Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing

Abstract

We propose an equivalence class of nonstationary Gaussian stochastic processes defined in the wavelet domain. These processes are characterized by means of wavelet multipliers and exhibit well-defined time-dependent spectral properties. They allow one to generate realizations of any wavelet spectrum. Based on this framework, we study the estimation of continuous wavelet spectra, i.e., we calculate variance and bias of arbitrary estimated continuous wavelet spectra. Finally, we develop an areawise significance test for continuous wavelet spectra to overcome the difficulties of multiple testing; it uses basic properties of continuous wavelet transform to decide whether a pointwise significant result is a real feature of the process or indistinguishable from typical stochastic fluctuations. This test is compared to the conventional one in terms of sensitivity and specificity. A software package for continuous wavelet spectral analysis and synthesis is presented.

On the impact of impurities on the densification of polar firn

“…It is therefore perfectly suited for our application in which we seek to identify annual cycles and their dependence on firn depth. Local significance was tested against a red noise null hypothesis using Monte Carlo experiments (Maraun et al, 2007). However, one must note that local significance testing will result in spurious significant patches especially as adjacent areas in the wavelet sample spectrum are not independent (Maraun et al, 2007).…”

“…calculating the correlation without relative shift in depth of Ca++ relative to the density) to demonstrate that this procedure does not affect the conclusions. In a second step of our statistical analysis we estimated the wavelet sample spectrum using the Morley wavelet (sowas package, Maraun and Kurths, 2004;Maraun et al, 2007) to analyze the depth dependent behavior of density and chemistry in the frequency domain. Wavelet analysis is a common tool for analyzing localized variations of power within a data series (Torrence and Compo, 1998), even if the dominant modes of variability are non-stationary.…”

Wavelet analysis of correlation among Canary Islands octopus captures per unit effort, sea-surface temperatures and the North Atlantic Oscillation

“…The smoothing is performed by means of the convolution with a constant-length window function both in the time and scale Maraun and Kurths, 2004;Maraun et al, 2007). The numerator and denominator must be smoothed to some extent or WCO i (s) would be identically one (Maraun and Kurths, 2004).…”

Time series data for Canadian arctic vertebrates: IPY contributions to science, management, and policy

“…When two normalized time series oscillate in a simultaneous manner (and whether there is a phase difference or not between the series), high coherence is expected and one series can predict the other (Torrence and Compo 1998;Grinsted et al 2004). Lag period can then be derived with a phase spectrum (Maraun et al 2008). We conducted wavelet analysis and wavelet coherence in R (R Development Core Team 2010) using the sowas package (Maraun and Kurths 2004;Maraun et al 2008).…”

“…Lag period can then be derived with a phase spectrum (Maraun et al 2008). We conducted wavelet analysis and wavelet coherence in R (R Development Core Team 2010) using the sowas package (Maraun and Kurths 2004;Maraun et al 2008). We calculated a general estimate of large- scale climatic variation on our time series data by running wavelet coherence and phase spectrum between all the continuous time series with the North-Atlantic Oscillation (NAO).…”