density dependence in the net population growth rate, account for the observed population dynamics of the rodent community dominated by lemmings (Lemmus lemmus) in an alpine Norwegian core habitat between 1970 and 1997, and predict the observed absence of rodent peak years after 1994. These local rodent dynamics are coherent with alpine bird dynamics both locally and over all of southern Norway, consistent with the influence of large-scale fluctuations in winter conditions. The relationship between commonly available meteorological data and snow conditions indicates that changes in temperature and humidity, and thus conditions in the subnivean space, seem to markedly affect the dynamics of alpine rodents and their linked groups. The pattern of less regular rodent peaks, and corresponding changes in the overall dynamics of the alpine ecosystem, thus seems likely to prevail over a growing area under projected climate change.

441 citations

01 Jan 1993
TL;DR: In this paper, it was shown that 1/f processes are optimally represented in terms of orthonormal wavelet bases, and the wavelet expansion's role as a Karhunen-Loeve-type expansion was developed.
Abstract: The 1/f family of fractal random processes model a truly extraordinary range of natural and man-made phenomena, many of which arise in a variety of signal processing scenarios. Yet despite their apparent importance, the lack of convenient representations for 1/f processes has, at least until recently, strongly limited their popularity. In this paper, we demonstrate that 1/f processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency domain characterization for 1/f processes, we develop the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes. As an illustration of potential, we show that wavelet based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes

314 citations

Journal ArticleDOI
TL;DR: In this paper, the authors use wavelet analysis to study business cycle synchronization across the EU-15 and the Euro-12 countries and find that the French business cycle has been leading the German business cycle as well as the rest of Europe.

308 citations

Journal ArticleDOI
TL;DR: A detailed overview of a wide range of surrogate types is provided, which include Fourier transform based surrogates, which have since been developed to test increasingly varied null hypotheses while characterizing the dynamics of complex systems, including uncorrelated and correlated noise, coupling between systems, and synchronization.

268 citations


Cites result from "Analysing multiple time series and ..."

  • ...A similar approach is also used in [146] to confirm associations in ecological time series....

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References
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Book
01 May 1992
TL;DR: This paper presents a meta-analyses of the wavelet transforms of Coxeter’s inequality and its applications to multiresolutional analysis and orthonormal bases.
Abstract: Introduction Preliminaries and notation The what, why, and how of wavelets The continuous wavelet transform Discrete wavelet transforms: Frames Time-frequency density and orthonormal bases Orthonormal bases of wavelets and multiresolutional analysis Orthonormal bases of compactly supported wavelets More about the regularity of compactly supported wavelets Symmetry for compactly supported wavelet bases Characterization of functional spaces by means of wavelets Generalizations and tricks for orthonormal wavelet bases References Indexes.

16,073 citations


"Analysing multiple time series and ..." refers background or methods in this paper

  • ...Wavelet analysis (Daubechies 1992) is a time scale and/or time–frequency decomposition of the signal that overcomes these problems and provides a powerful tool for analysing non-stationary, aperiodic and noisy signals often found in ecological time series (Torrence & Compo 1998)....

    [...]

  • ...This is because this methodology enables description of the variability of a time series in both time and frequency domains, and it can cope with aperiodic components, noise and transients (Daubechies 1992, Lau & Weng 1995, Torrence & Compo 1998)....

    [...]

Journal ArticleDOI
TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
Abstract: Introduction Preliminaries and notation The what, why, and how of wavelets The continuous wavelet transform Discrete wavelet transforms: Frames Time-frequency density and orthonormal bases Orthonormal bases of wavelets and multiresolutional analysis Orthonormal bases of compactly supported wavelets More about the regularity of compactly supported wavelets Symmetry for compactly supported wavelet bases Characterization of functional spaces by means of wavelets Generalizations and tricks for orthonormal wavelet bases References Indexes.

14,157 citations

Journal ArticleDOI
TL;DR: In this article, a step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino-Southern Oscillation (ENSO).
Abstract: A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino–Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of change...

12,803 citations


"Analysing multiple time series and ..." refers background or methods in this paper

  • ...Wavelet analysis (Daubechies 1992) is a time scale and/or time–frequency decomposition of the signal that overcomes these problems and provides a powerful tool for analysing non-stationary, aperiodic and noisy signals often found in ecological time series (Torrence & Compo 1998)....

    [...]

  • ...This is because this methodology enables description of the variability of a time series in both time and frequency domains, and it can cope with aperiodic components, noise and transients (Daubechies 1992, Lau & Weng 1995, Torrence & Compo 1998)....

    [...]

  • ...The pointwise testing approach, used by Torrence & Compo (1998), relies on a parametric bootstrap to assess the significance of areas....

    [...]

Book
01 Jan 2017
TL;DR: In this paper, simple descriptive techniques for time series estimation in the time domain forecasting stationary processes in the frequency domain spectral analysis bivariate processes linear systems state-space models and the Kalman filter non-linear models multivariate time series modelling some other topics.
Abstract: Simple descriptive techniques probability models for time series estimation in the time domain forecasting stationary processes in the frequency domain spectral analysis bivariate processes linear systems state-space models and the Kalman filter non-linear models multivariate time series modelling some other topics.

3,694 citations